Geometry and Mechanics of Defects in Structured ...

Geometry and Mechanics of Defects in Structured Surfaces

by

Ayan Roychowdhury

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY KANPUR KANPUR, INDIA May 12, 2017

Geometry and Mechanics of Defects in Structured Surfaces

A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY by

Ayan Roychowdhury

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY KANPUR KANPUR, INDIA May 12, 2017

To Ma and Baba

Our examination of the continuum problem contributes to critical epistemology’s investigation into the relations between what is immediately (intuitively) given and the formal (mathematical) concepts through which we seek to construct the given in geometry and physics. Hermann Weyl, from the Preface of The Continuum — A Critical Examination of the Foundations of Analysis, Dover, 1987.

Contents Acknowledgements

v

Abstract

vi

List of Publications

vii

List of Figures

viii

1 Introduction

1

1.1

Non-Euclidean nature of defects in 3-dimensional bodies . . . . . . . . . . . . . . . . . .

4

1.2

Non-Euclidean nature of defects in structured surfaces . . . . . . . . . . . . . . . . . . .

14

1.2.1

Local defects in structured surfaces . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.2.2

Global defects in structured surfaces . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.3

Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.4

Novel contributions of the present work . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2 Geometric Characterization of Defects 2.1

32

Characterization of defects in 3-dimensional solids . . . . . . . . . . . . . . . . . . . . .

33

2.1.1

Material response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.1.2

Material G-structure, material connection and material metric . . . . . . . . . . .

35

2.1.3

The material space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.1.4

Material torsion tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

i

2.2

2.1.5

Material curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.1.6

Decomposition of the curvature tensor . . . . . . . . . . . . . . . . . . . . . . . .

40

2.1.7

Material non-metricity tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.1.8

Unambiguous definition of a metric tensor field . . . . . . . . . . . . . . . . . . .

43

2.1.9

Parallel transport of the inner product and its path dependence . . . . . . . . . .

44

Characterization of defects in structured surfaces . . . . . . . . . . . . . . . . . . . . . .

46

2.2.1

Geometry on ω induced from the non-Riemannian structure on M: the material space

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.2.2

Non-metricity of the material connection: metric anomalies . . . . . . . . . . . .

48

2.2.3

Torsion of the material connection: dislocations . . . . . . . . . . . . . . . . . . .

50

2.2.4

Curvature of the material connection: disclinations . . . . . . . . . . . . . . . . .

52

3 Symmetries, Conservation Laws, and Representations of Defects 3.1

3.2

Compatibility of the geometric objects on the material space of 3-dimensional bodies . .

58

3.1.1

Conservation laws of material inhomogeneities . . . . . . . . . . . . . . . . . . .

59

3.1.2

Representation of metric anomalies in materially flat 3-dimensional solids . . . .

59

3.1.3

Irrotational metric anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.1.4

Quasi-plastic strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.1.5

Semi-metric geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.1.6

Quasi-plastic deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Compatibility of the geometric objects on the material space of structured surfaces . . .

72

3.2.1

Consequences of the first Bianchi-Padova relation . . . . . . . . . . . . . . . . . .

72

3.2.2

Consequences of the second Bianchi-Padova relation . . . . . . . . . . . . . . . .

74

3.2.3

Consequences of the third Bianchi-Padova relation . . . . . . . . . . . . . . . . .

74

3.2.4

Consequences of the fourth Bianchi-Padova relation . . . . . . . . . . . . . . . .

75

4 Strain Incompatibility in 3-dimensional Elastic Solids 4.1

57

Induced Riemannian structure

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

77 78

4.2

Strain incompatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.3

Metric anomalies associated with locally compatible elastic strain . . . . . . . . . . . . .

79

5 Strain Incompatibility in Structured Surfaces 5.1

5.2

81

Strain measures and local strain compatibility . . . . . . . . . . . . . . . . . . . . . . . .

82

5.1.1

Local strain compatibility conditions for Kirchhoff-Love shells . . . . . . . . . . .

88

Global strain compatibility for Kirchhoff-Love shells . . . . . . . . . . . . . . . . . . . .

88

5.2.1

Topological classification of 2-dimensional manifolds embeddable in R3 . . . . . .

89

5.2.2

Global integrability of the isometric embedding problem of 2-dimensional manifolds in R3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.3

Local strain incompatibility arising from local defects

95

5.4

Global strain incompatibility relations for a Kirchhoff-Love shell arising from global defects 97 5.4.1

Disc with holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.4.2

Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.4.3

Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.4.4

Twisted bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

6 Internal Stress Field and Natural Shape of Defective Structured Surfaces

100

6.1

Kinematics of Kirchhoff-Love shells with small strain accompanied by moderate rotation 102

6.2

Strain incompatibility relations for sufficiently thin Kirchhoff-Love shells with small strain accompanied by moderate rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3

6.4

6.2.1

Local incompatibility relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.2

Global incompatibility relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Material response and equilibrium equations for sufficiently thin Kirchhoff-Love shells . 112 6.3.1

Pure bending of thin elastic isotropic shells . . . . . . . . . . . . . . . . . . . . . 114

6.3.2

Combined bending and stretching of thin elastic isotropic shells . . . . . . . . . . 114

F¨ oppl-von K´ arm´ an equations with incompatible elastic strain for shells with arbitrary reference geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 iii

6.4.1

2-dimensional solid crystals with edge dislocations, wedge disclinations, and pure in-surface metric anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4.2 6.5

Growing biological surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Shape equation of disclinated isotropic fluid films . . . . . . . . . . . . . . . . . . . . . . 119

7 Organicum geometriae: Concluding Remarks and Future Scope

121

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

iv

Acknowledgements First and foremost, I thank my teacher, and thesis supervisor, Dr. Anurag Gupta, not just for the academic guidance that he has tirelessly offered me throughout my journey at IIT Kanpur, but also for the space he provided me with, warm and safe, where my intellect has had the privilege to break out of its bounds, flourish, and evolve in many dimensions. If this thesis has anything rewarding towards the advancement of the subject it concerns, it is solely the reflection of his own vision; the shortcomings are all mine. I am indebted to Professor Marcelo Epstein for all those illuminating discussions we had during my winter sojourn at Calgary. The clarity that I have in the fundamentals of the subject is due to his wonderful and lucid expository books and articles. I would also like to thank Professor Lev Truskinovsky for the detailed e-mail correspondence on various novel ideas in mechanics that have helped me push the limit of my present knowledge. I am also in debt to all my teachers in school and college, whose names these few pages would fail to accommodate, and to Dr. Anirban Ghosh, who was kind enough to share some of his profound musical knowledge during my short, intermittent lessons on the Sitar. Vivek (Tewary), Sandeep, Sankalp, Arijeet, Vijju, Ganesh, Rahul, Manab, Digen, Rajat, Anish, Animesh, Anup Da, Mousumi Di, Ashok Da, Santanu, Vivek (Sahu), Mamon Didi, Chhorda, Poka, Rangada, Reeti, Tina, Arnab, Amrita, Saurabh Da, Arunabha Da, Rikhi, Rayan, Vinit, Kaushika, Manish, and Akshay—I can only offer you my love for all the warm moments spent together at home and away. I am for ever indebted to my parents, whom I dedicate this thesis to: these words are here because they are there.

Ayan Roychowdhury, IIT Kanpur, February 10, 2017. v

Abstract The term structured surface is used to represent a variety of 2-dimensional structures ranging from 2-dimensional crystals, with intrinsic translational, rotational, and metrical order, to thin sandwiched structures, and liquid crystalline membranes and shells, with intrinsic crystalline order or without. The notion of structured surface also extends to the so-called Cosserat surfaces, which are used to model a hierarchy of plate and shell theories for thin elastic structures abundant in structural engineering applications. Our aim is to study geometry and mechanics of defects in structured surfaces. The defects are anomalies within the local arrangement of entities in an ordered structure where the order is usually defined in o terms of translational, rotational, and metrical symmetries of the underlying material. Examples of these anomalies are omnipresent in nature, e.g., edge dislocations, wedge disclinations, and point imperfections such as vacancies and self-interstitials on graphene sheets, or twist disclinations on lipid membranes. These local anomalies are known to generate internal stress field within the structured surface and also to relax the defective structured surfaces by acquiring certain natural shapes. Besides these local anomalies, structured surfaces are also known to contain a variety of global anomalies that affect the inherent topology of the surface, and act as additional sources to internal stress field. In this thesis, we will establish the non-Euclidean character of continuously distributed local and global anomalies over materially uniform, elastic structured surfaces and pose, within the same differential geometric setting, certain boundary-value-problems for determining internal stress fields and natural shapes of defective structured surfaces. We present a general framework for the boundaryvalue-problems for a large class of continuously defective structured surfaces under the Kirchhoff-Love deformation constraint that can undergo moderately large deformation while maintaining small insurface stretching everywhere. Upon simplification, under various assumptions, this framework reduces down to forms used in the recent literature on 2-dimensional matter, e.g., the shape equations for continuously defective thin isotropic fluid films, and F¨oppl-von K´arm´an equations for continuously defective thin elastic isotropic shells.

vi

Publications related to the thesis

1. A Roychowdhury and A Gupta. Dislocations, disclinations, and metric anomalies as sources of strain incompatibility in thin shells. (under review) 2. A Roychowdhury and A Gupta. On structured surfaces with defects: geometry, strain incompatibility, internal stress, and natural shapes (arXiv preprint arXiv:1702.03737). (under review) 3. A Roychowdhury and A Gupta. Non-metric connection and metric anomalies in materially uniform elastic solids. Journal of Elasticity, 126:1–26, 2017. 4. A Roychowdhury and A Gupta. Material homogeneity and strain compatibility in thin elastic shells. Mathematics and Mechanics of Solids, doi:10.1177/1081286515599438, 2015. 5. A Roychowdhury and A Gupta. Geometry of defects in solids. Directions, 14:32–52, 2013 (arXiv preprint arXiv:1312.3033).

vii

List of Figures 1.1

Formation of a positive edge dislocation in a cubic Bravais lattice: (a) a perfect lattice, and (b) insertion of an atomic half plane to create a positive edge dislocation. Formation of a negative edge dislocation in a hexagonal Bravais lattice: (c) a perfect lattice, and (d) removal of an atomic half plane to create a negative edge dislocation. . . . . . . . .

1.2

Closure failure of the Burgers parallelogram in dislocated cubic crystals: (a) edge dislocation, and (b) screw dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

8

9

Formation of a negative and a positive disclination in a cubic and a hexagonal Bravais crystal, respectively. (a) Removal of a 90◦ wedge from a cubic Bravais lattice, and (b) the resulting −90◦ disclination. (c) Insertion of a 60◦ wedge into a hexagonal Bravais lattice, and (d) the resulting +60◦ disclination. . . . . . . . . . . . . . . . . . . . . . . .

1.4

Parallel transport of a tangent vector along a closed loop in a disclinated crystal: (a) negative disclination, and (b) positive disclination. . . . . . . . . . . . . . . . . . . . . .

1.5

11

Zero-dimensional defects in a cubic Bravais lattice: (a) A vacancy, (b) a self-interstitial, and (c) a foreign interstitial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.6

10

12

Schematic diagram of a split-interstitial in fcc lattice. The original interstitial, located at A, is unstable and relaxes into a dumbbell shaped split-interstitial A0 B 0 (reproduced from [39]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7

12

Schematic diagrams of a (a) tetra, and a (b) penta vacancy in Copper in their stable configurations (reproduced from [124]). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

13

1.8

(a) A single wedge disclination of Frank angle 2π in a nematic membrane, located at O, such that d(θ1 , θ2 ) = cos θe1 + sin θe2 , where θ is the polar angle θ := tan−1 (θ2 /θ1 ). (b) A single twist disclination of Frank angle 2π in a nematic shell, such that d(θ1 , θ2 ) = cos θe1 − sin θe3 . (c) A transverse loop characterizing an effective 2-dimensional representation of the 3-dimensional distribution of disclinations within a thin multi-layered structure made up of some directed media.

1.9

. . . . . . . . . . . . . . . . . . . . . . . . .

15

Isolated edge dislocation in a thin multi-layered structure. The marks on the surface represent lattice points which may carry identical atoms as well as directors. . . . . . . .

17

1.10 (a) Incompatible surface growth of a plate. (b) Differential growth across the thickness of a thin multi-layered structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.11 Making a cylinder, a torus and a M¨obius strip from a plane sheet of paper. The cylinder is multiply-connected and orientable, so is the torus, whereas the M¨obius strip is multiplyconnected but non-orientable. A global defect in the M¨obius strip appears in the form of a twist disclination of Frank angle π around C6 . It characterizes its non-orientability.

19

1.12 Formation of global topology-preserving anomalies on a torus by removal or insertion of a wedge, and subsequent joining of the two open lips. (a) Removal of a wedge W with parallel faces A0 and B 0 from the torus, and subsequent joining of the open lips A and B produces a global edge dislocation with Burgers vector b. (b) Insertion of a wedge W with non-parallel faces A0 and B 0 into the torus, by joining the lips A with A0 , and B with B 0 produces a global anomaly consisting of a global dislocation mixed (edge and screw) type, with Burgers vector b, and a global wedge disclination with Frank tensor Ω. (c) Cutting the torus along a vertical irreducible loop, such as C5 as shown in Figure 1.11, would produce the two lips A and A0 infinitesimally separated from each other. Then upon twisting the torus with a rotation Ω, as shown, such that the points 1, 2 etc. on the left lip A merge with the points 10 , 20 etc. on right lip A0 , a global twist disclination will be produced with Frank tensor Ω; its defect line coincides with the circular centerline of the torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

21

1.13 Formation of global topology-preserving anomalies on a torus by removing a ring around the outer equatorial circle, and subsequent joining of the two open lips. The type of the global dislocation or the global disclination is determined by the direction of the global Burgers vector b and the global Frank tensor Ω. . . . . . . . . . . . . . . . . . . . . . .

22

1.14 Volterra construction associated with a local negative wedge disclination on a toroidal crystalline surface. With every local wedge disclination at P , an associated antipodal disclination P 0 of the same type will be generated; the antipodal disclination P 0 is located at the point where the defect line of the disclination at P cuts the torus. Note that two small discs have also been removed around the pointed tips of the wedge W in order to avoid the singularity, which characterize the inherent topological nature of the local disclinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1

22

(a) Two tangent vectors at the material point X ∈ B, with components ui and v i , are parallelly transported along each other. The closure failure of the resulting infinitesimal parallelogram is characterized by the components T˜jk i of the torsion tensor. (b) Change in length and angle between the tangent vectors under parallel transport due to nonmetricity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

39

˜ measures the stretching (a) The symmetric part of Ω(·, ·), characterized by the tensor ζ, ˜ part of the change brought about by Ω. Here, v is a principal vector of ζn. (b) The ˜ measures the purely rotational part skew part of Ω(·, ·), characterized by the tensor Θ, of the change brought about by Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

41

(a) Change in angle between two tangent vectors to U due to non-zero Qµαβ . (b) Change in angle between two vectors along a transverse curve due to non-zero Q3ij . (c) Change in length of a transverse vector along a surface curve due to non-zero Qµi3 . . . . . . . .

2.4

49

(a) Closure failure of an infinitesimal in-surface parallelogram due to the Tαβ i components of the torsion tensor. (b) Closure failure of an infinitesimal transverse parallelogram due to the Tα3 i components of the torsion tensor.

x

. . . . . . . . . . . . . . . . . .

51

2.5

(a) The symmetric part Ωij(kl) , characterized by the third-order tensor ζ, measures the stretching in v ∈ TX V brought about by curvature of the material space. Here, v is a principal direction of the second order tensor ζn, where n is the unit normal to the infinitesimal area element δA bounded by the loop C. (b) For C completely lying within U , the skew part Ωαβ[ij] , characterized by the vector field Θi Ai , measures the purely rotational part of the change in v = v i Ai brought about by the curvature tensor. (c) When C is transverse to U , the skew part Ωα3[ij] , characterized by the second-order tensor field Θαq Aα ⊗ Aq , measures the purely rotational part of the change in v = v i Ai brought about by the curvature tensor.

3.1

. . . . . . . . . . . . . . . . . . . . . . . . . . .

53

(a) General irrotational metric anomalies change the orientation, shape and size of a cube along a curve, whereas (b) isotropic metric anomalies, which are also irrotational, change the orientation and size, but not the shape, of a cube along a curve. . . . . . . .

3.2

Mappings between the tangent spaces of various configurations and spaces associated with the material manifold B, see Section 3.1.6 for details. . . . . . . . . . . . . . . . . .

5.1

92

Disc with a hole after making a cut. The global Frank tensor Ω and the global Burgers vector b as measures of global incompatibility of the surface. . . . . . . . . . . . . . . .

6.1

67

Reducible and irreducible loops on a generic multiply connected 2-dimensional manifold ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

64

98

Kinematics of the elastic-plastic decomposition of the total deformation measures in Kirchhoff-Love shells. The only non-trivial disclinations are represented by Θ3 , and hence we have a well defined normal over the surface in the natural configuration.

xi

. . . 103

Chapter 1

Introduction The aim of this thesis is to study geometry and mechanics of defects in structured surfaces. The term structured surface is used to represent a variety of 2-dimensional structures such as 2-dimensional crystals, with intrinsic translational, rotational, and metrical order (colloidosomes, carbon nanotubes, graphene etc.); thin sandwiched structures; liquid crystalline membranes and shells, with intrinsic crystalline order (single-layer viral capsids) or without (nematic membranes, single layers in smectics and cholesterics); and Cosserat surfaces, which are used to model a hierarchy of plate and shell theories for thin elastic structures abundant in structural engineering applications. The defects are anomalies within the local arrangement of entities in an ordered structure where the order is usually defined in terms of translational, rotational, and metrical symmetries of the underlying material. These anomalies are omnipresent in nature, e.g., 2-dimensional materials such as graphene are known to contain edge dislocations (translational anomalies), wedge disclinations (rotational anomalies), and point imperfections (metric anomalies) such as vacancies and self-interstitials; on the other hand, twist disclinations are commonly observed in lipid membranes. More examples are given in a following section as well as in an extensive review of the subject in [12, 95]. The phenomena of thermal deformation and biological growth can also be categorized as those leading to metric anomalies. Many of the superior physicochemical properties of the 2-dimensional defective structures can be attributed to the internal stress fields resulting from the distribution of defects [137], and also, unlike 3-dimensional bodies, due

1

to their lower dimensionality, to their ability to relax by acquiring a variety of natural (stress-free) shapes, for instance, the wavy edges of growing leaves [77, 78], the topological corrugations present on human brain [122], helical strands of DNA [30], among other examples. The present work is concerned with the central problem of formulating a general theory that takes under its ambit the geometric characterization of these multifarious 2-dimensional defective structures and also the determination of their internal stress fields and deformed shapes. Non-Euclidean differential geometry has been established to provide the necessary mathematical infrastructure to describe the geometric nature of defects in 3-dimensional solids, as well as to provide a rightful setting to discuss the related issues of strain incompatibility and residual stress distribution [2, 4, 10, 21, 25, 64, 67, 96]. Despite this success in 3-dimensions, the problem in lower dimensional structures is relatively less developed, primarily due to the complex interplay between the embedding geometries in the physical space, and the unavoidable non-linearities involved in the deformation as well as the constitutive response of 2-dimensional matter. We note the initial attempts made by Eshelby [40, 41] where analytical solutions for internal stress were derived for linearly elastic plates containing isolated screw and edge dislocations. This work was extended in several directions by Chernykh [17] and Nabarro [90, 92], among others [88, 111]. A theory of continuous distribution of defects over thin structures was first developed by Povstenko [101, 102] by drawing analogies from the non-Euclidean description of continuous distribution of defects in 3-dimensional elastic bodies. Povstenko introduced the notions of in-surface dislocations (characterized by the in-surface torsion tensor), disclinations (characterized by the in-surface curvature tensor), and metric anomalies (characterized by the insurface non-metricity tensor). He also provided the non-linear conservation laws for all the in-surface defect density fields as direct consequences of the Bianchi-Padova relations in 2-dimensions. The local compatibility conditions for strain fields in geometrically linear and non-linear shells, and Cosserat surfaces, are extensively discussed in existing works [32, 63, 80, 103]. The presence of defects, however, introduces incompatibility in strain fields. The local strain incompatibility equations for both linear elastic plates and von-K´arm´an plates, with a class of in-surface anomalies, have been derived by Zubov and Derezin [27, 28, 136, 139, 140]; these also include solutions of certain special 2

boundary-value-problems for determining stress and natural shape of the defective plate. The local strain incompatibility equations have also appeared in the recent works on non-Euclidean elastic plates [31,58], and growth and morpho-elasticity of thin biological materials [29,73,74,77,78,84,85]. Without explicitly incorporating defect densities, these works consider a non-Euclidean metric, representing the distribution of plastic/growth strain field, and use the Riemannian curvature of this metric, which is the measure of strain incompatibility, along with the strain decomposition, to pose boundary-valueproblems primarily for determining natural shapes. The concepts of material uniformity, material symmetry, and inhomogeneity in elastic Cosserat surfaces, following the pioneering works of Noll [96] and Wang [126], are also firmly established [33, 34, 37,38,127,128], although these works have neither attempted to describe the inhomogeneity distribution in terms of the curvature and non-metricity (the notion of torsion does appear in some of these works), nor have they discussed the relevant issue of strain incompatibility. A theory of materially uniform, inhomogeneous (dislocated) thin elastic films, derived from a 3-dimensional uniform, inhomogeneous (dislocated) elastic body, has been recently proposed by Steigmann [118], and applied to determining the natural shapes of plastically deformed thin sheets [24]. Finally, we mention the extensive work in the condensed matter physics literature on topologically defective (‘geometrically frustrated’) 2dimensional crystals and liquid crystalline surfaces [12,13,16,86,95,107], which, in contrast to the local theories mentioned above, have taken a distinguished local-global (geometrical-topological) standpoint in describing the nature of defects. There is a clear lack of a complete non-Euclidean geometric characterization of continuously distributed material defects in 2-dimensional structured continua. While these certainly have analogous descriptions in the 3-dimensional theory, there is a considerable richness in the description of the allowable defects as well as their geometrical properties for the 2-dimensional structure. Additionally, there are no derivations of local and global strain incompatibility relations for sufficiently general kinematic and constitutive response as afforded by most of the known 2-dimensional materials. With this in mind, we present a theory, within the natural setting of non-Euclidean differential geometry, that on one hand unifies the several seemingly different streams of research discussed above, and also provides 3

a rigorously constructed, sufficiently general framework for studying a large range of problems of geometry and mechanics of defective structured surfaces. In particular, we give a complete non-Euclidean characterization of all the translational, rotational, and metric anomalies in structured surfaces, derive the imposed restrictions from Bianchi-Padova relations, and establish general local and global strain incompatibility relations. To illustrate our theory, we consider the specific case of Kirchhoff-Love shells and provide a framework, involving kinematics, additive decomposition, incompatibility relations, and balance laws, for posing complete boundary-value-problems for determining internal stress and deformed shapes for a class of 2-dimensional continuously defective structures undergoing small stretch but moderately large deformation. A detailed discussion of the novel contributions made in this thesis is given in Section 1.4. Due to their relatively simpler construction, non-Euclidean nature of defects in 3-dimensional elastic bodies will be useful to study first, before going into the details of their counterparts in structured surfaces. In the present introductory chapter, we will first demonstrate the non-Euclidean character of defects in 3-dimensional elastic crystalline solids with simple illustrative examples in Section 1.1, which will be followed by the demonstration of the non-Euclidean nature of defects in structure surfaces in the next Section 1.2. We will then present a brief outline of the thesis in Section 1.3, followed by a list of novel contributions made in this thesis in Section 1.4.

1.1

Non-Euclidean nature of defects in 3-dimensional bodies

Crystalline solids constitute a class of ordered media whose microstructure appears as a highly regular pattern of lattice points; order is usually defined in terms of the translational, rotational and metrical symmetries of the underlying material. They are in general replete with local anomalies or defects which destroy the crystalline order. Defects can be present both naturally in a crystal or can appear as a result of external influence, e.g., during thermal activation, irradiation, or plastic deformation. A defective solid can have significantly different physical properties (mechanical, electrical, chemical, optical etc.) in comparison to an ideal solid; for instance, the plastic nature of metals is essentially

4

governed by the mechanics of dislocations and grain boundaries, and the optical nature of liquid crystals by the mechanics of disclinations. In a 3-dimensional crystalline solid body, defects can be broadly classified based on their dimensionality. Therefore, we have zero-dimensional defects (or point defects) in the form of vacancies (missing atoms), self-interstitials (extra atoms of the same kind) etc.; one-dimensional defects in the form of dislocations and disclinations; two-dimensional defects such as grain boundaries, phase boundaries, domain walls, stacking faults, and free surfaces; and three dimensional defects in the form of precipitates and inhomogeneities.1 In the present section, our purpose is to illustrate zero and one-dimensional defects in a 3-dimensional Bravais lattice and subsequently motivate their non-Euclidean character. For visual clarity, we have used 2-dimensional models in all the illustrations. The defects also render the crystal internally stressed. In each of the illustrations, defects are introduced by disturbing the minimum energy configuration of the perfectly ordered crystal; this has been done by either adding or removing an extra atomic halfplane, a wedge, and a single particle (of same or different constitution) into the ordered lattice and then joining of the open cuts afterwards to create a dislocation (a translational anomaly), disclination (a rotational anomaly) and a point defect (a metric anomaly), respectively. These surgeries immediately give rise to internal stress fields, much like permanent damage. To relax the defective crystal from its internally stressed state, it has to be cut into pieces; for instance a single cut suffices to relax the dislocated crystal in Figure 1.1(b). These pieces, in general, do not fit together to form a continuous crystal in a 3-dimensional Euclidean space. They, however, would fit together in some non-Euclidean space. This can be seen clearly by considering a single dislocation inside a plate. If the plate is constrained to remain flat even after introducing the dislocation then it would develop internal stresses; 1

For a general treatment of point defects in solids see [121]; for distortion and stress fields associated with point defects,

as well as for precipitates and inhomogeneities, see the pertinent papers of J D Eshelby, e.g., in [83]. For an extensive application of dislocation theory to diverse areas of physics, see [91]. For disclinations and other defects in liquid crystals and magnetic media see [60]. For an excellent perspective on the theory and application of defects from varied disciplines of mathematics (topology, geometry), physics (condensed matter, statistical mechanics), geology (glacier flow, earthquakes), and biology (membranes, cells) see the rich collection of expository articles in [42].

5

however if left unconstrained the plate would bend to come to a natural or stress free state. Hence to relax stresses the plate has to necessarily leave the 2-dimensional Euclidean space and occupy a 2-dimensional Riemannian space embedded in a 3-dimensional space. We will use these insights to develop, in the following chapters, a general theory of defects in solids where the defect distributions will be assumed to be continuous. When distributed continuously, the defects do not exist in isolation but instead are smeared over to give rise to effective fields of defect densities. The continuous defect density fields then directly correspond to various fundamental non-Euclidean geometric objects on the material space of the defective body, for instance, the density of dislocations, disclinations and metric anomalies (point defects, thermal deformation, biological growth etc.) correspond, respectively, to the torsion tensor, Riemann-Christoffel curvature tensor and non-metricity tensor of the material space. Remark 1.1.1. (Continuous distribution of defects.) The density of defects in real crystals is usually very high. For instance, the total length of dislocations in a plastically deforming crystalline solid is about 108 − 1010 mm/mm3 . In the (macroscopic) continuum scale, it is therefore reasonable to talk about a continuous distribution of defects, such that the number of defects tends to infinity as the lattice spacing gets closer to zero so as to preserve the total defect content (e.g. the total Burgers vector). This is analogous to keeping mass per unit volume fixed while reducing the lattice spacing to recover a continuum from a discrete lattice structure. For more details on the continuizing process, see Kr¨ oner [69, pp. 61-117]. A continuous density necessarily requires us to admit defect strengths of arbitrarily small magnitude. This requirement can impose a restriction on the possibility of continuizing the defect distribution. For instance, a density of dislocations in a crystalline solid can always be continuized because the magnitude of Burgers vector can be as small as the infinitesimal lattice spacing. On the other hand, a distribution of disclinations can never be continuized within a crystalline solid. This is due to the fact that the strength of disclinations in crystals can take only finite values of rotation available from the crystal’s symmetry class. Hence it does not make sense to talk about disclination density in a solid crystal, primary reason being this energy blow up, cf. Anthony [5]. The notion of a continuous density of disclinations is valid for 2-dimensional crystalline structures as

6

well as for 3-dimensional liquid crystals and spin structures; this, in case of 2-dimensional crystals, is because of the availability of an extra dimension where they can relax the excess formation energy of the disclinations, and, in case of Cosserat 3-dimensional bodies such as liquid crystals and spin structures, the availability of the extra degrees of freedom in the configuration space (cf. [12,60]). This is an appropriate place to mention a lesser known result, but significant nevertheless, that dislocation density is not a well defined field for isotropic solids. Its non-uniqueness is derived from the freedom available in imposing an arbitrary additional rotation field (see [96]). We do not make any attempt to provide even an elementary introduction to defect theory in the following discussion; it can be easily accessed from several excellent texts. Instead, we restrict ourselves to demonstrate their non-Euclidean nature.

Dislocations.

Dislocations are the most important defects in crystals; they govern not only the

deformation and the strength of crystalline solids but also their growth behavior. A dislocation can be created by inserting (or removing) a planar array of atoms in a perfect lattice. For example, in the perfect Bravais crystals shown in Figures 1.1(a) and 1.1(c), a semi-infinite atomic plane has been inserted and removed, respectively. The edge of this plane, which is a straight line piercing in and out of the paper ad infinitum (seen as a point in these figures), is identified as a linear defect called the dislocation. A dislocation is essentially a translational defect; its introduction into the lattice, in effect, translates all the lattice points above the ‘slip’ plane by one lattice spacing (this is clearly visualized when the crystal is cut into two halves along the slip plane). In doing so, it maintains both the orientation of the lattice as well as the size of lattice spacing. These facts are now illustrated geometrically. We start by drawing a parallelogram around the dislocation, see Figure 1.2. Two tangent vectors a1 and a2 , at a lattice point close to the dislocation line, are transported parallelly along a2 and a1 to yield tangent vectors a01 and a02 , respectively. The parallelogram formed by these four vectors is open. The deficiency, denoted as b in Figure 1.2, is called the Burgers vector of the dislocation; b lies within the lattice plane for an edge dislocation, and transverse to it for a screw dislocation. A parallelogram constructed 7

additional semi-infinite atomic plane

slip plane

(a)

(b)

semi-infinite plane removed

(c)

(d)

Figure 1.1: Formation of a positive edge dislocation in a cubic Bravais lattice: (a) a perfect lattice, and (b) insertion of an atomic half plane to create a positive edge dislocation. Formation of a negative edge dislocation in a hexagonal Bravais lattice: (c) a perfect lattice, and (d) removal of an atomic half plane to create a negative edge dislocation. around a regular lattice point, away from the dislocation, will always close. Drawing an analogy with the notion of torsion tensor of a non-Riemannian manifold which renders infinitesimal parallelograms open, we are thus led to associate dislocation with the torsion in the material space equipped with a particular affine connection (see Chapter 2 for details). The connection between dislocation density and the torsion tensor of the material space was first pointed out by K Kondo [64], and then later, independently by B A Bilby et al. [10]. On the other hand the fact that a dislocation preserves both the orientation and the spacing of the lattice has following geometrical consequences. Any tangent vector, when translated parallelly along a closed curve around the dislocation, remains invariant; thus

8

θ2 O

b

O

θ1

b

(a)

(b)

Figure 1.2: Closure failure of the Burgers parallelogram in dislocated cubic crystals: (a) edge dislocation, and (b) screw dislocation. the affine connection, which otherwise has a non-zero torsion, is curvature-free. Secondly, the length of a tangent vector remains unchanged as it traverses the lattice forcing the covariant derivative of the metric, with respect to the affine connection, to vanish identically. To summarize, the affine connection related to dislocation has a non-vanishing torsion but zero curvature and non-metricity.

Disclinations.

A disclination can be formed by inserting (or removing) a material wedge in (or out

of) a perfect crystal, with the wedge angle equal to any of the rotational symmetry angles of the crystal. For example a −90◦ disclination in a cubic Bravais crystal is obtained, as seen in Figure 1.3 (b), when a 90◦ wedge is removed from the perfect crystal and the two lips of the cut are joined thereafter. On the other hand, a +60◦ disclination can be formed in a hexagonal crystal, as shown in Figure 1.3 (d), after inserting a 60◦ wedge into the perfect crystal. The line which forms the edge of the wedge is a linear defect known as disclination. Disclination is an anomaly in the rotational order of the crystal. It changes the coordination number (number of closest neighbors) of the lattice points lying on the disclination line. For instance the coordination number of the red dot in Figure 1.3 decreases (increases) for a negative (positive) disclination. Introduction of a disclination into a perfect lattice brings about an orientational defect but preserves the translation order as well as preserves the lattice spacing. This is illustrated in Figure 1.4 where we observe that the parallel transportation of a tangent vector from a lattice point, along a

9

(a)

(b)

(c)

(d)

Figure 1.3: Formation of a negative and a positive disclination in a cubic and a hexagonal Bravais crystal, respectively. (a) Removal of a 90◦ wedge from a cubic Bravais lattice, and (b) the resulting −90◦ disclination. (c) Insertion of a 60◦ wedge into a hexagonal Bravais lattice, and (d) the resulting +60◦ disclination. closed curve, yields a different tangent vector. The (signed) angle of deficiency can be characterized in terms of a vector ω, called the Frank’s vector. Evidently, the Frank vector corresponds to the Riemann-Christoffel curvature tensor associated with the material space of the crystal; hence, the latter is a natural measure of the distribution of disclinations. This identification was first made by K H Anthony [2]. However, due to its translation preserving property, a parallelogram always closes around a disclination. The non-metricity is again zero due to preservation of lattice spacing. As a result the affine connection associated with disclinated space should have a non-zero Riemann-Christoffel 10

initial 90◦ final

final 60◦ initial

(a)

(b)

Figure 1.4: Parallel transport of a tangent vector along a closed loop in a disclinated crystal: (a) negative disclination, and (b) positive disclination. curvature but vanishing torsion and non-metricity.

Metric anomalies. Three type of zero-dimensional defects (point defects) are shown in Figure 1.5. A vacancy defect is created when a particle is missing from its regular residing place in the ordered lattice arrangement. An interstitial defect results when an extra constituent atom is present in the interparticle space of the ordered arrangement. A substitutional defect is created when a foreign particle, of a different constitution, comes to reside in the inter-particle space. All of these point defects occur frequently in crystalline solids and are significantly important during processes of thermal activation (at high temperatures) and irradiation, as well as for rate mediated phenomenon like creep. They often accumulate around higher dimensional defects (such as dislocations and grain boundaries), affecting their mobility and, subsequently, influencing the strength and life-time of the crystalline solid. Introduction of point defects preserve both the translational and orientational order of the lattice, while affecting the lattice spacing. Therefore, if the crystalline solid is cut to release its internal stress, and unlike dislocated/disclinated crystals mentioned above infinitely many cuts would be required presently, then different unit cells of the lattice would relax to different sizes. This is analogous to the presence of a thermal gradient within the solid [67, pp. 300-304], or biological growth [110,131]. Hence,

11

extra particle

missing particle

(a)

foreign particle

(b)

(c)

Figure 1.5: Zero-dimensional defects in a cubic Bravais lattice: (a) A vacancy, (b) a self-interstitial, and (c) a foreign interstitial.

A

A0

B

B0

Figure 1.6: Schematic diagram of a split-interstitial in fcc lattice. The original interstitial, located at A, is unstable and relaxes into a dumbbell shaped split-interstitial A0 B 0 (reproduced from [39]). presence of these metric anomalies always changes the local metrical properties (length of vectors, angle between vectors) of the crystal. The affine connection related to metric anomalies should therefore have vanishing torsion and curvature, so as to maintain translation and rotation symmetry, but nontrivial non-metricity. This identification was first provided by E Kr¨oner [66]. Foreign interstitials and substitutionals (Figure 1.5(c)), however, belong to the class of materially non-uniform bodies where the constitutive property changes from point to point; they are outside the scope of the present work as we will restrict ourselves to materially uniform bodies only. Within the category of metric anomalies, we are, in particular, interested to model distribution of anisotropic metric anomalies. It is well known that stable configurations of clusters of point defects form exotic anisotropic shapes [56, 57, 124]. In these works, divacancies have been found to be more mobile than single vacancies and clusters of trivacancies in Copper stronger, with increased binding

12

A D

A

0

0

A

A0

0

A

A

C0 B0

A0

B0 B0

B

B

(a)

(b)

Figure 1.7: Schematic diagrams of a (a) tetra, and a (b) penta vacancy in Copper in their stable configurations (reproduced from [124]). energy, against separation into single vacancies. As reported by Kiritani et al. [56, 57], high density of small vacancy clusters in the form of stacking-fault tetrahedra dominate the plastic deformation of thin foils of fcc materials under high strain-rate without any intervention from dislocations. Anisotropic cluster formation is usually more stable than free standing spherically symmetric point defects. For example, as shown in Figure 1.6, an intrinsic interstitial atom in fcc lattice relaxes into a split-interstitial in order to achieve stability. The interstitial atom at the original position A pushes the atom at B towards right yielding a dumbbell shaped defect A0 B 0 in the stable state. Rather than modelling point defects as spherically symmetric objects, it is only appropriate to consider point defect density as a distribution of infinitesimal rod-like dumbbell structures in the crystalline body. The individual dumbbells display transverse isotropic symmetry about their axes. In Figure 1.7, stable configurations of tetra and penta-vacancies in fcc Copper are depicted. The stable configurations are of octahedral and decaoctahedral shape for tetra vacancies, and of octahedral and bi-tetrahedral shape for penta vacancies. A continuous description of these clusters of vacancies requires an anisotropic representation of metric anomalies. Another example of elementary anisotropic point defects was described by Kr¨ oner [68, 70] in the form of point stacking faults. In this case, the individual point stacking faults are themselves anisotropic. Further instances of anisotropic metric anomalies are provided by finite thermal

13

deformation in crystalline materials [8] and bulk growth in biological materials [131]. In these cases, the thermal expansion or the growth coefficient is described by an anisotropic, symmetric second order tensor. It is prudent to emphasize here that the anisotropy we are referring to is the anisotropy associated with the structural symmetry of the distributed inhomogeneity (metric anomalies in the present case), which is independent of the symmetry (anisotropic or otherwise) of the material response function. For instance, a materially uniform body with isotropic material response can contain a distribution of anisotropic point defects such as those shown in Figures 1.6 and 1.7.

1.2

Non-Euclidean nature of defects in structured surfaces

We will now revisit some classical examples of isolated local anomalies, and global topology-transforming and topology-preserving defects, in structured surfaces. The primary difference between local and global anomalies are that the defect (“singulrity”) line of the local anomaly pierces through the surface, whereas the defect line of a global anomaly falls outside the surface. The examples are presented with an intent to emphasize the non-Euclidean geometric nature of the defects as is incorporated in the subsequent development. In particular, in order to bring these 2-dimensional multifarious defective structures under a uniform mathematical gamut, our central strategy will be to embed a suitable 2-dimensional manifold, a mathematical representation of the defective structured surface, into a 3dimensional non-Riemannian geometric space. This idea emerges naturally as we proceed through the following rudimentary illustrations.

1.2.1

Local defects in structured surfaces

Disclinations.

The rotational anomalies in a structured surface appear in the form of disclinations.

Depending on the nature of the material nature of the surface, rotational order can be present due to intrinsic crystallinity of the surface (such as in colloidosomes, single-layer viral capsids, carbon nanotubes, and graphene) or due to an extrinsic orientation field (such as in nematic membranes, single layers in smectics, and cholesterics) [12]. As a result, we distinguish between rotational order, or lack

14

C

θ3

θ2

2π O

C

θ3

θ2

2π

θ1

O

(a)

θ1

(b) transverse loop

mid-surface

(c)

Figure 1.8: (a) A single wedge disclination of Frank angle 2π in a nematic membrane, located at O, such that d(θ1 , θ2 ) = cos θe1 + sin θe2 , where θ is the polar angle θ := tan−1 (θ2 /θ1 ). (b) A single twist disclination of Frank angle 2π in a nematic shell, such that d(θ1 , θ2 ) = cos θe1 − sin θe3 . (c) A transverse loop characterizing an effective 2-dimensional representation of the 3-dimensional distribution of disclinations within a thin multi-layered structure made up of some directed media.

15

thereof, appearing intrinsically and extrinsically in a surface. We also note that unlike disclinations in 3-dimensional crystalline solids, which have large formation energy and hence are rarely observed, disclinations in 2-dimensional crystals are omnipresent since the surface can now relax the energy by escaping into the third dimension. Isolated disclinations in structured surfaces without intrinsic crystalline order are shown in Figures 1.8(a,b). The rotational order is here present due to a director field distribution, denoted by d(θ1 , θ2 ), over a planar domain parametrized by Cartesian coordinates (θ1 , θ2 ). The director field in Figure 1.8(a) is restricted to lie strictly in the θ1 θ2 -plane; it may represent a deformed configuration of a nematic membrane or a single layer in the cholesteric phase of some liquid crystalline material. In contrast, the directors in Figure 1.8(b) are allowed to orient themselves transversely to the plane; this can model either a lipid monolayer where the director orientation represents the orientation of individual lipid molecules, or a single layer of molecules in the smectic A or C phase [59]. In nematics, smectics, and cholesterics, d is identifiable with −d due to the mirror symmetry about the mid-orthogonal plane of the director axis. The lack of intrinsic crystalline order (translational and rotational), within the plane, in these examples can be primarily attributed to viscous relaxation [60]. Disclinations in such structured surfaces can be characterized by the signed angle through which the director rotates upon circumnavigating along a loop over the surface. The Frank vector ω of the disclination is a precise measure of this signed angle. A disclination is of wedge or twist type depending on whether ω is transverse or tangential, respectively, to the surface. The disclination in Figure 1.4(a) is of wedge type with Frank vector 2πe3 and the one in Figure 1.8(b) is of twist type with Frank vector 2πe2 . Here, the triple {e1 , e2 , e3 } denote the standard basis of the Cartesian coordinate system (θ1 , θ2 , θ3 ). Note that the wedge disclination line in Figure 1.8(a) is along the θ3 axis, whereas the twist disclination line in Figure 1.8(b) is the loop C itself. Disclinations can also appear in surfaces with intrinsic crystalline order, e.g., an ordered arrangement of lattice cites where the directors are attached in viral capsids or hexagonal lattice structure of the carbon atoms in graphene sheets. As illustrated in Figure 1.4, circumnavigating along a loop encircling the disclination, a lattice vector rotates through an angle which is an integral multiple of one of the rotational symmetry angles of the lattice. The wedge disclination located at O, in the 2-dimensional hexagonal lattice in Figure 1.4, is 16

characterized by its Frank vector ω = (π/3)e3 . Material surfaces can also possess twist disclinations in the form of local intrinsic orientational anomalies, which correspond to breaking of the reflectional symmetries of the 2-dimensional material with the local tangent plane of the surface as the mirror plane, e.g., hemitropic plates [37,115]. They are represented mathematically as ill-defined (multi-valued) local orientation field over the surface. Note that, in order to quantify the disclinations discussed so far, the loop of circumnavigation is restricted always within the surface. All the disclinations shown in Figures 1.8(a,b) and Figures 1.4, as well as the intrinsic orientational anomalies discussed above, are quantified using an in-surface loop C. The case otherwise can appear in 2-dimensional homogenized models of thin 3-dimensional multi-layered structures, e.g., a stack of few monolayers of smectics or cholesterics, thin multi-walled nanotubes, or a thin slice of some 3-dimensional oriented media. In these structures, disclinations may appear over the representative base surface (often the ‘mid- surface’ of the layered structure) as the homogenized or effective rotational anomaly of all the distributed disclinations across the thickness of the thin structure. In describing these disclinations, the loop of circumnavigation must be taken transversely to the base surface, see Figure 1.8(c). Depending on the direction of the resulting vector of angular mismatch, these disclinations may either be of wedge or twist type. Sandwiched layer

b

Figure 1.9: Isolated edge dislocation in a thin multi-layered structure. The marks on the surface represent lattice points which may carry identical atoms as well as directors.

Dislocations.

The translational anomalies are represented by dislocations. The nature of dislocations

in 2-dimensional matter is analogous to that in 3-dimensional materials. Isolated edge and screw surface dislocations are shown in Figures 1.2(a) and 1.2(b), respectively, within a 2-dimensional cubic lattice

17

along with the Burgers parallelograms. The Burgers vector, defined as the closure failure of the Burgers parallelogram, is tangential to the surface of the lattice in the former case and transverse in the latter. In these examples, the dislocations appear essentially due to the breaking of the intrinsic translational symmetries of the 2-dimensional matter. On the other hand, in thin multi-layered structures or thin slices of oriented media, dislocations may be present, irrespective of the crystallinity of the material, as a result of either an order-mismatch of individual layers within the stack or as a homogenized or effective limit of all the distributed dislocations within the 3-dimensional slice. The Burgers parallelogram is, naturally, transverse to the representative mid-surface of the stack, in contrast to the examples shown in Figures 1.2(a,b). The precise type of these dislocations, edge or screw, can be determined from the direction of the Burgers vector. An edge dislocation in a layered medium is shown in Figure 1.9, arising due to the presence of a sandwiched semi-infinite layer between two infinite layers of material [71, Ch. VI].

differential growth across the thickness in a multi-layered structure

wavy edges due to surface growth in a finite plate

(a)

(b)

Figure 1.10: (a) Incompatible surface growth of a plate. (b) Differential growth across the thickness of a thin multi-layered structure. Metric anomalies.

The metric anomalies bring about ambiguity in the (local) notion of “length”

and “angle” over the surface. Metric anomalies are generated due to intrinsic point imperfections such as vacancies and self-interstitials, see Figure 1.5(a,b), as well as a result of in-surface thermal

18

C4 vf

vi C3 C5

C2

C6

C1

uf π ui

Figure 1.11: Making a cylinder, a torus and a M¨obius strip from a plane sheet of paper. The cylinder is multiply-connected and orientable, so is the torus, whereas the M¨obius strip is multiply-connected but non-orientable. A global defect in the M¨obius strip appears in the form of a twist disclination of Frank angle π around C6 . It characterizes its non-orientability. deformation and biological growth, see Figure 1.10(a). If the distance between the constituent entities in a lattice structure is measured by counting lattice steps, the presence of point defects, such as a vacancy or a self-interstitial, clearly introduces ambiguity in this step counting [67]. Apart from these pure in-surface metric anomalies, differential growth (or thermal deformation) across the thickness direction within a thin multi-layered structure may result in transverse metric anomalies within an appropriately homogenized 2-dimensional theory, see Figure 1.10(b).

1.2.2

Global defects in structured surfaces

Instances of global anomalies can be found in a variety of 2-dimensional complex structures; during self assembly of certain copolymers in colloidosomes, toroidal micelles have been found to be energetically

19

more favourable over spherical or cylindrical topologies within a critical range of certain physical parameters, and when phase transformation occurs from the unstable spherical or cylindrical to the stable toroidal topology (driven by some internal or external agency), one or more global defects must be introduced in each spherical/cylindrical droplets of the unstable phase to achieve the new topology [55, 99]. Excellent illustrations of global defects in cylindrical, toroidal and M¨obius crystals are provided by Harris [50–52]. In Figure 1.11, a cylinder has been shown to be developed from a planar rectangular sheet of paper by identifying its two opposite edges. This process introduces a set of irreducible loops, such as C3 in the above diagram, over the cylinder. The irreducible loops cannot be continuously shrunk to points on the surface. There are two mutually non-homotopic sets of irreducible loops on a torus, e.g., C4 and C5 in Figure 1.11, meaning that irreducible loops belonging to two different sets cannot be continuously deformed into one another without leaving the surface. Transformation of the cylinder into a M¨ obius strip is achieved by introducing a twist: when the tangent vector uI is parallelly transported along the irreducible loop C6 with respect to the inherited Euclidean connection from R3 , as illustrated in Figure 1.11, it returns to the orientation uF that makes an angle π with the initial orientation uI . This twisting can be also described in terms of a global twist disclination whose defect line falls outside the surface (in this example, it coincides with the axis of the cylinder). The global twist disclination, clearly, changes the orientability of the original structure (M¨obius strip is non-orientable while cylinder is orientable). Similarly, transforming a cylinder into a truncated cone can be described by introducing a global wedge disclination into the cylinder, with the disclination line coinciding with the cylindrical axis. In the present thesis, however, we will not study such topological transformations introduced by global anomalies just described; instead, we will focus on the global anomalies that preserve the original topology of the surface. These global topology-preserving anomalies are illustrated in Figures 1.12 and 1.13 for a toroidal surface. In Figure 1.12, formation of (a) a pure global edge dislocation, (b) a combined global defect consisting of a global dislocation of mixed (edge and screw) type and a global wedge disclination, and (c) a global twist disclination, are demonstrated by removal or insertion of 20

b b 1 1

A

2

B

A

B

2

Ω

10

W

A0

20

10

W

20

B0

B0

A0

(a)

(b)

A A0 1 0 2 1 20

Ω

(c)

Figure 1.12: Formation of global topology-preserving anomalies on a torus by removal or insertion of a wedge, and subsequent joining of the two open lips. (a) Removal of a wedge W with parallel faces A0 and B 0 from the torus, and subsequent joining of the open lips A and B produces a global edge dislocation with Burgers vector b. (b) Insertion of a wedge W with non-parallel faces A0 and B 0 into the torus, by joining the lips A with A0 , and B with B 0 produces a global anomaly consisting of a global dislocation mixed (edge and screw) type, with Burgers vector b, and a global wedge disclination with Frank tensor Ω. (c) Cutting the torus along a vertical irreducible loop, such as C5 as shown in Figure 1.11, would produce the two lips A and A0 infinitesimally separated from each other. Then upon twisting the torus with a rotation Ω, as shown, such that the points 1, 2 etc. on the left lip A merge with the points 10 , 20 etc. on right lip A0 , a global twist disclination will be produced with Frank tensor Ω; its defect line coincides with the circular centerline of the torus. a wedge and subsequent joining of the two open lips. In Figure 1.13, the global defects are created by removing a ring around the outer equatorial circle of the torus, and subsequent joining of the two lips. In every case, the defect line of the global anomaly fall outside the surface, unlike the local anomalies whose defect lines always cross the surface. This is the primary characteristic of global anomalies. We have the measure b, the global Burgers vector, for the global dislocations, and Ω, the global Frank tensor, for the global disclinations. These constructions of global anomalies on torus 21

Ω

A

1

b

A0

10

Figure 1.13: Formation of global topology-preserving anomalies on a torus by removing a ring around the outer equatorial circle, and subsequent joining of the two open lips. The type of the global dislocation or the global disclination is determined by the direction of the global Burgers vector b and the global Frank tensor Ω. are analogous to the well-known Volterra construction of dislocations and disclinations in cylindrical surfaces [125]. In this respect, we would like to point out a novel feature of local wedge disclinations

W

P P0

Figure 1.14: Volterra construction associated with a local negative wedge disclination on a toroidal crystalline surface. With every local wedge disclination at P , an associated antipodal disclination P 0 of the same type will be generated; the antipodal disclination P 0 is located at the point where the defect line of the disclination at P cuts the torus. Note that two small discs have also been removed around the pointed tips of the wedge W in order to avoid the singularity, which characterize the inherent topological nature of the local disclinations.

22

in surface crystals with compact and closed (no boundary) topologies, such as spherical and toroidal crystals [12], which has not been emphasized in the existing literature. This is related to the associated Volterra construction of these local defects, which we have illustrated for a toroidal surface in Figure 1.14. With every local wedge disclination at P , there is an associated antipodal disclination of the same type, located at a point where the defect line of the original disclination cuts the surface. Formation of the antipodal disclination is inevitable due to the closed, compact topology of the torus. This is also true for each of the 12 negative wedge disclinations on a spherical crystal, which are necessarily required on every spherical topology [12]. The simple examples described above are sufficient to motivate the non-Euclidean nature of the defects. Recall that, in order to quantify disclinations, we required circumnavigation of a vector along a loop and rotational mismatch between the initial and the final orientation of the vector. These notions correspond, respectively, to parallelly transporting a vector with respect to an affine connection and to a Riemann-Christoffel curvature associated with the affine connection. The Frank vector ω uniquely characterizes the Riemann-Christoffel curvature tensor [2]. The affine connection has to be necessarily non-Euclidean, since the director fields leading to disclinations are clearly not parallel in the Euclidean sense. Moreover, as the directors may point outside the surface, a differential geometric description of disclinations in structured surfaces would necessarily require embedding the surface into a 3-dimensional space with a specific non-Euclidean connection. In the case of dislocations, the closure failure of the Burgers parallelogram is analogous to the notion of torsion of an affine connection over a manifold which characterizes closure failures of infinitesimal parallelograms [10, 64]. Finally, the metric anomalies are characterized by the non-metricity tensor, which quantifies the non-uniformity of the metric tensor with respect to an appropriate affine connection [4].

1.3

Outline of the thesis

With the above examples as motivation, we rigorously develop the non-Euclidean characterization of material defects in 3-dimensional solids and structured surfaces in Chapter 2. In §2.1, we introduce

23

the notion of material space for 3-dimensional material bodies by attributing a metric and a nonmetric affine connection, with non-zero torsion and curvature, to the 3-dimensional body manifold of a materially uniform simple elastic solid. The metric and the connection are both constructed from a given distribution of inhomogeneities in the body and an assumed constitutive response. In doing so, we extend the formulation of Noll [96] and Wang [126], which is restricted to a dislocated material body, where the material connection and metric are derived solely from the constitutively determined material uniformity field. We describe the geometrical significance of torsion, curvature and non-metricity, and relate them to distributions of dislocations, disclinations and metric anomalies, respectively. The main result in this section is the development of the notion of metrical disclinations and their relation with metric anomalies. The 3-dimensional theory is then used in §2.2 to develop the corresponding theory for structured surfaces. We introduce the notion of material space for structured surfaces—a 2-dimensional manifold embedded in the 3-dimensional physical space (the 2-dimensional body manifold), equipped with a non-Riemannian connection (the material connection), and a metric (the material metric)— which is our prototype for continuously defective structured surfaces. To make the model sufficiently general so as to take both the in-surface and out-of-surface defects into account, we consider the material connection and metric as being appropriately induced from a 3-dimensional connection and metric of a suitable 3-dimensional neighbourhood of the 2-dimensional body manifold. Drawing analogies between the fundamental geometric objects on the material space and the illustrations in Section 1.2, we then identify the components of the tensors of non-metricity, torsion, and Riemann-Christoffel curvature of the material connection with various components of the distributed metric anomalies, dislocations, and disclinations (also, intrinsic orientational anomalies), respectively. Torsion, curvature and non-metricity of a geometric space cannot be arbitrary due to geometric constraints, expressed through certain differential identities, known as Bianchi-Padova relations. In Chapter 3, we use these identities to derive the non-linear conservation laws satisfied by the defect density fields; in particular, we obtain the classical results that in absence of metric anomalies, dislocation lines must end on disclinations, and disclination lines cannot end inside the body. Moreover, the metrical disclinations introduced in §2.1 can appear only in a non-metric space and are related to path 24

dependence of the inner product of tangent vectors. Unlike the well known rotational disclinations, which are the only kind of disclinations possible in metric-compatible spaces, metrical disclinations are not fundamental line defects in materially uniform simple elastic solids. A distribution of rotational disclinations is also unfeasible in crystalline solids due to their unrealistically high elastic energy. Motivated by these concerns, we look for simplified representations of non-metricity in the absence of curvature in material space. This is equivalent to requiring distant material parallelism for crystalline solids. In §3.1, we focus on obtaining rigorous results of representations for non-metricity tensor for a 3-dimensional zero curvature space. Towards this end, we use the third Bianchi-Padova relation to obtain a necessary and sufficient representation of non-metricity in terms of a symmetric second order tensor. This also leads us to introduce the auxiliary material space which inherits the affine connection from the material space but has a metric such that the non-metricity vanishes identically. In particular, we recover the quasi-plastic strain framework, proposed by Anthony [4], now established on firm geometrical grounds. We also discuss non-metricity in the context of semi-metric geometry (which with zero torsion is called Weyl geometry). We show that the non-metricity tensor therein necessarily has an isotropic form, given in terms of a scalar field, when curvature of the space is identically zero. As a result, the Weyl geometry framework in its standard form, where the Weyl co-vector form is exact leading to an isotropic form of non-metricity [133,134], is insufficient to model anisotropic metric anomalies. We, further, propose a novel representation of metric anomalies in 3-dimensional crystalline solids in terms of a second order tensor (we call it quasi-plastic deformation) such that the total deformation gradient (with respect to a fixed reference configuration) can be multiplicatively decomposed into an elastic and a plastic deformation. Moreover, the plastic deformation is further decomposed multiplicatively in terms of a dislocation induced deformation and the quasi-plastic deformation. Such a framework is amenable to analytical and numerical solutions of boundary-value-problems for (internal) stress and displacement fields for elastic solids having a continuous distribution of dislocations and metric anomalies. The representation of non-metricity in terms of quasi-plastic deformation also allows us to consider a broader range of metric anomalies than what is afforded by quasi-plastic strain framework. In the following §3.2, we apply these 3-dimensional results to obtain the consequences of 25

the Bianchi-Padova identities in structured surfaces. We derive certain symmetries which imply interdependence of two distinct families of disclinations in structured surfaces, various conservation laws for the distributed defects, notable among which are the linearized conservation laws for the out-of-surface family of dislocations and disclinations, inferring that they must always form loops or leave the surface, and a simplified representation for metric anomalies in absence of disclinations. In Chapter 4, we introduce the Riemannian structure on the material space of 3-dimensional defective bodies induced by the material metric, and pose the boundary-value-problem for the determination of internal stress field due to a given defect distribution by identifying the 3-dimensional material metric with the elastic strain field. We also obtain the general form of the non-metricity tensor which corresponds to a zero stress field in the absence of dislocations, disclinations and any external source of stress. We derive a closed form solution to this problem in a linearized situation, assuming non-metricity and elastic strain to be small and of the same order. In Chapter 5, we apply the induced Riemannian structure developed in Chapter 4 to explore the issue of material homogeneity and local and global strain compatibility in continuously defective structured surfaces. In order to use the 3-dimensional results, we construct a 3-dimensional metric field using intrinsic strain measures of a Cosserat surface. This construction is fairly general, and includes consideration of the director shear and normal deformation. We also derive the relationship of the strain incompatibility fields with various local and global (topology-preserving) defect densities which is a precursor to the boundary-value-problems for internal stress and shape. In the penultimate Chapter 6, we formulate the boundary value problems for various internal stress measures and deformed shapes of a certain class of continuously defective structured surfaces under the Kirchhoff-Love deformation constraint and for small strain, moderate rotation. In order to do so, we postulate an additive decomposition of the total in-surface and bending strain fields into respective elastic-plastic parts, under a separation of order of these strain measures. This decomposition is motivated by the theory of small strain accompanied by moderate rotation for sufficiently thin structured surfaces undergoing moderately large deformation. Though stricter than the complicated fully nonlinear theory, this assumption still contains complex non-linearities, and is much more general than the 26

infinitesimal deformation, linear theories. Moreover, as a direct consequence of the global compatibility relations, we obtain a Ces` aro integral formula like expression for constructing the displacement field of from the given strain fields. Upon simplification, under various assumptions, this framework reduces down to forms used in the recent literature on 2-dimensional matter, e.g., F¨oppl-von K´arm´ an equations for continuously defective thin elastic isotropic shells, and the shape equations for continuously defective thin isotropic fluid films. In the final Chapter 7, we make concluding remarks and provide several future scopes of the thesis.

1.4

Novel contributions of the present work

Here, we present a list of the novel contributions made in this thesis. 1. In Chapter 2, (a) We have extended the notion of material inhomogeneity of a 3-dimensional material body, initially introduced in the seminal works of Noll [96] and Wang [126], by including metric anomalies and disclinations, in addition to dislocations, within the 3-dimensional material structure. In the presence of these general defects densities, the 3-dimensional material connection and metric are determined not only by the material uniformity field, but also by the defect density fields, in contrast to Noll and Wang’s theory of a dislocated body where they are determined solely in terms of the material uniformity field. (b) We have introduced a novel defect density, which we call metrical disclinations, related to the change in length of a material vector during parallel transportation along loops. Unlike the well-known rotational disclinations, these metrical disclinations are intertwined to the metrical properties of the body, and vanish in the absence of metric anomalies. We characterize these disclinations in terms of a symmetric part of the material curvature tensor, while the corresponding skew part provides a measure for the standard rotational disclinations.

27

(c) We have given a complete geometric characterization of material defects in structured surfaces in terms of the components of various fundamental tensors defined on the material space of the surface. The defect densities include in-surface metric anomalies (e.g., point defects, in-surface thermal deformation and growth), out-of-surface metric anomalies (e.g., differential tangential growth across the thickness of a biological surface), in-surface edge and screw dislocations, out-of-surface edge and screw dislocations (e.g., in layered and multiwalled structures), in-surface wedge and twist disclinations, out-of-surface disclinations such as those associated with multi-layered media, and metrical disclinations. 2. In Chapter 3, (a) As a consequence of the third Bianchi-Padova relation, we have derived a necessary and sufficient representation of metric anomalies in 3-dimensional bodies free of disclinations, which we call irrotational metric anomalies, in terms of a symmetric second-order tensor field. This representation provides a geometric justification of the quasi-plastic strain framework introduced by Anthony [4]. Our framework also leads to an unambiguous interpretation of the deviatoric part of the quasi-plastic strain in terms of the anisotropic metric anomalies; such an understanding is missing from the existing literature. (b) The quasi-plastic strain formulation, however, is difficult to implement practically for the problems where finite deformations are introduced by metric anomalies such as finite thermal strain or bulk growth. To overcome this, we have introduced a quasi-plastic deformation framework that allows us, in the absence of disclinations, to decompose the plastic distortion tensor multiplicatively into two independent parts responsible for distortions due to dislocations and metric anomalies. Our framework gives a precise geometric justification for such decompositions which have already appeared in literature (see, for example, [22]), while providing rigorous conditions under which such decompositions remain valid. This construction directly gives way to modelling anisotropic metric anomalies resulting into finite incompatible distortions. 28

(c) As consequences of the Bianchi-Padova relations on the structured surface, we have derived various symmetries and conservation laws for the surface defect distributions. These include (i) interdependence of two distinct families of surface disclinations in the absence of outof-surface dislocations, (ii) standard conservation laws for the out-of-surface dislocations and disclinations in a linearized setting that imply that they must always form loops or leave the surface, and (iii) representation of the surface metric anomalies in the absence of disclinations. 3. In Chapter 4, we have derived the incompatibility relations for the elastic strain fields over a 3dimensional solid for a given distribution of disclinations, dislocations and metric anomalies. The strain incompatibility relations with the provided generality have not reported anywhere in the present literature. Within a linearized setting, we have also derived a closed form representation of stress-free distribution of metric anomalies in absence of other defects. 4. In Chapter 5, (a) We have presented a novel proof of the local strain compatibility relations for various 2dimensional strain measures of a single director shell, adopted to represent the kinematics of a general structured surface, by appropriately constructing the material metric within a thin 3-dimensional material neighbourhood of the structured surface from the 2-dimensional strain fields. Our 2D strain measures include the effects of director tilt and thickness distension, along with tangential and normal bending. The local strain compatibility conditions when the director field is tangential to the surface (for instance, in nematic elastomers) are also discussed. This construction straightforwardly leads to the local strain incompatibility relations in a following section where the presence of defects are also considered. (b) One of the central contributions of this thesis is the formulation of global compatibility conditions for the 2-dimensional strain measures of a Kirchhoff-Love shell. We have derived these conditions for all the topologies attainable by a 2-dimensional manifold embeddable in the 3-dimensional Euclidean space R3 , namely, open disc, disc with holes, 2-sphere, possibly 29

with a finite number of handles attached (e.g., torus which has one handle), and twisted bands, e.g., M¨ obius strip. (c) We have derived the local strain incompatibility relations for a general single director shell due to the presence of distributed local anomalies, including in- and out-of-surface dislocations, disclinations and metric anomalies. Local strain incompatibility relations for elastic shells with this generality have not reported anywhere in the present literature. Under the Kirchhoff-Love deformation constraint, these general relations can be reduced to results widely used in the present literature [95]. (d) Within the Kirchhoff-Love setting, we have derived the global strain incompatibility relations for all the topologies attainable by the structured surface embeddable in R3 , owing to the presence of certain global/topological anomalies. These global anomalies arise primarily as a response to the topological properties of the surface, for example, multiply connectedness or twistedness. They have not been considered in the existing literature. These global/topological barriers to strain compatibility can provide additional insights to study the behavior of defective spherical, toroidal, or M¨obius crystalline surfaces, presently lacking in the relevant frameworks [12]. 5. In Chapter 6, (a) We have posed the complete boundary value problems to determine the internal stress field and deformed shape of a structured surface under the Kirchhoff-Love kinematic assumption. The boundary value problems consists of (i) the local and global incompatibility relations for the plastic strain fields for given distributions of local and global anomalies, (ii) additive decomposition of the total strain fields into elastic-plastic parts based on suitable assumptions of separated order, (iii) equilibrium equations satisfied by the surface stresses and moments, and (iv) constitutive relations. (b) These problems are then simplified to derive the F¨oppl-von-K´arm´an (F-v-K) equations for shells with arbitrary reference geometry. Such shells are capable of undergoing moderately 30

large rotations, while maintaining small in-surface stretching, in order to screen the elastic energy of the distributed defects. As a novel application of the general F-v-K equations, we have discussed the role of global strain incompatibility due to the presence of global topological defects in 2-dimensional crystals, where the base surface may assume any topology that it can attain in 3-dimensions, thus providing a generalization of the frameworks used in the present 2-dimensional condensed matter physics research [12, 95]. Also, within our generalized framework, these existing works can be extended to include more enriched class of local defects in 2-dimensional matter, such as twist disclinations, or defects in thin multi-walled structures. (c) As another novel application of our general F-v-K equations, we have introduced a secondorder growth model for biological surfaces to account for the tangential differential growth along the thickness direction—which gives rise to a non-trivial plastic bending strain field, and as a consequence, a screening mechanism for the mean curvature—that can play a key role in determining shape and stability of growing biological surfaces. The existing relevant work [77, 78] is only for shallow shells (i.e., moderately curved reference geometry), and considers only a pure in-surface growth model (which screens only the Gaussian curvature) while ignoring the role of growth curvature, and the resulting mean curvature screening effect altogether.

31

Chapter 2

Geometric Characterization of Defects1 The aim of this chapter is to develop the necessary mathematical paraphernalia to characterize material defects in terms of various fundamental geometric objects on the material space, which is a suitable nonRiemannian differential manifold associated with a continuously defective material body. In the first Section 2.1, we discuss the 3-dimensional theory, which is followed by the theory of structured surfaces in Section 2.2. Novel contributions made in Section 2.1 include a method of construction of the material connection and the material metric associated with the material space of a 3-dimensional materially uniform, simple, elastic solid with continuous distribution of dislocations, disclinations and metric anomalies. This generalizes the proposal of Noll [96] and Wang [126] for a 3-dimensional dislocated solid. We also develop the notion of a novel defect, which we call metrical disclinations, associated with the change in length of crystallographic vectors under parallel transport along loops. In Section 2.2, we give a complete characterization of all the possible topological and metrical anomalies, both in- and outof-surface, in 2-dimensional material surfaces (see Table 2.1). We do so by appropriately constructing a thin 3-dimensional material neighbourhood about the surface and identifying the components of the restrictions of various fundamental geometric objects on this 3-dimensional neighbourhood to the embedded surface. 1

Section 2.1 of this chapter appeared in [110].

32

2.1

Characterization of defects in 3-dimensional solids

Our prototype for the theory of a continuous material body is a connected, compact 3-dimensional differential manifold B which can be covered with a single chart. B is classically known as the body manifold for the 3-dimensional body [96,126] and the points in B, designated by X, are called material points. We assume the manifold structure on B to be sufficiently differentiable as the context demands and use a holonomic curvilinear coordinate system θi to label the material points X ∈ B. Lowercase Roman indices (i, j, p etc.) take values from the set {1, 2, 3}, an lowercase Greek indices (α, β, γ etc.) from the set {1, 2}; Einstein’s summation convention holds over repeated indices unless specified otherwise. From its manifold structure, B naturally inherits the Euclidean properties of R3 , including the Euclidean inner product (denoted by ·). Let Gi be the natural basis vector field of the coordinate system θi , Gij := Gi · Gj be the components of the Euclidean metric tensor with respect to the coordinates θi , [Gij ] := [Gij ]−1 , and Gi := Gij Gj the natural co-vector basis field. With the tangent space TX B of B at X as the underlying vector space, we denote Lin and InvLin+ as the set of all second order tensors and invertible second order tensors with positive determinant, respectively, Sym and Sym+ as the set of all second order symmetric and symmetric positive definite tensors, respectively, Skw as the set of all second order skew symmetric tensors, U nim as the set of all second order tensors with determinant equal to 1 and Orth+ as the set of all proper orthogonal second order tensors (i.e., rotations). We denote the identity tensor field over the manifold B by I := Gij Gi ⊗ Gj = Gij Gi ⊗ Gj . The inverse of an invertible tensor is indicated by a superscript (−1) while the transpose is denoted by a superscript T . We use the shorthand notation (·),i for the ordinary partial derivative

2.1.1

∂(·) . ∂θi

Material response

We restrict our consideration to materials classically known as simple elastic solids (without heat conduction). The constitutive response function for such a material is given by a mapping ˆ : Sym+ × B → R+ , W 33

(2.1)

known as the strain energy density function. Here, R+ denotes the set of non-negative real numbers. In addition, the body is assumed to be materially uniform, i.e., the material at any two points of the body is the same.2 More formally, for every pair X, Y ∈ B, there exists a second order tensor K XY : TX B → TY B, with det K XY > 0 (det denotes the determinant operator), such that ˆ (K T h K XY , X) = W ˆ (h, Y ) W XY

(2.2)

is satisfied for all h ∈ Sym+ .3 It can be easily shown that the set of values of K XY satisfying (2.2), for fixed X, Y ∈ B, satisfies the group axioms, namely, closure, associativity, identity, and invertibility, and, hence, forms a group KXY [96]. Moreover, the material symmetry group at X, defined as GX := KXX , in order to conform to the mass consistency condition, must satisfy GX ⊆ U nim [96]. Fixing a material point X 0 ∈ B in the materially uniform body, we can define a field K(X) := K X 0 X that satisfies ˆ (h, X) WX 0 (K T (X) h K(X)) = W

(2.3)

ˆ (·, X 0 ). K is known as the for all h ∈ Sym+ , where WX 0 : Sym+ → R+ is defined as WX 0 (·) := W material uniformity field (or the material transplant) with respect to the material point X 0 [36,96,126]. Since the body is assumed to be materially uniform, the choice of the material point X 0 is arbitrary. This renders the constitutive response function independent of any explicit dependence on material points, as is clear from the expression (2.3). The body is called a materially uniform solid, if we can choose X 0 such that GX 0 ⊆ Orth+ .4 In the present work, we restrict ourselves to materially uniform 2

Example of a materially non-uniform body would be functionally graded materials where the material constitution

changes across the body. For a differential geometric theory of such bodies, see [35]. 3 ˆ (·, X), for X ∈ B, is customarily assumed to be InvLin+ , the space where The domain of the partial function W the deformation gradients reside [96, 126], which, under the Principle of Material Frame Indifference, gets restricted to its subset Sym+ . In presence of certain material inhomogeneities (e.g., disclinations), a well-defined element in InvLin+ may not exist to appear in the constitutive function. Our treatment bypasses this limitation, as it is always guaranteed ˆ (·, X) as an argument. This well-defined element could be, in that a well-defined element in Sym+ exists to appear in W our context, any of the standard measures of strain. 4

Apart from the point symmetry group G, which essentially describes rotational symmetries, the material structure

presently under consideration possesses, due to its expanse in the Euclidean 3-space, spatial translational symmetries.

34

simple elastic solids.

2.1.2

Material G-structure, material connection and material metric

The material uniformity field K appearing in (2.3) is, in general, multi-valued due to non-trivial symmetry groups at X as well as at X 0 . A fiber bundle can be constructed by attaching the values of K(X) at respective X ∈ B, giving rise to the material G-structure [36, 126]. It has been shown that the material G-structure is a principal fiber bundle, with structure group (which is the same as the standard fiber) G := GX 0 (see, for example, [36, 126]). The domain of K, given a fixed degree of differentiability C k , may not span the whole material manifold B. The material G-structure is hence, in general, non-trivial. Remark 2.1.1. Note that the fiber at a point X ∈ B in the material G-structure, which consists of the all values of the material uniformity field K(X), is a diffeomorphic image of the standard fiber G ⊆ Orth+ under mappings whose linear parts have non-trivial, but positive, determinant. Hence, any generic fiber of the material G-structure can be represented by the general linear group of invertible second order tensors in R3 (see, for example, [36, Section 9.3]). For geometric constructions on the material G-structure, it can be equipped with an arbitrary affine connection and a metric. However, in order to formulate a geometric theory of the underlying material structure, we choose only a particular affine connection L and a particular metric g out of these infinite possibilities, as informed by the inhomogeneities present in the material structure, if any. The fundamental geometric objects associated with L and g, namely, the torsion tensor T, the curvature tensor Ω and the non-metricity tensor Q can be naturally identified with the densities of dislocations, disclinations and metric anomalies, respectively, as we will see in the following. Once these identifications are made, and the defect densities in a given material body are known in terms of these fundamental geometric objects over B, the connection L and the metric g can be constructed from these geometric objects by solving a system of PDEs. This system is constituted of the respective defining equations of the tensors T, Ω and Q in terms of L and g (see, respectively, the equations (2.7),

35

(2.9) and (2.14) below). For instance, the governing equation (2.9) for L, for an appropriately specified tensor Ω (the density of disclinations), is a system of first order non-linear PDEs. From a result by Talvacchia [123, Theorem 7] on the existence of an affine connection over a U nim-principal bundle5 with 3-dimensional base manifold whose curvature tensor is prescribed a priori as a generic real analytic function, and from the fact that the material G-structure is a principal subbundle of a U nim-principal bundle (since G ⊆ Orth+ ⊂ U nim) with the 3-dimensional base manifold B, it follows that a solution L exists, provided that Ω is analytic which we assume to be the case here. The torsion tensor obtained from this affine connection L has to be equal to an appropriately specified tensor T (the density of dislocations), see (2.7). On the other hand, the governing equation (2.14) for g is a system of first order nonhomogeneous linear PDEs, with appropriately given functions Q (the density of metric anomalies) and L. If we assume Q and the previously obtained L, as just discussed, to be analytic, then a unique analytic solution g indeed exists as a consequence of the Cauchy-Kowalevski and Holmgren’s existence and uniqueness theorems for first order linear system of PDEs with analytic coefficients and data.6 The affine connection L, thus constructed, is called the material connection, and the metric g, the material metric. The “material” nature of this connection and metric is clear from the above discussion; the fundamental geometric objects they yield, viz. the torsion, curvature and non-metricity, represent various inhomogeneity measures present in the material structure of the body. Remark 2.1.2. The above construction of material connection and material metric generalizes the original idea of Noll [96] and Wang [126], which is purely constitutive in nature and is applicable to dislocated material bodies free from any curvature and metric anomalies. The general form of a zerocurvature connection is determined solely in terms of an invertible second order tensor field which, in turn, determines the metric compatible with the connection (see Section 3.1.6 for details); Noll 5

A U nim-principal bundle is a principal fiber bundle whose structure group is the group U nim.

6

Within the realm of the classical solutions of the PDEs that we are considering here, the existence and uniqueness

theorems of Cauchy-Kowalevski and Holmgren do not extend to the class of smooth functions which are not analytic; in this context, we would like to refer to the well-known Lewy’s example that demonstrates a linear PDE with smooth coefficients which has no solution [75].

36

identified this second order tensor with the material uniformity field K. In the presence of curvature and metric anomalies, material connection and metric can no longer be derived from the material uniformity field alone but require additional information which comes from a given distribution of curvature and metric anomalies. In fact, with whatever connection and metric one is adorning the material G-structure, they should be compatible with the underlying constitutive nature of the body and the contained inhomogeneities.

2.1.3

The material space

Manifold B, equipped with the material connection L and the material metric g, forms the material space [96]. We denote the material space by the triple (B, L, g) and the coefficients of L and the components of g, with respect to the embedded coordinate system θi , by Likj and gij , respectively. The raising and lowering of indices of components of tensorial objects are performed with respect to the purely covariant components gij and the purely contravariant components g ij , where [g ij ] := [gij ]−1 . The covariant differentiation of a quantity with respect to L is denoted by the subscript (·);i , for example, ui;j := ui,j + Lijk uk , ui;j := ui,j − Lkji uk etc.,

(2.4)

where all the components are with respect to the coordinates θi . Given a tangent vector with components ui0 at the initial point of a curve C := {C i (s)Gi (s) ∈

B s ∈ [0, 1]}, a materially constant tangent vector field ui (C k (τ )) is constructed by solving the linear ODE

dui (τ ) = −Likj (τ )uj (τ )C˙ k (τ ), dτ

ui (τ = 0) = ui0 .

(2.5)

In other words, a materially constant vector field on a curve, by definition, has zero directional covariant derivative with respect to the material connection throughout the curve. This process of obtaining a materially constant vector field, starting with the vector ui0 , is also known as parallelly transporting the given vector ui0 along C with respect to the connection L. A curve C whose tangent vectors are all materially constant or parallel is called a geodesic [113, Chapter 1].

37

It is evident from our construction of the material connection L and the material metric g, as discussed previously, that the fundamental geometric objects of the material space (B, L, g) provide natural measures for various material inhomogeneities contained within the body. Importantly, the material inhomogeneity densities remain unaffected by a superposed compatible deformation of the body. In other words, material space is a geometric space where an internal observer would be able to detect only configurational changes in the body (i.e. those arising out of defects) but would otherwise fail to observe any compatible deformations incurred by the body as a result of external loading, etc. [67].

2.1.4

Material torsion tensor

The third order torsion tensor T associated with the material connection L is a mapping T : TX B × TX B → TX B,

(2.6)

which is bilinear and skew with respect to its arguments. Its components T˜jk i with respect to the coordinates θi are defined as [112] T˜jk i := Li[jk] .

(2.7)

Here, the square bracket in the subscript indicates the skew part of the field with respect to the enclosed indices (whereas a round bracket is used to indicate the symmetric part). Torsion tensor measures the closure failure of an infinitesimal parallelogram in the material manifold, and it is one of the fundamental geometric objects on the material space. The construction of such a parallelogram is illustrated in Figure 2.1(a).

Torsion inhomogeneities: Torsion tensor of the material manifold is the natural measure for density of dislocations in the material body (first identified by Kondo [64] and, later, independently by Bilby et al. [10]). Dislocations are one of the fundamental line defects in materially uniform simple elastic solids; they are associated with the translational symmetries of the underlying material structure. This identification is evident from the similar nature of the two objects, viz. the closure failure of infinitesimal

38

2T˜jk iuk v j ui + Lijk uk v j v i + Likj v j uk

vi ui X

B

B

(a)

(b)

Figure 2.1: (a) Two tangent vectors at the material point X ∈ B, with components ui and v i , are parallelly transported along each other. The closure failure of the resulting infinitesimal parallelogram is characterized by the components T˜jk i of the torsion tensor. (b) Change in length and angle between the tangent vectors under parallel transport due to non-metricity. parallelograms in the material space and the closure failure of the Burgers circuit [67]. The second order axial tensor of torsion, which has components α ˜ ij := ε˜imn T˜mn j , is called the dislocation density 1

1

tensor. Here, ε˜ijk := g − 2 eijk , and for later use, we define ε˜ijk := g 2 eijk , where eijk = eijk is the 3-dimensional permutation symbol and g := det[gij ]. The diagonal components of the matrix [α ˜ ij ] measure the density of edge dislocations and the off-diagonal components measure the density of screw dislocations [67].

2.1.5

Material curvature tensor

The fourth order Riemann-Christoffel curvature tensor Ω of the material connection L is a mapping Ω : TX B × TX B → Lin,

(2.8)

which is bilinear and skew with respect to its arguments. Its components Rjiq p with respect to the coordinates θi are defined as [112] ˜ jiq p := Lp − Lp + Lhiq Lp − Lhjq Lp . Ω iq,j jq,i jh ih 39

(2.9)

The curvature tensor Ω measures, in the linear approximation, the change that a tangent vector suffers under parallel transport along an infinitesimal loop. It is our second fundamental geometric object on the material space. ˜ klji of Ω, by lowering the fourth index with the We also define the purely covariant components Ω material metric gij , as ˜ klji := gip Ω ˜ klj p . Ω

(2.10)

˜ klq p = −Ω ˜ lkq p and Ω ˜ klij = −Ω ˜ lkij . It follows from the definition that Ω

2.1.6

Decomposition of the curvature tensor

˜ klij as [102] It is useful for our present objective to decompose the components Ω ˜ klij = ε˜pkl ε˜qij Θ ˜ pq + ε˜pkl ζ˜ij p , Ω

(2.11)

˜ pq and ζ˜ij p are defined as where Θ ˜ ij[kl] ˜ pq := 1 ε˜pij ε˜qkl Ω Θ 4



1 ˜ ijkl = ε˜pij ε˜qkl Ω 4



1 ˜ kl(ij) . and ζ˜ij p := ε˜pkl Ω 2

(2.12)

˜ := Θ ˜ pq Gp ⊗ Gq characterizes the skew part and the third order tensor The second order tensor field Θ field ζ˜ := ζ˜ij p Gp ⊗ Gi ⊗ Gj characterizes the symmetric part of the tensor field Ω(·, ·) ∈ Lin.7 The geometric interpretation of the symmetric and skew part is illustrated in the following. Let us consider an infinitesimal planar loop γ inside B originating and terminating at X. A tangent vector v at X, when parallelly transported along γ, suffers a change δv which, in the linear approximation, can be characterized by a second order tensor β := βij Gi ⊗ Gj , i.e., δv = βv, where βij is given by [112] βij 7

:= −

δA ˜ ˜ pq np − δA ζ˜ij p np . Ωklij ε˜rkl nr = −δA ε˜qij Θ 2

(2.13)

˜ is often called the Einstein tensor. This is from the context of general theory of The symmetric part of the tensor Θ

˜ identically vanishes, as the geometry of the spacetime continuum in Einstein’s theory relativity where the skew part of Θ is torsion free and metric compatible.

40

n γ δA

n γ

v

˜ pq npv j Gi ⊥ v δv ≈ −δA˜ εqij Θ

δA

δv ≈ −δAζij pnpv j Gi k v

B

v0

v

v0

B

(a)

(b)

˜ measures the stretching Figure 2.2: (a) The symmetric part of Ω(·, ·), characterized by the tensor ζ, ˜ part of the change brought about by Ω. Here, v is a principal vector of ζn. (b) The skew part of ˜ measures the purely rotational part of the change brought about Ω(·, ·), characterized by the tensor Θ, by Ω. Here, δA is the area of the infinitesimal flat surface bounded by γ and n := nr Gr its unit normal. The ˜ pq np in the above expression is skew with an axial vector wm = Θ ˜ pm np δA. first term Wij := −δA ε˜qij Θ It represents the amount of rotation with respect to the axis Gp , for a fixed p, given by three Euler angles ˜ is the measure of a small rotation about the axis n. The second term Sij := −δA ζ˜ij p np ˜ pq . Hence, Θ Θ ˜ as measures is symmetric; it represents a stretching, with the three principal values of the tensor ζn of the stretch along its three principal directions (Figure 2.2(a,b)). As an example, let us assume first that ζ˜ = 0 and the coordinates θi are orthonormal (i.e., Gi = Gi ) ˜ 3q . locally at a point X. Let the infinitesimal loop γ be such that n(X) = G3 . Then, βij = −δA ε˜qij Θ ˜ 3q Gi = Choose v = G1 . The deviation, after parallel transport along γ, is given by δv = −δA ε˜qi1 Θ ˜ 33 G2 − δA Θ ˜ 32 G3 . Since δv has no component along v, it is evident that v has suffered a rotation δA Θ ˜ = 0, θi s and γ as above, and v i as the principal ˜ 32 and Θ ˜ 33 . Next, assume that θ characterized by Θ direction of ζ˜ij p np = ζ˜ij 3 with the principal value λ, i.e., ζ˜ij 3 v j = λvi . The deviation after parallel transport along γ is now given by δv = −δA ζ˜ij 3 v j Gi = −δA λv. Clearly, there is a stretching of the vector along its original direction.

41

Curvature inhomogeneities:

Curvature inhomogeneities are known as disclinations. As is evident

from the above discussion, there are two independent sources that might lead to disclinations: the ˜ The Θ-disclinations ˜ and the third order tensor ζ. ˜ second order tensor Θ are pure rotational anomalies ˜ was first made in the material structure. Identification of rotational disclinations with the tensor Θ by Anthony [2]. Rotational disclinations are the second kind of fundamental line defects which our material structure (i.e., materially uniform simple elastic solid) allows.8 They are associated with the rotational symmetry group G of the material. The pure rotation that a vector suffers under parallel ˜ transport along a loop in the material space due to the presence of Θ-disclination lines piercing this loop necessarily belongs to G. A non-zero ζ˜ measures the distribution of another kind of disclinations in the material structure. The disclinations characterized by the tensor ζ˜ are not fundamental line defects in the present material structure under consideration (see Footnote 8). As already seen, these are related to the stretching of vectors under parallel transport along loops, and hence, are associated with the metrical properties ˜ of the material space. We simply call them ζ-disclinations or metrical disclinations. We will shortly prove that metrical disclinations cannot exist in a metric compatible manifold (they require a certain kind of non-metricity to exist). Materials with more enriched symmetry groups, for which generalized Volterra processes exist, can indeed sustain these metrical disclinations as fundamental line defects, as has been observed in the context of general relativity [62]. The absence of disclinations whatsoever is classically known as distant material parallelism; in that case, the associated material space is necessarily materially flat, as the Riemann-Christoffel curvature Ω of L will identically vanish. Under distant material parallelism, crystallographic vector fields can ˜ or ζ˜ will lead be unambiguously defined over the whole material space. Non-zero values of either Θ to deviation from distant material parallelism. For an unambiguous definition of crystallinity at every point, distant material parallelism is a required condition (see also Section 3.1.2 in Chapter 3). 8

This is with reference to Weingarten’s classical theorem [129] in linear elasticity and the subsequent construction of

elementary dislocations and disclinations by Volterra [125] as the fundamental line singularities in a linear elastic solid. The same construction also holds in non-linear elasticity, cf. [138, Chapter 1], [132] and [26].

42

2.1.7

Material non-metricity tensor

˜ kij Gi ⊗ Gj ⊗ Gk of the material manifold is defined as The (third order) non-metricity tensor Q := Q the negative of the covariant derivative of the material metric g with respect to the material connection L [112]: ˜ kij := −gij;k = −gij,k + Lp gip + Lp gjp , Q ki kj

(2.14)

where the negative sign in the definition is conventional. It measures how the measuring scale for length and angle, i.e., the material metric, varies over the material space. It forms the third fundamental geometrical object (see Figure 2.1(b)).

2.1.8

Unambiguous definition of a metric tensor field

Let us consider an infinitesimal parametric loop C := {C i (s)Gi (s) ∈ B s ∈ [0, 1], C k (0) = C k (1)},

starting and ending at the origin of the coordinate system θi , i.e., C k (0) = 0, and let the metric tensor

0 . We would like to at the base point s = 0 of this loop be given as some appropriate functions gij 0 remains invariant under parallel transport along the loop and, consequently, investigate whether gij 0 along the loop gives rise to a metric tensor field on the material space. To proceed, we transport gij

by solving the PDE ˜ kij gij,k − Lpkj gip − Lpki gjp = −Q

(2.15)

˜ kij and Lp . The above PDE follows from the definition (2.14). At an along C, with known functions Q ij arbitrary position s on the loop we have gij (s) =

0 gij

−

Z

s

0

˜ kij (τ ) C˙ k (τ ) dτ + Q

Z

0

s

Lpki (τ ) gpj (τ ) C˙ k (τ ) dτ

+

Z

0

s

Lpkj (τ ) gip (τ ) C˙ k (τ ) dτ. (2.16)

˜ kij (τ ) and gij (τ ) within first order in C k (τ ) as Let us expand Q ˜ kij (τ ) ≈ Q ˜ kij (0) + Q ˜ kij,m (0) C m (τ ), Q 0 0 0 ˜ kij (0) C k (τ ) + Lp (0)gip gij (τ ) ≈ gij −Q C k (τ ) + Lpki (0)gjp C k (τ ). kj

43

(2.17a) (2.17b)

F after After some careful calculations, it can be shown that the above relations give us the final value gij

the parallel transport along C to be F gij

≈

0 gij



p ˜ ˜ kij,m + Lp Q ˜ ˜ mk(ij) − Q + Ω ki mpj + Lkj Qmpi




 I C m dC k . (0) [mk]

(2.18)

C

The expression in the square bracket is identically zero from the third Bianchi-Padova relation (3.7) (see Section 3.1.2 in Chapter 3). Hence, the definition of the metric tensor is always unambiguous (path independent) in spite of the presence of non-metricity in the space, and this is the very reason why the material metric exists. But the inner product which comes via this unambiguous metric is not unambiguous, as we will see next.

2.1.9

Parallel transport of the inner product and its path dependence

Let us calculate how the inner product g(u, v) = gij ui v j of two tangent vectors, with components ui and v j , changes under parallel transport along the small parametric loop C as defined above (note that u and v are not necessarily tangential to the surface). According to the definition (2.14), upon parallel transport to a generic point s on C, the inner product between the said vectors is given by i j

Z

s

(gij ui v j ),k (τ ) C˙ k (τ ) dτ  Z s j = gij ui v j (0) + gij,k ui v j + gij ui,k v j + gij ui v,k (τ ) C˙ k (τ ) dτ Z0 s 
˜ kij + Lp gpj + Lp gip ui v j = gij ui v j (0) + −Q ki kj 0  +gij (ui − Li up )v j + gij ui (v j − Li v p ) (τ ) C˙ k (τ ) dτ i j

gij u v (s) = gij u v (0) +

0

;k

= gij ui v j (0) −

kp

Z

;k

s

kp

˜ kij (τ )ui (τ )v j (τ ) C˙ k (τ ) dτ Q

(2.19)

0

j ˙k since ui;k C˙ k ≡ 0 and v;k C ≡ 0 throughout C. This simple calculation, which can also be found in [120],

yields the well-known result that inner product of arbitrary tangent vectors on a non-metric space is preserved under parallel transport if and only if non-metricity Q vanishes identically.

44

Let us now expand ui (τ ) and v j (τ ) around τ = 0, within first order in C k (τ ), as ui (τ ) ≈ ui (0) − Limp (0)up (0)C m (τ ) and

(2.20a)

v j (τ ) ≈ v j (0) − Ljmq (0)v q (0)C m (τ ),

(2.20b)

keeping in mind that the fields ui (τ ) and v j (τ ) are materially parallel. Using the approximations (2.17a) and (2.20) into (2.19), we obtain, within second order in C m (τ ), Z s i j i j i j ˜ kij u v gij u v (s) ≈ gij u v (0) − Q C˙ k (τ ) dτ τ =0 0  Z p p i j ˜ ˜ Qkij,m (0) − Lmi (0)Qkjp (0) − Lmj (0)Qkip (0) −u v

s

C m (τ )C˙ k (τ )dτ.(2.21)

0

τ =0

Hence, for the loop C,   I p ˜ p ˜ i j i j i j ˜ gij u v ≈ gij u v − u v Qkij,m − Lmi Qkjp − Lmj Qkip C m (τ ) dC k ,

(2.22)

respect to the indices mk appears in the above expression, i.e.,   p ˜ p ˜ i j i j i j ˜ gij u v ≈ gij u v − u v Qkij,m − Lmi Qkjp − Lmj Qkip

F

I

I

C

I

where the subscripts F and I denote the final and initial values of the respective expressions. Since H m H k k = − m C (τ ) dC C C C (τ ) dC , only the skew part of the expression within the square bracket with

F

I

I

[mk] I

which can be rewritten as   i j i j i j ˜ kij,m + Lp Q ˜ mjp + Lp Q ˜ mip gij u v ≈ gij u v − u v Q ki kj F

I

I

[mk] I

I

C m (τ ) dC k ,

(2.23)

I

C m (τ ) dC k .

(2.24)

C

C

Consequently, due to non-vanishing of the expression within the square bracket for a general non˜ kij , parallel transport of the inner product depends on the path. For path indemetricity tensor Q pendence of the inner product, vanishing of this expression is necessary and sufficient. Also, as we will see in the next Chapter, the third Bianchi-Padova relation implies that this expression is directly proportional to the tensor ζ˜ij k . Hence, we have Proposition 2.1.1. Inner product of arbitrary tangent vectors on a non-metric space is path independent under parallel transport if and only if   p ˜ p ˜ ˜ Qkij,m + Lki Qmjp + Lkj Qmip

[mk]

45

=0

(2.25)

identically. This condition is equivalent to the vanishing of ζ˜ identically over B. In particular, a tangent vector in the material space, under parallel transport along loops, does not change its length if and only if ζ, i.e., the distribution of metrical disclinations, vanishes identically.

Non-metric inhomogeneities: The non-metricity tensor of the material space measures the nonuniformity of the material metric over the body manifold, thus, quantifying the density of a variety of metric anomalies: (i) Point defects (intrinsic interstitials, vacancies, point stacking faults etc.) change the local notion of length by distorting the lattice spacings, hence, are naturally identifiable with the material non-metricity; (ii) Non-uniform thermal strain or bulk material growth may inflate/deflate/shear volume elements in the material and, hence, associable with material non-metricity; (iii) Magnetostrictive strain locally changes the orientation of the magnetization vector and can be associated to the non-metricity tensor in ferromagnetic materials, cf. [4].

2.2

Characterization of defects in structured surfaces

The mathematical prototype for structured surfaces is a connected, compact 2-dimensional manifold ω, possibly with boundary, which is embeddable (as a topological submanifold) in R3 . Examples of such manifolds, in the orientable category, are 2-dimensional sphere, sphere with a finite number of handles added, twisted bands with 2nπ twists for integers n etc., and in non-orientable category, twisted bands with (2n + 1)π twists for integers n, e.g., a M¨obius band for which n is zero. We can add boundaries to these manifolds by removing a finite number of open discs. The condition of embeddability in R3 precludes Klein bottle like surfaces and real projective planes. Our prototype manifold ω is topologically characterized by its orientability, twistedness, Euler characteristic, the number of open discs removed, i.e., the boundaries, and other topological invariants. We will call ω the body manifold for the structured surface. A fundamental theorem in differential topology (Tubular Neighbourhood Theorem [14, Theorem 11.4]) guarantees the existence of a tubular neighbourhood M := {y ∈ R3 | dist(ω, y) < ,  > 0} of ω in R3 , for sufficiently small . Here, dist(ω, y) denotes the minimum

46

Euclidean distance of ω from y. As a bounded open set in R3 , M naturally admits a manifold structure, with ω as an embedded submanifold. Existence of M induces a vector bundle (the normal bundle) structure over ω [14], which entails a vector field d : ω → R3 defined over ω. Our choice of ω, naturally endowed with a director field d, is therefore appropriate for modelling structured surfaces. The differential structure, and all the fields to be defined over ω and M, including d, is assumed to be as smooth as the context demands. Our strategy for characterizing material defects on a structured surface is to first equip M with a geometrical structure by associating with it a metric and an affine connection. This is then used to induce an appropriate non-Riemannian geometrical structure over ω, where various fundamental geometric objects, such as non-metricity, torsion, and curvature, are interpreted as defect density measures. The induced metric and connection on ω is sufficient to encode all the information about the material structure of the structured surface.

2.2.1

Geometry on ω induced from the non-Riemannian structure on M: the material space

Let the 3-dimensional embedding manifold M be equipped with an affine connection L and metric g. Consider a chart (V, θi ) of M with U := V ∩ ω 6= ∅ such that the coordinates θα defined over V lie along U with ζ := θ3 ≡ 0 at U . Such a coordinate system θi is called adapted to U ⊂ ω. The restriction of the natural basis vector fields Gi over V to U will be denoted by Ai , i.e., Ai (θα ) := Gi (θα , ζ = 0), hence A3 is transverse to U . The coefficients of L and the covariant components of g are denoted by Lijk and gij , respectively, with respect to Gi . The covariant derivative of a sufficiently smooth vector field u = ui (θi )Gi : V → TX V , X ∈ V , with respect to L, is denoted by ui;j := ui,j + Lijk uk .

(2.26)

The notation ∇ is used for the surface covariant derivative of a tangent vector field v = v α (θα )Aα : U → TY U , Y ∈ U , with respect to the projection of L on U , i.e., a connection with coefficients Lµαν ζ=0 , µ ∇α v µ := v,α + Lµαν ζ=0 v ν . 47

(2.27)

Here, the subscript (·),i denotes ordinary partial derivative with respect to θi . A vector field u along a curve over V is called parallel with respect to L if, and only if, its covariant derivative along the curve vanishes identically. The body manifold ω, equipped with connection L and metric g from the embedding space M, forms the material space (ω; L, g) of the structured surface. We will call L the material connection and g the material metric. The “material” nature of these mathematical objects is due to the fact that the geometric quantities derived from L and g, when restricted to ζ = 0, represent various material inhomogeneities or defects within the material structure of the structured surface. As we will see immediately below, the non-metricity tensor is a measure of distributed metric anomalies, the torsion tensor is a measure of distributed translational anomalies (dislocations), and the Riemann-Christoffel curvature tensor is a measure of distributed orientational anomalies (disclinations). Most importantly, we assume L to be such that the non-metricity, torsion, and curvature tensors associated with L are uniform in the ζ coordinate and equal to their respective values at ζ = 0, i.e., at U ⊂ ω. This assumption alludes to the applicability of our model to thin multi-layered structures, or thin slices of defective media, represented as homogenized 2-dimensional surfaces. It should also be noted that we are only looking now at local defects and not the ones which could arise out of various topological anomalies for multiply connected and non-orientable surfaces.

2.2.2

Non-metricity of the material connection: metric anomalies

The third order non-metricity tensor of the material space, measuring non-uniformity of the metric ˜ kij defined as in (2.14). We assume g with respect to the connection L, has covariant components Q ˜ kij (θα , ζ) = Q ˜ kij (θα , 0) =: Qkij (θα ). The pure in-surface components Qαµν provide measure for that Q the distributed surface metric anomalies, whereas components Qkij , with either of k, i or j taking the value 3, indicate the presence of out-of-surface metric anomalies, e.g., thickness-wise growth. A nonzero Qαµν lead to variation in angle between tangent vectors during parallel transport with respect to the projected connection Lαβγ ζ=0 , see Figure 2.3(a). Indeed, the inner product g ζ=0 (u, w) = aαβ uα v β

of two tangent vectors u = uα Aα and v = v α Aα , where aαβ (θα ) := gαβ (θα , ζ = 0), changes under 48

M

uαAα v α Aα

U

ω

(a)

transverse curve M

M

u3A3

uαAα v α Aα

U

U

ω

ω

(b)

(c)

Figure 2.3: (a) Change in angle between two tangent vectors to U due to non-zero Qµαβ . (b) Change in angle between two vectors along a transverse curve due to non-zero Q3ij . (c) Change in length of a transverse vector along a surface curve due to non-zero Qµi3 .

49

parallel transport with respect to Lαβγ ζ=0 from the initial point C α (0) to any generic point C α (s),

along some parametrized curve C = C µ (s)Aµ (θα (s)) lying over U , by the amount α β

α β

Z

s

(aαβ uα v β ),µ (τ ) C˙ µ (τ ) dτ Z s Qµαβ (θα (τ ))uα (τ )v β (τ ) C˙ µ (τ ) dτ. = −

aαβ u v (s) − aαβ u v (0) =

0

(2.28)

0

β Here, we have used, uα;µ ζ=0 C˙ µ ≡ 0 and v;µ C˙ µ ≡ 0 throughout C, as they are parallelly transported ζ=0 fields along C, where C˙ µ (s) denotes the ordinary derivative of C µ (s) with respect to its argument.

In structured surfaces, as we have earlier discussed in Section 1.2, this variation in inner product, characterized above in terms of a non-trivial Qαµν , may arise from a distribution of point imperfections in the arrangement of molecules or atoms over the surface, e.g., vacancies and self-interstitials in 2dimensional crystals, inserting (or removing) a lipid molecule into (or out of) a crystalline arrangement of identical molecules over a monolayer, thermal deformation of the surface, biological growth of cell membranes, leaves etc. The remaining components Q3ij = −gij;3 ζ=0 and Qµi3 = −gi3;µ ζ=0 measure

the non-uniformity of the material metric in the ζ-direction, i.e., along the thickness of the structured surface, and the change in length of transverse vectors along the surface, respectively, see Figures 2.3(b) and 2.3(c). These provide faithful representations for differential growth along the thickness in thin multi-layered structures discussed in Section 1.2 and illustrated in Figure 1.10(b).

2.2.3

Torsion of the material connection: dislocations

Consider two tangent vectors v 1 = v1i Ai , v 2 = v2i Ai at some point Y on U . Translating v 1 parallelly along v 2 and v 2 along v 1 with respect to L, we obtain the vectors v 01 = v 1 + Lijk ζ=0 v1k v2j Ai and v 02 = v 2 + Likj ζ=0 v1k v2j Ai ,

(2.29)

respectively. The closure failure of the parallelogram is given by (see Figure 2.4) b = v 2 + v 01 − v 1 − v 02 = 2Tjk i (θα ) v1k v2j Ai ,

(2.30)

where the functions Tjk i (θα ) := Li[jk] ζ=0 50

(2.31)

M

M

v 01 = v 1 + Li3αv1αv23Ai v 01 = v 1 + Liβαv1αv2β Ai v 2 = v2αAα

v 1 = v1αAα

v 2 = v23A3

b = 2Tαβ iv1β v2αAi

b = 2T3β iv1β v23Ai

U

v 1 = v1αAα

U v 02 = v 2 + Liα3v1αv23Ai

v 02 = v 2 + Liαβ v1αv2β Ai

ω

ω

(a)

(b)

Figure 2.4: (a) Closure failure of an infinitesimal in-surface parallelogram due to the Tαβ i components of the torsion tensor. (b) Closure failure of an infinitesimal transverse parallelogram due to the Tα3 i components of the torsion tensor. constitute the components of the third-order torsion tensor (anti-symmetric in the lower indices) over U . Let T˜jk i := Li[jk] . We assume that T˜jk i (θα , ζ) = T˜jk i (θα , 0) (recall definition (2.7)), which in turn is same as Tjk i (θα ). Associated with the torsion tensor, we have the second-order axial tensor 1 αij (θα ) := εikl (θα )Tkl j (θα ). 2

(2.32)

1

Here, εijk (θα ) := g − 2 eijk , where eijk = eijk is the 3-dimensional permutation symbol and g := 1 det[gij ζ=0 ]. For later use, we define εijk (θα ) := g 2 eijk . The components αij provide measures for a variety of dislocation distributions over the structured surface. Taking v13 = v23 = 0 (i.e., v 1 and v 2

tangential to U , see Figure 2.4(a)), and comparing with Figures 1.2(a,b), it is immediate that 1 J α := α3α = εµν3 Tµν α 2

(2.33)

represent a distribution of in-surface edge dislocations and 1 J 3 := α33 = εµν3 Tµν 3 2 51

(2.34)

a distribution of in-surface screw dislocations (cf. [101, 102]). Next, taking v13 = v2α = 0 (i.e., v 1 tangential and v 2 transverse to U , see Figure 2.4(b)), and comparing with Figure 1.9, it is evident that the components αµk := 21 ε3αµ T3α k represent the out-of-surface dislocations in thin multi-layered oriented media such as those discussed in Section 1.2.

2.2.4

Curvature of the material connection: disclinations

˜ klj i of the fourth order Riemann-Christoffel curvature tensor of the material conThe components Ω ˜ klj i measure, in the linear approximation, nection L are given by the definition (2.9). The functions Ω the change that a vector, v ∈ TX V , X ∈ V , suffers under parallel transport with respect to L along an infinitesimal loop C based at X and lying within V : 1˜ i j δv ≈ − Ω klj (X)v 2 i

I

θk dθl ,

(2.35)

C

where v i are the components of the initial vector with respect to the basis Gi (X); the integral

H

C

θk dθl =

represents the infinitesimal area bounded by the loop C. The above formula in fact holds true for any ˜ klji by general loop (not necessarily infinitesimal) C in V . We define the purely covariant components Ω ˜ klji := gip Ω ˜ klj p . Clearly, Ω ˜ klj i = −Ω ˜ lkj i and lowering the fourth index with the material metric gij as Ω ˜ klij = −Ω ˜ lkij . Moreover, as we did for non-metricity and torsion tensors, we assume Ω ˜ klij (θα , ζ) = Ω ˜ klij (θα , 0) =: Ωklij (θα ). Ω It is useful to decompose the components Ωklij (θα ) into skew and symmetric parts [102] Ωklij = εpkl εqij Θpq + εpkl ζij p ,

(2.36)

1 1 Θpq := εpij εqkl Ωijkl and ζij p := εpkl Ωkl(ij) 4 2

(2.37)

where

are components of the second-order tensor field Θ = Θpq Ap ⊗ Aq and the third-order tensor field ζ = ζij k Ai ⊗ Aj ⊗ Ak . They represent, respectively, the skew part and the symmetric part of Ωklij with respect to the last two indices. A geometric interpretation of these two fundamental tensors is as follows (see Figure 2.5). Let the infinitesimal loop C in (2.35) be based at X ∈ U ⊂ ω. Then the change 52

in-surface loop M

M

v0 δA C

v

δv = −δA ζij pnp v j Ai

C

X

δA

X

v0 v

δv = −δA εijk Θk v j Ai

U

U

ω

ω

(a)

(b)

M

C

transverse loop

v0 δA X

v

δv = −δA εijk Θαk nαv j Ai

U

ω

(c)

Figure 2.5: (a) The symmetric part Ωij(kl) , characterized by the third-order tensor ζ, measures the stretching in v ∈ TX V brought about by curvature of the material space. Here, v is a principal direction of the second order tensor ζn, where n is the unit normal to the infinitesimal area element δA bounded by the loop C. (b) For C completely lying within U , the skew part Ωαβ[ij] , characterized by the vector field Θi Ai , measures the purely rotational part of the change in v = v i Ai brought about by the curvature tensor. (c) When C is transverse to U , the skew part Ωα3[ij] , characterized by the second-order tensor field Θαq Aα ⊗ Aq , measures the purely rotational part of the change in v = v i Ai 53 brought about by the curvature tensor.

δv that a vector v ∈ TX V undergoes when parallelly transported along C, in the linear approximation, can be characterized by a second-order tensor β = βij Ai ⊗ Aj , i.e., δv = βv, where βij := −

δA Ωklij εrkl nr = −δA εqij Θpq np − δA ζij p np . 2

(2.38)

Here, δA is a measure of the infinitesimal area bounded by C and n = nr Ar its unit normal. The first term Wij := −δA εqij Θpq np in the above expression is skew with axial vector wq = Θpq np δA. It represents the rotation that v has experienced under parallel transport about the axis Ap , for each fixed p, probed by the three Euler angles Θpq . Thus, Θ is the measure of the rotation of v about the axis n. The second term Sij := −δA ζij p np , on the other hand, is symmetric; it represents a stretching, with the three principal values of the tensor ζn = ζij p np Ai ⊗ Aj as measures of the stretch along their respective (linearly independent) principal directions. The tensor ζ can be shown to be related to the metrical properties of M as it gives rise to a smeared out anomaly within the material structure which causes elongation or shortening of material vectors under parallel transport along loops (see [110] for details), as shown in Figure 2.5(a). We will assume ζ ≡ 0 in rest of the paper since, at present, we do not know of any defects in 2-dimensional materials which they would otherwise represent. Some consequences of this assumption will be discussed in the next section. The curvature tensor Ωklij is then fully characterized in terms of the non-trivial independent components Ω[kl][ij] , i.e., the second-order tensor Θ. We distinguish between two families of local rotational anomalies characterized by Θ. Consider, first, the infinitesimal loop C completely lying within U , see Figure 2.5(b). Then, the i and j indices in Ωijkl can assume only values 1 and 2, and the resulting angular mismatch after parallel transport of arbitrary vectors is characterized by three fields 1 Θq (θα ) := Θ3q (θα ) = ε3αβ εqkl Ωαβkl . 4

(2.39)

These provide a measure for the distributed rotational anomalies within the material structure of the base manifold ω. Drawing analogy with Figure 1.4, it is clear that the out-of-surface component Θ3 provides a measure for the density of distributed wedge disclinations over the structured surface, see 54

Figures 1.3(b,d) and 1.8(a), irrespective of its crystallinity, whereas the in-surface components Θµ characterize either the distributed intrinsic orientational anomalies, in case of intrinsically crystalline surfaces, or distributed twist disclinations, in case of directed surfaces (as shown in Figure 1.8(b)). Next, we consider C, based at X ∈ U , to lie transversely to U , see Figure 2.5(c). Then one of the indices i and j in Ωijkl will take the value 3, and the resulting angular mismatch after parallel transport of arbitrary vectors is characterized by the remaining six independent components of Θ: 1 Θαq = εαµ3 εqkl Ωµ3kl . 4

(2.40)

Recalling our discussion in Section 1.2 on disclinations in thin multi-layered structures of oriented media, see also Figure 1.8(c), we conclude that these components provide a measure for a variety of homogenized/effective rotational anomalies of the distributed disclinations across the thickness of the multi-layered structured surface. Out of these six functions, Θ11 and Θ22 are of wedge type, and Θ12 , Θ21 , and Θα3 are of twist type. As we will see shortly, these functions are in fact dependent on each other in very thin monolayer structures where the dislocation densities αµk vanish altogether. We have summarized the set of all defect densities in Table 2.1.

55

Geometric objects

Defect densities

Qµαβ

In-surface metric anomalies; Figures 1.5(a,b), 1.10(a) and 2.3(a)

Q3ij and Qαi3

Out-of-surface metric anomalies; Figures 1.10(b) and 2.3(b,c)

J µ := α3µ

In-surface edge dislocations; Figures 1.2(a) and 2.4(a)

J 3 := α33

In-surface screw dislocations; Figures 1.2(b) and 2.4(a)

αµk

Out-of-surface dislocations; Figures 1.9 and 2.4(b)

Θ3 := Θ33

In-surface wedge disclinations; Figures 1.3(b,d), 1.8(a) and 2.5(b)

Θµ := Θ3µ

In-surface twist disclinations or intrinsic orientational anomalies; Figures 1.8(b) and 2.5(b)

Θµk

Disclinations associated with transverse loops; Figures 1.8(c) and 2.5(c)

ζij k

Metrical disclinations; Figure 2.5(a)

Table 2.1: Non-Riemannian geometric objects on ω and the defects they characterize in structured surfaces.

56

Chapter 3

Symmetries, Conservation Laws, and Representations of Defects1 Certain differential identities satisfied by the torsion, curvature and non-metricity of a geometric space, known as the Bianchi-Padova relations, will be used in this chapter to derive interesting consequences on various defect density fields, both in 3-dimensional bodies (§3.1), and in structured surfaces (§3.2). Novel contributions, in the theory of defective 3-dimensional bodies, include a necessary and sufficient representation of the metric anomalies in bodies free of disclinations (i.e., when the material curvature Ω identically vanishes, thus, making the body manifold B materially flat) in terms of a symmetric second order tensor, that coincides with the quasi-plastic strain framework of Anthony [2], and a finite deformation framework of quasi-plastic deformation which is amenable to direct numerical implementations in the studies of finite thermal deformation of crystals and bulk growth in tissues. Our framework immediately leads to a geometric representation for the anisotropic metric anomalies, both in 3-dimensional bodies and structured surfaces. Another novel contribution of the present chapter is to collect the implications of Bianchi-Padova relations for various defect densities over structured surfaces. These are in the form of various symmetries of the distributed surface disclinations, and non-linear conservation laws for the out-of-surface dislocations and disclinations (see Table 3.1). 1

Section 3.1 of this chapter appeared in [110].

57

3.1

Compatibility of the geometric objects on the material space of 3-dimensional bodies

The tensors of non-metricity, torsion, and curvature of a non-Riemannian space cannot be arbitrary ˜ k[ij] = 0, T˜(ij) k = 0, and Ω ˜ (ij)kl = 0, which due to geometric restrictions. Besides the restrictions Q follow from their definitions, they satisfy the following system of differential relations, known as the Bianchi-Padova relations [112, p. 144]: ˜ [ijk] l + 4T˜[ij p T˜k]p l , 2T˜[jk l ;i] = Ω ˜ [jk|l| p ;i] = 2T˜[ij q Ω ˜ k]ql p and Ω ˜ [j|kl|;i] = T˜ij p Q ˜ pkl − Ω ˜ ij(kl) . Q

(3.1a) (3.1b) (3.1c)

In the above expressions, anti-symmetrization with respect to three indices is defined as 1 A[nml]··· ··· := (Anml··· ··· + Alnm··· ··· + Amln··· ··· − Almn··· ··· − Anlm··· ··· − Amnl··· ··· ). 6

(3.2)

The enclosed indices within two vertical bars in the subscript are to be exempted from anti-symmetrization. Clearly, A[αβµ]··· ··· ≡ 0 and A[nnl]··· ··· ≡ 0 (no summation on n). The first Bianchi-Padova relation (3.1a) is obtained by alternation of the indices jiq in (2.9). The second relation (3.1b) is the first order in˜ jiq p , whereas the last tegrability condition of (2.9), considered as a PDE in Lijk , given the functions R relation (3.1c) follows from the formula for the second skew covariant derivative of the material metric, i.e., gkl;[ij] . Additionally, there is a fourth Bianchi-Padova relation [112, p. 145], purely algebraic in ˜ ijkl nature, based on the following identity satisfied by the components of any fourth-order tensor Ω ˜ (ij)kl ≡ 0: with Ω
˜ ijkl − Ω ˜ klij = − 3 Ω ˜ [jik]l + Ω ˜ [jlk]i + Ω ˜ [lik]j + Ω ˜ [ijl]k + Ω ˜ kj(li) + Ω ˜ ik(lj) + Ω ˜ jl(ik) + Ω ˜ li(jk) + Ω ˜ lk(ji) + Ω ˜ ij(lk) . Ω 2

(3.3)

˜ ijkl − Ω ˜ klij After substituting relations (3.1a) and (3.1c) into (3.3), it boils down to an expression for Ω ˜ kij , T[jk l ;i] and Q ˜ [j|kl|;i] . For a torsion-free, metric-compatible connection (i.e., a in terms of T˜ij k , Q ˜ ijkl = Ω ˜ klij . Levi-Civita connection), this implies the familiar symmetry Ω 58

3.1.1

Conservation laws of material inhomogeneities

The Bianchi-Padova relations can be written in terms of various defect density tensors, namely, the ˜ kij , ˜ ij and ζ˜ij p , the dislocation density α ˜ ij and the density of metric anomalies Q disclination densities Θ as (cf. [102]) 1 mk ˜ n ˜ nm + ε˜ijm α α ˜ ;iik = ε˜kmn Θ ˜ ij α ˜ mk + ζ˜m mk + α ˜ Qmn , 2 ˜ mk Q ˜ mn n + 1 Θ ˜ mn Q ˜ mn k − 1 ε˜kmn ζ˜mi p Q ˜ pn i , ˜ ik = ε˜ijm α ˜ mk + 1 Θ Θ ˜ ij Θ ;i 2 2 2 1 ˜ mn n + Θ ˜ mn Q ˜ m(k p ε˜r)np − Q ˜ m(k p ζ˜r)p m and ζ˜kr i ;i = ε˜ijm α ˜ ij ζ˜kr m + ζ˜kr m Q 2 ˜ jmn;i = 2ζ˜mn k − α ˜ pmn . ε˜ijk Q ˜ kp Q

(3.4a) (3.4b) (3.4c) (3.4d)

These relations represent the conservation laws for various defect densities. Clearly, in the absence of metric anomalies, ζ˜ = 0 and the above relations reduce down to the conservation laws for the ˜ ij : dislocation density α ˜ ij and the disclination density Θ ˜ nm + ε˜ijm α α ˜ ;iik = ε˜kmn Θ ˜ ij α ˜ mk and

(3.5a)

˜ ik ˜ mk . Θ ˜ijm α ˜ ij Θ ;i = ε

(3.5b)

˜ ij to be small and of the same order, we recover the well-known conservation laws Assuming α ˜ ij and Θ ˜ nm and α ˜ ;iik = ε˜kmn Θ

(3.6a)

˜ ik = 0, Θ ;i

(3.6b)

which state the classical result that in absence of metric anomalies, dislocation lines must end on disclinations (within the body) and disclination lines cannot end inside the body [2].

3.1.2

Representation of metric anomalies in materially flat 3-dimensional solids

For anisotropic elastic solids (e.g., crystalline materials), the (rotational) symmetry group G is discrete. In the continuum limit of such a material from its discrete state, the translational symmetry parameters, which have the order of magnitude of one lattice parameter, get infinitesimally small. But the rotational 59

symmetries (the elements of G) remain finite in the continuum limit. The rotation that a tangent vector ˜ suffers under parallel transport along a loop in the material space that encircles the Θ-disclination lines necessarily belongs to the symmetry group G. If the loop encircles infinite number of these disclination lines, as is the case when a continuous distribution of disclinations is present, the resulting rotational deviation can go unbounded, leading to an unbounded elastic energy. Due to such unrealistic energy ˜ penalty, a continuous distribution of Θ-disclinations has never been observed in crystalline materials ˜ = 0 in anisotropic having discrete rotational symmetry groups. It is therefore reasonable to assume Θ simple elastic materials [5]. Moreover, as the metrical disclinations lead to ambiguity in the definition of inner product field and are not fundamental line defects within the scope of the present class of materials being considered, we will also assume ζ˜ to be identically zero. Remark 3.1.1. Single disclinations of finite energy may exist in 3-dimensional crystals, e.g., a thin whisker. In that case, the singular wedge/screw disclination density should be treated as a Dirac measure, as is done in classical works on the elasticity of Volterra dislocations. The fundamental solution of the equations (in the sense of distributions) would then lead to the determination of energy, shape etc. of the whisker. Also, partial disclinations and dislocations usually act as sites of termination of 2-dimensional defects, such as twin boundaries, stacking faults etc. A dipole consisting of wedge disclinations of opposite signs is geometrically and energetically equivalent to a single edge dislocation (as in the works of J C M Li [76], for example). Such dipoles usually concentrate in arrays to form scars, grain boundaries etc. A distribution of dipoles of partial disclinations, in that sense, can be treated as a distribution of dislocations of appropriate Burgers vectors. However, polymers are 3D Cosserat media, and are outside the scope of the present constitutive assumption for the 3-dimensional body made the in thesis. As in any other directional media, disclinations are the primary defects, and may indeed form loops. Under this assumption of absolute distant material parallelism, we will now discuss three possible models of the non-metricity tensor in the following.

60

3.1.3

Irrotational metric anomalies

The third Bianchi-Padova relation (3.1c) can be rewritten as p ˜ ˜ jkl,i + Lp Q ˜ Q jk ipl + Ljl Qipk

or equivalently




[ji]

˜ ij(kl) =Ω


1 qij ˜ ˜ ipl + Lp Q ˜ ipk = ζ˜kl q . ε˜ Qjkl,i + Lpjk Q jl 2

(3.7)

(3.8)

˜ kmj = 0 is the trivial solution of the last equation for ζ˜km p = 0, but there exist nonClearly, Q ˜ = 0, we seek the most general form of the non-metricity tensor that uniquely trivial solutions. For Θ ˜ corresponds to ζ˜ = 0. In the absence of Θ-disclinations, such a form of non-metricity is necessary to maintain absolute distant material parallelism throughout the body. We have ˜ = 0, then the necessary condition for ζ˜ = 0 is Proposition 3.1.1. If B is simply connected and Θ ˜ kij = −2˜ ˜ := q˜ij Gi ⊗ Gj : B → Sym such that Q ˜ is that there exists a tensor field q qij;k . Moreover, if q ¯ := g − 2˜ such that g q is positive definite, then the above condition is also sufficient. Note that the sufficient condition involves the material metric g. Towards proving this proposition, we will need the following theorem. For our purpose, sufficient regularity of the respective fields can be assumed. The notations used for various functional spaces are standard. Theorem 3.1.1. (Fundamental existence theorem for linear differential systems [82]) Let U be a simply connected open subset of Rp whose geodesic diameter is finite, and let q ≥ 1 be an integer. Let there be matrix fields Aα ∈ L∞ (U, M q ), Bα ∈ L∞ (U, M p ) and Cα ∈ L∞ (U, M p×q ) such that Aα,β + Aα Aβ = Aβ,α + Aβ Aα ,

(3.9a)

Bα,β + Bα Bβ = Bβ,α + Bβ Aα and

(3.9b)

Cα,β + Cβ Aα + Bα Cβ = Cβ,α + Cα Aβ + Bβ Cα

(3.9c)

are satisfied in D0 (U, M q ), D0 (U, M p ) and D0 (U, M p×q ), respectively. In this theorem, the Greek indices α, β varies over 1, 2, . . . , p. Then there exists a matrix field Y ∈ W 1,∞ (U, M p×q ) that satisfies Y,α = Y Aα + Bα Y + Cα in D0 (U, M p×q ). 61

(3.10)

Proof of Proposition 3.1.1: In the above theorem, choose p = q = 3, U = B, Ak = Bk = [Likj ] and ˜ = 0 and ζ˜ = 0 and the ˜ kij ]. Then the conditions (3.9a) and (3.9b) amount to Ω ˜ ijq p = 0, i.e. Θ Ck = [Q
p ˜ ˜ ˜ jkl,i + Lp Q ˜ condition (3.9c) yields Q jk ipl + Ljl Qipk [ij] = 0 which, due to (3.8), is equivalent to ζ = 0. Hence, as a consequence of the above theorem, there exists a sufficiently regular matrix field [˜ qij ] such that ˜ kij = −2˜ Q qij,k + 2Lpki q˜pj + 2Lpkj q˜ip = −2˜ qij;k .

(3.11)

˜ kij = Q ˜ k(ij) implies the symmetry of the matrix field [˜ The symmetry Q qij ]. ˜ kij = −2˜ To prove the sufficiency, we insert Q qij;k in (3.7) to obtain ˜ jikm q˜m l − Ω ˜ jilm q˜m k = Ω ˜ ij(kl) . −Ω

(3.12)

˜ = 0, Ω ˜ jikm = Ω ˜ ji(km) = Here, q˜m k := g im q˜ik . For the time being, let us assume that ζ˜ = 6 0. For Θ ε˜pji ζ˜km p , (3.12) reduces to ε˜pji (ζ˜km p q˜m l + ζ˜lm p q˜m k ) = −˜ εijp ζ˜kl p ,

(3.13)

1 1 (˜ qml + gml )ζ˜m k p + (˜ qmk + gmk )ζ˜m l p = 0. 2 2

(3.14)

which implies

We now find out the conditions on the matrix [˜ qij ] such that the 3 × 3 matrix [ζ˜i j p ], for each p, is identically zero. Recalling that [gij − 2˜ qij ] is symmetric, there exists a basis in which it is a diagonal matrix. In that basis, [gij − 2˜ qij ] can be written as diag(a1 , a2 , a3 ), where ai s are the three real eigenvalues of [gij − 2˜ qij ]. Moreover, let us denote by [bij ] := [ζ˜i j p ] for some fixed p. The last expression

62

boils down to 

              diag(2a1 , a1 + a2 , a1 + a3 , a1 + a2 , 2a2 , a2 + a3 , a1 + a3 , a3 + a2 , 2a3 )              

b11 b12 b13 b21 b22 b23 b31 b32 b33





                            =                          



0    0    0     0     . 0     0    0     0    0

(3.15)

˜ to vanish identically is that the determinant of the 9×9 diagonal matrix The condition for [bij ], hence ζ, in the above expression is non-zero, i.e., a1 a2 a3 (a1 + a2 )2 (a2 + a3 )2 (a3 + a1 )2 6= 0.

(3.16)

Hence, if [˜ qij ] is such that the eigenvalues ai s of [gij − 2˜ qij ] satisfies (3.16) everywhere, then ζ˜ vanishes identically. For a positive definite [¯ gij ] := [gij − 2˜ qij ], (3.16) is always satisfied.



˜ kij = −2˜ ˜ satisfies the necessary and sufficient conMetric anomalies characterized by Q qij;k , where q ditions of Proposition 3.1.1, are called irrotational, because they uniquely correspond to the vanishing of material curvature. Only irrotational metric anomalies are allowed under our present assumption of ˜ is a comabsolute distant parallelism in a materially uniform simple elastic solid. The tensor field q plete measure of irrotational metric anomalies. It induces a field of orthonormal triple of eigenvectors ˜ a = aa, q ˜ b = bb and q ˜ c = cc. {a, b, c}, corresponding to a field of eigenvalues a, b and c, respectively: q ˜ , as well as its field of eigenpairs {a, a} etc. can be restricted to a curve. Then, we The tensor field q have a field of rectangular parallelepiped formed by the triple of orthogonal vectors {aa, bb, cc} over this curve (see Figure 3.1(a)). This field of rectangular parallelepiped is uniquely defined over the whole material space because of distant material parallelism. The parallelepiped returns to its initial shape, size and orientation after circumnavigation along a loop. The formula (2.28) for inner product of 63

B

B

(a)

(b)

Figure 3.1: (a) General irrotational metric anomalies change the orientation, shape and size of a cube along a curve, whereas (b) isotropic metric anomalies, which are also irrotational, change the orientation and size, but not the shape, of a cube along a curve. arbitrary vectors under parallel transport, in presence of purely irrotational metric anomalies, reduces down to (up to leading order) gij u v (s) ≈ gij u v (0) + 2(˜ qij;k u v ) i j

3.1.4

i j

i j

τ =0

Z

s

C˙ k (τ ) dτ.

(3.17)

0

Quasi-plastic strain

¯ can be used to define For irrotational metric anomalies, the positive definite symmetric tensor field g ¯ ), equipped with the original material connection L and the metric an auxiliary material space (B, L, g ¯ . The non-metricity of this auxiliary material space vanishes identically by definition: g¯ij;k = gij;k − g ˜ kij + Q ˜ kij = 0. The curvature is identically zero for both (B, L, g) and (B, L, g ¯ ). Auxiliary 2˜ qij;k = −Q material space is a geometric space derived from the material space, with identical torsion and curvature fields, whose non-metricity is identically zero. ˜ (calling it quasi-plastic strain) to study metric anomalies when Anthony [2, 3] (cf. [43]) used q absolute distance parallelism is maintained throughout the material space. The fundamental geometric ˜ , as we discussed above, was absent in their work. The terminology reasoning behind the existence of q ¯ ), i.e., q ˜ is the difference between the respective metric ˜ = 12 (g − g “strain” is clear from the relation q 64

tensors of the material space and the auxiliary material space. ˜ characterizing irrotational Remark 3.1.2. (Isotropic metric anomalies.) The second order tensor q metric anomalies has the unique decomposition q˜ij = λgij + qij ,

(3.18)

˜ (λgij is called the spherical/isotropic part), and qij is the deviatoric where λ := 31 q˜k k is the trace of q ˜ is purely isotropic, i.e., q˜ij = λgij . Then, it part of q˜ij , i.e., qk k = 0. Let us consider the case when q ˜ kij = −µ,k gij , where µ := ln(1 + 2λ). In this case, the formula (3.17) for is straightforward to obtain Q the inner product of arbitrary vectors under parallel transport along a curve C reduces to   Z s gij ui v j (s) ≈ gij ui v j (0) 1 + µ,k τ =0 C˙ k (τ ) dτ .

(3.19)

0

Hence, orthogonal vectors always remain orthogonal under parallel transport. Since all the eigenvalues of an isotropic tensor are equal, the cube formed by the eigenpairs {λa, λb, λc}, where {a, b, c} are any triple of orthonormal vectors forming the eigenspace of q (any orthonormal triple of vectors forms the eigenspace of an isotropic tensor), inflates/deflates as one moves along the curve C (see Figure 3.1(b)). The auxiliary material space for isotropic metric anomalies is conformal to the original material space, ¯ = (1 + 2λ)g. because g Remark 3.1.3. (Regularity of the induced field of parallelepipeds.) The one parameter fields of eigen˜ (s) along a parametric curve C always posses the same values of the one parameter tensor field q ˜ (s) [72, 104]. If all the eigenvalues are simple throughout C, then the correspondregularity as that of q ˜ . This is also true if the multiplicity of ing field of eigenvectors has the same regularity as that of q all the eigenvalues remains constant throughout the curve. In case of isotropic metric anomalies, the multiplicity is 3 throughout C; hence, the field of cubes has the same regularity as the function µ,k (s) along C.

3.1.5

Semi-metric geometry

For a second representation of metric anomalies we look into semi-metric geometry. In semi-metric ˜ kmj is given in terms of a vector Q ˜ k as Q ˜ kmj = Q ˜ k gmj (semi-metric geometry with zero geometry, Q 65

˜ kij , when plugged into the third Bianchi-Padova torsion is called Weyl-geometry) [112]. This form of Q relation (3.7), reduces it to ˜ [j,i] gkm = Ω ˜ ij(km) . Q

(3.20)

˜ = 0 and the domain B is simply connected, (3.20) implies, from Poincar´e’s lemma, that Ω ˜ ij(km) = 0 If Θ ˜ i = φ,i [48]. Define ψ := exp φ − 1. Then, if and only if there exists a function φ : B → R such that Q φ,i =

ψ,i ψ+1 ;

hence, g¯ij;k = 0 where g¯ij := (ψ +1)gij , i.e., we recover the isotropic representation discussed

in Remark 3.1.2. Moreover, using the Helmholtz representation (decomposition of a vector field into a ˜ j as Q ˜ j = φ,j + gij ε˜imn qn;m , with g mn qn;m = 0 curl free and a divergence free part) of the vector field Q (cf. [53]), we have the following ˜ = 0, the curl free part Proposition 3.1.2. In semi-metric geometry, with B simply connected and Θ ˜ k , expressed as φ,k for some scalar function φ, and the divergence free part of Q ˜ k , characterized of Q by the vector field qi as above, uniquely correspond to ζ˜ = 0 and ζ˜ = 6 0, respectively. The concept of non-metricity, as well as its semi-metric form (in fact, with vanishing torsion), was introduced by Hermann Weyl [130, pp. 121-125] in an attempt to unify gravity with electromagnetism. The semi-metric form of non-metricity preserves the ratio of the magnitude of two vectors during parallel transport along a curve. Indeed (cf. [120]), let u = gij ui uj and v = gij v i v j be the squared lengths of two tangent vectors at any point on a curve C. Then, upon parallel transport along an ˜ k dC k and infinitesimal segment dC k , the changes in u and v, by (2.19), are given by ∆u = −u Q ˜ k dC k , respectively. Hence, ∆v = −v Q     u ∆u u u u ˜ = − 2 ∆v = − − Qk dC k = 0. ∆ v v v v v

(3.21)

˜ i = φ,i , has been used by Miri and Rivier [87] and more recently by Semi-metric geometry, with Q Yavari and Goriely [133, 134] to model isotropic metric anomalies in the context of a distribution of ˜ = ζ˜ = 0, the semispherically symmetric point defects. It is clear from Proposition 3.1.2 that, with Θ metric model can represent only isotropic metric anomalies. In this context, the scope of quasi-plastic strain model discussed previously is larger and physically amenable in representing anisotropic metric anomalies. 66

Euclidean Space E3

F

κr (B)

¯ K

κt(B)

¯ H

K

H

(B, L, g)

(B, L, g¯ ) Q Auxiliary Material Space

Material Space

Curvature R = 0 Torsion T 6= 0 Non-metricity Q = 0

Curvature R = 0 Torsion T 6= 0 Non-metricity Q 6= 0

Non-Euclidean Space

Figure 3.2: Mappings between the tangent spaces of various configurations and spaces associated with the material manifold B, see Section 3.1.6 for details.

3.1.6

Quasi-plastic deformation

¯ ), defined in Section 3.1.3, inherits the affine connection L (with The auxiliary material space (B, L, g ¯ such that its nonzero curvature) from the material space (B, L, g) but has a different metric field g metricity vanishes identically, i.e., g¯ij;k = 0. With both curvature and non-metricity identically zero, ¯ ) can support only the auxiliary material space can still have non-trivial torsion. Therefore (B, L, g dislocations as possible sources of inhomogeneity. In this scenario, according to a classical result in differential geometry (cf. [109, Theorem 2.1]), there exists a sufficiently smooth invertible tensor field

67

¯ := H ¯ ij Gi ⊗ Gj over B such that H ¯ li,j and ¯ −1 )pl H Lpij = (H ¯ T H. ¯ ¯=H g

(3.22a) (3.22b)

¯ > 0. The tensor H ¯ maps ¯ is positive definite by construction. We assume det H The auxiliary metric g the tangent spaces of the auxiliary material space to the tangent spaces of the current configuration κt (B) ⊂ E3 (see Figure 3.2). Here, E3 denotes the 3-dimensional Euclidean point space. We assume that there exists a sufficiently smooth tensor field in InvLin+ , Q := Qi j Gi ⊗ Gj , over B which maps the tangent spaces of the material space to the tangent spaces of the auxiliary material space, such that ¯ Q. g = QT g

(3.23)

The tensor Q is the third representation of metric anomalies discussed in this paper. We call it quasiplastic deformation for reasons that will be discussed below. The preceding assumption is tantamount to the existence of a well-defined material uniformity field and a crystallographic basis field over the material space. Indeed, substituting (3.22b) in (3.23) allows us to write material metric as g = H T H,

(3.24)

¯ where H = HQ is a sufficiently smooth tensor field in InvLin+ which maps the tangent spaces of the material space to the tangent spaces of the current configuration of the body (see Figure 3.2). Consider a fixed reference configuration κr (B) ⊂ E3 and let F ∈ InvLin+ be the deformation gradient tensor which maps tangent spaces in κr to those in κt (see Figure 3.2). The field h ∈ Sym+ introduced in Section 2.1.1 is of the form F T F . Recalling the discussion in Section 2.1.1, and using (3.24), we can construct a well-defined smooth material uniformity field K ∈ InvLin+ such that (see Figure 3.2) H = F K.

(3.25)

The tensors H and K are conventionally called the elastic and plastic deformation tensor, respectively. The tensors H and K −1 are usually denoted as F e and F p , respectively, in the plasticity literature. 68

¯ ∈ InvLin+ With the above mappings in place, we can define an auxiliary plastic deformation tensor K ¯ = F K. ¯ The multiplicative decomposition such that H ¯ K = KQ

(3.26)

of the plastic deformation tensor follows immediately (see Figure 3.2). With the existence of Q we can also construct an unambiguous crystallographic vector field g i = HGi such that gij = g i · g j . The motivation for introducing Q is clear from the multiplicative decompositions (3.25) and (3.26). Consider the case when the material space (B, L, g) has only metric anomalies and is therefore free of ¯ ) is free of any inhomogeneity, and will be a dislocations. Then, the auxiliary material space (B, L, g ¯ = I and H ¯ =F connected subset of E3 . We can identify it with the reference configuration κr , i.e., K identically over the whole domain. The tensor Q then determines the plastic distortion K, hence the terminology quasi-plastic deformation. On the other hand, when the material space is dislocated and also has metric anomalies, the components of the torsion tensor can be calculated from (3.22a) as ¯ −1 )pl H ¯ l[i,j] . Tij p = (H

(3.27)

It is clear that the information about dislocation density in the material space is contained only in ¯ (or equivalently of K) ¯ . The tensor Q can then be understood to conthe incompatibility of H tain information about the metric anomalies, as described in the following paragraph. The proposed framework can be seen as a generalization of a version of the fundamental theorem of Riemannian geometry in the context of continuum theory of material defects, as stated and proved in Roychowdhury and Gupta [109, Theorem 2.1], by including metric anomalies into consideration. In the absence of non-metricity we recover Theorem 2.1 in [109]. ˜ kij = −gij;k with (3.23), it is straightforward to relate the Combining equations g¯ij;k = 0 and Q non-metricity tensor of the material space to the quasi-plastic deformation tensor as   −1 p ˜ Qkij = 2 (Qip;k ) (Q ) j

,

(3.28)

(ij)

where Qij := gip Qp j . However, the quasi-plastic deformation Q cannot be an arbitrary tensor. According to the third Bianchi-Padova relation (3.1c), it has to necessarily satisfy the following second 69

˜ ˜ and ζ: order non-linear PDE in order to conform to the vanishing of Θ 

−1

(Q

p

)ip;[mk] Q

j

p

+Q

i;[m

−1

(Q

p

−1 q

)|jp|;k] − Tmk Qiq;p (Q

)

j



= 0.

(3.29)

(ij)

˜ ijp q = 0. The indices This equation has been obtained by substituting (3.28) into (3.1c) and imposing Ω enclosed within the vertical bars | · | are exempt from any symmetrization or anti-symmetrization operation. We should emphasize that a description of metric anomalies in terms of the quasi-plastic deformation tensor Q is an alternative model for irrotational metric anomalies in crystalline solids as proposed in Section 3.1.3. While the present representation allows us to obtain elegant multiplicative decompositions of total and plastic deformations, it comes at the cost of satisfying conditions (3.29). The quasi-plastic strain model in Section 3.1.3 is free from such constraints but provides no basis for multiplicative decompositions. Both of these representations can be used to model anisotropic metric ¯ in Proposition 3.1.1, we can obtain a relation anomalies. By comparing (3.23) with an equation for g ˜: between Q and q Q−T gQ−1 = g − 2˜ q.

(3.30)

˜ these provide only six (nonlinear) equations to be solved for Q. This is illustrated clearly For a given q in the linearized setting of Remark 3.1.4 below. In the absence of metric anomalies it is reasonable to ˜ = 0 (so that g ¯ is identical to g) and Q = I (so that the auxiliary material space is identical to take q the material space). A multiplicative decomposition framework, such as that provided by Equations (3.25) and (3.26), is useful for analytical and numerical studies of displacement boundary-value-problems of inhomogeneous solids. We point out here that the decomposition of the total plastic distortion into parts associated with dislocations and metric anomalies is not new (see [21, p. 373] and [22]). The present formulation provides a rigorous geometric setting in which such a decomposition can be justified in absence of disclinations. Moreover, representation of metric anomalies by Q allows us to take into account more general distortional defect densities such as those arising in a distribution of point stacking faults [68,70]. We summarize the above discussion in the following

70

Proposition 3.1.3. If the non-metricity Q is given in terms of the quasi-plastic deformation Q by ˜ = ζ˜ = 0, (3.28) such that (3.29) is satisfied in order to conform to distant material parallelism, i.e., Θ then there exist multiplicative decompositions (3.25) and (3.26) of the total deformation gradient into an elastic and a plastic part, and further of the plastic part into a term which relates to dislocations and other to metric anomalies. Remark 3.1.4. (Linearization of quasi-plastic deformation.) Consider the linearizations ˆ + w and Q≈I +q g ≈ I + 2,

(3.31a) (3.31b)

ˆ ∈ Sym, w ∈ Skw and  ∈ Sym such that they are all of the same order. Also assume the quasiwhere q ˜ to be infinitesimally small of the order of q ˆ . Substituting the above approximations plastic strain q ˆ with q ˜ . The tensor w is left into (3.30), and collecting the leading order terms, we can identify q undetermined. The two frameworks therefore coincide in a linearized formulation. Remark 3.1.5. (Anisotropic distribution of point defects.) In the introduction, we referred to certain clusters of point defects in crystals which form exotic shapes in their stable equilibrium configurations (Figures 1.6 and 1.7). A continuous distribution of such anisotropic point defects can be represented in terms of quasi-plastic deformation tensor Q for a suitable symmetry class. The symmetry class of Q corresponds to the structural symmetries of the shape of the point defect clusters distributed throughout the body; it is the point symmetry group of each clusters in E3 . For example, for the case of split-interstitials (as shown in Figure 1.6), each cluster is transversely isotropic with axis along the dumbbell. We can read off the transversely isotropic representation form for Q from the table provided in Section 4 of Lokhin and Sedov [79]. This table contains forms of various invariant tensors for all the crystal symmetry classes. If the transverse isotropy axis field is given by k(X), then Q has the representation Q = A(X) Gi ⊗ Gi + B(X) k ⊗ k in terms of the two scalar fields A(X) and B(X) and the unit vector field k(X). Note that the scalar field B(X) captures the anisotropic part of Q. Remark 3.1.6. (Anisotropic thermal deformation.) Thermal deformation can be modelled as metric anomalies in the material manifold [67]. For modelling thermal deformation, the appropriate form 71

of quasi-plastic deformation is Q = A∆T , where A is the symmetric tensorial coefficient of thermal expansion and ∆T is the temperature change. Anisotropic deformation is characterized by appropriate forms of A chosen from the table provided in Lokhin and Sedov [79] for the specific symmetry class under consideration.2

3.2

Compatibility of the geometric objects on the material space of structured surfaces

The first three Bianchi-Padova relations, restricted to a surface, have been considered previously by Povstenko [102], but without studying any of the implications, some of which are noted below.

3.2.1

Consequences of the first Bianchi-Padova relation

Equation (3.1a) is non-trivial only when at least one of the indices i, j and k assume the value 3, since otherwise A[αβµ] ≡ 0. Recalling our assumption that that T˜ij k is uniform with respect to the ζ coordinate, (3.1a) then reduces to 4∇[β T|3|α] l = −(Ωαβ3 l + Ω3αβ l − Ω3βα l ) − 4(Tαβ µ T3µ l + T3α p Tβp l − T3β p Tαp l ).

(3.32)

We recall that, in the above expression, the torsion components Tαβ µ are uniquely characterized by the in-surface edge dislocation densities J α , whereas the components Tα3 i are characterized by the dislocation densities αµk . If we assume that the structured surface is sufficiently thin with no dislocations associated with the transverse Burgers parallelograms, i.e., αµk ≡ 0 (the in-surface dislocations J i can 2

It may be noted that the classical theory of 3-dimensional nonlinear thermo-elasticity, see, for example, [46], does

not require the decomposition the total deformation gradient into elastic-thermal parts (represented by F = KQ, with ¯ ≡ I). However, the present formulation of thermal distortions as metric anomalies puts the nonlinear thermo-elasticity K within the unified framework of continuous defects. Moreover, in deriving 2-dimensional theories from a 3-dimensional thin thermo-elastic solid structure, one would require an infinite number of temperature fields in order to make such theories exact, much like the many director “exact” shell models. Our treatment of thermo-elasticity clearly avoids such complications.

72

still be present), then (3.32) simplifies into a system of algebraic equations: Ωαβ3l = Ω3βαl − Ω3αβl .

(3.33)

For l = 3, we obtain Ω3βα3 = Ω3αβ3 , since Ωαβ33 = 0 (from ζ ≡ 0). This is equivalent to Ωβ3α3 = Ωα3β3 , or Θαβ = Θβα .

(3.34)

For l = µ, (3.33) can be rewritten as Ωαβ3µ = Ω3βαµ − Ω3αβµ , or equivalently Θ3µ = Θµ3 .

(3.35)

Combining the above two relations we can therefore infer that, for vanishing αµk , the disclination density tensor Θ is symmetric. Moreover, due to (2.39), Θµ = Θµ3 , i.e., the pure in-surface disclination densities Θµ (which may either characterize densities of twist disclinations in directed surfaces or intrinsic orientational anomalies in hemitropic surfaces) should be identical to the wedge disclination densities Θµ3 associated with transverse loops, e.g., in multi-layered surfaces as discussed in Section 1.2; in particular, they should vanish in sufficiently thin structured surfaces, e.g., in 2-dimensional crystals, where both Θµ3 and αµk will be absent. We note that, in contrast, for 3-dimensional solids, the symmetry of the disclination density tensor is implied only under vanishing of the full torsion and the non-metricity tensor. It is worthwhile to reemphasize that the assumption αµk ≡ 0 is realistic only in sufficiently thin structures (biological membranes, graphene sheets, etc.), which, otherwise, can contain surface edge and screw dislocations (characterized by J i ). On the other hand, if we consider multi-layered or moderately thin structures of oriented media, where the assumption of vanishing αµk is no longer physical, and assume that they do not contain any disclinations and metric anomalies, and also that J i and αµk are small (of the same order), then (3.32) yields ∇µ αµk = 0.

(3.36)

This is a conservation law for the αµk -type dislocations enforcing that they must always form loops or leave the surface. In either case, whether the αµk -dislocations are absent or not, there is no restriction 73

on the distribution of in-surface dislocations J i . This again is in contrast to 3-dimensional solids, where the first Bianchi-Padova relation provides a conservation law for all dislocation densities [102, 110].

3.2.2

Consequences of the second Bianchi-Padova relation

Equation (3.1b), in the absence of both αµq -type dislocations and metric anomalies (Qijk ≡ 0), in addition to ζ ≡ 0, reduces to a simple conservation law ∇µ Θµk = 2ε3µν J µ Θνk ,

(3.37)

to be satisfied by disclinations characterized by Θµk , as well as Θkµ owing to the symmetries Θµk = Θkµ (Equations (3.34) and (3.35)), and surface edge dislocations. Assuming that J α and Θµk = Θkµ are small, and of the same order, we obtain

∇µ Θµk = ∇µ Θkµ = 0.

(3.38)

These are linear conservation laws for the respective disclinations, requiring their lines to either form loops or leave the surface. Note that there is no restriction on Θ3 (wedge disclinations), in contrast to what one would expect for 3-dimensional solids.

3.2.3

Consequences of the third Bianchi-Padova relation

From our discussion in Section 3.1.2, we recall that, in the absence of disclinations (i.e., Ωijkl = 0) over a simply connected U (hence V ), if the matrix field g¯ij := gij − 2˜ qij is positive definite for symmetric ˜ kij = −2qij;k is the only solution to the third Bianchi-Padova relation over V . functions q˜ij = q˜ji , then Q As the density of metric anomalies is assumed to be uniform with respect to the ζ coordinate, we will interpret this representation of the metric anomalies in absence of disclinations over simply connected patches over ω as Qkij (θα ) = −2˜ qij;k ζ=0 .

(3.39)

In absence of disclinations, the positive definite symmetric matrix field g¯ij can be used to define an ¯ ), equipped with the original material connection L but a metric g ¯. auxiliary material space (ω, L, g 74

The non-metricity of the auxiliary material space vanishes identically by definition. The second-order tensor field q := qµν Aµ ⊗ Aν , where qµν (θα ) := q˜µν (θα , ζ = 0), characterizing pure in-surface metric anomalies in the absence of disclinations, has the unique decomposition qµν = λaµν + qµν ,

(3.40)

where λ := 21 q µ µ = 12 aµα qαµ is the trace of q and qµν is the deviatoric part of qµν (i.e., qµ µ = 0). The first term represents isotropic metric anomalies and the second represents anisotropic metric anomalies. When q is purely isotropic, i.e., qµν = λaµν , it is straightforward to obtain Qαµν = −µ,α aµν , where µ := ln(1 + 2λ). The surface metric of the auxiliary material space for isotropic metric anomalies is, hence, conformal to the surface metric of the original material space, a ¯µν = (1 + 2λ)aµν . This formulation is readily applicable to model a wide variety of real-life surface metric anomalies such as 2-dimensional anisotropic biological growth, thermal expansion, distributed point defects, etc.

3.2.4

Consequences of the fourth Bianchi-Padova relation

The fourth Bianchi-Padova relation imposes interdependence on the disclination density measures Θpq , out of which the interdependence between the two distinct families of disclinations characterized by Ωαβij and Ωα3ij , derived in the following, are in particular interesting. Assuming ζ ≡ 0, the insurface components of (3.3) require Ωαβµν − Ωµναβ ≡ 0, since A[αβµ]··· ≡ 0, which is the trivial relation Θ33 = Θ33 . Next, if we also assume that the metric anomalies are absent, i.e., Qkij ≡ 0, then (3.3), together with (3.1a), yields
Ωαjµ3 − Ωµ3αj = −3 T[3µ|α|;j] − 2T[j3 i Tµ]iα + T[αµ|j|;3] − 2T[3α i Tµ]ij + T[j3|µ|;α] − 2T[αj i T3]iµ .

(3.41)

Here, Tijp := Tij k gkp . After substituting Tijk;3 ≡ 0, as per our assumption on L, and T3α i ≡ 0, or equivalently αµk ≡ 0, the right hand side of the above relation vanishes identically, thereby enforcing the symmetries Ωαβµ3 − Ωµ3αβ = 0 and Ωα3µ3 − Ωµ3α3 = 0.

(3.42)

In terms of disclination densities, these are, respectively, Θ3µ = Θµ3 and Θνµ = Θµν . Interestingly, we reached the same conclusion from the first Bianchi-Padova relation. We will of course obtain a 75

non-trivial consequence of the fourth Bianchi-Padova identity whenever αµk 6= 0. The consequences of all the Bianchi-Padova relations, as discussed above, are summarized in Table 3.1. Symmetries and conservation laws Implications on defect densities from Bianchi-Padova relations with ζij k ≡ 0 The two distinct families of disclinations

αµk ≡ 0 ⇒ Θij = Θji

Θ3i and Θαi are dependent on each other

{Θij ≡ 0, Qkij ≡ 0, and αµk , J i small} ⇒

αµk -dislocations either form

∇µ αµk = 0

loops or leave the surface

{αµk ≡ 0, Qkij ≡ 0, J α and Θµk = Θkµ small}

Disclinations associated with the transverse

⇒ ∇µ Θµk = ∇µ Θkµ = 0

loops, either form loops or leave the surface

On simply connected domains on ω, with Θij ≡ 0, Qkij = −2˜ qij;k ζ=0

Non-metricity Qkij can be represented in terms of a symmetric second-order tensor

Table 3.1: Symmetries, conservation laws, and representations of defect density fields imposed by the Bianchi-Padova relations.

76

Chapter 4

Strain Incompatibility in 3-dimensional Elastic Solids1 In this chapter, we introduce the Riemannian structure on the material space of a 3-dimensional body induced by its material metric g. Differential geometry provides a natural relationship between the respective Riemann-Christoffel curvatures of this Riemannian structure and the non-Riemannian structure discussed previously, which is a precursor to the boundary-value-problem for the internal stress field in 3-dimensional bodies. By identifying the material metric g with elastic strain, the said curvature relationship yields the local incompatibility relations for the elastic strain field, with known defect densities. We also derive a representation for metric anomalies on a material body free of dislocations and disclinations that results in zero incompatibility of the elastic strain, hence, vanishing internal stress field. 1

This chapter appeared in [110].

77

4.1

Induced Riemannian structure

By alternating various indices in the definition (2.14) of the non-metricity tensor, the coefficients of the material connection can be written as [112, p. 141] ˜ ij p , Lpij = Γpij + W

(4.1)

where 1 Γpij := g pn (gni,j + gnj,i − gij,n ), 2

(4.2a)

˜ ij p := C˜ij p + M ˜ ij p , W

(4.2b)

C˜ij p := g pk − T˜ikj + T˜kji − T˜jik ),

(4.2c)


˜ ikj − Q ˜ kji + Q ˜ jik , and ˜ ij p := 1 g pk Q M 2

(4.2d)

T˜ijp := T˜ij k gkp . The functions Γpij are the coefficients of the Levi-Civita connection of the metric gij , whereas the functions C˜ij p form the components of the contortion tensor [96]. It is straightforward to ˜ ijpl and R ˜ ijpl of the Riemannderive the following identity relating the purely covariant components Ω Christoffel curvatures of the material connection Lpij and the Levi-Civita connection Γpij , respectively (cf. [112, p. 141]): ˜ ijpl = R ˜ ijpl + 2W ˜ [j|pl|;i] + 2W ˜ [j|p| m Q ˜ i]ml − 2W ˜ [i|ml| W ˜ j]p m + 2T˜ij m W ˜ mpl Ω

(4.3)

˜ ijpl = Ω ˜ ijpl − 2∂˜[i W ˜ j]pl − 2W ˜ [i|ml| W ˜ j]p m , R

(4.4)

or equivalently

˜ ijp := W ˜ ij k gkp ; ∂˜ denotes covariant differentiation with respect to the Levi-Civita connection where W Γpij . The skew part of (4.4) (or (4.3)) with respect to indices pl yields the third Bianchi-Padova iden˜ ij[pl] = 0 by definition). The symmetric part, on the other hand, provides a tity (3.1c) (recall that R system of non-linear PDEs for the material metric gij , given various material inhomogeneity measures ˜ ijp q , material torsion T˜ij p and material non-metricity Q ˜ kij . In fact, in terms of material curvature Ω ˜ ijpl = 0 which, provided that B is simply conin the absence of inhomogeneities, (4.4) is reduced to R nected, yields the classical fundamental theorem of Riemannian geometry, i.e., there exists a sufficiently 78

smooth diffeomorphism χ : B → R3 such that g = (Gradχ)T Gradχ. Here Grad denotes the covariant differentiation with respect to the Levi-Civita connection of the Euclidean metric Gij .

4.2

Strain incompatibility

The above discussion leads us to the important interpretation of the material metric g in terms of the ˜ ijpl can be identified elastic strain tensor E := 21 (g − I). Consequently, the Riemannian curvature R ˜ ijpl in the absence of material with the local incompatibility of the elastic strain field Eij . Vanishing of R inhomogeneities in a simply connected body results in the existence of an elastic deformation field χ over
simply connected open sets such that E = 12 (Gradχ)T Gradχ − I [25,67].2 These local conditions are

also sufficient for strain compatibility of simply connected bodies. For the global existence of the elastic deformation map over a multiply connected body, we need additional compatibility conditions on E, besides the vanishing of the Riemannian curvature point-wise. These global compatibility conditions, which we will not discuss here, are given by vanishing of certain integrals of functions of E over irreducible loops inside the body. See [132, 138] for more details.

4.3

Metric anomalies associated with locally compatible elastic strain

˜ ijpl = 0, and zero dislocation Under the assumption of absolute distant material parallelism, i.e., Ω density, i.e., T˜ij p = 0, (4.4) yields ˜ ijpl = −2∂˜[i M ˜ j]pl − 2M ˜ [i|ml| M ˜ j]p m . R

(4.5)

˜ ijk = 0, R ˜ ijpl = 0, which would imply a vanishing internal In absence of metric anomalies, i.e., M stress field if there are no external sources of stress. The contrary however is not true. We can have ˜ ijpl = 0. In rest of this section our aim a non-trivial distribution of metric anomalies which lead to R ˜ kij , which when substituted into (4.5) gives is to obtain the general form of the non-metricity tensor Q 2

Recall, from our discussion of quasi-plastic deformation in Section 3.1.6, the tensor H which is now identified with

Gradχ in the absence of material inhomogeneities.

79

˜ ijpl = 0, under the assumption that both elastic strain Eij and Q ˜ kij are small and are of the same R ˜ ijpl = 0, (4.5) can be linearized to obtain order. Under such an assumption, with R ˜ j]pl = 0 ∂˜[i M

(4.6)

˜ j(pl) = 2Q ˜ jpl and M ˜ j[pl] = Q ˜ [pl]j . Taking the symmetric part of (4.6) with respect to pl, we with M obtain ˜ j]pl = 0. ∂˜[i Q

(4.7)

The torsion and curvature corresponding to both the connections Lpij and Γpij are identically zero. The last equation then implies, from Poincar´e lemma, assuming B to be simply connected, that there exists a symmetric matrix field [Spl ] over B such that ˜ ipl = ∂˜i Spl . Q

(4.8)

˜ [pl]j − ∂˜j Q ˜ [pl]i = 0, after substituting On the other hand, the skew part of (4.6) with respect to pl, i.e., ∂˜i Q from (4.8), yields ∂˜ip Slj − ∂˜jp Sli − ∂˜il Spj + ∂˜jl Spi = 0.

(4.9)

This relation implies that there exists a sufficiently smooth vector field Al over B, such that Spl = ∂˜(p Al) .

(4.10)

˜ ipl = ∂˜i(p Al) Q

(4.11)

Hence,

is the most general form of stress-free distribution of metric anomalies under the assumptions made above. We summarize the result as Proposition 4.3.1. In absence of dislocations and disclinations in a simply connected material body B, if the elastic strain and non-metricity tensor are assumed to be small and of the same order, then there exists a sufficiently smooth vector field Aj over B such that the stress-free distribution of metric ˜ kij = ∂˜k(i Aj) . anomalies is given by Q

80

Chapter 5

Strain Incompatibility in Structured Surfaces1 We begin this chapter by introducing the notion of strain for a structured surface. The complete set of strains represent essentially the kinematical nature of the shell theory that is being employed to describe the kinematics of structured surfaces. The strain fields also provide us with fundamental variables for construction of constitutive responses of the continuum. Once the strain fields are fixed, we look for the necessary and sufficient (compatibility) conditions for the existence of local and global isometric embedding of the surface in the 3-dimensional Euclidean space R3 . Existence of such an isometric embedding is synonymous to the existence of a global sufficiently smooth bijective deformation map which results into the given strain fields. We first present a novel proof of the nonlinear local strain compatibility relations already derived by Epstein [32]. Towards this, we construct the material metric g in the tubular neighbourhood M of ω from the strain fields given on ω. The local compatibility then follows by requiring that this g is flat, i.e., curvature free. Next, we restrict to the Kirchhoff-Love constraint, and derive the global compatibility conditions as certain loop integrals of functions of the 2-dimensional strain measures. We also discuss the role of Gauss-Bonnet theorem in global strain compatibility. Finally, we discuss how various local defect densities become sources of local strain 1

An earlier version of this chapter appeared in [109].

81

incompatibility precluding the existence of the local isometric embedding, and, when local defects are absent, i.e., the strain fields are locally compatible, how the presence of certain global anomalies precludes the existence of a global isometric embedding. This will then set the stage for posing complete boundary-value-problems for internal stress distribution and natural shapes of defective structured surfaces, as will be discussed in the next chapter. Most of the results appearing in this chapter have not appeared elsewhere.

5.1

Strain measures and local strain compatibility

Let us assume that there exist an isometric embedding R : ω → R3 of ω into R3 . Let Aα := R,α and N :=

A1 ×A2 |A1 ×A2 | .

The first and second fundamental forms associated with this embedding are therefore

Aαβ = Aα · Aβ and Bαβ = −N ,β · Aα , respectively, over a local open neighbourhood U ⊂ ω. We consider the following sufficiently smooth fields, defined over R(U ), as descriptors of strain on the structured surface: (i) a symmetric tensor Eαβ , representing the in-surface strain field for measuring the local changes in length and angle; (ii) a tensor Λαβ for transverse bending strains; (iii) two vectors ∆α and Λα for measuring transverse shear and normal bending strains, respectively; and (iv) a scalar ∆ for normal expansion. We now pose the central question for conditions of local strain compatibility. Given sufficiently smooth strain fields (i)-(iv) over a fixed local isometric embedding R(U ) of a 2-dimensional ω, with first and second fundamental forms Aαβ (θα ) and Bαβ (θα ), respectively, what are the conditions to be satisfied for there to exist a sufficiently smooth local isometric embedding r : U ⊂ ω → R3 , with first and second fundamental forms aαβ and bαβ suitably constructed out of the

82

given fields, along with a sufficiently smooth director field d : r(U ) → R3 , so that the equations 1 1 Eαβ = (aα · aβ − Aα · Aβ ) = (aαβ − Aαβ ), 2 2 ∆α = d · aα − N · Aα = dα , ∆ = d · a3 − N · N = d3 − 1, Λαβ = d,β · aα − N ,β · Aα = ∂β dα − d3 bαβ + Bαβ , and Λβ = d,β · a3 − N ,β · N = d3,β + dµ bµβ , are satisfied on U ? Here aα := r ,α , a3 :=

a1 ×a2 |a1 ×a2 | ,

(5.1a) (5.1b) (5.1c) (5.1d) (5.1e)

while ∂ denotes covariant derivative with respect

to the surface Christoffel symbols induced by the metric aαβ on the deformed base configuration. Clearly, the strain fields measure deformation of the structured surface from its reference configuration (R(U ), N (U )) to the deformed configuration (r(U ), d(U )). The necessary and sufficient conditions, to be satisfied by the given strain fields, so that a local deformed configuration of the structured surface does exist such that equations (5.1) are satisfied, are called local strain compatibility conditions. These are nothing but the integrability conditions for r and d, as inferred from the system of PDEs in (5.1). Such compatibility conditions in the context of thin shells have been derived earlier by Epstein [32] and more recently by the present authors [109]. The discussion below follows the latter. The local strain compatibility conditions, over a simply-connected open set W ⊂ U , are given by aαβ := Aαβ + 2Eαβ is positive-definite, ∆ 6= −1,

(5.2a) (5.2b)

Λβ − ∆,β − ∆µ bµβ = 0,

(5.2c)

Λ[αβ] − ∂[β ∆α] = 0,

(5.2d)

∂1 b21 − ∂2 b11 = 0,

(5.2e)

∂1 b22 − ∂2 b12 = 0, and

(5.2f)

K1212 − (b212 − b11 b22 ) = 0,

83

(5.2g)

where bαβ :=

∂(β ∆α) − Λ(αβ) + Bαβ ∆+1

(5.3)

is symmetric and K1212 is the only independent component of the Riemann-Christoffel curvature of the projected Levi-Civita connection on U . We assume ∆ 6= −1 for (5.3) to be a valid definition. This would physically mean that directors are nowhere tangential to the base surface (see also Remark 5.1.1). Equations (5.2e), (5.2f) and (5.2g) are the well-known Codazzi-Mainardi and Gauss equations for aαβ and bαβ . Whenever these conditions are satisfied by the strain fields, there exists a sufficiently smooth local isometric embedding r : W → R3 , with first and second fundamental form given by aαβ and bαβ , respectively, and a director field d : r(U ) → R3 given by dα = ∆α , d3 = ∆ + 1, such that the PDEs (5.1) are identically satisfied everywhere on W . We now prove this result. Using the given strain fields Eαβ , Λαβ , Λα , ∆α and ∆, we construct a material metric g with components gαβ := aαβ + ζ Pαβ + ζ 2 Qαβ , gα3 = g3α := ∆α + ζ Uα , g33 := V,

(5.4)

where aαβ is as defined in (5.2a) and Pαβ := 2(Λ(αβ) − Bαβ ),

(5.5a)

Qαβ := aσγ (Λσα − Bσα )(Λγβ − Bγβ ) + Λα Λβ ,

(5.5b)

Uα := aσγ ∆σ (Λγα − Bγα ) + Λα (∆ + 1), and

(5.5c)

V := aαβ ∆α ∆β + (∆ + 1)2 .

(5.5d)

In the above, [aαβ ] := [aαβ ]−1 exists if we assume Eαβ to be such that aαβ is positive-definite. Note that, since ω is bounded and gij is continuous in θα and ζ, gij will be positive-definite on V := U ×(−, ) ⊂ M for sufficiently small . Our result is valid for this sufficiently small  and we a priori construct M such that  conforms to this small value throughout. For a technical discussion on the issue of smallness of  and positive definiteness of gij , refer to the proof of Theorem 2.8-1 in [19]. The ‘sufficient thinness’ of the structured surface is encoded in the definition (5.4) which describes how the 2-dimensional strain fields can be used to construct a 3-dimensional metric on the tubular neighbourhood M of ω. The parameter 84

 can be thought of as a physical length scale inherent to the description of the structured surface, e.g., thickness of a shell structure or the length of the individual molecules (not necessarily transverse to the surface) in lipid membranes. The 3-dimensional metric g is of second-order in the transverse coordinate ζ and this dependence brings out the non-locality in the kinematics of the structured surface, taking into account the transverse shear and normal distortion of the attached directors. The form of the metric in (5.4) is a generalization of the metric with components gαβ (θα , ζ) = aαβ − 2ζbαβ + ζ 2 aµν bµα bνβ , gα3 = g3α = 0, g33 = 1

(5.6)

defined in the proof of Theorem 2.8-1 in [19], which was otherwise restricted to Kirchhoff-Love theory (i.e., Λα = ∆α = ∆ = 0), where bαβ := −Λ(αβ) + Bαβ , and hence ignored any normal distortion or transverse shearing effect of the directors. The coefficients of the Levi-Civita connection of the metric (5.4), defined by Γqij := g pq Γijp where [g pq ] := [gpq ]−1 and Γijp := 12 (gip,j + gjp,i − gij,p ), can be calculated as 1 1 Γ333 = 0, Γ33ρ = Uρ − V,ρ , Γ3ρ3 = Γρ33 = V,ρ , 2 2   1 Γ3ρσ = Γρ3σ = ∆[σ,ρ] + Pρσ + ζ U[σ,ρ] + Qρσ , 2   1 Γρσ3 = ∆(σ,ρ) − Pρσ + ζ U(σ,ρ) − Qρσ , and 2     ζ2 ζ Pρδ,σ + Pσδ,ρ − Pσρ,δ + Qρδ,σ + Qσδ,ρ − Qσρ,δ . Γρσδ = sρσδ + 2 2

(5.7a) (5.7b) (5.7c) (5.7d)

The local strain compatibility conditions are the conditions for the embedding space M to be Euclidean, ˜ ijkl of the metric (5.4) to become identically zero. However, i.e., the Riemann-Christoffel curvature R as shown in [109], in order to ensure compatibility of the 2-dimensional strain fields, it is enough to ˜ ijkl impose that R = Rijkl (θα ) = 0. The curvature Rijkl has six independent components such that ζ=0

Rijkl = 0 if and only if R1212 = 0, R12σ3 = 0, and Rρ3σ3 = 0. After some manipulations, it can be shown that R1212 = K1212 − (b212 − b11 b22 ),

(5.8)

where the functions Kβαµν := aρν (Γραµ,β −Γρβµ,α +Γδαµ Γρβδ −Γδβµ Γραδ ) constitute the covariant components of the Riemann-Christoffel curvature of the projected Levi-Civita connection Γµαν ζ=0 on U . These, 85

by definition, have the symmetries Kαβµν = −Kαβνµ = Kµναβ and, hence, have only one independent component K :=

1 αβ µν 4 ε ε Kαβµν ,

the Gaussian curvature induced by the surface metric aαβ , where

1

εαβ := a− 2 eαβ and eαβ = eαβ denotes the 2-dimensional permutation symbol; also, a := det[aαβ ]. It is easily seen that K1212 = 4aK. Consequently, R1212 = 0, in conjunction with (5.8), can be used to infer (5.2g), which is the single independent Gauss’ equation satisfied by aαβ and bαβ . Further, we can evaluate R1213

     2  = a ∆β K2121 − b12 − b11 b22 − (∆ + 1) ∂2 b11 − ∂1 b12 and 2β

1β

R1223 = −a ∆β



(5.9)

    2  K1212 − b12 − b11 b22 − (∆ + 1) ∂2 b21 − ∂1 b22 .

(5.10)

Substituting (5.2g) in (5.9) and (5.10), the condition R12σ3 = 0 yields (5.2e) and (5.2f), which are the two independent Codazzi-Mainardi equations satisfied by aαβ and bαβ . The Gauss and CodazziMainardi equations satisfied over a simply-connected domain W ⊂ ω ensure the existence of a local isometric embedding r : W → R3 with first fundamental form aαβ and second fundamental form bαβ . Finally, we calculate αβ



1 bβ(ρ Iσ) + eβ(ρ Iσ) J − bρσ Iβ 2

Rρ3σ3 = (∆ + 1) I(ρ|σ) − Λ(ρ Iσ) − a ∆α   αβ 2 αµ βν + a (∆ + 1) + a a ∆µ ∆ν (J)2 eαρ eβσ , where αγ

Iβ := Λβ − ∆,β − ∆α a bγβ


εαβ ∂[β ∆α] − Λ[αβ] . and J := 2(∆ + 1)

 (5.11)

(5.12)

The condition Rρ3σ3 = 0 is therefore a set of three coupled first order homogeneous non-linear partial differential algebraic equations (PDAE) for three unknowns Iα and J. It has been shown previously [109] that the only physically meaningful solution is the trivial set Iα = 0 and J = 0; it can be shown that the non-zero solutions are unstable under generic perturbations (see [109]), hence we discard them. These equalities are equivalent to (5.2c) and (5.2d), respectively. They ensure the existence of a welldefined director field d : r(W ) → R3 , defined by dα = ∆α and d3 = ∆ + 1 (see (5.1b) and (5.1c)), which satisfies (5.1d) and (5.1e) identically over any simply-connected open set W ⊂ ω. This finishes our proof. 86

Remark 5.1.1. (Strain fields in structured surfaces with tangential director field.) When the director fields are everywhere tangential to their respective base surfaces, we choose the reference director field D to be some known tangent vector field over R(ω) (rather than the normal field N ). The relations (5.1) are now replaced by 1 1 Eαβ = (aα · aβ − Aα · Aβ ) = (aαβ − Aαβ ), 2 2 ∆α = d · aα − D · Aα = dα − Dα , ¯ β Dα , and Λαβ = d,β · aα − D ,β · Aα = ∂β dα − ∇ Λβ = d,β · n − D ,β · N = dµ bµβ − Dµ Bβµ ,

(5.13a) (5.13b) (5.13c) (5.13d)

¯ denotes the covariant derivative with respect to the induced Levi-Civita connection by the where ∇ metric Aαβ on the reference embedding R(ω). The integrability conditions, obtained from the above PDEs, for unknown r and d, given Aαβ , Bαβ , Dα , and the strain fields, provide the local strain compatibility conditions. To derive local compatibility relations, we note that the metric of the deformed surface is completely determined by (5.13a), aαβ := Aαβ + 2Eαβ , with Eαβ such that aαβ is positivedefinite; this is same as before. However, we no longer have a straight forward formula for the functions bαβ , contrary to the case with non-tangential directors. As a candidate for the second fundamental form of the deformed surface, we choose any bαβ that solves the algebraic equation (∆µ + Dµ )bµβ = Λβ + Dµ Bβµ ,

(5.14)

which is arrived after eliminating dα between (5.13b) and (5.13d). The Codazzi-Mainardi and Gauss’ equations involving aαβ and bαβ provide the first set of strain compatibility conditions, ensuring the existence of a local embedding r : U ⊂ ω → R3 with metric and curvature given by aαβ and bαβ , respectively, modulo isometries of R3 . The other strain compatibility condition is given by ¯ β Dα , Λαβ = ∂β (∆α + Dα ) − ∇

(5.15)

obtained by eliminating dα between (5.13b) and (5.13c). This ensures the existence of a tangential director field d : r(U ) → R3 such that (5.13) are satisfied. 87

5.1.1

Local strain compatibility conditions for Kirchhoff-Love shells

The Kirchhoff-Love constraint d = n simplifies the formalism of the last section. Local strain compatibility for Kirchhoff-Love shells, as discussed by Ciarlet [19], can be recovered from the general conditions discussed above by requiring Λα = ∆α = ∆ = 0. We now have the following formulae: aαβ := Aαβ + 2Eαβ , bαβ := Bαβ − Λ(αβ) . The local compatibility conditions, under the said restrictions, are then given by Eαβ is such that [aαβ ] := [Aαβ + 2Eαβ ] is positive definite, K1212 + [b11 b22 − b212 ] = 0,

(5.16b)

∂2 b11 + ∂1 b12 = 0,

(5.16c)

−∂2 b21 + ∂1 b22 = 0, and Λ[αβ] = 0.

5.2

(5.16a)

(5.16d) (5.16e)

Global strain compatibility for Kirchhoff-Love shells

In this section, we will restrict to the Kirchhoff-Love kinematic assumption, i.e., ∆ = ∆α = Λα ≡ 0, as just discussed. Thus, we have aαβ := Aαβ + 2Eαβ , and bαβ := Bαβ − Λ(αβ) . We now look for the global compatibility of the strain fields Eαβ and Λαβ , namely, the necessary and sufficient conditions on these strain fields such that there exists a sufficiently smooth global deformation map (isometric embedding) r : ω → R3 , such that the following equations are satisfied everywhere on R(ω) identically: 1 1 Eαβ = (aα · aβ − Aα · Aβ ) = (aαβ − Aαβ ), and 2 2 Λ(αβ) = n,β · aα − N ,β · Aα = −bαβ + Bαβ ,

(5.17a) (5.17b)

where aα := r ,α , and n := a1 × a2 /|a1 × a2 |. This physically means that the given strain fields can be constructed from the global deformation r everywhere on the global reference configuration R(ω). If the strain fields satisfy the local compatibility conditions (5.16), we know that r exists only locally 88

on every simply connected open subset U of ω. For global compatibility, these simply connected open patches must fit together nicely to form the globally isometric image of ω in R3 . Hence, the global conditions must involve the global or topological properties of ω; in other words, the global conditions are topology specific. We will now discuss the topological classification of 2-dimensional manifolds embeddable in R3

5.2.1

Topological classification of 2-dimensional manifolds embeddable in R3

All the 2-dimensional topological manifolds which are embeddable in R3 are homeomorphic to one of the following topological category of surfaces [6, Section 7.1]: 1. disc with no holes, 2. disc with holes, 3. the 2-dimensional sphere S2 , 4. S2 with a finite number g of attached handles, and 5. twisted bands, with either even or odd number of twists. The first four categories, and the bands with even number of twists, belong to the orientable class; twisted bands with odd number of twists are non-orientable, e.g., M¨obius band which has a single twist. Moreover, categories 1 and 3 are simply connected manifolds, whereas the rest are all multiply connected. Note that we have discarded the possibility of Klein Bottle or projective plane like structures, which are non-orientable 2-dimensional manifolds non embeddable in R3 .

5.2.2

Global integrability of the isometric embedding problem of 2-dimensional manifolds in R3

The global strain compatibility problem, synonymous with the issue of global isometric embedding of a 2-dimensional manifold into R3 , is tantamount to the global integrability of the following system of

89

nonlinear first-order PDEs, with the vector field r : R(ω) → R3 as the unknown variable, r ,α · r ,β = aαβ .

(5.18)

The solution of the isometric embedding problem, if it exists, can be highly non-unique [49]. For instance, the sphere S2 with an appropriately specified metric aαβ is realizable in the physical space R3 in an infinitely large number of ways, i.e., shapes. Each shape is associated with a fixed second fundamental form bαβ . The problem (5.18) is associated with strain compatibility of membranes which are structured surfaces with zero bending rigidity. The history and solution of this problem for various orientable topological 2-dimensional manifolds are discussed in [49]. However, the structured surfaces that we are interested in have finite bending rigidity. Hence, we will look into the more specialized class of isometric embedding problems where both aαβ and bαβ are specified. So we replace the system of PDEs (5.18) with r ,α · r ,β = aαβ and

(5.19a)

n,β · r ,α = −bαβ ,

(5.19b)

where n has been defined previously as a function of r. Remark 5.2.1. Note that the least regularity required by a classical solution to the problem (5.18) is C 1 , whereas for the problem (5.19) it is C 2 . The lesser regularity of the first problem can produce wildly oscillating solutions (discontinuous second derivative but once continuously differentiable) which cannot be afforded by the second problem, for example, there exists a C 1 global isometric embedding of the flat torus (i.e., a torus with a curvature free metric) into R3 where the second fundamental form bαβ does not exist anywhere on the embedding [11] (the embedding has a C 1 fractal structure). In the present case, we are demanding at least C 2 regularity on the solution. To investigate the existence and uniqueness of a global solution to the system (5.19), we will divide the non-linear second-order system into two separate linear first-order systems. Geometrically, this means that the compatibility is sought at two levels: at the first, we seek the existence of a well-defined

90

sufficiently smooth tangent space field that carries the specified metric everywhere; at the next, we seek the existence of a sufficiently smooth surface that fits the tangent space distribution.2

Problem 1:

In the first problem, we ask whether there exists a unique, at least C 1 , frame field

y := [a1 , a2 , n]T , with aα · aβ = aαβ and n = a1 × a2 /|a1 × a2 |, that satisfies the Gauss and Weingarten relations aα,β = Γµβα aµ + bβα n and n,β = −bαβ aα everywhere on ω. In other words, we look for the existence and uniqueness of the first-order linear system [97, 98] y,α = Aα y, such that aα · aβ = aαβ and n = a1 × a2 /|a1 × a2 |  1  Γα1   Aα :=  Γ1α2   −b1α

(5.20)

are satisfied satisfied everywhere on R(ω), where  2 Γα1 bα1    , (5.21) Γ2α2 bα2    −b2α 0

with Γταβ denoting the coefficients of the Levi-Civita connection induced by the metric aαβ (with a little abuse of the correct notation Γταβ |ζ=0 ). Problem 2: If there indeed exists a unique solution to the problem (5.20), then we ask whether the frame field {a1 , a2 }T is globally integrable, i.e., whether it spans the tangent spaces of an at least C 2 surface (isometric embedding) r : R(ω) → R3 . In other words, we look for the existence and uniqueness of the first order linear system [97, 98] r ,α = aα .

(5.22)

Solution to Problem 1: Towards constructing a solution to the first problem (5.20), following [97, 98], we restrict the PDE system to a system of linear first-order ODEs, dy(s) = A|γ (s)y(s), ds 2

(5.23)

The Frobenius theorem generalizes this second level compatibility to the necessary and sufficient integrability condi-

tions of a differentiable distribution on a sufficiently smooth manifold (see, for example, [36, p. 201]). Also, the two-level compatibility has been generalized by Gromov into a general technique to geometrically solve a certain class of underdetermined system of PDEs, famously known as the h-principle [47].

91

ω

C

η

X0

X

γ

Figure 5.1: Reducible and irreducible loops on a generic multiply connected 2-dimensional manifold ω. along curves γ on the reference configuration, parametrized by their arc length s (see Figure 5.1). Here, α

A|γ (s) := Aα (s) dγds(s) , where {γ α (s)Aα (s) ∈ R(ω), s ∈ [0, 1]} is an arc length parametrization of γ. Given an appropriate initial value y0 := y(X 0 ) = [a01 , a02 , n0 ]T at some generic point X 0 ∈ R(ω), i.e., a0α · a0β = aαβ (X 0 ) and n0 = a01 × a02 /|a01 × a02 |, the existence and uniqueness theorem of first-order linear ODEs (with variable coefficients) guarantees a unique solution to the above system over the full interval [0, 1]. Moreover, the solution automatically satisfies aα · aβ = aαβ and n = a1 × a2 /|a1 × a2 | everywhere on the interval [0, 1] [98]. The actual solution can be written in terms of a product integral (also called the Volterra Integral Formula) as [98] y(s) =

s Y

(γ)eA|γ (s ) ds y0 , 0

0

(5.24)

0

where s Y 0

(γ)eA|γ (s ) ds := I + 0

0

Z

0

s

A|γ (s0 ) ds0 +

Z

0

s  Z s0 0

 A|γ (s00 ) ds00 ds0 + · · · .

(5.25)

The value of the field y at any arbitrary point X can be calculated from y0 via the above integral along any curve on the reference configuration connecting X 0 to X, such as η = {η α (t)Aα (t) ∈ R(ω), t ∈ [0, 1] η(0) = γ(0), η(1) = γ(1)}, as shown in Figure 5.1. The everywhere single-valuedness of the field

y depends on the topological properties of ω as is discussed next.

92

a.

If ω is belongs to one of the first four topological categories of manifolds mentioned in Section 5.2.1, then y along γ (s = 1) = y along η (t = 1) = y0 ; hence, we have 1 Y

(γ)eA|γ (s ) ds = 0

0

s=0

1 Y 0 0 (η)eA|η (t ) dt ,

(5.26)

t=0

or, equivalently, Y

(C)eA|C (r) dr = I,

(5.27)

where C := η −1 ◦ γ is the loop (parametrized by its arc length r) obtained by traversing from X 0 to X(s = 1) along the path γ, then traversing from X(s = 1) = X(t = 1) to X 0 along the reverse path of η. This condition for a single-valued y must hold for every loop C over the reference configuration. If C is an irreducible loop, i.e., it encircles a hole and cannot be continuously shrunk to a point on the surface (as shown in Figure 5.1), then (5.27) is an necessary and sufficient integral condition to be satisfied by the fields aαβ and bαβ (hence, Eαβ and Λ(αβ) ). For a reducible loop, which can be continuously shrunk to any point on the surface, we can use Stokes theorem to convert the above line integrals into surface integrals over simply connected patches of the reference configuration. Then, the condition (5.27), for arbitrarily small reducible loops C, is equivalent to the local condition [97, 98] Aα,β + Aα Aβ = Aβ,α + Aβ Aα .

(5.28)

It is straightforward to show that the above equations reduce to the Gauss and Codazzi-Mainardi system satisfied by aαβ and bαβ at every point on the reference configuration, i.e., we recover the previously discussed local compatibility conditions (5.16) [97, 98]. Hence, the integrability condition for the first system is the integral relation (5.27) to be satisfied for every irreducible loop C (if there is any) on the reference configuration, if ω belongs to one of the first four topological categories of manifolds.

b.

Consider ω be homeomorphic to the M¨obius band, i.e., a twisted band with a single twist. In

that case, the field y must have a non-uniqueness of rotation Q := diag[−1, 1, −1] upon traversing an irreducible loop C encircling the hole. For a band with n twists, the rotational non-uniqueness would 93

be Qn . Hence, for ω belonging to the fifth category of twisted loops, e.g., a band with n twists, then the necessary and sufficient global compatibility conditions for the integrability of the first problem is Y (C)eA|C (r) dr = Qn .

(5.29)

Solution to Problem 2: Given that the above integrability conditions hold for the first problem, we look for the necessary and sufficient conditions for the existence of a global solution to the problem (5.22). For some fixed r(X 0 ) := r 0 , the value r(X) can be constructed from the quadrature [97, 98] r(s) = r 0 +

Z

s

aα (s0 )dγ α (s0 )

(5.30)

0

along a curve γ connecting X 0 to X, parametrized by its arc length s. Here, the vectors aα (s0 ) are known from the solution (5.24) in terms of Eαβ , Λ(αβ) , and y0 . Then, for a twice continuously differentiable r over R(ω), the integral appearing in the above expression must be path independent, i.e.,

I

aα (s)dC α (s) = 0

(5.31)

C

for every loop C on the reference configuration. The local implications of this condition for infinitesimal reducible loops C are contained in (5.28) [97,98]. For irreducible loops C on the reference configuration, if any, (5.31) is an additional integral necessary and sufficient condition to be satisfied by the fields Eαβ and Λ(αβ) in order to ensure the existence of a sufficiently smooth r. It is not difficult to show that the r thus constructed is unique up to rigid body motions in R3 . Finally, note that the skew part Λ[αβ] of the bending strain does not play any role in the compatibility of Kirchhoff-Love shells; hence, we can assume that Λ[αβ] ≡ 0. Whereas the first global compatibility condition (5.27) ensures the existence of a single-valued surface deformation gradient, the second global compatibility condition (5.31) ensures the existence of a single-valued displacement field. Global strain compatibility conditions similar to (5.27) and (5.31) have appeared in [132, 138] in the context of 3-dimensional nonlinear elasticity. The global conditions derived here in the context of Kirchhoff-Love shells are novel. We summarize our discussion of the present Section 5.4 in the following 94

Proposition 5.2.1. (Global strain compatibility conditions for Kirchhoff-Love shells) For sufficiently smooth strain fields Eαβ and Λαβ given on the reference configuration R(ω), there exists a sufficiently smooth global isometric embedding r : R(ω) → R3 so that it satisfies (5.17) everywhere on R(ω), if and only if, besides the local necessary and sufficient compatibility conditions (5.16), the given strain fields also satisfies the following conditions: (i) If ω belongs to one of the first four topological categories of 2-dimensional manifolds in Section 5.2.1, then

I Y A|C (r) dr (C)e = I and aα (s)dC α (s) = 0

(5.32)

C

must be satisfied for every irreducible loop (if there is any) over R(ω). (ii) If ω is a twisted band with n twists, the condition (5.32)1 should be replaced by Y

(C)eA|C (r) dr = Qn ,

(5.33)

where Q := diag[−1, 1, −1], and the second condition (5.32)2 remains as it is. The uniqueness of the embedding r is guaranteed up to rigid motions in R3 .

5.3

Local strain incompatibility arising from local defects

It is well-known that distributed defects within the material structure are inherent sources of strain incompatibility and, hence, internal stress [67]. Continuous distribution of material anomalies gives rise to non-trivial strain fields over a structured surface which are, in general, incompatible, which means that the fields aαβ and bαβ , constructed out of the strain fields that solve the strain incompatibility relations, do not correspond to the first and second fundamental form of any realizable isometric embedding of ω into R3 , not even locally. Hence, all the local and global strain compatibility conditions must be violated in presence of defects. The local strain incompatibility relations are derived from the 3-dimensional curvature relations

95

(4.4) restricted to U ⊂ ω, i.e., ζ = 0: Rαβµν = Ωαβµν − 2∂[α Wβ]µν − 2W[α|iν| Wβ]µ i ,

(5.34a)

Rαβµ3 = Ωαβµ3 − 2∂[α Wβ]µ3 − 2W[α|i3| Wβ]µ i , and

(5.34b)

Rα3µ3 = Ωα3µ3 − ∂α W3µ3 − 2W[α|i3| W3]µ i .

(5.34c)

The 2-dimensional independent strain incompatibility measures Rαβµν , Rαβµ3 and Rα3µ3 appearing on the left hand side are defined by the expressions (5.8)-(5.11) in terms of the 2-dimensional strain fields, ˜ ij k (θα , ζ = and on the right hand side appear the source terms of various defect densities: Wij k (θα ) := W 0) are defined in terms of dislocation densities and metric anomalies as Wij k := Cij k + Mij k , where the components Cij k (θα ) = C˜ij k (θα , ζ = 0) of contortion tensor are algebraic functions of the dislocation ˜ ij k (θα , ζ = 0) are algebraic functions of densities J i and αµk , and the components Mij k (θα ) = M the densities of metric anomalies Qkij , see the relations (4.2).3 After appropriate substitutions using equations (5.8)-(5.11), these relations reduce to K1212 − [b212 − b11 b22 ] = gΘ3 − 2∂[1 W2]12 − 2W[1|i2| W2]1 i ,

(5.35)

      a2β ∆β K1212 − b212 − b11 b22 − (∆ + 1) ∂2 b11 − ∂1 b12 = −gΘ2 − 2∂[1 W2]13 − 2W[1|i3| W2]1 i , (5.36) 1β



−a ∆β K1212 −



b212 −b11 b22





  −(∆+1) ∂2 b21 −∂1 b22 = gΘ1 −2∂[1 W2]23 −2W[1|i3| W2]2 i and (5.37) 

 1 (∆ + 1) ∂(σ Iρ) − Λ(ρ Iσ) − a ∆α bβ(ρ Iσ) + eβ(ρ Iσ) J − bρσ Iβ 2   + aαβ (∆ + 1)2 + aαµ aβν ∆µ ∆ν (J)2 εαρ εβσ = ερ3ν εσ3µ Θνµ − ∂ρ W3σ3 αβ

−2W[ρ|i3| W3]σ i .

(5.38)

The above are the local strain incompatibility relations for a continuously defective structured surface in their full generality. In the absence of dislocations and metric anomalies, i.e., when Wij k ≡ 0, clearly, the 3

It should be noted that the components Ωµ3αβ do not appear in any of the equations (5.34). This is because, according

to the fourth Bianchi-Padova relation, they can be written in terms of Ωαβµ3 and other defect measures, and hence are not independent quantities.

96

density of wedge disclinations Θ3 act as the single source to the incompatibility of the Gauss equation (5.35), while the densities of twist disclinations/intrinsic orientational anomalies Θµ = Θµ3 are the single source terms to the incompatible Codazzi-Mainardi equations (5.36) and (5.37); the symmetric disclination density fields Θµν are sources to non-trivial Iα and J. In many applications, to follow in the next chapter, we will restrict attention to sufficiently thin structured surfaces, e.g., purely disclinated nematic membranes, monolayer bio-membranes, 2-dimensional crystals etc., the Θµν -disclinations and αµk -dislocations are naturally absent. We will discuss further simplifications of (5.35)-(5.38) in Section 6.2 under some realistic assumptions of smallness/vanishing of certain strain fields and defect densities. We will also obtain certain forms of these relations that have already appeared in the literature. Finally, note that, the disclination densities Θµ3 seem to be absent from the above relations. This is so because they are not independent but expressible in terms of Θµ and other defect densities as a consequence of the fourth Bianchi-Padova relation (see Footnote 3). The local strain incompatibility relations for Kirchhoff-Love shells easily follow by substituting ∆ = ∆α = Λα = 0, hence, Iα ≡ 0, in the above relations:

K1212 − [b212 − b11 b22 ] = aΘ3 − 2∂[1 W2]12 − 2W[1|i2| W2]1 i ,

5.4

(5.39)

−∂2 b11 + ∂1 b12 = −aΘ2 − 2∂[1 W2]13 − 2W[1|i3| W2]1 i ,

(5.40)

−∂2 b21 + ∂1 b22 = aΘ1 − 2∂[1 W2]23 − 2W[1|i3| W2]2 i , and

(5.41)

aαβ (Λ[αβ] )2 εαρ εβσ = ερν εσµ Θνµ − ∂ρ W3σ3 − 2W[ρ|i3| W3]σ i .

(5.42)

Global strain incompatibility relations for a Kirchhoff-Love shell arising from global defects

Local defects are not the only source to the incompatibility of the strain fields. For example, it is well known that a linear temperature distribution would produce a compatible strain field in a disc without holes, but it can produce an incompatible strain if there are holes in the disc. This is an example of

97

n a2 a1

Ω b

a2

n a1

Figure 5.2: Disc with a hole after making a cut. The global Frank tensor Ω and the global Burgers vector b as measures of global incompatibility of the surface. a global metric anomaly. We have seen in Section 1.2.2 that there can be global topology-preserving anomalies, such as global dislocations and disclinations, on surface crystals. These global defects act as additional source to the incompatibility of the strain fields, hence, residual stress. In this section, we will write down the global strain incompatibility relations for all the topologies attainable by the surface embeddable in R3 . These relations are obtained by violating the global compatibility conditions discussed previously.

5.4.1

Disc with holes

For a globally incompatible strain, the field y may be multi-valued, i.e., upon circumnavigating along the irreducible loop C, y(s = 1) 6= y0 . Hence, there exists a second order rotation tensor Ω(C), dependent on the loop C, such that y(s = 1) = Ω(C)y0 , which translates into Y

(C)eA|C (r) dr = Ω(C).

(5.43)

Here, Ω(C) is a measure of global strain incompatibility (see Figure 5.2). We will call the loop dependent tensor Ω(C) the global Frank tensor associated with the loop C (recall the discussion in Section 1.2.2). The orthogonality of Ω(C) is due to the fact that there are no distributed metrical disclinations on the surface ζ ≡ 0, hence, inner product is preserved during parallel transport [110]. 98

There is a second source of global incompatibility coming from the violation of the second global compatibility relation, due to a non-zero global Burgers vector b(C) associated with the loop C: I

aα (s)dC α (s) = b(C).

(5.44)

C

The vector b(C) is the second measure of global incompatibility (see Figure 5.2, and recall the discussion in Section 1.2.2). It can be easily shown that in absence of local anomalies, Ω(C) and b(C) become independent of the loop C. Then we can drop the argument C, and call Ω and b, simply, the global Frank tensor and the global Burgers vector of the surface, respectively.

5.4.2

Torus

For a toroidal topology, the global incompatibility relations are given by the relations of the kind (5.43) and (5.44), but with two distinct global Frank tensor Ω1 , Ω2 , and two distinct global Burgers vector b1 and b2 , associated with the two mutually non-homotopic class of loops C1 and C2 , respectively, as discussed previously (recall the discussion in Section 1.2.2, and Figures 1.12 and 1.13). The global strain incompatibility relations for the torus T (with genus 1) are 1. 2.

Q H

(C1 )eA|C1 (r) dr = Ω1 ,

C1

5.4.3

aα (s)dC α (s) = b1 ,

Q H

(C2 )eA|C2 (r) dr = Ω2 , and

C2

aα (s)dC α (s) = b2 .

Sphere

There are no global incompatibility relation for the sphere, since it is compact, closed and simply connected, hence there are no irreducible loops.

5.4.4

Twisted bands

For a twisted band with n twists, the global incompatibility relations are given by 1. 2.

Q H

C

(C)eA|C (r) dr = ΩQn , and aα (s)dC α (s) = b.

99

Chapter 6

Internal Stress Field and Natural Shape of Defective Structured Surfaces A central problem in the mechanics of solids is, for a given distribution of material defects, to determine the internal stress field and the deformed shape of the defective body with respect to a fixed reference configuration. The notion of defects is to be understood in the sense of material anomalies, as discussed in Section 2.2, which lead to an inhomogeneous material response in an otherwise materially uniform body. In particular, if we assume stress to be purely elastic in origin, then, in general, there is no oneto-one mapping from the current configuration of the defective body, which is realized as a connected set in the physical space, to its defect-free natural (stress-free) state. This means that the natural state of the defective material body cannot be realized as a connected set in the physical space. It also entails an incompatible elastic strain field, which appears as the energetic dual of stress, with sources of incompatibility derived from various defect densities. The absence of an elastic deformation map also implies that there is no one-to-one (plastic deformation) map which connects the reference configuration to the natural state. A plastic strain field, whose incompatibility is again related to defect densities, can be derived from the difference of metric tensors associated with the natural and the reference configurations. The formulation of a well-posed boundary-value-problem for stress and deformed shape, for given defect densities, requires a prescription on how the strain fields - total,

100

elastic, and plastic - are all related to each other. The problem of relating the three configurations (reference, natural, and current) is usually addressed by assuming a multiplicative decomposition of the total deformation gradient into elastic and plastic distortion tensors. The total deformation gradient tensor is the gradient of the total deformation map and yields a compatible total strain tensor. The elastic and plastic distortion tensors map tangent spaces from the natural configuration to the current configuration and from the reference configuration to the natural configuration, respectively. However, in the presence of disclination density, the elastic and plastic distortion tensors are not well-defined [109]. The ambiguity arises due to the rotational part of the tensors becoming multi-valued. Nevertheless, the multiplicative decomposition can be used for isotropic materials where both elastic and plastic rotations do not play any role in the final boundary-value-problem [24]. The need for multiplicative decomposition is also circumnavigated if we assume an additive decomposition of the total strain into elastic and plastic counterparts. In such a situation, we do not require the notion of elastic and plastic distortion tensors at all. For 3dimensional elastic solids, the additive decomposition of strain is essentially based on the smallness of both deformation and plastic strain (to the same order). The resulting theory is necessarily applicable to small deformation problems [25]. On the other hand, an additive decomposition of strains, with the notion of strain as defined in the beginning of Section 5.1, in the context of 2-dimensional structured surfaces is less restrictive. It in fact allows for moderately large rotations in the deformation while keeping small strains [93]. This is important for structured surfaces since, unlike 3-dimensional bodies, they are very much likely to accommodate internal stresses by escaping into the third dimension via moderately large rotations. The nature of the assumed additive decomposition, which allows for a separation of order of the in-surface stretching and the bending mode of deformation for structured surfaces, will be discussed in detail in the following. The decomposition of strain field gives way to formulating the boundary-value-problem. We consider a reference configuration for the structured surface where directors are aligned with the normal; the case where directors are tangential can be treated following Remark 5.1.1. The plastic strains satisfy the local incompatibility equations (5.35)-(5.38), and the global incompatibility relations in Section 5.4 101

for specific topologies of the surface. The elastic strains will satisfy a different form of incompatibility equations with the reference configuration replaced by current configuration in the derivation of these equations. For this difference, they are much more involved since the current configuration is itself unknown (with directors not necessarily coinciding with the normal); we do not use incompatibility relations for elastic strain in our framework. The (plastic) strain incompatibility relations are combined with the additive decomposition, the constitutive laws (for relating elastic strains with stresses and moments), and the equilibrium equations, to yield the full boundary-value-problem for the determination of stress field and natural shape of the structured surface for a given distribution of defects. We will proceed to do so in the following under the Kirchhoff-Love deformation constraint on the director field, requiring them to coincide with the local normal field in the current configuration. This is done only in order to present a simplified theory, while postponing further generalizations to future works.

6.1

Kinematics of Kirchhoff-Love shells with small strain accompanied by moderate rotation

Following Section 5.1, we consider the fixed reference configuration of the Kirchhoff-Love structured surface to be given by a local isometric embedding R : U → R3 , where U is a simply-connected open set of ω; also, as before, we take (θα , ζ) as the adapted coordinates on U . The tangent spaces of R(U ) are spanned by the natural base vectors Aα = R,α . The first and second fundamental forms associated with the reference surface are given by Aαβ = Aα · Aβ and Bαβ = −N ,α · Aβ , respectively, where N :=

A1 ×A2 |A1 ×A2 |

is the local unit normal. We will assume the adapted coordinates

ˆ α on the tangent (θα , ζ) to be convected by deformation of the surface. The natural base vectors A ˆ : U → R3 , a different isometric embedding of U , are given by spaces of the current configuration R ˆα = R ˆ ,α . The first and second fundamental forms associated with the current configuration are A ˆα · A ˆ β and B ˆ ,α · A ˆ β , respectively, where N ˆ := ˆαβ = −N Aˆαβ = A

ˆ 1 ×A ˆ2 A ˆ 1 ×A ˆ2 | . |A

The reference and the

ˆαβ ) individually current configurations are shown in Figure 6.1. The pairs (Aαβ , Bαβ ) and (Aˆαβ , B satisfy the Gauss and Codazzi-Mainardi equations owing to the existence of the isometric embeddings

102

ˆ α ⊗ Aα F=A

E = Eαβ Aα ⊗ Aβ = 12 (Aˆαβ − Aαβ )Aα ⊗ Aβ

Reference configuration

ˆαβ + Bαβ )Aα ⊗ Aβ Λ = Λαβ Aα ⊗ Aβ = (−B

N A2

A1

θ2

R(θα )

R3

ˆ α) R(θ

e3

Aα = R,α Aαβ = Aα · Aβ

e2 e1

Bαβ = −N ,β · Aα

θ2

ˆ1 A θ1 ˆα = R ˆ ,α A ˆα · A ˆβ ˆ Aαβ = A

ˆ ,β · A ˆα ˆαβ = −N B

e Ee = Eαβ aα ⊗ aβ E e := 1 (Aˆαβ − aαβ )

p Ep = Eαβ Aα ⊗ Aβ

p Eαβ := 12 (aαβ − Aαβ )

Λp =

ˆ N ˆ2 A

θ1

Λpαβ Aα

Current configuration

αβ e

Λ =

⊗ Aβ

ζ

Λp(αβ) := −bαβ + Bαβ

θ1

θ2

U

Natural configuration

⊗ aβ

ˆαβ + bαβ Λe(αβ) := −B

d=n O

2

Λeαβ aα

aα are defined modulo rotational symmetries at O

Metric gij Connection Lkij aαβ := gαβ ζ=0

Non-Euclidean space

Figure 6.1: Kinematics of the elastic-plastic decomposition of the total deformation measures in Kirchhoff-Love shells. The only non-trivial disclinations are represented by Θ3 , and hence we have a well defined normal over the surface in the natural configuration. ˆ The total surface deformation gradient tensor maps the tangent spaces of R(U ) to those R and R. ˆ ) and is given by F = A ˆ α ⊗ Aα . The total surface strain and the total bending strain tensors, of R(U ˆαβ +Bαβ )Aα ⊗Aβ , defined as E = Eαβ Aα ⊗Aβ = 21 (Aˆαβ −Aαβ )Aα ⊗Aβ , and Λ = Λαβ Aα ⊗Aβ = (−B respectively, measure the relative first and second fundamental forms of the current configuration with respect to the reference configuration of the structured surface. Other strain measures, introduced in the beginning of Section 5.1, are identically zero under the Kirchhoff-Love constraint (which imposes the director field to coincide with the unit normal field).

103

The elastic surface strain tensor Ee and the elastic bending strain tensor Λe are defined as energetic dual of surface stress tensor and bending moment tensor, respectively, see Section 6.3. On the other hand, the plastic surface strain and and the plastic bending strain tensors can be defined as Ep = p p := 12 (aαβ −Aαβ ) and Λp(αβ) := −bαβ +Bαβ . Eαβ Aα ⊗Aβ and Λp = Λpαβ Aα ⊗Aβ , respectively, where Eαβ

Also, aαβ and bαβ are, respectively, the first and (non-Riemannian) second fundamental form of U in the material space (ω, L, g); both of them are determined from the metric and the connection of the material space. Here, and henceforth, we will use superscripts e and p to denote elastic and plastic, variables respectively; in particular, they should not be read as indices. We note that it is only in the absence of disclinations and intrinsic orientational anomalies that there exist well-defined crystallographic vector fields aα := Fp Aα over the tangent spaces of local intermediate configuration U , where Fp is the (single-valued) plastic distortion field (an analogous description can be given in terms e aα ⊗ aβ and of elastic distortion field). The elastic strain tensors can then be written as Ee = Eαβ e := 1 (A α αβ ˆαβ − aαβ ) and Λe ˆ Λe = Λeαβ aα ⊗ aβ , where Eαβ 2 (αβ) := −Bαβ + bαβ . Here, a := a aβ are the

dual crystallographic vector fields on the material space, with [aαβ ] := [aαβ ]−1 . The crystallographic vector fields aα are no longer well-defined in the presence of rotational anomalies. We now discuss the additive decomposition of total strain tensors in elastic and plastic parts. Introduce a small parameter  := h/R, where h is the maximum thickness of the structured surface and R is the minimum radius of curvature that U can assume in all possible deformations. Let E, Ep , Ee and their first and second spatial derivatives be of order O(), and Λ, Λp , Λe and their first 1

spatial derivatives be of order O( 2 ). Here, following Landau’s notation, for f : R → Rk , we write f (s) = O(s) if and only if there exist positive constants M and δ such that ||f (s)||Rk ≤ M |s| for all |s| < δ. Following Naghdi and Vongsarnpigoon [93], we emphasize that the resulting theory, where the surface and bending strains follow these separated orders, allows for small surface strain accompanied by moderate rotation. We postulate that the following decompositions for the total surface and bending

104

strains hold: E = Ee + Ep of order O(),

(6.1a)

1

Λ = Λe + Λp of order O( 2 ).

(6.1b)

The first decomposition, which is of order O(), is the standard additive decomposition for small 1

strains used commonly in small deformation theories. The second decomposition, of order O( 2 ), is non-standard and captures moderately large rotation relative to in-surface stretching. It can be shown 1

that the resulting deformation, which is of order O( 2 ), is more general than infinitesimally small deformation theory of structural shells and, at the same time, stricter than a fully non-linear finite 1

ˆ α ≈ aα , up to order O( 2 ) (cf. [93]), we have deformation theory [93]. Also, as Aα ≈ A p e Eαβ = Eαβ + Eαβ of order O(), 1

Λαβ = Λeαβ + Λpαβ of order O( 2 ).

(6.2a) (6.2b)

Note that, as [Λαβ ] is symmetric, necessarily Λp[αβ] = −Λe[αβ] . These approximated decompositions with the mentioned order hold for sufficiently thin structured surfaces where the bending mode dominates over surface stretching for a given loading (internal or external).

6.2

Strain incompatibility relations for sufficiently thin KirchhoffLove shells with small strain accompanied by moderate rotation

6.2.1

Local incompatibility relations

We assume that disclination densities with components Θµν , Θµ3 and Θµ , and dislocation densities with components αµk , are identically zero. This is reasonable under the Kirchhoff-Love constraint, where director field coincides with the orientation field, and if we restrict ourselves to sufficiently thin structured surfaces. The allowable defects are then the in-surface wedge disclinations Θ3 , the surface screw and wedge dislocations J i , and the metric anomalies Qkij . For a non-trivial wedge disclination density Θ3 , and with other disclinations absent, the rotational ambiguity in the crystallographic basis 105

vector fields aα always falls within the rotational symmetry group of the base material at the respective points on U . As a result, under the considered assumption on the nature of allowable defects, the fields aα are well-defined with known rotational ambiguity. The normal n at each point in the natural configuration is then well-defined. The Kirchhoff-Love constraint simplifies the formalism of Section 5.3. We now revisit the strain incompatibility relations derived therein for the plastic strain tensor. The corresponding result for the elastic strain tensor is more involved since the reference configuration, used in case of plastic strains, has to be replaced with current configuration, which is itself unknown. The material metric has a simple block diagonal form, given in (5.6) but now in terms of the plastic strain components, which can be deduced from (5.4) using Λpα = ∆pα = ∆p = 0. Also, from (5.12), we can infer that Iα = 0 and J = −εαβ Λp[αβ] . The local strain incompatibility relations (5.35)-(5.38), under these considerations, and with only Θ3 , J i , and Qkij as non-trivial defect measures, are reduced to K1212 + [b11 b22 − b212 ] = aΘ3 − 2∂[1 W2]12 − 2W[1|i2| W2]1 i ,

(6.3)

−∂2 b11 + ∂1 b12 = −2∂[1 W2]13 − 2W[1|i3| W2]1 i ,

(6.4)

−∂2 b21 + ∂1 b22 = −2∂[1 W2]23 − 2W[1|i3| W2]2 i , and

(6.5)

aαβ εαρ εβσ a−1 (Λp[12] )2 = −∂ρ M3σ 3 − (Cρα 3 + Mρα 3 )(C3σ α + M3σ α ) 3 α 3 −Mρ3 3 Mσ3 + M3α 3 Wρσ + M33 3 Wρσ ,

(6.6)

where, recall that, the tensor with components Wijk and Wij k ( = g mk |ζ=0 Wijm ) is given in terms of the contortion and non-metricity tensors, see (4.2), as Wij k = Cij k + Mij k . The components of the contortion tensor, defined in (4.2c), take a simple form, for αµk = 0 and metric given by (5.6), as collected below: C3β 3 = Cβ3 3 = C33 i = C3β3 = Cβ33 = C33i = 0, C3β α = Cβ3 α = aαν ενβ J 3 ,

(6.7a)

C3βα = Cβ3α = Cαβ 3 = Cαβ3 = εαβ J 3 ,

(6.7b)


Cαβµ = J σ aσβ εµα + aσα εµβ + aσµ εαβ , and Cαβ µ = aµν Cαβν . 106

(6.7c)

1

Here, εαβ := a 2 eαβ . On the other hand, the tensor associated with non-metricity has components 1 1 M33 3 = M333 = Q333 , M33α = (2Q3α3 − Qα33 ), M33 α = aαβ M33β , 2 2 1 M3α 3 = Mα3 3 = M3α3 = Mα33 = Qα33 , 2 1 M3α β = Mα3 β = aβν (Q3να − Qνα3 + Qα3ν ), 2 1 Mαβ 3 = Mαβ3 = (Qα3β − Q3βα + Qβα3 ) 2 1 Mαβµ = (Qαµβ − Qµβα + Qβαµ ), and Mαβ µ = aµν Mαβν . 2

(6.8a) (6.8b) (6.8c) (6.8d) (6.8e)

These are to be substituted into (6.3)-(6.6) to obtain the incompatibility relations in terms of defect densities. As discussed above, the only non-trivial disclination density is Θ3 , and the only non-trivial dislocation densities are J i ; there are no restrictions on the non-metricity densities. Note that the in-surface metric anomalies do not appear in (6.6). Therefore, in the absence of out-of-surface met1

ric anomalies, the right side of (6.6) reduces to aαβ εαρ εβσ (J 3 )2 which implies that |Λ[12] | = a 2 |J 3 |. The skewness of plastic bending strain is then completely characterized in terms of in-surface screw dislocations.

Assumptions on the order of defect densities:

Motivated from the assumed order of in-surface

and bending strain, we assume that the density of in-surface wedge disclinations Θ3 , the densities of edge dislocations J α up to their first spatial derivatives, be of order O(), and the density of the 1

screw dislocation J 3 , up to its first spatial derivatives, be of order O( 2 ). We further assume that the densities of the pure in-surface metric anomalies Qµαβ , along with their spatial derivatives up to first order, to be of order O(), and the densities of metric anomalies characterized by Qkij , with at least one of the indices k, i or j taking the value 3, along with their first spatial derivatives, to be of order 1

O( 2 ). The identical order of the in-surface strain and the in-surface defects, and of the bending strain and out-of-surface defects, have another physical justification: it puts a limit on the magnitudes of the defect density fields within which our constitutive assumptions on the material is valid. 2-dimensional solid crystals are known to undergo a transition to an intermediate hexatic phase when defect densities proliferate under some external agency [95, Chapter 6]. This is known in condesed matter literature as 107

defect mediated melting. The strain measures, Eαβ , Λαβ etc., within the present setup are incapable to model such phase transitions, and the above assumptions on the order of various defect densities can be seen as the threshold of the defect density magnitudes under which the material under consideration continues to retain its present phase.

Reduced local strain incompatibility relations: We will now reduce the incompatibility relations (6.3)-(6.6) under the kinematical assumption of small strain accompanied by moderate rotation, i.e., 1

p Eαβ = O() and Λpαβ = O( 2 ), and the assumption on the order of the defect densities. We note that p a = A(1 + 2tr(Aαµ Eµβ )) + o(), and

aαβ = Aαβ − 2E pαβ + o(),

(6.9a) (6.9b)

p where E pαβ := Aαρ Aβσ Eρσ = O(). Hence, we obtain

1 p τ Γταβ := aτ σ (aσβ,α + aσα,β − aαβ,σ ) = Γ¯ταβ + Hαβ + o(), 2

(6.10)

1 Γ¯ταβ := Aτ σ (Aσβ,α + Aσα,β − Aαβ,σ ), 2

(6.11)

where

Γ¯αβσ := Γ¯ταβ Aτ σ , are the surface Christoffel symbols on the reference configuration, and the functions p τ p , Hαβ := Aτ ν Hαβν

(6.12)

where p Hαβν

p p p := ∂¯α Eνβ + ∂¯β Eνα − ∂¯ν Eαβ ,

(6.13)

form the components of a tensor of order O(). Thus, we get the following expression by retaining terms up to order O(): p Kβαµν = K¯ βαµν + 2∂¯[β Hα]µν .

(6.14)

¯ and Here, ∂¯ denotes the surface covariant derivative on the reference configuration with respect to Γ, ¯ βαµν are the components of the Riemann-Christoffel curvature of Γ. ¯ The single independent relation K is p p p p p K1212 = K¯ 1212 + ∂¯1 H212 − ∂¯2 H112 = K¯ 1212 + ∂¯11 E22 + ∂¯22 E11 − 2∂¯2 ∂¯1 E12 .

108

(6.15)

We further obtain 2 b11 b22 − b212 = B11 B22 − B12 + {Λp11 Λp22 − (Λp(12) )2 }

(6.16)

up to order O(). Recalling that the pair (Aαβ , Bαβ ) satisfies the Gauss and Codazzi-Mainardi equations on the reference configuration, we obtain, up to order O(), p p p K1212 + [b11 b22 − b212 ] = ∂¯11 E22 + ∂¯22 E11 − 2∂¯2 ∂¯1 E12 + {Λp11 Λp22 − (Λp(12) )2 }.

(6.17)

Moreover, ∂α bµν

= −∂¯α Λp(µν) + ∂¯α Bµν

  p p p − Bµσ (∂¯ν Eσα + ∂¯α Eσν − ∂¯σ Eαν ) (µν) + o()

= −∂¯α Λp(µν) + ∂¯α Bµν + O().

(6.18)

−∂2 bµ1 + ∂1 bµ2 = ∂¯2 Λp(µ1) − ∂¯1 Λp(µ2) .

(6.19)

1

Hence, up to order O( 2 ),

The local incompatibility relations (6.3)-(6.6), under the assumptions on the strains and the defect densities, reduce to p p p ∂¯11 E22 + ∂¯22 E11 − 2∂¯2 ∂¯1 E12

√ +{Λp11 Λp22 − (Λp(12) )2 } = AΘ3 + 2 A Aσ[1 ∂¯2] J σ − 2∂¯[1 M2]12 − 2M[1|32| M2]1 3 , ∂¯2 Λp11 − ∂¯1 Λp(12) = ∂¯2 Λp(21) − ∂¯1 Λp22 =

√

A∂¯1 J 3 − 2∂¯[1 M2]13 − 2M[1|33| M2]1 3 ,

√ A∂¯2 J 3 − 2∂¯[1 M2]23 − 2M[1|33| M2]2 3 , and

(6.20) (6.21) (6.22)

3 3 A−1 Aαβ eαρ eβσ (Λp[12] )2 = A−1 Aαβ eαρ eβσ (J 3 )2 − Mρα 3 M3σ α − Mρ3 3 Mσ3 + M33 3 Mρσ .

(6.23)

6.2.2

Global incompatibility relations

The global incompatibility relations for the plastic strains can be written for various topologies of the surface, as discussed in Section 5.2. The primary integrand Aα , defined as in (5.21), that appears in 109

these integral incompatibility relations can be approximated, under the present assumptions on the in-surface and the bending strain fields, by retaining terms up to order O(): 

p p p 1ν ¯ 2ν ¯ ¯ p ¯ p ¯ p ¯ p ¯1 ¯2  Γα1 + A (∂ α Eν1 + ∂ 1 Eνα − ∂ ν Eα1 ) Γα1 + A (∂ α Eν1 + ∂ 1 Eνα − ∂ ν Eα1 ) Bα1 − Λα1   p p p p p p Aα :=  Γ¯1α2 + A1ν (∂¯α Eν2 + ∂¯2 Eνα − ∂¯ν Eα2 ) Γ¯2α2 + A2ν (∂¯α Eν2 + ∂¯2 Eνα − ∂¯ν Eα2 ) Bα2 − Λpα2   −A1β (Bαβ − Λpαβ ) −A2β (Bαβ − Λpαβ ) 0



   .  

(6.24)

We can use the above expression, and further approximations, to write down the reduced global incompatibility relations in a straightforward way. However, in the present section, we will derive a more useful version of the global incompatibility relations which would help to bring out the physical interpretation of various components of the global Burgers vector b and the global Frank tensor Ω. In Section 5.2, the global incompatibility relations were derived in two steps: first, writing down the incompatibility of the field aα (i.e., its existence and uniqueness), and then, the incompatibility of the deformation map r. In practical applications, however, it is more appropriate to study the incompatibility of the associated deformation gradient F = aα ⊗ Aα + n ⊗ N . Note that, in the polar √ decomposition of the deformation gradient F = RU, the stretching part U = I + 2E is uniquely determined by the in-surface strain field Eαβ , and it is single-valued. Then, the incompatibility in F boils down to the incompatibility of its rotational part R. The incompatibility relation for the rotation tensor comes from the non-integrability of the following PDEs for R [93]:

RT R,α = Aα .

(6.25)

Here, the right hand side skew coefficient tensor Aα is a function of the given strain fields [93, Equation 2.25]. Hence, R is uniquely determined by the strain fields, if and only if, the above system of PDEs is integrable. Under the present order assumptions on the strain fields, the approximated form of Aα up to order O() can be written directly from the formula [93, Equation 3.11] as Aα = −ATα = (Λαβ + Eµβ Bαµ )(Aβ ⊗ N − N ⊗ Aβ ) + (∂¯γ Eαβ − ∂¯β Eαγ )Aβ ⊗ Aγ . 110

(6.26)

On the other hand, the R has the unique representation R = I + Φ + Ψ in terms of a symmetric 1

tensor field Φ of order O() and a skew tensor field Ψ of order O( 2 ) [93, Equations 3.15–3.24]. The differential equations for these two tensors are obtain from (6.25) and (6.26) as1

Ψ,α = Λαβ (Aβ ⊗ N − N ⊗ Aβ ), and

(6.27)

Φ,α − ΨΨ,α = Eµβ Bαµ (Aβ ⊗ N − N ⊗ Aβ ) + (∂¯γ Eαβ − ∂¯β Eαγ )Aβ ⊗ Aγ .

(6.28)

Hence, the global incompatibility of R is determined by the non-integrability of the above equations for Φ and Ψ. These relations can be written as the path dependence of the integrals of the right hand side quantities: Ψ

I

Λαβ (Aβ ⊗ N − N ⊗ Aβ ) dC α , and IC  Eµβ Bαµ (Aβ ⊗ N − N ⊗ Aβ ) + (∂¯γ Eαβ − ∂¯β Eαγ )Aβ ⊗ Aγ ΩΦ (X) =

Ω (X) =

C

Z +Λαβ (A ⊗ N − N ⊗ A ) β

β

¯ X

X0

Λµν (A ⊗ N − N ⊗ A ) dC¯ µ ν

ν

(6.29)



dC α .

(6.30)

Here, C is an irreducible loop on the reference configuration. Note that the first equation is of order 1

O( 2 ), while the second one is of order O(). The tensors ΩΨ and ΩΦ are two independent constituents of the global Frank tensor Ω introduced in Section 5.2. It can be seen that the above global incompatibility relations for reducible loops C reduce to the local incompatibility relations (6.21), (6.22) and (6.20), respectively, by using Stokes theorem on the simply connected patches bounded by the reducible loops. The differential equation for the displacement field, up to order O(), is [93, Equation 3.41] u,α = (E + Φ + Ψ)Aα .

(6.31)

Hence, the corresponding global incompatibility relation can be written as the path dependence of the integral of the right hand side quantity: 1

In [93], Naghdi and Vongsarnpigoon have written a similar expression by retaining the in-surface strain gradient terms 1

Eαβ,γ which, unlike our theory, are assumed to be of order O( 2 ).

111

b(X) =

I  C

¯ Eαβ Aβ (X)

Z + +

¯ X

Λµν (Aν ⊗ N − N ⊗ Aν ) dC¯ µ

X0 ¯  X

Z

X0

Eµβ Bρµ (Aβ ⊗ N − N ⊗ Aβ ) + (∂¯γ Eρβ − ∂¯β Eργ )Aβ ⊗ Aγ

β

β

+Λρβ (A ⊗ N − N ⊗ A )

Z

¯ X

X0

¯ Λµν (A ⊗ N − N ⊗ A ) dC¯ µ ν

ν



.

  ρ ¯ ¯ dC¯ α dC Aα (X) (6.32)

Remark 6.2.1. (Construction of the displacement field from prescribed compatible strain fields:) When the global Burgers vector b on the left hand side of the equation (6.32) identically vanishes for every loop C in the surface, the strain fields Eαβ and Λαβ are compatible globally. Then displacement vector u, in that case, can be written in terms of the given compatible strain fields as Z X ¯ Eαβ Aβ (X) u(X) = X0

Z + +

Z

¯ X

Λµν (Aν ⊗ N − N ⊗ Aν ) dC¯ µ

X0 ¯  X

X0

Eµβ Bρµ (Aβ ⊗ N − N ⊗ Aβ ) + (∂¯γ Eρβ − ∂¯β Eργ )Aβ ⊗ Aγ

β

β

+Λρβ (A ⊗ N − N ⊗ A )

Z

¯ X

X0

¯ Λµν (A ⊗ N − N ⊗ A ) dC¯ µ ν

.

ν



  ρ ¯ ¯ dC Aα (X) dC¯ α (6.33)

Above is the Ces` aro integral formula for a Kirchhoff-Love surface undergoing small strain, but moderate rotation. This is a generalization of a similar formula obtained in [20] for geometrically linear shells (i.e., infinitesimally small deformation).

6.3

Material response and equilibrium equations for sufficiently thin Kirchhoff-Love shells

We assume that the structured surface is materially uniform, simple, and hyperelastic. Material uniformity requires that there exist locally undistorted states with respect to which the constitutive response 112

function (e.g., stress-strain relation) is independent of the material points on the surface; the stress-free natural configuration provides such an undistorted state. The response of a simple material is local in nature in the sense that the material response at a point depends only on the local state of deformation at that point. These two hypotheses, combined with the principle of material frame indifference, require that the isothermal material response of materially uniform and simple structured surfaces is expressible in terms of the elastic strain field with respect to the natural state of the surface. In particular, a hypere , Λ e , Λ e , ∆e , ∆e ) elastic material response is governed by a single scalar energy density function ψ(Eαβ α α αβ

per unit area of the surface in natural configuration. The energy density function will be further required to satisfy appropriate material symmetry restrictions [33,34,38,115,119,127]. Such 2-dimensional strain energy density functions have been established by techniques such as thickness-wise integration of a 3-dimensional material response [116,117], gamma-convergence [44,45], asymptotic expansion [18], etc. e , Λe ), The equilibrium equations of a Kirchhoff-Love shell, with strain energy density function ψ(Eαβ αβ

are [118]

where σ

βα

1 = 2

∂ˆα (σ µα + M βα Λµβ ) + ∂ˆβ M βα Λµα = 0 and

(6.34a)

(σ βα + M µα Λβµ )Λβα + ∂ˆαβ M βα = 0,

(6.34b)

  r  r  a ∂ψ ∂ψ 1 a ∂ψ ∂ψ βα and M = + , e + ∂E e 2 Aˆ ∂Λeαβ ∂Λeβα Aˆ ∂Eαβ βα

(6.35)

respectively, are the tangential surface stress and bending moment measures. Here, ∂ˆ denotes the covariant derivative on the current configuration and Aˆ := det[Aˆαβ ]. For a sufficiently thin isotropic e , Λe ) Kirchhoff-Love shell, dimensional analysis and representation theorems can be used to express ψ(Eαβ αβ

as [119]

  7 X e ψ(Eαβ , Λeαβ ) = Eh C(i1 , i2 ) + h2 in Dn (i1 , i2 ) ,

(6.36)

n=3

e aαβ , i := E e E e aαµ aβν , i := where E is the Young’s modulus of the shell material and i1 := Eαβ 2 3 αβ µν e Λe aαµ aβν )2 , i := a−1 (εαγ Λe E e a aσµ aβν )2 , and i := (Λeαβ aαβ )2 , i4 := Λeαβ Λeµν aαµ aβν , i5 := (Eαβ 6 7 µν αβ µν σγ e Λe Λe aρσ aαµ aβν . Here, C and D are dimensionless functions. We will now summarize certain Eαβ n ρσ µν

113

special forms of the above relations. Note, that only the symmetric part Λe(αβ) of the elastic bending strain contributes to the constitutive response. The skew part, as a consequence of the additive decomposition, is determined by the skew part of the plastic bending strain, Λe[αβ] = −Λp[αβ] .

6.3.1

Pure bending of thin elastic isotropic shells

The surface stress components σ αβ in the equilibrium equations (6.34) of a Kirchhoff-Love shell undergoing pure bending are to be interpreted as Lagrange multipliers σ ¯ αβ (θα ) associated with the dee = 0; these are determined a posteriori after solving the complete boundaryformation constraint Eαβ

value-problem [118]. The bending moment components Mαβ , with respect to an adapted Cartesian coordinate system θi , are determined from (6.35)2 and (6.36), M

αβ

  e αβ e µα νβ = D νΛµµ A + (1 − ν)Λ(µν) A A ,

(6.37)


where ν is the Poisson’s ratio of the shell material and D := Eh3 / 12(1 − ν 2 ) is the bending rigidity.

6.3.2

Combined bending and stretching of thin elastic isotropic shells

e = Under the assumption of small elastic surface strain and moderate elastic bending strain, i.e., Eαβ 1

O() and Λeαβ = O( 2 ), we have i1 = O(), i2 = O(2 ), i3 = O(), i4 = O(), i5 = O(2.25 ), i6 = O(2.25 ), and i7 = O(2 ). Neglecting the coupling term i7 , the 2-dimensional linear stress-strain and bending 1

moment-bending strain relations, up to O() and O( 2 ), respectively, are given by σ

αβ

    Eh e αβ e µα νβ αβ e αβ e µα νβ = νEµµ A + (1 − ν)Eµν A A and M = D νΛµµ A + (1 − ν)Λ(µν) A A . (1 − ν 2 )

(6.38)

6.4

F¨ oppl-von K´ arm´ an equations with incompatible elastic strain for shells with arbitrary reference geometry

In the classical F¨ oppl-von K´ arm´ an theory for thin elastic shells [81], the linearized version of the Kirchhoff-Love equilibrium equations (6.34), retained up to O(), are posed as the localized in-plane 114

and vertical force balance relations: ∂¯β σ αβ = 0, and σ αβ Λαβ + ∂¯αβ M αβ = 0.

(6.39a) (6.39b)

Equation (6.39a) is identically satisfied when the stress components σ αβ are expressed in terms of the 2-dimensional Airy stress function Φ(θα ) over the reference configuration as σ αβ = ε¯αµ ε¯βν ∂¯µν Φ. Equation (6.39b), after plugging in these expressions, the constitutive relation (6.38)2 , and the decomposition (6.2b), reduces to
D ν A¯µν A¯αβ + (1 − ν)A¯µα A¯νβ ∂¯αβ Λµν + ε¯αµ ε¯βν Λαβ ∂¯µν Φ = DΩp ,

(6.40)

where Ωp := [ν A¯µν A¯αβ + (1 − ν)A¯µα A¯νβ ]∂¯αβ Λp(µν) .

On the other hand, the compatibility relations for the total strain require ∂¯11 E22 + ∂¯22 E11 − 2∂¯12 E12 + Λ11 Λ22 − (Λ12 )2 = 0, and ∂¯1 Λµ2 − ∂¯2 Λµ1 = 0.

(6.41) (6.42)

In the first of the above equations, we use the additive in-surface strain decomposition (6.2a), the
e = 1 (1 + ν)A ¯αµ A¯βν − ν A¯αβ A¯µν σ µν , and inverse of the stress-strain relation (6.38)1 in the form Eαβ E

then write σ µν in terms of the Airy stress function, to obtain

 
µρ νσ 1 ¯ ¯ ¯ ¯ ¯ ¯ ∂ αβ , (1 + ν)Aαµ Aβν − ν Aαβ Aµν ε¯ ε¯ ∂ρσ Φ + Λ11 Λ22 − (Λ12 )2 = −λp , E

(6.43)

where [Cαβ , Dαβ ] := C11 D22 + C22 D11 − 2C12 D12 for scalar quantities Cαβ and Dαβ , and λp := p p p −2∂¯12 E12 + ∂¯22 E11 + ∂¯11 E22 . We call Ωp the total plastic curvature incompatibility, and λp , the

total plastic stretch incompatibility. Equations (6.40) and (6.43) constitute the F¨oppl-von K´ arm´ an equations for a shell with arbitrary reference geometry. The fields Ωp and λp can be determined from the strain incompatibility relations in terms of various defect density fields. In a global Cartesian reference frame Ei , if the local reference and current configurations of the ¯ ) = θα Eα + wE ˆ ) = θα Eα + wE surface are expressed in the Monge forms R(U ¯ 3 , and R(U ˆ 3 , respectively, 115

p p then A¯αβ = δαβ + w¯,α w¯,β , and Λαβ = w¯,αβ / 1 + (w¯,1 )2 + (w¯,2 )2 − wˆ,αβ / 1 + (wˆ,1 )2 + (wˆ,2 )2 . Equa-

tions (6.40) and (6.43), after plugging in these expressions, reduce to the following system of coupled partial differential equations


D ν A¯µν A¯αβ + (1 − ν)A¯µα A¯νβ ∂¯αβ Λµν   w¯,αβ wˆ,αβ αµ βν p +¯ ε ε¯ −p ∂¯µν Φ = DΩp , and 1 + (w¯,1 )2 + (w¯,2 )2 1 + (wˆ,1 )2 + (wˆ,2 )2  
µρ νσ 1 ¯ ¯ ¯ ¯ ¯ ¯ ∂ αβ , (1 + ν)Aαµ Aβν − ν Aαβ Aµν ε¯ ε¯ ∂ρσ Φ E  w¯,αβ wˆ,αβ 1 + p −p , 2 1 + (w¯,1 )2 + (w¯,2 )2 1 + (wˆ,1 )2 + (wˆ,2 )2  wˆ,αβ w¯,αβ p −p = −λp , 1 + (w¯,1 )2 + (w¯,2 )2 1 + (wˆ,1 )2 + (wˆ,2 )2

(6.44)

(6.45)

for the functions Φ(θα ) and w(θ ˆ α ), which determine, respectively, the internal stress field and shape of p and Λpαβ the defective material surface, for a given reference shape w(θ ¯ α ), and plastic strain fields Eαβ

(or the total plastic curvature and stretch incompatibilities Ωp and λp ). Finally, the above governing equations must be complemented by the global incompatibility relations for the plastic strain as discussed in Section 6.2.2.

Shallow shells: When the reference geometry is moderately curved, the functions w¯ and w, ˆ along 1

with their first and second spatial derivatives, are of order O( 2 ). Then, up to the same order, Λαβ = (w ¯ − w) ˆ ,αβ . We can also replace the reference covariant derivatives ∂¯ in all the above expressions with ordinary partial derivatives when retaining terms upto the leading order. Then the F¨ oppl-von K´arm´an equations (6.40) and (6.43) reduce to their standard form as given in [74, 78, 81]. However, the restrictions imposed by the global strain incompatibility relations that must accompany these local F¨oppl-von K´ arm´ an equations have not been considered anywhere in the literature.

116

6.4.1

2-dimensional solid crystals with edge dislocations, wedge disclinations, and pure in-surface metric anomalies

We assume that the density of screw disclinations and the densities of out-of-surface metric anomalies to be identically zero. With this, the local plastic strain incompatibility relations yield the following non-trivial expressions p p p ∂¯11 E22 + ∂¯22 E11 − 2∂¯2 ∂¯1 E12

√ +Λp11 Λp22 − (Λp12 )2 = AΘ3 + 2 A Aσ[1 ∂¯2] J σ − ∂¯1 M212 + ∂¯2 M112 ,

(6.46)

and ∂¯α Λpµβ − ∂¯β Λpµα = 0, with the skew part Λp[αβ] identically zero. As the above equations are underdep and Λpαβ , the solutions are non-unique. However, we can choose Λpαβ ≡ 0. termined for the variables Eαβ √ Hence, λp = AΘ3 + 2 A Aσ[1 ∂¯2] J σ − ∂¯1 M212 + ∂¯2 M112 , and Ωp ≡ 0. Hence, our formulation reduces

to the formulae used in [95, Chapter 6] to study 2-dimensional crystals carrying discrete disclinations, dislocations, and isotropic point defects (where the point defect density is given by Qµαβ = φ,µ Aαβ , with the scalar field φ(θα ) as the measure of the distributed point defects).

6.4.2

Growing biological surfaces

In growing biological surfaces, disclinations and dislocations are absent. Then, we can represent metric anomalies in terms of the symmetric quasi-plastic strain fields q˜ij : W × (−, ) → R as Qkij (θα ) = −2˜ qij;k ζ=0 over simply connected patches W ⊂ ω. We assume the following form of q˜ij (θα , ζ): 0 0 0 0 q˜αβ = qαβ − 2ζqαβ + ζ 2 Aµν qαµ qνβ , q˜α3 = q˜3α = 0, and q˜33 = 1,

(6.47)

0 (θ α ), along with their first derivatives, and q 0 (θ α ) are O() and where the symmetric functions qαβ αβ 1

O( 2 ), respectively. The above expression is motivated by the form of the material metric for KirchhoffLove shells with small in-surface strain accompanied by moderate rotations. We obtain Qµαβ = 0 (up to O()), Q 0 −2∂¯µ qαβ 3αβ = 4qαβ , and Qk33 = Qkα3 = Qk3α = 0. Accordingly, the functions 0 measure in-surface metric anomalies, e.g., surface growth, whereas q 0 qαβ αβ measure the tangential

117

differential surface growth along the thickness direction. The plastic strain incompatibility relations reduce to p p p ∂¯11 E22 + ∂¯22 E11 − 2∂¯2 ∂¯1 E12


0 0 0 0 0 0 2 +{Λp11 Λp22 − (Λp(12) )2 } = ∂¯11 q22 + ∂¯22 q11 − 2∂¯2 ∂¯1 q12 + 4 q11 q22 − (q12 ) ,

(6.48)

0 0 ∂¯2 Λp11 − ∂¯1 Λp(12) = 2(∂¯1 q12 − ∂¯2 q11 ), and

(6.49)

0 0 ∂¯2 Λp(21) − ∂¯1 Λp22 = 2(∂¯1 q22 − ∂¯2 q21 ),

(6.50)

0 with the following restrictions on the functions qαβ (which also give rise to a skew part of the plastic

bending strain): (Λp[12] )2 =

0 q0 4AAαβ q1α 1β

A22

=

0 q0 4AAαβ q2α 2β

A11

=−

0 q0 4AAαβ q1α 2β

A12

.

(6.51)

p 0 , and Λp 0 We readily make the identifications: Eαβ = qαβ (αβ) = −2qαβ . Hence, the plastic in-surface 0 , and the plastic bending strain field strain can be specified directly by the in-surface growth tensor qαβ 0 . Tangential differential growth along the thickness, by the tangential differential growth tensor qαβ

thus, acts as a source to the incompatible growth curvature field Ωp . Remark 6.4.1. (Plates:) If the reference configuration is flat, we get interesting consequences of the relations (6.51). We can choose the coordinate system θα to be the Cartesian coordinate system. 0 )2 + (q 0 )2 ), (Λp )2 = 4((q 0 )2 + (q 0 )2 ), and Then, the relation (6.51) leads to (Λp[12] )2 = 4((q11 12 22 12 [12] 0 (q 0 + q 0 ) = 0. The first two of these imply that q 0 = ±q 0 . According to the latter one, when q12 11 22 11 22 0 = −q 0 6= 0, q 0 may assume any non-zero value, e.g., in anisotropic tangential differential growth q11 22 12 0 = q 0 6= 0, q 0 = 0, a case of isotropic tangential differential growth along along thickness. For q11 22 12 0 | = 2|q 0 |. Finally, if q 0 = q 0 = 0, q 0 may assume any nonthickness, implying |Λp[12] | = 2|q11 22 11 22 12

zero value, representing the tangential differential growth of shear type along the thickness, such that 0 |. |Λp[12] | = 2|q12

As an immediate application, we look for conditions on the temperature field which would yield locally compatible thermal strain in thin plates. For isotropic thermal deformation in thin elastic 118

plates, q˜µν (θα , ζ) = qµν (θα ) = αT (θα )δµν , where α is the uniform thermal expansion coefficient and T is the change in temperature. Clearly, from (6.48), the temperature distribution T (θα ) that gives rise to locally compatible strain fields satisfies the 2-dimensional Laplace equation T,αα = 0 [7, Chapter 14].

6.5

Shape equation of disclinated isotropic fluid films

Fluid films cannot resist in-surface strain. We consider a thin isotropic incompressible fluid film with strain energy density of the form e W (Eαβ , Λeαβ )

e 2

= k(H ) − µ

q

 ˆ A/a − 1 ,

where k is a material parameter, H e is the trace of the elastic bending strain H e :=

(6.52) 1 ¯µν e 2 A Λµν ,

and

µ(θα ) is a constitutively undetermined Lagrange multiplier corresponding to the deformation constraint of incompressibility. The above is the standard Helfrich energy [54]. The explicit dependence of constitutive function W on θα in fact represents its relation to the local reference neighbourhoods through the reference fundamental forms Aαβ and Bαβ . The equilibrium equations with zero body force and couple are [1] ¯ H ˆ − H p ) + 2k(H ˆ − H p )(2H ˆ 2 − K) − 2k H( ˆ H ˆ − H p )2 + 2µH ˆ = 0, k ∆(

(6.53)

and µ,α = W,α (for fixed H e ), where we have already made the approximation imposed by the assumpˆ − H p ) for H e , where H p is the trace tion of moderately large bending strain, and also substituted (H ¯ denotes the Laplace-Beltrami of the plastic bending strain, H p := 21 A¯µν Λpµν . In the above equation, ∆ ¯ ˆ and K ˆ are, respectively, the trace and operator on the reference configuration, ∆(·) := ∂¯αβ (·)A¯αβ ; H ˆ := determinant of the total bending strain, H

1 ˆαβ 2 A Λαβ

ˆ := Λ11 Λ22 − (Λ12 )2 . Equation (6.53) , K

ˆ = H p and µ = 0 of the determines the shape of the fluid film for a given value of H p . The solution H above equation, implying a global minimum to the total energy, is ruled out in presence of disclinations over the film, since H p might not then correspond to any realizable surface isometrically embedded in R3 . The parameter µ is to be determined from the boundary data after the complete boundary-valueproblem has been solved. The source of incompatibility in the shape equation, namely, the function 119

H p , can be written in terms of the wedge disclination density Θ using the solution Λpαβ from the plastic incompatibility relations for pure bending: Λp11 Λp22 − (Λp12 )2 = AΘ3 , and ∂¯2 Λpµ1 − ∂¯1 Λpµ2 = 0.

(6.54) (6.55)

We can determine H p in terms of the solution wp of the inhomogeneous covariant Monge-Amp`ere p equation [∂¯αβ wp , ∂¯αβ wp ] = 2AΘ3 , where Λpαβ = w,αβ satisfies the last of the above equations identi-

cally. For shallow films (i.e., when the reference geometry is moderately curved), the shape equation p reduces to the simple form (k/2)w ˆ,ααββ + µw ˆ,αα = (k/2)w,ααββ , where wp is a solution to the standard p p inhomogeneous Monge-Amp`ere equation [wαβ , wαβ ] = 2AΘ3 .

120

Chapter 7

Organicum geometriae: Concluding Remarks and Future Scope1 . . . true geometry [is] a doctrine of space itself and not merely like Euclid, and almost everything else that has been done under the name of geometry, a doctrine of the configurations that are possible in space. Hermann Weyl [130, p. 102]. Up until early nineteenth century, the notion of geometry was unambiguously Euclidean. Based on fundamental objects such as points, straight lines and planes, and a set of elementary presuppositions (axioms) about their mutual relationship, the predominant focus in Euclidean geometry is to derive, in a logically consistent manner, “the configurations that are possible in space”.2 The “space itself” remains continuous (i.e., between any two points in space there are infinitely many points), locally flat (i.e., the solid angle around any point in space is same; it is 2π for a two-dimensional Euclidean plane and 4π for a three-dimensional Euclidean space), homogeneous (i.e., a body can move in space without changing its size and shape), and similar (i.e., a body can be reconstructed to any scale in another part 1

Parts of this chapter appeared in [108].

2

This basis for geometry remained “self-evident” for more than twenty-two centuries. An occasional discomfort was

caused by the parallel postulate of Euclidean geometry, but never to an extent of questioning the validity of the geometry.

121

of space).3 That, such is the true nature of space was an unshakeable belief held alike by philosophers, mathematicians and physicists. In particular, no scientific theory of the physical world was expected to be in discord with this structure of the space and hence with the propositions of Euclidean geometry. The pioneering non-Euclidean revolution was brought about by the Russian mathematician Nikolai Ivanovich Lobachevski—“what Copernicus was to Ptolemy, that was Lobachevski to Euclid” [23, p. 212]—who constructed a geometry by providing an alternative to Euclid’s parallel postulate. This in effect introduced the possibility of a geometrical space which is continuous, locally flat and homogeneous, but not similar; such spaces necessarily have a constant non-zero curvature (Euclidean space has zero curvature). Curvature characterizes the angular mismatch between a tangent vector and its parallelly transported image around a loop. Lobachevski’s construction led to spaces with constant negative curvature; the other possibility, of spaces with constant positive curvature, was suggested several decades later by Riemann. Most importantly, Lobachevski’s contribution exposed our fallibility of considering Euclidean geometry as the irreplaceable truth of nature. The second breakthrough in non-Euclidean geometries came from the German mathematician Georg Friedrich Bernhard Riemann. Inspired by Gauss’s theory of surfaces, Riemann considered spaces (of arbitrary dimension) which are continuous and locally flat, but not necessarily homogeneous and similar. He characterized them in terms of a metric field which generates a quadratic form representing the distance between infinitesimally closed points. For Riemannian spaces, knowing the metric function is sufficient to determine the parallel transport of vectors and the curvature of space; the latter no longer remaining a scalar constant. The ingenuity of Riemann was to interpret metric not as an a priori property of the space but instead as a characteristic of the “physical phenomenon” manifested in an otherwise formless space. Hence unlike both Euclidean spaces and Gaussian surfaces, where the metrical properties are fixed once for all, the Riemannian metric, and consequently the resulting 3

The concept of space was absent from Euclidean geometry until the insightful work of Descartes. The ancient geome-

ters took the nature of space for granted and busied themselves solely with understanding the character of geometrical figures which could occupy it. The four postulates about the nature of space that are mentioned here were given by Clifford [23, pp. 210-230]. In this work of great originality Clifford demonstrated the equivalence of these postulates with the axioms of Euclidean geometry.

122

geometry, is allowed to be derived from the “matter” filling the space and “the binding forces which act upon it” [105]. The geometry, rather than acting merely like a rigid skeleton in the background of a physical theory, was now organic and free to participate in it.4 The physical relevance of Riemannian geometry remained completely unappreciated for as long as seventy years, until after the appearance of Einstein’s theory of general relativity wherein the metrical structure of the four-dimensional spacetime continuum was identified with the gravitational field associated with the “matter” occupying the continuum. Fortunately, Riemann’s theory met with an all together different fate in the hand of mathematicians; by the time that Einstein’s theory made its appearance, it had already achieved maturity in the well established disciplines of tensor analysis on manifolds and fiber bundles. These ´ Cartan, Hermann Weyl, Charles post-Riemannian breakthroughs, thanks to the pioneering works of Elie Ehresmann, among others, essentially generalized and separated the notion of parallel transport of tensorial objects on fiber bundles from their metrical notion, and, along with curvature, led to two new independent fundamental geometric objects, namely, torsion and non-metricity, which measure, respectively, the closure failure of infinitesimal parallelograms and change in length of vectors during parallel transport. The most successful application of differential geometry on (complex) fiber bundles is found in the standard model of elementary particles of nature, also known as the gauge theory [94]. The success of non-Euclidean geometries with the relativity theory provided impetus to their application in other domains of mechanics including development of a elasticity theory of a continuous distribution of defects.5 Whereas it is the geometry of the four-dimensional space-time which is treated as non-Euclidean in a relativistic continuum, it is the geometry of the material space which is most 4

Aptly summarized by Weyl [130, p. 220], “this seals the doom of the idea that a geometry may exist independently

of physics”. 5

The theory of elasticity of solids, which acted as the precursor to the later field theories of electromagnetism and

relativity, has in turn benefitted significantly from them in the last century. Besides motivating a non-Euclidean framework, there are several other instances where these later field theories have assisted elasticity with fundamental breakthroughs. The force acting on an isolated defect in a solid body and the motion of a dislocation were, for instance, introduced as analogous to the force acting on a point charge and the equation of motion of particles in special relativity, respectively, cf. [83, p. 62, 107].

123

naturally described as non-Euclidean in a defective continua. Interestingly, while it is the “matter” which induces curvature in a relativistic space-time continuum, it is the presence of defects (apparent “lack of matter”) which brings about a curvature in the material space. It would be worthwhile, once again, to overemphasize the physicality of the material space: it can be thought of as a natural (or relaxed) configuration of the body where it is described only in terms of the intrinsic structure of the constituting matter. The material space of a crystalline solid, for instance, is the configuration obtained by relaxing the solid of all internal and external stresses. The relaxed configuration of a defective solid will not be a coherent body in the Euclidean space. The geometric experience of the imaginary beings living in the material space would be Euclidean in the absence of any defects, but non-Euclidean when the body is defective. These beings, otherwise insensitive to distortions caused by any external agency (such as load, temperature field, etc.), will recognize geometrical evolution of their space of existence only with appearance or disappearance of defects (cf. [67, p. 287] and [100, ch. 4]). To bring forth the appealing connection between non-Euclidean geometries and defects in solids, specifically in 2-dimensional material structures, was one of the central aims of this thesis. Drawing a correspondence between the nature of a defect and a specific geometric property of the material space not only illuminates the underlying structure of defects in solids but also provides an unambiguous way to represent defect densities within a physical theory.6 As we have seen, defect characterization can be used to calculate the internal stress field and deformed shape of an elastic solid. Moreover, it can also be used to represent hardening during plastic deformation (cf. [67,89]), or can serve as a macroscopic representative for the microstructure; it is a natural device to introduce microscopic length scales in the theory [89]. Most importantly, the kinetic laws for the dynamic evolution of the material body are conveniently expressed as partial differential equations for defect densities [89]. A geometrical study of defects can be motivated from these, among several other practical applications, if not only from the sheer elegance of its mathematical structure. 6

The connection between dislocation density and the torsion tensor from differential geometry was first pointed out by

Kondo [64], and independently by Bilby et al. [10]. Notable extensions were provided by Kondo and coworkers, cf. [65], Bilby and coworkers, cf. [9], and Noll [96]. An excellent review of the subject is given by de Wit [25]; see also Kl´eman [61].

124

We list our concluding notes in the following, while emphasizing the main contributions of this thesis. 1. The 3-dimensional theory of continuum defects, although well studied, has not been presented in the generality as it has been dealt with in this thesis. There are at least four results, listed next, which have not appeared anywhere in the literature: (i) Extension of the notion of material inhomogeneity of Noll to 3-dimensional bodies with continuously distributed dislocations, disclinations and metric anomalies: The material connection and the material metric in a defective 3-dimensional body, in general, are determined not just in terms of the material uniformity field of the body, as in Noll’s theory [96], but also in terms of the prescribed defect density fields. (ii) The notion of metrical disclinations: Metrical disclinations are generalized disclinations which do not appear in the classical Volterra construction of topological defects in simple 3dimensional solids (a related theoretical notion of generalized disclinations has however appeared recently in [135], and a similar notion has appeared in the context of general relativity in (2 + 1)dimensional spacetime continuum in [62]). Metrical disclinations are closely related to the metric anomaly distribution in the continuum; their presence in real materials is yet to be explored. (iii) Representation of anisotropic metric anomalies: Experimental observation of anisotropic conglomerations of point defects have been reported, which are discussed at length in the introduction; these anisotropic clusters cannot be modelled by semi-metric geometry (i.e., isotropic non-metricity tensor) which is the only available technique to model point defects in the present literature. The anisotropic description is further amenable in describing thermal deformation of anisotropic crystals and anisotropic bulk growth in tissues. (iv) Representation of general metric anomalies in absolute absence of disclinations (the irrotational metric anomalies): The irrotational metric anomalies, on the other hand, are shown, for the first time, to be necessarily and sufficiently represented by a second order symmetric tensor (Anthony’s quasi-plastic strain); moreover, we have also derived fundamental geometric 125

restrictions on the quasi-plastic deformation tensor (Equation (3.29)) in absence of disclinations, which is unique. Our results on metric anomalies have a direct bearing on the geometrical nature of thermal deformation and biological growth. Our results, therefore, also provide novel geometric insights in these disciplines. The concept of quasi-plastic deformation, and the multiplicative decomposition associated with it is used, without much justification, in areas such as finite thermoelasticity, growth, hygroelasticity, to name a few. Our work provides a rigorous geometric basis for the same, while highlighting the restrictions under which such a concept remains valid. 2. On the other hand, the continuum theory of defective structured surface in the thesis, although based on the 3-dimensional theory, is by no means a trivial specialization of the 3-dimensional theory. The novel results obtained for defective structured surfaces are as follows. (i) A complete classification and quantification of defects in structured surfaces: Ours is the first work, to best of our knowledge, to give such a broad classification of defects possible in thin structures. These structures include both solid and liquid crystalline membranes (and shells), biological surfaces, and laminar media. The central idea on which our classification scheme has been based is that: All the kinematical features of a continuously defective single director shell, such as the one adopted in the thesis to model structured surfaces, including the effects of director tilt and distension, can be appropriately mapped to the kinematics of a hypothetical continuously defective 3-dimensional simple solid. The tubular neighbourhood (introduced in Section 2.2) is the region occupied by this hypothetical 3-dimensional solid. The 2-dimensional body and the hypothetical 3-dimensional body are constitutively very different, and the former should not be seen as a limiting case, or a homogenized model, of the latter. Rather, the 3-dimensional solid should be seen as an arbitrary extension of the structured surface. In fact we have exploited this arbitrariness to fix the material connection of the 3-dimensional solid to be such that the material space has uniform curvature, torsion, and non-metricity across the thickness, equal to their respective values at the embedded surface. Homogenization of a defective 3D solid, simple or Cosserat, into a defective structured surface is not a trivial task, and the problem is still open.

126

(ii) Symmetries and conservation laws of defects in structured surfaces: As shown in Section 3.2, the symmetries and conservation laws necessarily satisfied by the 2-dimensional defect density fields appear as a non-trivial extension of the 3-dimensional theory; for instance, the disclination density tensor Θ is symmetric under much milder conditions than the 3-dimensional theory, and the in-surface wedge disclinations, and edge and screw dislocations do not follow any conservation law. Our theory also includes a novel representation of anisotropic metric anomalies in thin structures, such as anisotropic clusters of point defects in material membranes (see, [95, p. 231]), for which no model has appeared yet in the literature. (iii) Complete set of compatibility and incompatibility relations for structured surfaces: In Chapter 5, the 3-dimensional material metric is constructed from the given 2-dimensional strain fields in an absolutely novel way. The general theory retains many special features of a thin body, for example, transverse shear and thickness distension, which are non-Kirchhoff-Love notions; this special construction of 3-dimensional metric from the information of 2-dimensional strain fields is one of the central contributions of this thesis, one which directly leads to the local strain compatibility and incompatibility relations for general single director defective shells. We have also derived, for the first time, the global strain compatibility and incompatibility relations for Kirchhoff-Love shells, for all the topologies attainable by a surface in 3-dimensional Euclidean space, which is another novel finding of this work. The global sources of incompatibility include contributions from the global dislocations and disclinations. Though we have considered only topology-preserving global anomalies, our framework, most notably, is amenable to be extended to include topological transformations of material surfaces mediated by global defects. (iv) Novel insights into the theory of defective 2-dimensional crystals and growing biological surfaces: In Chapter 6, the specialization of our general framework developed in earlier chapters to formulate complete boundary-value-problems for material surfaces undergoing moderately large deformation while maintaining small in-surface stretching is absolutely novel; the posed problems give the internal stress distribution and the natural shape of a large class of ma-

127

terial surfaces where the defect distribution is known a priori. To the best our knowledge, these complete set of boundary-value-problems have appeared nowhere in literature in this generality. In particular, thickness-wise tangential differential growth of thin biological membranes appears as an incompatible second-order growth strain, which is highly non-trivial and does not have a 3-dimensional counterpart. Note, further, that Kirchhoff-Love shells can, in general, contain an enriched set of defects associated with the transverse loops (which are ignored in the “sufficiently thin” Kirchhoff-Love specialization in Section 6.2), e.g., twist disclinations in lipid bilayers and out-of-surface dislocations in thin multi-walled structures, which have not been considered in the literature yet.

Future directions As our theory is much more general than the ones which appear in the present literature (for example, the inclusion of anisotropic point defects, defective multi-layered structures, and thickness wise differential growth in biological membranes, with effects of transverse shear and thickness distention), it indeed has wider scope in understanding many already well-known phenomena with new insights, or in predicting novel physical phenomena. Our work is also amenable to more sophisticated plate and shell theories (beyond F¨ oppl-von-K´ arm´ an). An immediate future direction is to test this unique theory with numerical experiments which seem to be the only option to tackle the complex nonlinearities of the governing equations. Moreover, this work is readily extendable to model topological transformations mediated by various global defects as already pointed out. Another potential extension would be to describe geometry driven, inherently discontinuous physical phenomena, e.g., incompatibility induced microstructures, such as wrinkles and phase transformations in active structures [106], which necessarily requires a paradigm shift to generalized function spaces (measures) and consideration of non-convex material response. We defer these extensions to future studies.

128

Bibliography [1] A Agrawal and D J Steigmann. Modeling protein-mediated morphology in biomembranes. Biomechanics and Modeling in Mechanobiology, 8:371–379, 2008. [2] K H Anthony. Die theorie der disklinationen. Archive for Rational Mechanics and Analysis, 39:43–88, 1970. [3] K H Anthony. Nonmetric connexions, quasidislocations, and quasidisclinations: a contribution to the theory of non-mechanical stresses in crystals. In J A Simmons and R Bullough, editors, Fundamental Aspects of Dislocation Theory, volume 1, pages 637–649. Nat. Bur. Stand. (U.S.) Spec. Publ. 317, 1970. [4] K H Anthony. Die theorie der nichtmetrischen Spannungen in Kristallen. Archive for Rational Mechanics and Analysis, 40:50–78, 1971. [5] K H Anthony. Crystal disclinations versus continuum theory. Solid State Phenomena, 87:15–46, 2002. [6] M A Armstrong. Basic Topology. Springer-Verlag, 1983. [7] J R Barber. Elasticity. Springer Netherlands, 3rd edition, 2010. [8] T H K Barron. Generalized theory of thermal expansion of solids. In C Y Ho, editor, Thermal Expansion of Solids, pages 1–105. ASM International, 1998.

129

[9] B A Bilby. Continuous distributions of dislocations. In I N Sneddon and R Hill, editors, Progress in Solid Mechanics, Volume 1. North-Holland, Amsterdam, 1960. [10] B A Bilby, R M Bullough, and E Smith. Continuous distributions of dislocations: a new application of the methods of non–Riemannian geometry. Proceedings of the Royal Society at London A, 231:263–273, 1955. [11] V Borrelli, S Jabrane, F Lazarus, and B Thibert. Flat tori in three-dimensional space and convex integration. Proceedings of the National Academy of Sciences in the United States of America, 109:7218–7223, 2012. [12] M J Bowick and L Giomi. Two-dimensional matter: order, curvature and defects. Advances in Physics, 58:449–563, 2009. [13] M J Bowick and A Travesset. The geometrical structure of 2d bond-orientational order. Journal of Physics A: Mathematical and General, 34:1535–1548, 2001. [14] G E Bredon. Topology and Geometry. Springer-Verlag, New York, 1993. [15] P Cesana, P Plucinsky, and K Bhattacharya. Effective behavior of nematic elastomer membranes. Archive for Rational Mechanics and Analysis, 218:863–905, 2015. [16] K F Chernykh. Relation between dislocations and concentrared loadings in the theory of shells. Journal of Applied Mathematics and Mechanics (PMM), 23:359–371, 1959. [17] P G Ciarlet. Mathematical Elasticity, Volume III: Theory of Shells. North-Holland, Amsterdam, 2000. [18] P G Ciarlet. An introduction to differential geometry with applications to elasticity. Journal of Elasticity, 78-79:1–215, 2005. [19] P G Ciarlet, L Gratie, and M Serpilli. Explicit reconstruction of a displacement field on a surface by means of its linearized change of metric and change of curvature tensors. Comptes Rendus de l’Acad’emie des Sciences - Series I, 346:1113–1117, 2008. 130

[20] J D Clayton. Nonlinear Mechanics of Crystals. Springer Netherlands, 2011. [21] J D Clayton, D J Bammann, and D L McDowell. A geometric framework for the kinematics of crystals with defects. Philosophical Magazine, 85(33-35):3983–4010, 2005. [22] W K Clifford. Lectures and Essays. Macmillan and Co., 1886. [23] A de Feraudy, M Queguineur, and D J Steigmann. On the natural shape of a plastically deformed thin sheet. International Journal of Non-Linear Mechanics, 67:378–381, 2014. [24] R de Wit. A view of the relation between the continuum theory of lattice defects and nonEuclidean geometry in the linear approximation. International Journal of Engineering Science, 19:1475–1506, 1981. [25] D H Delphenich.

On the topological nature of Volterra’s theorem.

2011 (arXiv preprint:

arxiv1109:2012). [26] S Derezin. Gauss-Codazzi equations for thin films and nanotubes containing defects. In H Altenbach and V A Eremeyev, editors, Shell-like structures, pages 531–547. Springer-Verlag, Berlin Heidelberg, 2011. [27] S V Derezin and L M Zubov. Equations of nonlinear elastic medium with continuously distributed dislocations and disclinations. Doklady Physics, 44:391–394, 1999. [28] J Dervaux, P Ciarletta, and M B Amar. Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the F¨oppl-von K´arm´an limit. Journal of the Mechanics and Physics of Solids, 57:458 – 471, 2009. [29] E Efrati. Non-Euclidean ribbons. Journal of Elasticity, 119:251–261, 2015. [30] E Efrati, E Sharon, and R Kupferman. Elastic theory of unconstrained non-Euclidean plates. Journal of the Mechanics and Physics of Solids, 57:762–775, 2009.

131

[31] M Epstein. A note on nonlinear compatibility equations for sandwich shells and Cosserat surfaces. Acta Mechanica, 31:285–289, 1979. [32] M Epstein and M de Le´ on. Uniformity and homogeneity of elastic rods, shells and Cosserat three-dimensional bodies. Archivum Mathematicum, 32:267–280, 1996. [33] M Epstein and M de Le´ on. On uniformity of shells. International Journal of Solids and Structures, 35:2173–2182, 1998. [34] M Epstein and M de Le´ on. Homogeneity without uniformity: towards a mathematical theory of functionally graded materials. International Journal of Solids and Structures, 37:7577–7591, 2000. [35] M Epstein and M Elzanowski. Material Inhomogeneities and their Evolution, A Geometric Approach. Springer-Verlag, Berlin Heidelberg, 2007. [36] V A Eremeyev and W Pietraszkiewicz. Local symmetry group in the general theory of elastic shells. Journal of Elasticity, 85:125–152, 2006. [37] J L Ericksen. Uniformity in shells. Archive for Rational Mechanics and Analysis, 77:73–84, 1970. [38] J D Eshelby. Point defects. In P B Hirsch, editor, The Physics of Metals – Sir Nevill Mott 60th Anniversary Volume, pages 1–42. Cambridge University Press, 1975. [39] J D Eshelby. Boundary problems. In F R N Nabarro, editor, Dislocations in Solids, Vol. 1, pages 167–221. North-Holland, Amsterdam, 1979. [40] J D Eshelby and A N Stroh. Relation between dislocations and concentrared loadings in the theory of shells. Philosophical Magazine, 42:1401–1405, 1951. [41] R Balian et al., editor. Les Houches, Session XXXV, 1980 – Physique des d´efauts. North-Holland, New York, 1981.

132

[42] F Falk. Theory of elasticity of coherent inclusions by means of non-metric geometry. Journal of Elasticity, 11:359–372, 1981. [43] G Friesecke, R D James, M G Mora, and S M¨ uller. Derivation of nonlinear bending theory for shells from three-dimensional elasticity by gamma-convergence. Comptes Rendus Mathematique, 336:697–702, 2003. [44] G Friesecke, R D James, and S M¨ uller. A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Archive for Rational Mechanics and Analysis, 180:183–236, 2006. [45] A E Green and P M Naghdi. Thermoelasticity without energy dissipation. Journal of Elasticity, 31:189–208, 1993. [46] M Gromov. Partial Differential Relations. Springer-Verlag, 1986. [47] H G¨ unther and M Zorawski. On geometry of point defects and dislocations. Annalen der Physik, 46:41–46, 1985. [48] Q Han and J-X Hong. Isometric embedding of Riemannian manifolds in Euclidean spaces. American Mathematical Society, 2006. [49] W F Harris. Topological restriction on the distribution of defects in surface crystals and possible biophysical application. In J A Simmons, R de wit, and R Bullough, editors, Fundamental Aspects of Dislocation Theory, volume 1, pages 579–592. National Bureau of Standards Special Publication No. 317, Washington, 1970. [50] W F Harris. The geometry of disclinations in crystals. In M W Roberts and J M Thomas, editors, Surface and Defect Properties of Solids, volume 3, pages 57–92. The Royal Society of Chemistry, 1974. [51] W F Harris. Dislocations, disclinations and dispirations: distractions in very naughty crystals. South African Journal of science, 74:332–338, 1978. 133

[52] F W Hehl, J D McCrea, E W Mielkeand, and Y Ne’eman. Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance. Physical Reports, 258:1–171, 1995. [53] W Helfrich. Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift f¨ ur Naturforschung C, 28:693, 1973. [54] S Jain and F S Bates. On the origins of morphological complexity in block copolymer surfactants. Science, 300:460–464, 2003. [55] M Kiritani. Similarity and difference between fcc, bcc and hcp metals from the view point of point defect cluster formation. Journal of Nuclear Materials, 276:41–49, 2000. [56] M Kiritani, Y Satoh, Y Kizuka, K Arakawa, Y Ogasawara, S Arai, and Y Shimomura. Anomalous production of vacancy clusters and the possibility of plastic deformation of crystalline metals without dislocations. Philosophical Magazine Letters, 79:797–804, 1999. [57] Y Klein, E Efrati, and E Sharon. Shaping of elastic sheets by prescription of non-euclidean metrics. Science, 315:1116–1120, 2007. [58] M Kl´eman. Defect densities in directional media, mainly liquid crystals. Philosophical Magazine, 27:1057–1072, 1973. [59] M Kl´eman. Points, Lines and Walls: in liquid crystals, magnetic systems and various ordered media. John Wiley and Sons, 1983. [60] M Kl´eman and J Friedel. Disclinations, dislocations, and continuous defects: A reappraisal. Review of Modern Physics, 80:61–115, 2008. [61] C Kohler. Line defects in solid continua and point particles in (2 + 1)-dimensional gravity. Classical and Quantum Gravity, 12:2977–2993, 1995. [62] W T Koiter. On the nonlinear theory of thin elastic shells. Proceedings of the Koninklijke Nederlandse Akademie Van Wetenschappen Series B, 69:1–54, 1966. 134

[63] K Kondo. On the geometrical and physical foundations of the theory of yielding. Proceedings of the 2nd Japan National Congress for Applied Mechanics, held 1952, pages 41–47, 1953. [64] K Kondo, editor. Memoirs of the Unifying study of the Basic Problems in Engineering Sciences by means of Geometry, volume 1–4. Gakujutsu Bunken Fukyu-Kai, Tokyo,, 1955–1968. [65] E Kr¨ oner. Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Archive for Rational Mechanics and Analysis, 4:273–334, 1959. [66] E Kr¨ oner. Continuum theory of defects. In R Balian et al., editor, Les Houches, Session XXXV, 1980 – Physique des d´efauts, pages 215–315. North-Holland, New York, 1981. [67] E Kr¨ oner. The differential geometry of elementary point and line defects in Bravais crystals. International Journal of Theoretical Physics, 29:1219–1237, 1990. [68] E Kr¨ oner. Theory of crystal defects and their impact on material behaviour. In G Herrmann, editor, Modeling of Defects and Fracture Mechanics, pages 61–117. 1993. [69] E Kr¨ oner. Crystal lattice defects and differential geometry. Journal of Mechanical Behaviour of Materials, 5:233–246, 1994. [70] L D Landau and E M Lifshitz. Theory of Elasticity. Elsevier Ltd., 2005. [71] P D Lax. Linear Algebra and its Applications. John Wiley and Sons, Inc., 2007. [72] M Lewicka, L Mahadevan, and Md. Reza Pakzad. The F¨oppl-von K´arm´an equations for plates with incompatible strains. Proceedings of the Royal Society at London A, 467:402–426, 2011. [73] M Lewicka, L Mahadevan, and Md. Reza Pakzad. Models for elastic shells with incompatible strains. Proceedings of the Royal Society at London A, 470:20130604, 2014. [74] H Lewy. An example of a smooth linear partial differential equation without solution. Annals of Mathematics, 66:155–158, 1957. [75] J C M Li. Disclination model of high angle grain boundaries. Surface Science, 31:12–26, 1972. 135

[76] H Liang and L Mahadevan. Shape of a long leaf. Proceedings of the National Academy of Sciences in the United States of America, 106:22049–22054, 2009. [77] H Liang and L Mahadevan. Growth, geometry, and mechanics of a blooming lilly. Proceedings of the National Academy of Sciences in the United States of America, 108:5516–5521, 2011. [78] V V Lokhin and L I Sedov. Nonlinear tensor functions of several tensor arguments. Journal of Applied Mathematics and Mechanics, 27:597–629, 1963. [79] D J Malcolm and P G Glockner. Nonlinear sandwich shell and Cosserat surface theory. Journal of the Engineering Mechanics Division, 98:1183–1203, 1972. [80] E H Mansfield. The Bending and Stretching of Plates. Cambridge University Press, Cambridge, UK, 1989. [81] S Mardare. The fundamental theorem of surface theory for surfaces with little regularity. Journal of Elasticity, 73:251–290, 2003. [82] X Markenscoff and A Gupta, editors. Collected works of J D Eshelby, The Mechanics of Defects and Inhomogeneities. Springer, 2006. [83] J McMahon, A Goriely, and M Tabor. Nonlinear morphoelastic plates I: Genesis of residual stress. Mathematics and Mechanics of Solids, 16:812–832, 2011. [84] J McMahon, A Goriely, and M Tabor. Nonlinear morphoelastic plates II: Exodus to buckled states. Mathematics and Mechanics of Solids, 16:833–871, 2011. [85] N D Mermin. The topological theory of defects in ordered media. Review of Modern Physics, 51:591–648, 1979. [86] M Miri and N Rivier. Continuum elasticity with topological defects, including dislocations and extra-matter. Journal of Physics A: Mathematical and General, 35:1727–1739, 2002.

136

[87] W C Moss and W G Hoover. Edge-dislocation displacements in an elastic strip. Journal of Applied Physics, 49:5449–5451, 1978. [88] F N R Nabarro and M S Duesbery, editors. Dislocations in Solids, volume 11. North-Holland, 2002. [89] F R N Nabarro. Disclinations in surfaces. In J A Simmons, R de wit, and R Bullough, editors, Fundamental Aspects of Dislocation Theory, volume 1, pages 593–606. National Bureau of Standards Special Publication No. 317, Washington, 1970. [90] F R N Nabarro. Theory of Crystal Dislocations. Dover, 1987. [91] F R N Nabarro and E J Kostlan. The stress field of a dislocation lying in a plate. Journal of Applied Physics, 49:5445–5448, 1978. [92] P M Naghdi and L Vongsarnpigoon. A theory of shells with small strain accompanied by moderate rotation. Archive for Rational Mechanics and Analysis, 83:245–283, 1983. [93] M Nakahara. Geometry, Topology and Physics. Taylor and Francis Group, London, 2003. [94] D R Nelson. Defects and Geometry in Condensed Matter Physics. Cambridge University Press, 2002. [95] W Noll. Materially uniform bodies with inhomogeneities. Archive for Rational Mechanics and Analysis, 27:1–32, 1967. [96] W Pietraszkiewicz and M L Szwabowicz. Determination of the midsurface of a deformed shell from prescribed fields of surface strains and bendings. International Journal of Solids and Structures, 44:6163–6172, 2007. [97] W Pietraszkiewicz and C Vall´ee. A method of shell theory in determination of the surface from components of its two fundamental forms. Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 87:603–615, 2007.

137

[98] D J Pochan, Z Chen, H Cui, K Hales, K Qi, and K L Wooley. Toroidal triblock copolymer assemblies. Science, 306:94–97, 2004. [99] H Poincar´e. Science and Hypotheses. The Walter Scott Publishing Co. Ltd., 1905. [100] Y Z Povstenko. Continuous theory of dislocations and disclinations in a two-dimensional medium. Journal of Applied Mathematics and Mechanics, 49:782–786, 1985. [101] Y Z Povstenko. Connection between non-metric differential geometry and mathematical theory of imperfections. International Journal of Engineering Science, 29:37–46, 1991. [102] E Reissner. Linear and nonlinear theory of shells. In Y C Fung and E E Sechler, editors, Thinshell structures: Theory, experiment, and design, pages 29–44. Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1974. [103] F Rellich. Perturbation Theory of Eigenvalue Problems. Gordon and Breach Science Publishers Inc., 1969. [104] B Riemann. On the hypotheses which lie at the bases of geometry (1884), translated by W K Clifford. Nature, 8(183,184):14–17, 36, 37, 1873. [105] P Ronceray. Active contraction in biological fiber networks. PhD thesis, Universit´e Paris-Saclay, 2016. [106] R Rosso, E G Virga, and S Kralj. Parallel transport and defects on nematic shells. Continuum Mechanics and Thermodynamics, 24:643–664, 2012. [107] A Roychowdhury and A Gupta. Geometry of defects in solids. Directions, 14:32–52, 2013 (arXiv preprint arXiv:1312.3033). [108] A Roychowdhury and A Gupta. Material homogeneity and strain compatibility in thin elastic shells. Mathematics and Mechanics of Solids, 10.1177/1081286515599438, 2015.

138

[109] A Roychowdhury and A Gupta. Non-metric connection and metric anomalies in materially uniform elastic solids. Journal of Elasticity, 126:1–26, 2017. [110] K Saito, R O Bozkurt, and T Mura. Dislocation stresses in a thin film due to the periodic distributions of dislocations. Journal of Applied Physics, 43:182–188, 1972. [111] J A Schouten. Ricci-Calculus, an introduction to tensor analysis and its geometrical applications. Springer-Verlag, Berlin Heidelberg GMBH, 1954. [112] M Spivak. A Comprehensive Introduction to Differential Geometry, volume 2. Publish or Perish, Inc., Houston, Texas, third edition, 1999. [113] D J Steigmann. Fluid films with curvature elasticity. Archive for Rational Mechanics and Analysis, 150:127–152, 1999. [114] D J Steigmann. Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity. Journal of Elasticity, 111:91–107, 2013. [115] D J Steigmann. A model for lipid membranes with tilt and distension based on three-dimensional liquid crystal theory. International Journal of Non-Linear Science, 56:61–70, 2013. [116] D J Steigmann. Mechanics of materially uniform thin films. Mathematics and Mechanics of Solids, 20:309–326, 2015. [117] D J Steigmann and R W Ogden. Elastic surface-substrate interactions. Proceedings of the Royal Society at London A, 455:437–474, 1999. [118] P Steinmann. Geomatrical Foundations of Continuum Mechanics, An Application to First- and Second-Order Elasticity and Elasto-Plasticity. Springer-Verlag, Berlin Heidelberg, 2015. [119] A M Stoneham. Theory of Defects in Solids. Oxford, 1975. [120] T Tallinen, J Y Chung, F Rousseau, N Girard, J Lef`evre, and L Mahadevan. On the growth and form of cortical convolutions. Nature Physics, 12:588–593, 2016. 139

[121] J C Talvacchia. Prescribing the curvature of a principal bundle connection. PhD thesis, University of Pennsylvania, 1989. [122] G H Vineyard. General introduction. Discussions of the Faraday Society, 31:7–23, 1961. [123] V Volterra. Sur l’´equilibre des corps ´elastiques multiplement connexes. Annales scientifiques de ´ l’Ecole Normale Sup´erieure, 24:401–517, 1907. [124] C-C Wang. On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Archive for Rational Mechanics and Analysis, 27:33–94, 1967. [125] C-C Wang. Material uniformity and homogeneity in shells. Archive for Rational Mechanics and Analysis, 47:343–368, 1972. [126] C-C Wang and J J Cross. On the field equations of motion for a smooth, materially uniform, elastic shell. Archive for Rational Mechanics and Analysis, 65:57–72, 1977. [127] G Weingarten. Sulle superfici di discontinuit`a nella teoria della elasticit`a dei corpi solidi. Rendiconti della Reale Accademia dei Lincei, 10:57–60, 1901. [128] H Weyl. Space-Time-Matter. Dover, 1952. [129] A Yavari. A geometric theory of growth mechanics. Journal of Nonlinear Science, 20:781–830, 2010. [130] A Yavari.

Compatibility equations of nonlinear elasticity for non-simply-connected bodies.

Archive for Rational Mechanics and Analysis, 209:237–253, 2013. [131] A Yavari and A Goriely. Weyl geometry and the nonlinear mechanics of distributed point defects. Proceedings of the Royal Society A, 468:3902–3922, 2012.

140

[132] A Yavari and A Goriely. Non-metricity and the nonlinear mechanics of distributed point defects. In R J Knops, G Q Chen, and M Grinfeld, editors, Differential Geometry and Continuum Mechanics. Springer Proceedings in Mathematics and Statistics (PROMS), 2014. [133] C Zhang and A Acharya. On the relevance of generalized disclinations in defect mechanics. arXiv preprint 1701.00158, 2016. [134] E V Zhbanova and L M Zubov. The influence of distributed dislocations on large deformations of an elastic sphere. In K Naumenko and M Aßmus, editors, Advanced Methods of Continuum Mechanics for Materials and Structures, pages 61–76. Springer Singapore, 2016. [135] X Zou and B I Yakobson. An open canvas – 2D materials with defects, disorder, and functionality. Accounts of Chemical Research, 48:73–80, 2015. [136] L M Zubov. Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, 1997. [137] L M Zubov. von-K´ arm´ an equations for an elastic plate with dislocations and disclinations. Doklady Physics, 52:67–70, 2007. [138] L M Zubov. The linear theory of dislocations and disclinations in elastic shells. Journal of Applied Mathematics and Mechanics, 74:63–72, 2010.

141