Projector representation of isotropic linear elastic ...

Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 11 July 2017

Projector representation of isotropic linear elastic material laws for directed surfaces Marcus Aßmus∗ , Johanna Eisentr¨ager, and Holm Altenbach Otto von Guericke University Magdeburg, Institute of Mechanics, Chair of Engineering Mechanics, Universit¨atsplatz 2, 39106 Magdeburg Received 2 May 2017, revised 10 July 2017, accepted 11 July 2017 Published online xx xx xxxx Key words Oriented body, directed surface, polar continuum, linear elasticity, isotropy, projector representation. In the framework of linear elasticity, it is possible to use eigenspace projectors to describe the elasticity tensor, at least for special cases of material symmetries. A similar procedure is also advantageous in the context of directed surfaces. It is thus possible to introduce this representation to stiffness measures of thin-walled members. Hereby, we reduce our considerations to an elastic mid-surface, where the in-plane, out-of-plane, and transverse shear states are uncoupled, but superposed eventually. Thereby, we introduce a unique decomposition of the stiffness measures for the considered deformation states, while we limit our considerations to homogeneous materials without pronounced orientation dependency. For each of these three states, eigenvalues of the stiffness tensors are evaluated based on engineering material parameters. Finally, the whole procedure allows for the clear distinction of dilatoric and deviatoric portions in the constitutive equations. After all, a compact and mathematically easy-to-handle representation for the stiffness tensors with respect to in-plane, out-of-plane, and transverse shear state has been found. Thereby, we show correlations to classical representations as well as advantages due to the clarity of present scheme. Copyright line will be provided by the publisher

1 1.1

Introduction Background

In three-dimensional linear elasticity, eigenspace projectors can be used to describe the elasticity tensor if the material features particular symmetries. The whole procedure results in a compact, unique, and mathematically simple representation of the elasticity tensor. For these reasons, the procedure for three-dimensional continua, cf. [24], is applied to two-dimensional surfaces in this paper. The first ideas for this procedure can be dated back to the works of L ORD K ELVIN, as stated in [25]. Here, we follow the pathway drawn by RYCHLEWSKI [25], not in detail but in standpoint and programme. In §8 of his comprehensive treatise, RYCHLEWSKI presents some ideas on the transfer of a procedure of classical to plane elasticity, thus providing the starting point for our considerations. However, our scope is beyond the bounded considerations presented in [9, 10]. Two-dimensional surfaces can be introduced ab initio or derivativo. While the first one is known as geometrically exact but suffers from establishment and identification of the constitutive equations, the latter one is approximative due to inconsistencies of derivations from a three-dimensional parent continuum. This kind of two-dimensional continuum is enriched with independent rotational degrees of freedom, whereby it can be assigned to the group of directors’ theories, i.e. micromorphic, microstretch, and micropolar continua, according to the discourse in [11]. We deny arbitrary characteristics of such directors and restrict the enrichments to a set of orthonormal and inextensible directors, while the theory considers only one director attached to the surface. Considering these constraints on the directors’ theories, we derive a setup where the remaining director is a unit vector, undeformable but not necessarily normal to the surface. The surface under consideration is a planar subset E2 of the three-dimensional E UCLIDean space E3 in equivalence to what is known from classical or three-dimensional C AUCHY continua. In that sense, ‘shells’ are completely excluded from present consideration. An engineering analogy to the present approach are (initially flat) homogeneous plate theories, restricted to geometries L1 ≈L2 L3 , while Li ∀ i={1, 2, 3} is the dimension in the corresponding direction where L3 is the structural thickness. The introduced material surface should be purely elastic deformable and initially stress-free, while we confine our concern to small deflections and rotations. ∗

Corresponding author: e-mail: [email protected], Phone: +49 391 67-51985 Copyright line will be provided by the publisher

2

M. Aßmus, J. Eisentr¨ager, and H. Altenbach: Projector representation of material laws for directed surfaces

1.2

Notation

This paper applies a direct notation for tensors, whenever this is possible. Tensors of zeroth order (or scalars) are symbolised by italic letters (e.g. a), italic lowercase bold letters denote first-order tensors (or monads) (e.g. a = ai ei or b = bj ej ), secondorder tensors (or dyads) are designated by italic uppercase bold letters (e.g. A = Alm el ⊗ em or B = Bno en ⊗ eo ), and fourth-order tensors (or tetrads) are symbolised by italic uppercase bold calligraphic letters (e.g. A = Apqrs ep ⊗eq ⊗er ⊗es ), whereas E INSTEIN sum convention is applied. By default, latin indices run through the values 1, 2, and 3, while greek indices run through the values 1 and 2, cf. Appendix A.1. Deviations thereof are indicated in the text. In the following, basic operations for tensors used in this paper are introduced based on a Cartesian coordinate system and orthonormal bases, e.g. {ei }: • the scalar product a · b = ai bj ei · ej = ai bi = α

α ∈ R,

(1)

• the dyadic product a ⊗ b = ai bj ei ⊗ ej = C ,

(2)

• the composition of a second and a first-order tensor A · a = Alm ai el ⊗ em · ei = Ali ai el = d ,

(3)

• the composition of two second-order tensors A · B = Alm Bno el ⊗ em · en ⊗ eo = Alm Bmo el ⊗ eo = D ,

(4)

• the double scalar product between a fourth and a second-order tensor

A : B = Apqrs Bno ep ⊗ eq ⊗ er ⊗ es : en ⊗ eo = Apqrs Bsr ep ⊗ eq = F ,

(5)

• the double scalar product between two fourth-order tensors

A : B = Apqrs Btuvw ep ⊗ eq ⊗ er ⊗ es : et ⊗ eu ⊗ ev ⊗ ew = Apqrs Bsrvw ep ⊗ eq ⊗ ev ⊗ ew = G ,

(6)

• the cross product between two first-order tensors a × b = ai bj ei × ej = ai bj ijk ek = c ,

(7)

• the cross product between a second and a first-order tensor A × b = Alm bj el ⊗ em × ej = Alm bj mjk el ⊗ ek = J , where ijk is the permutation symbol   +1 if (i, j, k) is an even permutation of (1, 2, 3) , ijk = −1 if (i, j, k) is an odd permutation of (1, 2, 3) ,   0 if (i, j, k) is not a permutation of (1, 2, 3) .

(8)

(9)

In the sequel, the transposition of tensors is required. For second-order tensors, this is defined via a · A>· b = b · A · a. For fourth-order tensors, A : A>: B = B : A : A holds true. The nabla operator is defined as ∇ = eα ∂/∂Xα for twodimensional considerations. ∇ ·  is the divergence, and ∇ is the gradient of a tensor, where  holds true for every differentiable tensor field. The transposed gradient is defined as ∇>  = [∇]> , and ∇sym  = 1/2[∇ + ∇> ] is the symmetric part of the corresponding gradient, where  holds for all first-order tensors. Furthermore, every tensor A can be decomposed into its dilatoric Adil and deviatoric parts Adev . Since we are working on E2 , this is done via Adil = 1/2 [A : P ] P and Adev =A−Adil , while A = Aαβ eα ⊗ eβ holds true, and P = eα ⊗ eα is the unit tensor of the planar surface. Copyright line will be provided by the publisher

ZAMM header will be provided by the publisher

1.3

3

Structure of representation

The purpose of the present work is the introduction of the projection methodology for constitutive measures of five-parameter surface continua, what is based on following structure. In Sect. 2, we disclose the governing equations of the surface continuum in the spirit of Z HILIN [30]. Starting from the kinematics, we proceed to dynamic measures and equilibrium equations up to constitutive relations as presented customarily. In Sect. 3, the projection method for stiffness tensors of the surface continuum is discussed. Based on the constitutive measures introduced previously and exploiting their properties, we obtain a clear description of the constitutive laws based on the spectral representation. Concluding remarks as well as an outlook on possible future directions can be found in Sect. 4.

2

Governing equations

2.1 2.1.1

Kinematics Degrees of freedom

The two-dimensional plane surface is described by the orthonormal base vectors {eα , n}, whereas the vectors eα refer to two in-plane directions and n represents the normal vector of the surface. We consider five degrees of freedom for every material point of the two-dimensional surface in analogy to the work of Z HILIN [31], which is a more general case of R EISSNER’s plate theory [23], which in turn is a special case of the linear, isotropic C OSSERAT model, cf. [20]. These can be expressed as follows. u = v + wn

(10)

ψ = ψα eα = −ϕ2 e1 + ϕ1 e2

(11)

Herein, v = vα eα are in-plane displacements, w is the deflection, and ψ = ψα eα are circumferential out-of-plane rotations. All five degrees of freedom are assumed to be infinitesimal small. Considering the rotational degrees of freedom, we follow a more straightforward introduction given in PAL’ MOV [22], which slightly differs from Z HILIN [31]. ϕ = ϕα e α

(12)

Approaches to differential geometry in context of present problem are discussed in [30]. Overall, we can distinguish between physical reasonable and mathematical consistent approaches. The present treatise is dedicated to the latter one.

2.1.2

Deformation measures

We define the following measures of deformation of the plane, whereby we neglect higher order terms of in-plane displacement gradient ∇v and rotational gradient ∇ϕ. Results are the following infinitesimal measures. G = ∇sym v = Gαβ eα ⊗ eβ sym

K=∇

ϕ = Kαβ eα ⊗ eβ

g = ∇w + ϕ = gα

eα

(13) (14) (15)

Therein, G is the in-plane strain tensor, K is the curvature change tensor, and g is the transverse shear strain vector. G and K are symmetric what follows from Eqs. (10) and (12).

2.2 2.2.1

Kinetics Dynamic measures

Instead of introducing analogies of C AUCHY’s stress principle, postulate, and fundamental lemma at the surface, we refer the stress measure of a three-dimensional continuum to the surface. Here, the C AUCHY stress tensor T serves as basis. T = Tαβ eα ⊗ eβ + Tα3 eα ⊗ n + T3α n ⊗ eα + T33 n ⊗ n

(16) Copyright line will be provided by the publisher

4

M. Aßmus, J. Eisentr¨ager, and H. Altenbach: Projector representation of material laws for directed surfaces

The transformation to descend from three dimensions is done via over thickness integration of this stress tensor [16, 17]. We can derive so called stress resultants by applying the unit tensor of the planar surface P in different concatenations. Z N = P · T · P dX3 = Nαβ eα ⊗ eβ (17) h

Z X3 P · T · P dX3

L =

= Lαβ eα ⊗ eβ

(18)

h

Z (P · T − P · T · P ) dX3 = Qα eα ⊗ n = q ⊗ n

Q =

(19)

h

These stress resultants are conjugate to the deformation measures introduced in Sect. 2.1.2 and known as in-plane force tensor N , polar tensor of moments L, and transverse shear force vector q. Even though we call them force and moment tensors, the units are [N/mm] and [N]. The tensors N and L are symmetric. Main diagonal elements of N are in-plane normal forces, while off-diagonal elements represent in-plane shear forces. Main diagonal elements of L are bending moments, while off-diagonal elements represent twisting moments. Components of q are transverse shear forces. 2.2.2

Balance equations

We formulate the local form of E ULER’s laws of motion for deformable surfaces reduced to quasi-static problems. ∇ · (N + q ⊗ n) ∇ · (−L×n)

+ s + pn = o + q×n + m

=o

balance of forces

(20)

balance of moments

(21)

Herein, s=sα eα are tangentially distributed forces, p are lateral distributed forces, and m=mα eα are distributed moments. 2.3

Constitutive relations

We reduce our considerations to simple elastic materials, as originally defined by N OLL [21]. Here, the dynamic measures depend to a maximum on the first gradient of displacements, deflection, or rotations. The allowance for the zeroth gradient of rotations results from the reduction to two dimensions. N = F1 (G) ,

L = F2 (K) ,

q = F3 (g) .

(22)

Herein, Fi are the elastic material laws. Since we restrict our concern to small deflections and rotations, it is reasonable to approximate the relations between kinematic and dynamic measures by linear laws. Such linear functions between second or first-order tensors can be represented by fourth-order or second-order tensors, respectively. N = A: G

(23)

L = D: K

(24)

q =Z ·g

(25)

Above equations hold true for completely decoupled, but possibly superposed deformation states, i.e. stretching, bending, twisting, and shearing. The neglect of coupling terms is possible due to symmetry considerations of geometry and material properties in transverse direction. The introduced constitutive measures are the fourth-order in-plane stiffness tensor A, the fourth-order out-of-plane stiffness tensor D, and the second-order transverse shear stiffness tensor Z. Considering a material where all material directions are equivalent, the structure of these stiffness tensors is given as follows.

A = 2Gh P sym + (B − G) h P ⊗ P 3

h h P sym + (B − G) P ⊗ P 12 12 Z = κGh P

D = 2G

(26)

3

(27) (28)

Here we can identify two material parameters, one geometry parameter and one correction factor, while B = Y /2(1 − ν) is the surface bulk modulus, cf. Appendix A.3, G = Y /2(1 + ν) is the shear modulus, h ≡ L3 is the structural thickness, and κ is the shear correction factor. There, we use Y for YOUNG’s modulus and ν for P OISSON’s ratio. Furthermore, Copyright line will be provided by the publisher

ZAMM header will be provided by the publisher

5

P sym = 1/2 [eα ⊗ eβ ⊗ eα ⊗ eβ + eα ⊗ eβ ⊗ eβ ⊗ eα ] is the symmetric part of the fourth-order identity and P = eα ⊗eα is the second order identity. This depiction is in analogy to L AM E´ ’s representation of the constitutive tensor of C AUCHY or B OLTZMANN continua, even if we use engineering constants since there is a unique correlation in the case of isotropy, cf. [15]. Thereby, it is particularly evident that the first terms in Eqs. (26)–(28) represent deviatoric portions while the second terms of Eqs. (26) and (27) incorporating B − G represent a mixture of deviatoric and dilatoric portions. Furthermore, we can identify subsequent engineering interpretations. Yh = (B + G)h , 1 − ν2

DT =

DF =

Y h3 h3 = (B + G) , 2 12(1 − ν ) 12

DS =

κY h = κGh. 2(1 + ν)

(29)

Therin, DT is the tensional rigidity, DF is the flexural rigidity, and DS is the transverse shear rigidity. These analogies are in accordance with the works of K IRCHHOFF [13], R EISSNER [23], and M INDLIN [18] as can be found in classical textbooks in this field, e.g. [29]. 2.4

Properties of stiffness tensors

Restricting our considerations to completely decoupled, but possibly superposed deformation states, the strain energy function can be assumed as quadratic form.  1 G: A: G + K : D : K + g · Z · g 2

W =

(30)

The existence postulate of an elastic potential leads to the main symmetries of all three stiffness tensors [3] introduced in Eqs. (23), (24), and (25). Furthermore, A, D, and Z are positive-definite, which is equivalent to the positiveness of all eigenvalues λi > 0 (principal stiffnesses, also called K ELVIN moduli to honour W ILLIAM T HOMSON, 1st Baron K ELVIN) due to the symmetry of all stiffness tensors. To be exact, the symmetries of the fourth-order tensors H = {A, D} are as follows. A: H: B = B : H: A = A>:

A: H

H: A =

H H : A>

H = H>

Hαβγδ = Hγδαβ

main symmetry

Hαβγδ = Hβαγδ

left subsymmetry

Hαβγδ = Hαβδγ

right subsymmetry

This holds true for all second-order tensors A, B. However, the subsymmetries result from the symmetries of corresponding deformation and dynamic measures. Furthermore, the second-order transverse shear stiffness tensor Z is symmetric since the following expression is true for all first-order tensors b. Z·b = b ·Z

3

Z = Z>

Zαβ = Zβα

symmetry

Projection method

3.1

Spectral decomposition of stiffness tensors

Due to the symmetry properties of all three stiffness tensors, a unique eigenprojector representation through spectral decomposition is possible [27].

A=

2n X

λA i Pi ,

i=1

D=

2n X

λD i Pi ,

Z=

i=1

n X

λZ i Pi .

(31)

i=1

Z D In the general case, A and D have 2n ≤ 4 and Z has n ≤ 2 different real eigenvalues λA i , λi , and λi , respectively. Then and only then, n equals the number of dimensions under consideration. Using eigenbases, the projectors are defined as follows [12].

Pi =

2n X j=1

Cj ⊗ Cj ,

Pi =

n X

cj ⊗ cj .

(32)

j=1

Herein, C j are symmetric normed eigentensors of second-order and cj are normed eigenvectors. The eigenvalues always depend on the elastic parameters, while the eigentensors or -vectors respectively do not necessarily depend only on these parameters, cf. [25]. Copyright line will be provided by the publisher

6

M. Aßmus, J. Eisentr¨ager, and H. Altenbach: Projector representation of material laws for directed surfaces

3.2

Determination of projectors

The fourth-order projectors have the following properties ( Pi if j = i Pi : Pj = O if i 6= j while the properties of the second-order projectors are as follows. ( P i if j = i Pi· Pj = O if i 6= j

X

Pi = P sym ,

X

(33)

Pi = P

(34)

Herein, O and O are the second- and fourth-order zero tensors. These systems of projection operators are therewith idempotent, orthogonal, and complete. In the case of isotropy, the value of n reduces to 1, and the projectors map any tensor H into its dilatoric and deviatoric parts, cf. also Appendix A.2. 1/2 [H : P ] P = H dil P1 : H = P2 : H = H−1/2 [H : P ] P = H dev

(dilatations)

(35)

(area-preserving, distortions)

(36)

P1· g = g

(pure area-preserving deformations)

(37)

Here, one should note that the terms ‘dilatoric’ and ‘deviatoric’ are referring to two dimensions, e.g. two-dimensional dilatoric deformations describe uniform plane compression or expansion, respectively. The term H : P =Hαα is the trace of H, with H = {G, K}. Due to the equality of the eigenvalues of Z, only one projector remains, such that P 1 =P holds true. It is apparent that the first-order tensor g is irreducible. So P is idempotent only, what results in the fact that g represents pure distortions. Isotropic eigenprojectors fulfilling the projector rules (33) or (34), respectively, are as follows.

P1 = 1/2 P ⊗ P P2 = P sym− P1

(38)

P = eα ⊗ eα

(40)

(39)

As is apparent, the eigenspace of P coincides with its vector space, see Eq. (32). 3.3

Eigenvalues of stiffness tensors

With the aid of the eigentensors and eigenvectors defining the eigenprojectors, cf. Eq. (31), the eigenvalue problems of the stiffness measures can be solved.

A : C i = λA i Ci ,

D : C i = λD i Ci ,

Z · ci = λ Z i ci .

(41)

The isotropic eigenvalues are given as follows. Yh λA = 2Bh , 1 = 1−ν Y h3 h3 λD = 2B , 1 = 12(1 − ν) 12 κY h λZ = = κGh. 1 2(1 + ν)

Yh λA = 2Gh , 2 = 1+ν Y h3 h3 λD = 2G , 2 = 12(1 + ν) 12

(42) (43) (44)

The eigenvalues of A and D depend on three parameters: one geometrical parameter (h) as well as two material parameters (Y and ν). In addition, we can identify the up to now undetermined shear correction factor (κ) in the eigenvalue of Z. This parameter is introduced artificially to adjust the energetic mean of the transverse shear contribution in the sense of R EISSNER [23]. However, due to the clear split introduced in Eqs. (35)–(37), we can consider the eigenvalues as resistances Z D A D to specific deformations. λA 1 and λ1 are resistances to dilatations and λ2 , λ2 , as well as λ1 represent resistances to distortions. Furthermore, following correlations to Eq. (29) hold true.     1/2 λA + λA = D , 1/2 λD + λD = D , λZ (45) T F 1 = DS . 1 2 1 2 The tensional and flexural stiffnesses are the arithmetic means of the corresponding eigenvalues. This also holds true for the Z Z correlation between λZ i and DS . However, the averaging is omitted since λ1 and λ2 coincide in the isotropic case. Copyright line will be provided by the publisher

ZAMM header will be provided by the publisher

3.4

7

Projector representation

3.4.1

Stiffness tensors

Following restrictions hold true to obtain positive-definite stiffness tensors. −1 < ν < 1/2 ,

Y > 0,

κ > 0.

(46)

This allows the following representation for stiffness measures with isotropic properties based on Eq. (31). A A = λA 1 P1 + λ2 P2 ,

D D = λD 1 P1 + λ2 P2 ,

Z = λZ 1 P .

(47)

Z D A D Herein, λA 1 and λ1 are correlated with area changes of the surface, while λ2 , λ2 , and λ1 are correlated with a change in shape. Comparing stiffness tensors achieved in Eqs. (47) with the stiffness tensors typically used as given in Eqs. (26)–(28), we can identify a clear distinction of deformations portions within the newly derived representation. The procedure is also applicable to the compliance tensors −1 ∀  = {A, D, Z}, which allows us to write Eqs. (23)–(25) in inverse form.

1

1

1

2

A−1 = A P1 + A P2 , λ λ

1

1

1

2

D −1 = D P1 + D P2 , λ λ

Z −1 =

1 λZ 1

P.

Here, A−1: A = P sym , D −1: D = P sym , and Z −1· Z = P are valid. The inversions of the stiffness measures are possible since for all eigenvalues λ i 6= 0 ∀  = {A, D, Z} holds true. 3.4.2

Constitutive laws

The constitutive equations can now also be written as follows. N = 2Bh Gdil + 2Gh Gdev 3

(48)

3

h h K dil + 2G K dev 12 12 κh q = 2G g 2

L = 2B

(49) (50)

As is particularly evident, studying the structure of Eqs. (48) and (49), one can find an analogy to the constitutive law of classical three-dimensional elasticity as derived in [25]. In contrast to Eqs. (26)–(28), we have derived a representation with clear distinction of deformation portions.

4

Conclusion and outlook

In the foregoing treatise, we have derived a projector representation for two-dimensional isotropic material bodies. Thereby, three characteristic stiffness measures have been decomposed such that the constitutive laws obtained through this transformation highlight the two-dimensional dilatoric and deviatoric responses of deformation. However, in present context, it seems conceptually unfavourable to introduce the transverse shear terms with tensors of reduced order since it prevents a consistent representation for all stiffness measures of the surface. This is due to the fact that the force tensor (N + q ⊗ n) is not symmetric a priori. Alternatively, once the eigenvalues are determined by the non-trivial solution of the characteristic equations, cf. Eq. (41), it is possible to determine the eigenprojectors with S YLVESTER’s formula [28] or more generally with B UCHHEIM’s extension [7]. However, considering the elastic surface, fourth-order stiffness tensors have a maximum of four, and second-order stiffness tensors possess a maximum of two eigenvalues only. After all, the projector representation of stiffness measures simplifies the mathematical treatment such that it should be applied as often as possible. For three-dimensional bodies, such general representation as given in Eq. (31) exists for materials with anisotropies, too [14, 26]. This indicates that this procedure can also be transferred to anisotropic surfaces. Thereby, material symmetries known from three-dimensional continua partially conincide at the surface. This results in four disjoint types of physical symmetries, cf. [6, 25]. However, more complex descriptions are necessary when considering materials with pronounced orientation dependency since the projectors depend on the material parameters in this case, cf. [14]. Furthermore, one has to account for coupled deformation states as obvious with initially curved surfaces. Then, simple linear mappings as presented in Eqs. (23)–(25) are no longer possible. Copyright line will be provided by the publisher

8

M. Aßmus, J. Eisentr¨ager, and H. Altenbach: Projector representation of material laws for directed surfaces

Furthermore, B ERTRAM & O LSCHEWSKI [4] showed that this concept is also applicable to viscoelasticity, while C AROL , R IZZI & W ILLAM [8] have extended the concept to damage. Such improvements will be useful considering practical applications of elastic surfaces, as shown for example in [2]. Finally, there is a variety of differing treatments for directed surfaces, deviating from this consideration of a five-parameter theory for planar surfaces, cf. [1]. The application of the present concept to such continua would be useful for generalisation.

A A.1

Appendix Tensors on E2

In the present work, we make use of tensors on two-dimensional surfaces where Greek indices α, β, γ, δ are introduced which are running from 1 to 2 only. The variable NE=NDNP determines the number of elements of a tensor, while ND is the number of dimensions under consideration, and NP signifies the order of the tensor. However, since in the present case the number of elements differs significantly from those of three-dimensional continua, we repeat the structure of first, second, and fourth-order tensors and their coefficient matrices with respect to an orthonormal basis in the sequel. • fourth-order tensors (NE=16, ND=2, NP =4)  H1111  H  1121 [Hαβγδ ] =    H2111 H2121

H = Hαβγδ eα ⊗ eβ ⊗ eγ ⊗ eδ

H1112 H1122



H2112 H2122



 H1211 H1221  H2211 H2221

 H1212 H1222     H2212  H2222

• second-order tensors (NE=4, ND=2, NP =2) H = Hαβ eα ⊗ eβ

[Hαβ ] =

 H11 H21

H12 H22



• first-order tensors (NE=2, ND=2, NP =1)  h = hα eα A.2

[hα ] =

h1 h2



Validity of isotropic projection operators

In the sequel, we present the validity of expressions given in Eqs. (35)–(37). Thereby, we assume that H=H > holds true.

P1 : H =1/2P ⊗ P : H =1/2eα ⊗ eα ⊗ eβ ⊗ eβ : Hγδ eγ ⊗ eδ =1/2Hββ eα ⊗ eα =1/2 [H : P ] P

P2 : H =P sym : H − P1 : H =1/2eα ⊗ eβ ⊗ eα ⊗ eβ : Hγδ eγ ⊗ eδ + 1/2eα ⊗ eβ ⊗ eβ ⊗ eα : Hγδ eγ ⊗ eδ − 1/2 [H : P ] P =1/2Hβα eα ⊗ eβ + 1/2Hαβ eα ⊗ eβ − 1/2 [H : P ] P =1/2H > + 1/2H − 1/2 [H : P ] P =H − 1/2 [H : P ] P P · g =eα ⊗ eα · gβ eβ =gβ eα δαβ =gα eα = g Herein, δαβ is the K RONECKER symbol. ( 1 if α = β δαβ = 0 if α 6= β Copyright line will be provided by the publisher

ZAMM header will be provided by the publisher

A.3

9

Surface bulk modulus

The bulk modulus is defined as the ratio of an applied isostatic stress to the fractional volumetric change. Since our concern is restricted to an infinite thin two-dimensional surface, a third dimension is missing to generate a volume at all, so that we have to reformulate this measure. An adaption of the B IRCH -M URNAGHAN equation of state [5, 19] caused by the assumption of plane stress state, directly leads us to following expression. B = −A

dp dp =− dA dA/A

Herein, A is the area, and p denotes the pressure. We assume an isostatic stress state at the surface. T = −pP Thereby it is possible to reformulate B in terms of stress and strain. B=

Tm Gv

with Tm = 1/2 T : P = −p

(isostatic stress)

and Gv =

(relative area change)

G : P = G11 + G22

Using inverse H OOKE’s law of plane stress state, we can determine the strain components. G11 = G22 = −p/Y (1 − ν)

⇒

Gv = −

2p (1 − ν) Y

This results in the definition of the bulk modulus of the surface. B=

Y 2(1 − ν)

Acknowledgement We acknowledge the financial support by the German Research Foundation within the reserach training group 1554.

References [1] J. Altenbach, H. Altenbach, and V. A. Eremeyev, On generalized cosserat-type theories of plates and shells: a short review and bibliography, Archive of Applied Mechanics 80(1), 73–92 (2010). [2] M. Aßmus, J. Nordmann, K. Naumenko, and H. Altenbach, A homogeneous substitute material for the core layer of photovoltaic composite structures, Composites Part B: Engineering 112, 353–372 (2017). [3] G. Backhaus, Deformationsgesetze (Akademie Verlag, Berlin, 1983). [4] A. Bertram and J. Olschewski, Zur Formulierung anisotroper linear anelastischer Stoffgleichungen mit Hilfe einer Projektionsmethode, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f¨ur Angewandte Mathematik und Mechanik 73, T401–T403 (1993). [5] F. Birch, Finite Elastic Strain of Cubic Crystals, Physical Review 71(11), 809–824 (1947). [6] A. Blinowski, J. Ostrowska-Maciejewska, and J. Rychlewski, Two-dimensional Hooke’s tensors – isotropic decomposition, effective symmetry criteria, Archives of Mechanics 48(2), 325–345 (1996). [7] A. Buchheim, An extension of a theorem of Professor Sylvester relating to matrices, Philosophical Magazine 22(5), 173–174 (1886). [8] I. Carol, E. Rizzi, and K. Willam, An extended volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate, European Journal of Mechanics - A/Solids 21(5), 747–772 (2002). [9] G. Dzier˙zanowski and T. Lewi´nski, Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem, Archives of Mechanics 64(1), 21–40 (2012). [10] G. Dzier˙zanowski and T. Lewi´nski, Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions, Archives of Mechanics 64(2), 111–135 (2012). [11] A. C. Eringen, Microcontinuum Field Theories - I. Foundations and Solids (Springer, New York, 1999). [12] P. R. Halmos, Finite-Dimensional Vector Spaces (Springer, New York, 1958). ¨ [13] G. R. Kirchhoff, Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe, Crelles Journal f¨ur die reine und angewandte Mathematik 40, 51–88 (1850). [14] K. Kowalczyk-Gajewska and J. Ostrowska-Maciejewska, Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material, Engineering Transactions 57(3–4) (2014). [15] W. Lai, D. Rubin, and E. Krempl, Introduction to Continuum Mechanics, 4. edition (Butterworth-Heinemann, Oxford, 2009). Copyright line will be provided by the publisher

10

M. Aßmus, J. Eisentr¨ager, and H. Altenbach: Projector representation of material laws for directed surfaces

[16] A. Libai and J. G. Simmonds, Nonlinear elastic shell theory, Advances in Applied Mechanics 23, 271–371 (1983). [17] A. Libai and J. G. Simmonds, The nonlinear theory of elastic shells, The Nonlinear Theory of Elastic Shells (1998). [18] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics 18, 31–38 (1951). [19] F. D. Murnaghan, The Compressibility of Media under Extreme Pressures, Proceedings of the National Academy of Sciences of the United States of America 30(9), 244–247 (1944). [20] P. Neff, K. I. Hong, and J. Jeong, The Reissner-Mindlin plate is the Γ-limit of Cosserat elasticity, Mathematical Models and Methods in Applied Sciences 20(9), 1553–1590 (2010). [21] W. Noll, A mathematical theory of the mechanical behavior of continuous media, Archive for Rational Mechanics and Analysis 2(1), 197–226 (1958). [22] V. A. Pal’mov, On the Cosserat plate theory (in Russian), in: Dynamics and Strength of Machines – Collected Papers of the Leningrad Polytechnic Institute, , No. 386 (Leningrad Polytechnic Institute, 1982), pp. 3–8. [23] E. Reissner, On the theory of bending of elastic plates, Journal of Mathematics and Physics 23(1–4), 184–191 (1944). [24] J. Rychlewski, On Hooke’s law, Priklat. Mat. Mech. 48(3), 303–314 (1984). [25] J. Rychlewski, Unconventional approach to elasticity, Archives of Mechanics 47(2), 149–171 (1995). [26] J. Rychlewski and J. Zhang, Anisotropy degree of elastic materials, Archives of Mechanics 47(5), 697–715 (1989). [27] S. Sutcliffe, Spectral decomposition of the elasticity tensor, Journal of Applied Mechanics 59(4), 762–773 (1992). [28] J. J. Sylvester, On the equation to the secular inequalities in the planetary theory, Philosophical Magazine 16(5), 267–269 (1883). [29] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959). [30] P. A. Zhilin, Applied Mechanics - Fundamentals of the theory of shells (in Russian) (Publisher of the Polytechnic University, St. Petersburg, 2006). [31] P. A. Zhilin, Mechanics of deformable directed surfaces, International Journal of Solids and Structures 12(9), 635–648 (1976).

Copyright line will be provided by the publisher