Received , ;
Revised , ;
Accepted ,
DOI: xxx/xxxx
ARTICLE TYPE
A mathematical model for elasticity using calculus on discrete manifolds Ioannis Dassios*1 | Gary Oโ Keeffe2 | Andrey P Jivkov3 1 ESIPP,
University College Dublin, Ireland Department of Mathematics & Statistics, University of Limerick, Ireland 3 Mechanics of Physics of Solids Research Group, School of Mechanical Aerospace & Civil Engineering, The University of Manchester, UK 2 MACSI,
Correspondence *Ioannis Dassios Email:
[email protected]
Abstract We propose a mathematical model to represent solid materials with discrete lattices and to analyse their behaviour by calculus on discrete manifolds. Focus is given on the mathematical derivation of the lattice elements by taking into account the stored energy associated with them. We provide a matrix formulation of the non-linear system describing elasticity with exact kinematics, known as finite strain elasticity in continuum mechanics. This formulation is ready for software implementation, and may also be used in atomic scale models as an alternative to existing empirical approach with pair and cohesive potentials. An illustrative example, analysing a local region of a node, is given to demonstrate the model performance. KEYWORDS: discrete manifold, lattice model, steel microstructure, elasticity, energy, non-linear system
1
INTRODUCTION
Lattice models for deformation and fracture of solid materials were developed firstly for quasi-brittle materials, such as concretes and rocks (1), (7), and extended recently for elastic-plastic materials, such as structural steels (2). The benefit of modelling materials, treated as continua in classical mechanics, with discrete lattices is that the nucleation, growth and coalescence of discontinuities (cracks) become natural processes. As these are non-topological changes in the system, the classical solid mechanics, being a thermodynamic bulk theory, does not work. Additional benefit of the discrete approach is the possibility for introducing heterogeneities and local anisotropies in the modelled structure by appropriate spatial and directional variation of lattice properties, as well as natural and/or essential boundary conditions in what would be considered an interior in the continuum approach (since each lattice vertex is a boundary). A lattice, in the language of algebraic topology, is a 1-complex embedded in โ2 or โ3 , i.e. a graph with nodes (sites) equipped with Cartesian coordinates and edges (bonds) between some nodes. The first challenge is to ensure the elastic response of a graph is equivalent to the continuum response measured experimentally, i.e. to derive a link between properties of lattice elements, e.g. bond stiffness coefficients, and macroscopic properties. Isotropic materials, described by two macroscopic constants, can be represented exactly by 2๐ท graphs based on hexagonal structure, e.g. (3), (5), (9), and by 3๐ท graphs based on truncated octahedral structure (8), (12). Other regular 3๐ท graphs can represent cubic elasticity, described by three macroscopic constants (10). However, there is no general recipe for deriving properties of lattice elements of arbitrary non-regular graph, which is required to represent closely given material microstructure, and all the existing representations with regular lattices are derived for linearised kinematics (infinitesimal strain in continuum settings). One of the aims of this work is to develop an alternative description of lattice behaviour, which makes arbitrary graphs represent macroscopic elasticity with exact kinematics (finite strain in continuum settings).
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Ioannis Dassios ET AL
A mathematically rigorous analysis of graphs is based on discrete exterior calculus (DEC) (6). However, the analysis on graphs developed in this reference is applicable to physical problems, where the nodal unknown (a 0-cochain) is a scalar, i.e. temperature, pressure, concentration, etc, and its gradient is also a scalar over the edges (1-cochain). In mechanical problems the nodal unknown is a vector, e.g. nodal displacements in linearised kinematics or nodal coordinates in exact kinematics, which makes the problem critically different. One possibility for formulating elasticity is to use 3-complexes (11), which leads to a complexity similar to the one discretising continuum problems with finite elements for example. The main aim of this work is to investigate whether 1-complexes can still be used to describe elastic behaviour as computationally more efficient alternative to 3-complexes.
2
THE MODEL WITH CALCULUS ON DISCRETE MANIFOLDS
Consider a tessellation of a 3D region into cells, e.g. a Voronoi tessellation around a given set of ๐ points, representing a particular material microstructure (2). The tessellation is a 3-complex denoted by V and containing ๐ 3-cells (cells), ๐2 2-cells (faces), ๐1 1-cells (edges) and ๐0 0-cells (nodes). We explore the possibility to replace V with two 1-complexes (graphs), basic and complimentary, determined by V. The basic graph, denoted by G, has ๐ = ๐ nodes placed at 3-cellsโ centres, and ๐ = ๐2 edges connecting nodes whose 3-cells have common 2-cells. The connectivity of G is thus provided by the cell-face connectivity of V. The complementary graph, denoted by G* has ๐โ = ๐ = ๐2 nodes, placed at 2-cellsโs centres (which are also centres of G edges), ๐โ = ๐1 edges connecting nodes whose 2-cells have common 1-cells. The connectivity of G* is thus provided by the face-edge connectivity of V. To describe operations on G we introduce a matrix ๐ด = [๐๐๐ ]๐=1,2,...,๐ โ โ๐ร๐ , and a matrix ๐ = [๐ ๐๐ ]๐=1,2,...,๐ โ โ๐ร๐ , where ๐=1,2,...,๐ ๐=1,2,...,๐ โง 0, if node ๐ is not a node of edge ๐ โซ โช โช ๐๐๐ = โจ 1, if node ๐ is the first node of edge ๐ โฌ โช โ1, if node ๐ is the second node of edge ๐ โช โญ โฉ
{ and
๐ ๐๐ =
0, if node ๐ is not a node of edge ๐ 0.5, if node ๐ is a node of edge ๐
} .
These reflect the cell-face connectivity of V: ๐ด is the standard incidence matrix, providing a co-boundary operator for nodes (6), while ๐ is an averaging operator over nodes. Similar operators can be defined for G*, of which we will need only the ๐=1,2,...,๐โ ๐โ ร๐โ co-boundary operator ๐ดโ = [๐โ๐๐ ]๐=1,2,...,๐ constructed similarly to A but reflecting face-edge connectivity of V. โ โ โ Let โก ๐ถโ โค โก ๐ถ1 โค โข 1โ โฅ โข โฅ โ ๐ถ ๐ถ2 โฅ ๐ร3 โ โข โโ and ๐ถ = โข 2 โฅ โ โ๐ ร3 ๐ถ= โข โฎ โฅ โข โฎ โฅ โข ๐ถโ โฅ โข๐ถ โฅ โฃ ๐โฆ โฃ ๐โฆ denote discrete vector-valued functions over nodes of G and G*, respectively, where ] [ [ ] โ โ โ ๐ถ๐2 ๐ถ๐3 โ โ3 , ๐ถ๐ = ๐ถ๐1 ๐ถ๐2 ๐ถ๐3 โ โ3 , ๐ = 1, 2, ..., ๐ and ๐ถ๐โ = ๐ถ๐1
๐ = 1, 2, ..., ๐โ .
In order to apply ๐ด, ๐ and ๐ดโ on such functions we expand these operators by repetition of each raw and each column three โ โ times, so that from now on we will use the resultant ๐ด โ โ3๐ร3๐ , ๐ โ โ3๐ร3๐ and ๐ดโ โ โ3๐ ร3๐ . When ๐ด and ๐ดโ are applied to ๐ถ and ๐ถ โ they provide the gradient of these functions as vector-valued functions over edges of the respective graphs โก ๐1 โค โข โฅ ๐ ๐ด๐ถ = ๐ = โข 2 โฅ โ โ๐ร3 โข โฎ โฅ โข๐ โฅ โฃ ๐โฆ where
[ ] ๐๐ = ๐๐1 ๐๐2 ๐๐3 โ โ3 ,
๐ = 1, 2, ..., ๐
and
and
โก ๐โ โค โข 1โ โฅ โ ๐ โ โ โ ๐ด ๐ถ = ๐ = โข 2 โฅ โ โ๐ ร3 โข โฎ โฅ โข ๐โ โฅ โฃ ๐โฆ [ ] ๐โ๐ = ๐โ๐1 ๐โ๐2 ๐โ๐3 โ โ3 ,
๐ = 1, 2, ..., ๐โ .
For the purposes of developing exact kinematics, we associate ๐ถ and ๐ถ โ with the nodal coordinates in G and G*, respectively, in which case we note that ๐ถ โ = ๐๐ถ. Further ๐ and ๐โ represent coordinate differences associated with edges of G and G*, respectively, which provide edge lengths via |๐๐ |, ๐ = 1, 2, ..., ๐, and |๐โ๐ |, ๐ = 1, 2, ..., ๐โ . A boundary value problem is formulated by prescribing boundary conditions to the nodes of G. These can be either natural, i.e. prescribed external forces, or essential, i.e. prescribed new coordinates due to nodal displacements. As a result of these
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Ioannis Dassios ET AL
conditions, the geometry of the whole structure changes, so that the nodes of G and G* have new coordinates [ ] [ ] โ โ โ ๐๐2 ๐๐3 ๐๐ = ๐๐1 ๐๐2 ๐3 โ โ3 , ๐ = 1, 2, ..., ๐ and ๐๐โ = ๐๐1 โ โ3 , ๐ = 1, 2, ..., ๐โ , which we assume to be related by ๐ โ = ๐๐, i.e. the deformation of G* is ๐๐๐๐ข๐๐๐ by the deformation of G. This leads to new edge lengths calculated as |๐ฆ๐ |, ๐ = 1, 2, ..., ๐, and |๐ฆโ๐ |, ๐ = 1, 2, ..., ๐โ from โก ๐ฆ1 โค โข โฅ ๐ฆ ๐ฆ = ๐ด๐ = โข 2 โฅ โ โ๐ร3 โข โฎ โฅ โข๐ฆ โฅ โฃ ๐โฆ
and
โก ๐ฆโ โค โข 1โ โฅ โ ๐ฆ โ โ โ โ ๐ฆ = ๐ด ๐ = ๐ด ๐๐ = โข 2 โฅ โ โ๐ ร3 . โข โฎ โฅ โข ๐ฆโ โฅ โฃ ๐โ โฆ
(1)
Upon deformation, the edges of G and G* store elastic energy dependent on the length changes |๐ฆ๐ | โ |๐๐ |, ๐ = 1, 2, ..., ๐ and |๐ฆโ๐ | โ |๐โ๐ |, ๐ = 1, 2, ..., ๐โ , respectively. As a first approximation, we assume the edge energies to be quadratic functions of length changes within specified length intervals as follows ๐๐ = ๐พ๐ (|๐ฆ๐ | โ |๐๐ |)2 ,
๐0๐ โค |๐ฆ๐ | โ |๐๐ | โค ๐1๐ ,
๐๐โ = ๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 ,
๐ = 1, 2, ..., ๐
๐2๐ โค |๐ฆโ๐ | โ |๐โ๐ | โค ๐3๐ ,
and
(2)
๐ = 1, 2, ..., ๐โ ,
(3)
๐พ๐โ
where ๐พ๐ and are material parameters associated with the edges of G and G*, respectively, and the intervals [๐0๐ , ๐1๐ ] and [๐2๐ , ๐3๐ ] provide the limits for local physically linear behaviour. Taking into account that the deformation of G* is induced by ๐ โ = ๐๐, it is clear that the system unknowns (nodal coordinates) are associated with the nodes of G only and correspondingly the system reaction (edge forces) need to be associated with the edges of G only. To achieve this, we associate the stored energy of the system only with the edges of G, making the total energy of G-edge ๐ equal to the sum of its internal energy, ๐๐ , and half โ of the energies in the G*-edges incident with the G*-node centred at G-edge ๐. This can be written as 12 ๐ ๐๐โ , where the sum is over the G*-edges incident with the G*-node. Thus, the total stored energy associated with each edge of G is given by 1โ โ ๐๐ = ๐๐ + ๐ , ๐ = 1, 2, ..., ๐. (4) 2 ๐ ๐ The gradient of the total energy with respect to the change of edge length, provides the magnitude of the force in the G-edge, i.e. for |๐ฆ๐ | โ |๐๐ | we have ๐๐ , ๐ = 1, 2, ..., ๐. (5) |๐น๐ | = |๐ฆ๐ | โ |๐๐ | Finally, we restrict our present consideration to materials with non-polar behaviour by constraining the forces in G-edges to act ๐ฆ along the edges in the deformed state, i.e. ๐น๐ ||๐ฆ๐ , ๐ = 1, 2, ..., ๐. Hence, if ๐๐ = |๐ฆ๐ | , ๐ = 1, 2, ..., ๐ are the unit vectors along edges ๐ in G, the edge forces are calculated by |๐น | ๐น๐ = |๐น๐ |๐๐ = ๐ ๐ฆ๐ , ๐ = 1, 2, ..., ๐. (6) |๐ฆ๐ | This can be summarised for the whole G by { } |๐น๐ | |๐น1 | |๐น2 | , ,โฆ. โ โ๐ร๐ . (7) ๐น = ๐(|๐ฆ|)๐ฆ, where ๐(|๐ฆ|) = ๐๐๐๐ |๐ฆ1 | |๐ฆ2 | |๐ฆ๐ | [ ] Let ๐ต๐ = ๐ต๐1 ๐ต๐2 ๐ต๐3 โ โ3 , ๐ = 1, 2, ..., ๐, be the external forces at the nodes of G, either provided as natural boundary conditions, or arising as reactions to essential boundary conditions. Since the balance of angular momentum is automatically fulfilled at all nodes by ๐น๐ ||๐ฆ๐ , the equilibrium of the system with the boundary conditions is ensured by the balance of linear momentum at all nodes. This is given by ๐ด๐ ๐น = ๐ต, (8) where ๐ด๐ โ โ๐ร๐ is the transpose of the incidence matrix ๐ด, a boundary operator on edges of G. By substituting (7) into (8), we get ๐ด๐ ๐(|๐ฆ|)๐ฆ = ๐ต, (9) which incorporates the contribution of G* to the equilibrium via (4) and (5)). By substituting (1) into (9), we arrive at the general description of the system elasticity in terms of positions and forces of nodes in G [๐ด๐ ๐(|๐ด๐|)๐ด]๐ = ๐ต.
(10)
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Ioannis Dassios ET AL
The application of boundary conditions to the system (10) requires a separation of the nodal coordinate directions into two groups: directions with prescribed natural condition - a force component, which maybe zero (free boundary), and directions with prescribed essential condition - a new coordinate value which also maybe zero (fixed boundary). This separation can be represented by the following expressions for nodal positions and forces, and a correspondingly re-arranged incidence matrix [ ๐] [ ๐] [ ๐] ๐ ๐ต ๐ด ๐ร3 ๐ร3 ๐= โ โ , ๐ต = โ โ and ๐ด = โ โ3๐ร3๐ , ๐๐ ๐ต๐ ๐ด๐ where ๐ ๐ โ โ๐ and ๐ต ๐ โ โ๐ are vectors of the unknown coordinates and the known corresponding forces, ๐ ๐ โ โ๐ and ๐ต ๐ โ โ๐ are vectors of the known coordinates and the unknown corresponding forces, and ๐ + ๐ = 3๐ holds.
3
MAIN RESULTS
System (10) is a non-linear system. This is because of the diagonal matrix ๐(|๐ด๐ |) defined in (8). We provide the following Theorem: Theorem 1. Consider the non-linear system (10). Then (a) An effective linearization of the system is given by ฬ = ๐ต. ๐ด๐
(11)
ฬ with ๐พฬ = ๐๐๐๐[๐พฬ ๐ ]1โค๐โค๐ and for 0 < ๐ 0, ๐2๐ โฅ 0 โช [ โ๐โ โ 2 ] โช โช ๐พ๐ |๐๐ | ๐=1 [๐พ๐ ๐(๐ง+2)๐ ] โช ๐พ + 1 โ1 โช โ , ๐ > 0, ๐ โค 0 0๐ 2๐ ๐ง=0 ๐(1โ๐ง)๐ (๐0๐ +๐1๐ ) ๐ 2 ๐๐ง๐ +|๐๐ | โช โช [ โ๐โ โ 2 ] ๐พฬ ๐ = โจ โฌ. [๐พ ๐ ] โ โช ๐พ๐ + 1 1๐ง=0 ๐ง ๐=1 ๐ (๐ง+2)๐ โ ๐พ๐ |๐๐ | , ๐0๐ < 0, ๐2๐ โฅ 0 โช 2 2๐(1โ๐ง)๐ (๐(1โ๐ง)๐ +|๐๐ |) ๐๐ง๐ +|๐๐ | โช โช [ โ ] ๐โ โ 2 โช โช [๐พ ๐ ] ๐พ๐ |๐๐ | 1 โ1 ๐ (๐ง+2)๐ ๐=1 โช ๐พ๐ + 2 ๐ง=0 ๐ง ๐ (๐ +๐ ) โ ๐ +|๐ | , ๐0๐ < 0, ๐2๐ โค 0 โช (1โ๐ง)๐ 0๐ 1๐ ๐ง๐ ๐ โฉ โญ |๐น๐ | . |๐๐ |
For |๐ฆ๐ | = |๐๐ |, i.e. ๐0๐ = 0 or ๐1๐ = 0, we have ๐พฬ ๐ = (b) Let
[ ๐ดฬ ๐ดฬ = ฬ 11 ๐ด21
] ๐ดฬ 12 . ๐ดฬ 22
Where ๐ดฬ 11 โ โ๐ร๐ , ๐ดฬ 12 โ โ๐ร๐ , ๐ดฬ 21 โ โ๐ร๐ , ๐ดฬ 22 โ โ๐ร๐ . Then system (11) can be divided into the following subsystems ๐ดฬ 11 ๐ ๐ = ๐ต ๐ โ ๐ดฬ 12 ๐ ๐
(12)
๐ต ๐ = ๐ดฬ 13 ๐ ๐ + ๐ดฬ 22 ๐ ๐ .
(13)
and From the above systems only (12) has to be solved. Then ๐ ๐ can be replaced in (13) and ๐ต ๐ is easily computed. Proof. For the proof of (a) we consider (8), (10) and will seek optimal bounds for |๐น๐ | =
๐๐ , |๐ฆ๐ | โ |๐๐ |
whereby replacing (4) into the above expression we get |๐น๐ | =
๐๐ +
โ๐โ ๐=1
๐๐โ
|๐ฆ๐ | โ |๐๐ |
By using (2) and (3) we have |๐น๐ | =
1 2
๐พ๐ (|๐ฆ๐ | โ |๐๐ |)2 +
1 2
โ๐โ ๐=1
โ๐ = 1, 2, ..., ๐. From (5)
๐ = 1, 2, ..., ๐,
,
๐ = 1, 2, ..., ๐.
[๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 ]
|๐ฆ๐ | โ |๐๐ |
|๐น๐ | , |๐ฆ๐ |
,
๐ = 1, 2, ..., ๐,
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Ioannis Dassios ET AL
or, equivalently,
โ๐โ
1 |๐น๐ | = ๐พ๐ (|๐ฆ๐ | โ |๐๐ |) + 2 or, equivalently, |๐ | |๐น๐ | 1 = ๐พ๐ (1 โ ๐ ) + |๐ฆ๐ | |๐ฆ๐ | 2 From (2) and (3) we have that ๐0๐ โค |๐ฆ๐ | โ |๐๐ | โค ๐1๐ ,
๐=1
,
๐ = 1, 2, ..., ๐,
,
๐ = 1, 2, ..., ๐.
|๐ฆ๐ | โ |๐๐ | โ๐โ ๐=1
[๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 ]
|๐ฆ๐ |(|๐ฆ๐ | โ |๐๐ |)
๐ = 1, 2, ..., ๐
Hence we have
[๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 ]
๐2๐ โค |๐ฆโ๐ | โ |๐โ๐ | โค ๐3๐ ,
and
๐22๐ โค (|๐ฆโ๐ | โ |๐โ๐ |)2 โค ๐23๐ ,
if
๐2๐ โค 0,
๐พ๐โ ๐22๐ โค ๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 โค ๐พ๐โ ๐23๐ ,
if
๐2๐ โฅ 0,
0 โค ๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 โค ๐พ๐โ ๐๐๐{๐22๐ , ๐23๐ },
๐=1
0โค
[๐พ๐โ ๐22๐ ] โค
โ๐โ ๐=1
[๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 ] โค
โ๐โ
[๐พ๐โ (|๐ฆโ๐ | โ |๐โ๐ |)2 ] โค ๐=1
๐ = 1, 2, ..., ๐โ .
๐2๐ โฅ 0,
if
0 โค (|๐ฆโ๐ | โ |๐โ๐ |)2 โค ๐๐๐{๐22๐ , ๐23๐ }, or, equivalently, by multiplying the above expressions by ๐พ๐โ
or, equivalently, by summing โ๐โ
(14)
โ๐โ ๐=1
if
๐2๐ โค 0.
[๐พ๐โ ๐23๐ ],
if
๐2๐ โฅ 0, (15)
โ๐โ
[๐พ๐โ ๐๐๐{๐22๐ , ๐23๐ }], ๐=1
๐2๐ โค 0.
if
Furthermore, we have ๐0๐ + |๐๐ | โค |๐ฆ๐ | โค ๐1๐ + |๐๐ |, or, equivalently,
๐0๐ + |๐๐ | |๐๐ |
or, equivalently,
|๐๐ | ๐1๐ + |๐๐ |
or, equivalently, 1โ or, equivalently, ๐พ๐ (1 โ
|๐๐ | ๐0๐ + |๐๐ | |๐๐ |
๐0๐ + |๐๐ |
โค
โค
|๐ฆ๐ | |๐๐ | |๐๐ | |๐ฆ๐ |
โค1โ
|๐๐ | |๐๐ |
โค
|๐๐ | |๐ฆ๐ |
) โค ๐พ๐ (1 โ
๐1๐ + |๐๐ |
โค
๐0๐ + |๐๐ | โค1โ
|๐๐ | |๐ฆ๐ |
,
,
|๐๐ | ๐1๐ + |๐๐ |
) โค ๐พ๐ (1 โ
.
|๐๐ | ๐1๐ + |๐๐ |
).
(16)
Also from the above expressions 1 1 1 โค โค . ๐1๐ + |๐๐ | |๐ฆ๐ | ๐0๐ + |๐๐ |
(17)
Finally 1 ๐1๐
โค
1 |๐ฆ๐ |โ|๐๐ |
โค
1 , ๐0๐
if
๐0๐ > 0,
if
๐0๐ < 0.
(18) 1 |๐ฆ๐ |โ|๐๐ |
โ
2 ๐0๐ +๐1๐
ยฑ ๐,
Hence, for ๐0๐ > 0 and ๐2๐ โฅ 0, from (15), (16), (17) and (18), we have โ๐โ โ๐โ [๐พ๐โ ๐22๐ ] [๐พ๐โ ๐23๐ ] |๐๐ | |๐๐ | |๐น๐ | ๐=1 ๐=1 ๐พ๐ (1 โ )+ โค โค ๐พ๐ (1 โ )+ , ๐0๐ + |๐๐ | 2๐1๐ (๐1๐ + |๐๐ |) |๐ฆ๐ | ๐1๐ + |๐๐ | 2๐0๐ (๐0๐ + |๐๐ |) or, equivalently,
โ๐โ 1 โก [๐พ๐โ ๐2(๐ง+2)๐ ] ๐พ๐ |๐๐ | โค |๐น๐ | ๐=1 1 โโข โฅ ยฑ ๐, โ โ
๐พ๐ + |๐ฆ๐ | 2 ๐ง=0 โข 2๐(1โ๐ง)๐ (๐(1โ๐ง)๐ + |๐๐ |) ๐๐ง๐ + |๐๐ | โฅ โฃ โฆ
๐ = 1, 2, ..., ๐.
6
Ioannis Dassios ET AL
For ๐0๐ > 0 and ๐2๐ โค 0, from (15), (16), (17) and (18), we have โ๐โ โ๐โ [๐พ๐โ ๐22๐ ] |๐น๐ | [๐พ๐โ ๐23๐ ] |๐๐ | |๐๐ | ๐=1 ๐=1 ๐พ๐ (1 โ )+ โค โค ๐พ๐ (1 โ )+ , ๐0๐ + |๐๐ | ๐1๐ (๐0๐ + ๐1๐ ) |๐ฆ๐ | ๐1๐ + |๐๐ | ๐0๐ (๐0๐ + ๐1๐ ) or, equivalently,
โ โ โ 2 1 โก ๐ ๐พ๐ |๐๐ | โค |๐น๐ | 1 โ โข ๐=1 [๐พ๐ ๐(๐ง+2)๐ ] โฅ ยฑ ๐, ๐ = 1, 2, ..., ๐. โ
๐พ๐ + โ |๐ฆ๐ | 2 ๐ง=0 โข ๐(1โ๐ง)๐ (๐0๐ + ๐1๐ ) ๐๐ง๐ + |๐๐ | โฅ โฃ โฆ For ๐0๐ < 0 and ๐2๐ โฅ 0, we have โ๐โ [๐พ๐โ ๐23๐ ] |๐๐ | |๐๐ | |๐น๐ | ๐=1 )โค โค ๐พ๐ (1 โ )+ , ๐พ๐ (1 โ ๐0๐ + |๐๐ | |๐ฆ๐ | ๐1๐ + |๐๐ | 2๐0๐ (๐0๐ + |๐๐ |) or, equivalently,
โ๐โ 1 โก [๐พ๐โ ๐2(๐ง+2)๐ ] ๐พ๐ |๐๐ | โค |๐น๐ | ๐=1 1 โโข โฅ ยฑ ๐, ๐ = 1, 2, ..., ๐. ๐ง โ
๐พ๐ + โ |๐ฆ๐ | 2 ๐ง=0 โข 2๐(1โ๐ง)๐ (๐(1โ๐ง)๐ + |๐๐ |) ๐๐ง๐ + |๐๐ | โฅ โฃ โฆ For ๐0๐ < 0 and ๐2๐ โค 0, we have โ๐โ [๐พ๐โ ๐23๐ ] |๐๐ | |๐๐ | |๐น๐ | ๐=1 ๐พ๐ (1 โ )โค โค ๐พ๐ (1 โ )+ , ๐0๐ + |๐๐ | |๐ฆ๐ | ๐1๐ + |๐๐ | ๐0๐ (๐0๐ + ๐1๐ ) or, equivalently,
โ๐โ 1 โก [๐พ๐โ ๐2(๐ง+2)๐ ] ๐พ๐ |๐๐ | โค |๐น๐ | ๐=1 1 โโข โฅ ยฑ ๐, ๐ง โ
๐พ๐ + โ |๐ฆ๐ | 2 ๐ง=0 โข ๐(1โ๐ง)๐ (๐0๐ + ๐1๐ ) ๐๐ง๐ + |๐๐ | โฅ โฃ โฆ
๐ = 1, 2, ..., ๐.
It is easy to observe that for |๐ฆ๐ | = |๐๐ |, i.e. ๐0๐ = 0 or ๐1๐ = 0, and from (8) we have ๐พฬ ๐ = and a similar one can be found in (2). The proof is completed.
|๐น๐ | . |๐๐ |
The proof of (b) is trivial
For an alternative numerical method with iterations to solve the non-linear system (10), we provide the following Theorem. Theorem 2. An iterative method for solving the non-linear system (10) is ๐บ๐ ๐๐+1 = ๐ต๐+1 . Where
[ ๐๐ =
] ๐๐๐ , ๐๐
[ ๐ต๐ =
(19) ] ๐ต๐ . ๐ต๐๐
๐บ๐ = ๐ด๐ ๐(|๐ฆ๐,๐ |)๐ด and
[
] โ๐โ [๐พ๐โ (๐2๐ + ๐3๐ )2 ] ๐=1 1 ๐(|๐ฆ๐,0 |) = ๐๐๐๐ ๐พ๐ (1 โ )+ 2|๐๐ | + ๐0๐ + ๐1๐ 2 (2|๐๐ | + ๐0๐ + ๐1๐ )(๐0๐ + ๐1๐ ) 2|๐๐ |
. 1โค๐โค๐
Furthermore let
[
] ๐บ๐11 ๐บ๐12 ; ๐บ๐21 ๐บ๐22 Then, a solution of (10) through a first iteration of (19) is given by ๐บ๐ =
๐1๐ = (๐บ0(11) )โ [๐ต ๐ โ ๐บ0(12) ๐ ๐ ] and [ ]
๐ต1๐ = ๐บ0(21) (๐บ0(11) )โ [๐ต ๐ โ ๐บ0(12) ๐ ๐ ] + ๐บ0(22) ๐ ๐ . If ๐บ0(11) is invertible then (๐บ0(11) )โ is its inverse and when ๐บ0(11) is singular then (๐บ0(11) )โ is its pseudoinverse. Proof. We seek a method of solution for the non-linear system (10) [๐ด๐ ๐(|๐ด๐ |)๐ด]๐ = ๐ต.
(20)
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Ioannis Dassios ET AL
Based on the special structure of ๐ด๐ ๐(|๐ด๐ |)๐ด we may use the Picard method and construct the matrix difference equation [๐ด๐ ๐(|๐ด๐๐ |)๐ด]๐๐+1 = ๐ต๐+1 or, equivalently, by replacing ๐ฆ๐,๐ = ๐ด๐๐
[๐ด๐ ๐(|๐ฆ๐,๐ |)๐ด]๐๐+1 = ๐ต๐+1 , or, equivalently, for ๐บ๐ = ๐ด๐ ๐(|๐ฆ๐,๐ |)๐ด we arrive at (19). Note that [ ๐] [ ๐] ๐ต ๐๐ ๐๐ = , ๐ต = , ๐ ๐๐ ๐ต๐๐ because ๐ ๐ , ๐ต ๐ are considered known while ๐ ๐ , ๐ต ๐ are unknown. Since ๐0๐ โค |๐ฆ๐ | โ |๐๐ | โค ๐1๐ for ๐ = 1, 2, ..., ๐ and ๐ +๐ โ2|๐โ | ๐ +๐ โ2|๐ | ๐2๐ โค |๐ฆโ๐ | โ |๐โ๐ | โค ๐3๐ for ๐ = 1, 2, ..., ๐โ , we may use |๐ฆ๐,0 | = 0๐ 1๐2 ๐ and |๐ฆโ๐,0 | = 2๐ 3๐2 ๐ as an initial approximation. Then ๐บ0 = ๐ด๐ ๐(|๐ฆ๐,0 |)๐ด, where { } |๐น๐,๐ | . ๐(|๐ฆ๐,0 |) = ๐๐๐๐ |๐ฆ๐,๐ | ๐=1,2,...,๐ or, equivalently { } โ๐โ โ โ โ 2 |๐๐ | 1 ๐=1 [๐พ๐ (|๐ฆ๐ | โ |๐๐ |) ] ๐(|๐ฆ๐,0 |) = ๐๐๐๐ ๐พ๐ (1 โ )+ . |๐ฆ๐ | 2 |๐ฆ๐ |(|๐ฆ๐ | โ |๐๐ |) ๐=1,2,...,๐ or, equivalently ๐(|๐ฆ๐,0 |) = { } โ๐โ โ โ 2 2|๐๐ | 1 ๐=1 [๐พ๐ (๐2๐ + ๐3๐ โ 2|๐๐ |) ] ๐๐๐๐ ๐พ๐ (1 โ )+ . ๐0๐ + ๐1๐ โ 2|๐๐ | 2 (๐0๐ + ๐1๐ โ 2|๐๐ |)(๐0๐ + ๐1๐ ) ๐=1,2,...,๐ Hence for [ 11 12 ] ๐บ0 ๐บ0 , ๐บ0 = ๐บ021 ๐บ022 a first iteration is given by ๐บ0 ๐1 = ๐ต1 , or, equivalently, [ 11 12 ] [ ๐ ] [ ๐ ] ๐ต ๐บ0 ๐บ0 ๐1 = . ๐๐ ๐บ021 ๐บ022 ๐ต1๐ or, equivalently, ๐บ0(11) ๐1๐ + ๐บ0(12) ๐ ๐ = ๐ต ๐ ] [ ๐บ0(21) (๐บ0(11) )โ [๐ต ๐ โ ๐บ0(12) ๐ ๐ ] + ๐บ0(22) ๐ ๐ = ๐ต1๐ and from here we arrive easily at (20). The proof is completed. For the non-linear system (10), let ๐ด๐ ๐(|๐ด๐ |)๐ด = ๐บ(๐) = Then for
] ๐๐ ๐= , ๐๐ [
we have ๐ ๐ = (๐บ(11) (๐))โ [๐ต ๐ โ ๐บ(12) (๐)๐ ๐ ],
[
] ๐บ11 (๐) ๐บ12 (๐) . ๐บ21 (๐) ๐บ22 (๐) [
] ๐ต๐ ๐ต= , ๐ต๐
[ ] ๐ต ๐ = ๐บ(21) (๐) (๐บ(11) (๐))โ [๐ต ๐ โ ๐บ(12) (๐)๐ ๐ ] + ๐บ(22) (๐)๐ ๐ ,
and consequently the iterative method (19) converges for { } { } [ ] โ โ โ โ โโ (๐บ(11) (๐))โ [๐ต ๐ โ ๐บ(12) (๐)๐ ๐ ] โ < 1, โโ ๐บ(21) (๐) (๐บ(11) (๐))โ [๐ต ๐ โ ๐บ(12) (๐)๐ ๐ ] + ๐บ(22) (๐)๐ ๐ โ < 1. โ โ โ โ With โโ
โ we denote an induced norm. The numerical method (19) can be more efficient than the linearization in Theorem 1 if the above expressions hold, and only after many iterations. However, besides these restrictions, the method in Theorem 1 is more useful because it has low memory requirements and avoids matrix factorizations.
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Ioannis Dassios ET AL
FIGURE 1 Numerical Example: On the left the graph with the initial positions and edges and on the right the graph with the final positions and edges
๐
๐ถ๐
๐ถ๐โ
๐๐
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
(0, 0,1) (1,0,0) (0,-1,0) (-1,0,0) (0,1,0) (0,0,-1)
(-0.5,0,0.5) (0,0.5,0.5) (0.5,0,0.5) (0,-0.5,0.5) (-0.5,-0.5,0) (-0.5,0.5,0) (0.5,0.5,0) (0.5,-0.5,0) (-0.5,0,-0.5) (0,0.5,-0.5) (0.5,0,-0.5) (0,-0.5,-0.5)
(1,0,1) (0,-1,1) (-1,0,1) (0,1,1) (1,-1,0) (-1,-1,0) (-1,1,0) (1,1,0) (1,0,-1) (0,-1,-1) (-1,0,-1) (0,1,-1)
|๐ | โ๐ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2
๐โ๐ (-0.5,-0.5,0) (-0.5,0.5,0) (0.5,0.5,0) (0.5,-0.5,0) (0,0.5,0.5) (0,-0.5,0.5) (0.5,0,0.5) (-0.5,0,0.5) (0,-0.5,0.5) (0,0.5,0.5) (-0.5,0,0.5) (0.5,0,0.5) (0,-1,0) (-1,0,0) (0,1,0) (1,0,0) (-0.5,0,0.5) (0,-0.5,0.5) (0,0.5,0.5) (-0.5,0,0.5) (0.5,0,0.5) (0,0.5,0.5) (0,-0.5,0.5) (0.5,0,0.5) (-0.5,-0.5,0) (-0.5,0.5,0) (0.5,0.5,0) (0.5,-0.5,0) (-1,0,0) (0,1,0) (-1,0,0) (0,1,0) (0,0,1) (0,0,1) (0,0,1) (0,0,1)
|๐โ๐ | โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 1 1 1 1 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 1 1 1 1 1 1 1 1
TABLE 1 Initial nodes, edges, and edge lengths.
Numerical example Figure 1 illustrates a three-dimensional representation of the system modelled for demonstration purposes. In this example, we assume a lattice of a set of ๐ = 6 nodes which are connected through ๐ = 12 edges. A second set of ๐โ = ๐ = 12 nodes are placed at the centres of the 12 primary edges and these secondary nodes are connected by ๐โ = 36 secondary edges. Table 1 provides coordinate values for nodes i.e., ๐ถ๐ , ๐ถ๐โ , and vector values for edges, i.e., ๐๐ , ๐โ๐ . We then assume that a force ๐น is applied to the primary edges. The forces, ๐น๐ , the material coefficients, i.e., ๐พ๐ , ๐พ๐โ , and the bounds of (2) and (3), i.e., ๐0๐ , ๐1๐ , ๐3๐ , ๐4๐ , are given in Table 2. We solve the non-linear system (10) numerically by implementing the results of Theorem 1 in MATLAB, and this yields: the new coordinate values for the primary and secondary nodes, and vector values for the primary and secondary edges, i.e., ๐๐ , ๐๐โ , and ๐ฆ๐ , ๐ฆโ๐ , respectively (all of these values are provided in Table 3). The script used to generate this numerical example is available for use under an open source licence (4). We note that, although this script (more specifically, the method used to solve (12)) performs well for the numerical example in this paper, it is not
9
Ioannis Dassios ET AL ๐
๐0,๐
๐1,๐
๐2,๐
๐3,๐
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
0.2121 0.2121 0.2121 0.2121 0.2121 0.2121 0.2121 0.2121 0.2121 0.2121 0.2121 0.2121
7.0711 7.0711 7.0711 7.0711 7.0711 7.0711 7.0711 7.0711 7.0711 7.0711 7.0711 7.0711
0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1500 0.1500 0.1500 0.1500 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.1061 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15
3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 5 5 5 5 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 3.5355 5 5 5 5 5 5 5 5
๐พ๐ ร 104 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2 โ 2
๐พ๐โ ร 104 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 1 1 1 1 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 โ 1โ 2 1 1 1 1 1 1 1 1
๐น๐ (0,0,0.1586) (0.1760,0,0) (0,-0.1954,0) (-0.2169,0,0) (0,0.2407,0)
TABLE 2 Given forces on nodes and material properties.
๐ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
๐๐ (-0.3652,0.4054,1.835) (5.0234,0.3122,0.4175) (-0.2813,-1.5274,0.4175) (-6.707,0.3122,0.4175) (-0.2813,2.3709,0.4175) (0,0,-1)
๐๐โ (-3.5361,0.3588,1.1262) (-0.3232,1.3881,1.1262) (2.3291,0.3588,1.1262) (-0.3232,-0.5610,1.1262) (-3.4941,-0.6076,0.4175) (-3.4941,1.3416,0.4175) (2.3711,1.3416,0.4175) (2.3711,-0.6076,0.4175) (-3.3535,0.1561,-0.2913) (-0.1406,1.1855,-0.2913) (2.5117,0.1561,-0.2913) (-0.1406,-0.7637,-0.2913)
๐ฆ๐ (6.3418,0.0931,1.4175) (-0.0839,-1.9656,1.4175) (-5.3886,0.0931,1.4175) (-0.0839,1.9328,1.4175) (6.4257,-1.8396,0) (-6.4257,-2.0587,0) (-5.3047,2.0587,-0.0000) (5.3047,1.8396,0) (6.7070,-0.3122,-1.4175) (0.2813,-2.3709,-1.4175) (-5.0234,-0.3122,-1.4175) (0.2813,1.5274,-1.4175)
|๐ฆ๐ | 6.4990 2.4248 5.5727 2.3983 6.6839 6.7475 5.6901 5.6146 6.8623 2.7766 5.2289 2.1027
๐ฆโ๐ (-3.2129,-1.0293,0) (-2.6523,1.0293,0) (2.6523,0.9198,0) (3.2129,-0.9198,0) (-0.042,0.9664,0.7087) (-0.0420,-0.9828,0.7087) (3.1709,0.0466,0.7087) (-2.6943,0.0466,0.7087) (-0.042,-0.9828,0.7087) (-0.042,0.9664,0.7087) (-2.6943,0.0466,0.7087) (3.1709,0.0466,0.7087) (0,-1.9492,0) (-5.8652,0,0) (0,1.9492,0) (5.8652,0,0) (-3.3535,0.1561,0.7087) (-0.1406,-0.7637,0.7087) (-0.1406,1.1855,0.7087) (-3.3535,0.1561,0.7087) (2.5117,0.1561,0.7087) (-0.1406,1.1855,0.7087) (-0.1406,-0.7637,0.7087) (2.5117,0.1561,0.7087) (-3.2129,-1.0293,0) (-2.6523,1.0293,0) (2.6523,0.9198,0) (3.2129,-0.9198,0) (-5.8652,0,0) (0,1.9492,0) (-5.8652,0,0) (0,1.9492,0) (-0.1826,0.2027,1.4175) (-0.1826,0.2027,1.4175) (-0.1826,0.2027,1.4175) (-0.1826,0.2027,1.4175)
|๐ฆโ๐ | 3.3737 2.8451 2.8073 3.3419 1.1992 1.2124 3.2495 2.7863 1.2124 1.1992 2.7863 3.2495 1.9492 5.8652 1.9492 5.8652 3.4311 1.0513 1.3883 3.4311 2.6144 1.3883 1.0513 2.6144 3.3737 2.8451 2.8073 3.3419 5.8652 1.9492 5.8652 1.9492 1.4435 1.4435 1.4435 1.4435
TABLE 3 Final nodes, edges, and edge lengths.
optimal for larger networks: as the size of system increases, ๐ดฬ becomes more sparse, therefore ๐ดฬ 11 , ๐ดฬ 12 , ๐ต ๐ , and ๐ ๐ should be stored as sparse matrices to speed up MATLABโs โ\โ function.
Conclusions In this article we proposed a mathematical model of elasticity with exact kinematics, where the intrinsically discrete structure of materials is represented by two graphs and analysed using calculus on discrete manifolds. By making the kinematics of one of the graphs induced by the kinematics of the other, we derived the governing equations of elasticity where the deformations of both graphs contribute energy to the system, but the reaction of the system is only via forces in the edges of latter graph. This provides a single non-linear system of governing equations, for which we offered linearisation, computational implementation, and a
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Ioannis Dassios ET AL
simple demonstration of the model at work. The numerical example shown is for a particular simple arrangement of neighbours of a single vertex, but the model is applicable to any spatial arrangement of vertices and their connectivity, e.g. an arrangement arising from a Voronoi tessellation of a domain with randomly distributed nodes as described in Section 2. The model requires extensive testing with larger lattices to compare with experimentally measured elastic behaviour of various materials, which is a subject of ongoing work. We anticipate that the model can be used for atomic scale simulations as an alternative to the currently used in molecular dynamics interactions based on empirical pair and cohesive potentials. Our expectation is based on the following observation. The original atomic scale models used only pair potentials, e.g. Lenard-Jones or Morse, suitable for neutral atoms and molecules, and were later complemented by cohesive terms, e.g. in embedded atom and Finnis-Sinclair models, for describing the behaviour of crystalline lattices [new citation]. The latter were still further modified to include angular dependences of the cohesive potentials in order to capture the complex interaction of a single metal atom with its neighbourhood. The calibration of such potential functions is increasingly difficult and involves a great degree of empiricism. Our proposition is that the method described here will allow for capturing the complex linear and angular interactions by the combination of energies stored in the primary and complementary lattices. Testing this proposition is one of the avenues for future work on the model developed here. In addition, we plan to compare the two-graph approach with an approach based on a 3-complex description of solids, which is still under development. It will be important to demonstrate to what extent the two-graph approach is an approximation to the โexactโ 3-complex solution, in order to justify the applicability of the former as computationally more efficient.
Acknowledgments I. Dassios is supported by Science Foundation Ireland (Strategic Partnership Programme Grant Number SFI/15/SPP/E3125). G. J. OโKeeffe acknowledges the support of the Irish Research Council (GOIPG/2014/887). AP Jivkov acknowledges gratefully the financial support of EPSRC via grant EP/N026136/1 โGeometric Mechanics of Solidsโ.
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