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A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus

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A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus Pieter D. Boom∗, Odysseas Kosmas1, Lee Margetts2, Andrey Jivkov3 Department of MACE, University of Manchester, George Begg Building, Manchester, M1 3BB, UK Abstract A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vector- valued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitu- tive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes. Keywords: Discrete Exterior Calculus, Elastic Materials arXiv:2104.06000v1 [math-ph] 13 Apr 2021 ∗Corresponding author (+44-7547-975379) Email address: [email protected] (Pieter D. Boom) [email protected] [email protected] [email protected] Preprint submitted to International Journal of Solids and Structures April 14, 2021 1. Introduction Solids have discrete structures at a number of length scales, from atomic to polycrys- talline, and their macroscopic behaviour emerges from finite rearrangements of these struc- tures [12]. Analyses of discrete structures are performed by length-scale-specific mathemat- ical formulations and corresponding numerical methods [16]. Examples include molecular dynamics at the atomic scale [20], particle-based methods, such as peridynamics [15] and discrete elements [23], and lattice-based methods [19, 13] at longer length scales. In many cases, the particle- and lattice-based methods are calibrated in such a way that the behaviour of the particle or lattice assembly matches a prescribed macroscopic (continuum) behaviour. Such calibrations are challenging and sometimes not possible for irregular arrangements of the assembly elements. However, the benefit of the discrete formulations is that existing, as well as emerging and evolving, material and geometric discontinuities are naturally captured and represented, which is beyond the capabilities of classical continuum formulations. In this respect, it will be beneficial to develop a generic mathematical formulation of deformation of finite discrete systems, from which the methods mentioned above can be either derived or calibrated. A natural starting point is the Discrete Exterior Calculus (DEC), which describes inte- gration and differentiation through the topology and geometry of finite discrete cell com- plexes and their dual [9, 8]. DEC can be seen as a generalisation of the smooth exterior calculus, that satisfies the fundamental theorem of calculus (generalised Stoke's theorem) and Poincar´e'sLemma. In DEC properties are intrinsically linked to the geometry of the cell components - vertices, edges, faces, volumes. For example, displacement are defined at vertices [m=m0], strain occurs along edges [m=m1], stress acts through faces [N=m2], and forces are in volumes [N=m3]. A similar example for electromagnetism is presented in Gillette [6]. Physical interactions are defined by the topology of the complex, how the com- ponents are connected. For example, volumes interact through shared faces and conversely volumetric properties change as a result of the sum of fluxes through their bounding faces. Mathematically, the connections define exterior derivatives and enable common vector cal- 2 culus operations to be mimicked, such as gradient, curl, divergence and Laplacian. More formal connections to vector fields can be made through the use of maps called musical isomorphisms [1]. These properties make DEC an ideal platform to develop descriptions of physical pro- cesses which are fundamentally discrete. To date, DEC has been successfully applied to describe physical phenomena such as incompressible fluid flow [11], flow in porous media [10], and electromagnetism [4]. There have also been efforts to develop DEC formulations of elasticity. Yavari [22] outlined one such formulation, but without specifying how to prescribe the material constitutive relations. Angoshtari and Yavari [2] later presented a geometric description of incompressible linearised elasticity using DEC to identify a divergence-free solution space and recover the pressure field. However, strain is computed from the gradient of an interpolated displacement field, rather than using a discrete exterior derivative. DEC has also being used to inform finite-element approaches, for example Gillette and Bajaj [7]; however, this still relies on discretizing continuum descriptions of elasticity. To date no complete DEC formulation for elastic deformations has been proposed, a mandatory first step in extending DEC into modelling dissipative processes in discrete structures. The aim of this work is to make this first step by presenting a geometric formulation of linear elasticity based on DEC. Brief reviews of elasticity and discrete exterior calculus are given in Sections 2 and 3, respectively. The two are combined in Section 4 to build a description of linear elasticity using DEC. The kinematics are defined on the primal complex and forces on the dual complex, similar to Yavari [22], with the constitutive law applied through musical isomorphisms. Numerical results are presented for several classical problems in Section 5. The development is discussed and conclusions drawn in Section 6. 2. Elasticity The mechanics of an elastic body are defined by Newton's second law, providing equations for the balance of linear and angular momentum. The balance of linear momentum for a 3 continuum body Ω can be written as r · τ~(~x) + f~(~x) =ρ ~a~(~x); forx ~ 2 Ω; (1) whereτ ~ is a stress tensor45, f~ are volumetric body forces, andρ ~a~ are the forces due to acceleration. In this article we will be concerned with static problems, where the inertial forces are zero,ρ ~a~ = 0. The deformation of the the body is defined as a map between a reference configuration X~ and the deformed configurationx ~ = X~ +u ~, whereu ~ is the displacement field. Stress can be derived from the deformation through a constitutive law, such as Hooke's law or the Mooney-Rivlin model for neo-Hookean solids. The simplest relationship is for a linear elastic isotropic material. In this case the strain tensor, under the infinitesimal strain assumption, reduces to 1 ~ = ru~ + (ru~)T ; (2) 2 and stress is defined through the linear constitutive relationship τ~ = λ tr(~)I~+ 2µ,~ (3) where I~ is the identity tensor, λ is Lam´e'sfirst constant, and µ is the shear modulus. We assume in this article that these constants do not vary as a function of either spatial location or time. The symmetry of the stress tensor satisfies the continuum requirement for balance of angular momentum. The resulting governing equation for the balance of linear momentum becomes well-posed with the addition of essential (Dirichlet) and natural (Neumann) conditions on the boundary @Ω of the body. Traditionally, the natural conditions take the form τ~ · n~i = t~i 2 @Ωi (4) wheren ~i is the outward pointing unit normal of the boundary surface patch @Ωi, and t~i is prescribed surface traction. 4 ~ In this article vector fields are marked with tildes to distinguish them from discrete quantities. 5Solid mechanics and discrete exterior calculus commonly use symbol σ to represent stress and simplicies, respectively. Therefore, in this article the stress tensor field will be denotedτ ~ and σ used for simplicies. 4 3. Discrete exterior calculus This section is intended to give a brief introduction to discrete exterior calculus (DEC). The theory and notation presented are based on Refs. [5, 9, 8]. Readers familiar with DEC can proceed to the next section, where DEC is applied to the equations of linear elasticity. Calculus describes change through integration and differentiation. The familiar vector calculus is concerned with the behaviour of fields in three-dimensional Euclidean space. Smooth exterior calculus generalises vector calculus to differential manifolds in arbitrary dimensions [1]. Likewise, discrete exterior calculus is an extension to oriented piecewise flat manifolds [8]. The specific formulation used presently is based on simplicial complexes in arbitrary dimensions [9]. DEC satisfies critical identities such as Stoke's theorem and Poincar´e'sLemma, and provides analogues to the gradient, divergence, and curl. 3.1. Primal simplicial complex A solid body Ω is represented as an oriented three-dimensional simplicial complex K. An example for a circular cylinder is shown in Figure 1. A simplicial complex is composed of k k-dimensional simplices σ = fv0; : : : ; vkg, where the vi identify the vertices of each simplex, and the order of the elements vi defines an orientation. The boundary of each k-simplex is formed by (k − 1)-simplices whose orientations
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