arash_yavari's blog

In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton's principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers.

Compatibility equations of elasticity are almost 150 years old. Interestingly they do not seem to have been rigorously studied for non-simply-connected bodies to this date. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible even if the standard compatibility equations ("bulk" compatibility equations) are satisfied.

In this paper, a closed-form solution is presented for bending analysis of shape memory alloy (SMA) beams.

In the theory of dislocations, the Burgers vector is usually defined by referring to a crystal structure. Using the notion of affine development of curves on a differential manifold with a connection, we give a differential geometric definition of the Burgers vector directly in the continuum setting, without making use of an underlying crystal structure.

In this paper we obtain the residual stress field of a nonlinear elastic solid with a spherically-symmetric distribution of point defects. To our best knowledge, this is the first nonlinear solution for point defects since the linear solution of Love in the 1920s.

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold - where the body is stress free - is a Weitzenbock manifold, i.e. a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan's moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions.

In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem we consider the particular case of determining the residual stress field of a cylindrically-symmetric distribution of parallel wedge disclinations. We first use the tools of differential geometry to construct a Riemaniann material manifold in which the body is stress-free. This manifold is metric compatible, has zero torsion, but has non-vanishing curvature.

In this paper we make a connection between covariant elasticity based on covariance of energy balance and Lagrangian field theory of elasticity with two background metrics. We use Kuchar's idea of reparametrization of field theories and make elasticity generally covariant by introducing a "covariance field", which is a time-independent spatial diffeomorphism. We define a modified action for parameterized elasticity and show that the Doyle-Ericksen formula and spatial homogeneity of the Lagrangian density are among its Euler-Lagrange (EL) equations.

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