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# Geometric elasticity

## On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

Sun, 2014-12-07 17:14 - arash_yavariThe problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially-symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity.

## Differential Complexes in Continuum Mechanics

Wed, 2014-09-24 11:50 - arash_yavariWe study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motions of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of the displacement gradient and the existence of stress functions on non-contractible bodies.

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## Nonlinear elastic inclusions in isotropic solids

Fri, 2013-09-13 11:07 - arash_yavariWe introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space.

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## A Geometric Structure-Preserving Discretization Scheme for Incompressible Linearized Elasticity

Wed, 2013-03-06 01:38 - arash_yavariIn 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.

## A Geometric Theory of Thermal Stresses

Mon, 2009-11-30 13:00 - arash_yavariIn this paper we formulate a geometric theory of thermal stresses.

Given a temperature distribution, we associate a Riemannian

material manifold to the body, with a metric that explicitly

depends on the temperature distribution. A change of temperature

corresponds to a change of the material metric. In this sense, a

temperature change is a concrete example of the so-called

referential evolutions. We also make a concrete connection between

our geometric point of view and the multiplicative decomposition

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