User login

You are here

Geometric elasticity

Arash_Yavari's picture

Differential Complexes in Continuum Mechanics

We 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.

Arash_Yavari's picture

Nonlinear elastic inclusions in isotropic solids

We 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.

Arash_Yavari's picture

A Geometric Structure-Preserving Discretization Scheme for Incompressible Linearized Elasticity

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.

Arash_Yavari's picture

A Geometric Theory of Thermal Stresses

In 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

Subscribe to RSS - Geometric elasticity

More comments

Syndicate

Subscribe to Syndicate