arash_yavari's blog

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground.

Ericksen's problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold.

In this paper we formulate the problem of elastodynamic transformation cloaking for Kirchhoff-Love plates and elastic plates with both in-plane and out-of-plane displacements. A cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic plate (virtual problem) to that of an anisotropic and inhomogeneous elastic plate with a hole surrounded by a cloak that is to be designed (physical problem). For Kirchhoff-Love plates, the governing equation of the virtual plate is transformed to that of the physical plate up to an unknown scalar field.

In this paper, we formulate a theory for the coupling of accretion mechanics and thermoelasticity. We present an analytical formulation of the thermoelastic accretion of an infinite cylinder and of a two-dimensional block.

In nonlinear elasticity, universal deformations are the deformations that exist for arbitrary strain-energy density functions and suitable tractions at the boundaries. Here, we discuss the equivalent problem for linear elasticity. We characterize the universal displacements of  linear elasticity: those displacement fields that can be maintained by applying boundary tractions in the absence of body forces for any linear elastic solid in a given anisotropy class.

In this paper we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to nonlinear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold.

In this book chapter we discuss some applications of algebraic topology in elasticity. This includes the necessary and sufficient compatibility equations of nonlinear elasticity for non-simply-connected bodies when the ambient space is Euclidean. Algebraic topology is the natural tool to understand the topological obstructions to compatibility for both the deformation gradient F and the right Cauchy-Green strain C. We will investigate the relevance of homology, cohomology, and homotopy groups in elasticity.

A new family of mixed finite element methods --- compatible-strain mixed finite element methods (CSFEMs) --- are introduced for three-dimensional compressible and incompressible nonlinear elasticity. A Hu-Washizu-type functional is extremized in order to obtain a mixed formulation for nonlinear elasticity. The independent fields of the mixed formulations are the displacement, the displacement gradient, and the first Piola-Kirchhoff stress. A pressure-like field is also introduced in the case of incompressible elasticity.

The 55th Meeting of the Society for Natural Philosophy: Microstructure, defects, and growth in mechanics will be from September 13-15, 2019 at Loyola University Chicago.

http://webpages.math.luc.edu/55SNP.html

A very limited number of openings to give Roundtable (25 min) talks are available. Special consideration will be given to young researchers. Two nights of lodging will be funded for these speakers. If you are interested in giving a Roundtable talk, you must submit an abstract.

The 55th Meeting of the Society for Natural Philosophy: Microstructure, defects, and growth in mechanics
September 13-15, 2019, Loyola University Chicago

http://webpages.math.luc.edu/55SNP.html