arash_yavari's blog

Usually the multiplicative decomposition of deformation gradient in finite plasticity is (incorrectly) attributed to Lee and Liu (1967). This short note discusses the origins of this idea, which go back to the late 1940s. We explain that the first explicit mention of this decomposition appeared a decade earlier in the work of Bilby, et al. (1957) and Kröner (1959). While writing this note I found out that Bruce Bilby passed away a couple of years ago at the age of 91.

We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first-Piola Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work done by the prescribed boundary displacements. This condition is very similar to the classical compatibility equations for the linear strain.

Dear Colleagues:

I thought the following recent paper by Prof. Neff may be of interest to some of you.

http://arxiv.org/abs/1505.02203

This paper discusses the natural appearance of the  Hencky-logarithmic strain tensor together with the Hencky strain energy, which can be motivated from some purely geometrical (kinematical) arguments based on the geodesic distance on the general linear group of all invertible tensors GL(n).

The 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.

In this paper we formulate a geometric theory of nonlinear thermoelasticity that can be used to calculate the time evolution of the temperature and thermal stress fields in a nonlinear elastic body. In particular, this formulation can be used to calculate residual thermal stresses. In this theory the material manifold (natural stress-free configuration of the body) is a Riemannian manifold with a temperature-dependent metric. Evolution of the geometry of the material manifold is governed by a generalized heat equation.

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.

Differential geometry in simple words is a generalization of calculus on some curved spaces called manifolds. An n-manifold is a space that locally looks like R^ⁿ but globally can be very different. The first significant application of differential geometry happened to be in Einstein’s theory of general relativity.

Dear friends:

I’d like to bring to your attention three recently published books that I have read. The following are a few words about each book in the order that I read them.

1) “Vito Volterra” by A. Guerraggio and G. Paoloni
http://www.amazon.com/Vito-Volterra-Angelo-Guerraggio/dp/3642272622/ref…

The geometric formulation of continuum mechanics provides a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects, or other effects leading to residual stresses. The central idea is to assume that the material manifold, prescribing the reference configuration for a body, has an intrinsic, non-Euclidean, geometric structure. Residual stresses then naturally arise when this configuration is mapped into Euclidean space.

Dear friends:



Prof. Patrizio Neff has asked me to post the attached recent paper on Logarithmic Strain. It looks quite interesting (I haven’t read it yet).



Regards,

Arash