## You are here

# Arash_Yavari's blog

## The Twist-Fit Problem: Finite Torsional and Shear Eigenstrains in Nonlinear Elastic Solids

Fri, 2015-10-02 10:47 - Arash_YavariEigenstrains in nonlinear elastic solids are created through defects, growth, or other anelastic effects. These eigenstrains are known to be important as they can generate residual stresses and alter the overall response of the solid. Here, we study the residual stress fields generated by finite torsional or shear eigenstrains. This problem is addressed by considering a cylindrical bar made of an incompressible isotropic solid with an axisymmetric distribution of shear eigenstrains.

- Read more about The Twist-Fit Problem: Finite Torsional and Shear Eigenstrains in Nonlinear Elastic Solids
- 6 comments
- Log in or register to post comments
- 3508 reads

## The Weak Compatibility Equations of Nonlinear Elasticity and the Insufficiency of the Hadamard Jump Condition for Non-Simply Connected Bodies

Mon, 2015-09-28 00:23 - Arash_YavariWe derive the compatibility equations of L2 displacement gradients on non-simply-connected bodies. These compatibility equations are useful for non-smooth strains such as those associated with deformations of multi-phase materials. As an application of these compatibility equations, we study some configurations of different phases around a hole and show that, in general, the classical Hadamard jump condition is not a sufficient compatibility condition.

## On the origins of the idea of the multiplicative decomposition of the deformation gradient

Tue, 2015-09-22 17:55 - Arash_YavariUsually 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.

- Read more about On the origins of the idea of the multiplicative decomposition of the deformation gradient
- 6 comments
- Log in or register to post comments
- 3625 reads

## On the Compatibility Equations of Nonlinear and Linear Elasticity in the Presence of Boundary Conditions

Mon, 2015-08-10 00:48 - Arash_YavariWe 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.

## A new paper on Hencky-logarithmic strain by Prof. Neff

Sat, 2015-05-16 17:46 - Arash_YavariDear 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).

- Read more about A new paper on Hencky-logarithmic strain by Prof. Neff
- Log in or register to post comments
- 1109 reads

## 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.

## Geometric nonlinear thermoelasticity and the time evolution of thermal stresses

Fri, 2014-12-05 16:57 - Arash_YavariIn 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.

- Read more about Geometric nonlinear thermoelasticity and the time evolution of thermal stresses
- Log in or register to post comments
- 1141 reads

## 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.

- Read more about Differential Complexes in Continuum Mechanics
- Log in or register to post comments
- 984 reads

## Geometry, topology, and solid mechanics

Mon, 2014-08-04 07:26 - Arash_YavariDifferential 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.

- Read more about Geometry, topology, and solid mechanics
- 18 comments
- Log in or register to post comments
- 14806 reads

## Three interesting, recent books

Thu, 2014-06-26 16:20 - Arash_YavariDear 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=...

- Read more about Three interesting, recent books
- 3 comments
- Log in or register to post comments
- 2881 reads

## The Geometry of Discombinations and its Applications to Semi-Inverse Problems in Anelasticity

Wed, 2014-06-11 10:46 - Arash_YavariThe 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.

- Read more about The Geometry of Discombinations and its Applications to Semi-Inverse Problems in Anelasticity
- Log in or register to post comments
- 2821 reads

## A recent paper on Logarithmic Strain by Prof. Patrizio Neff

Fri, 2014-05-23 20:10 - Arash_YavariDear 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

- Read more about A recent paper on Logarithmic Strain by Prof. Patrizio Neff
- Log in or register to post comments
- 1633 reads

## Non-Metricity and the Nonlinear Mechanics of Distributed Point Defects

Thu, 2014-05-01 12:09 - Arash_YavariWe discuss the relevance of non-metricity in a metric-affine manifold (a manifold equipped with a connection and a metric) and the nonlinear mechanics of distributed point defects. We describe a geometric framework in which one can calculate analytically the residual stress field of nonlinear elastic solids with distributed point defects. In particular, we use Cartan's machinery of moving frames and construct the material manifold of a finite ball with a spherically-symmetric distribution of point defects.

- Read more about Non-Metricity and the Nonlinear Mechanics of Distributed Point Defects
- Log in or register to post comments
- 1252 reads

## PhD Position in Geometric Mechanics at Georgia Tech

Sun, 2013-09-22 14:34 - Arash_YavariI am looking for a new Ph.D. student to work on discretization of nonlinear elasticity using geometric and topological ideas. Requirements for this position are a strong background in solid mechanics and some background in differential geometry and analysis. If interested please email me your CV.

- Read more about PhD Position in Geometric Mechanics at Georgia Tech
- Log in or register to post comments
- 2042 reads

## 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.

- Read more about Nonlinear elastic inclusions in isotropic solids
- Log in or register to post comments
- 1613 reads

## 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.

- Read more about A Geometric Structure-Preserving Discretization Scheme for Incompressible Linearized Elasticity
- 3 comments
- Log in or register to post comments
- 3351 reads

## Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Sun, 2013-02-03 11:15 - Arash_YavariCompatibility 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.

- Read more about Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies
- 14 comments
- Log in or register to post comments
- 4470 reads

## On superelastic bending of shape memory alloy beams

Wed, 2013-01-23 15:49 - Arash_YavariIn this paper, a closed-form solution is presented for bending analysis of shape memory alloy (SMA) beams.

- Read more about On superelastic bending of shape memory alloy beams
- 1 comment
- Log in or register to post comments
- 2352 reads

## 12thU.S. National Congress on Computational Mechanics (US-NCCM12)

Thu, 2013-01-10 15:31 - Arash_Yavari- Read more about 12thU.S. National Congress on Computational Mechanics (US-NCCM12)
- Log in or register to post comments
- 2731 reads

## Affine Development of Closed Curves in Weitzenbock Manifolds and the Burgers Vector of Dislocation Mechanics

Fri, 2012-09-14 13:47 - Arash_YavariIn 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.

## Weyl Geometry and the Nonlinear Mechanics of Distributed Point Defects

Wed, 2012-07-25 09:00 - Arash_YavariIn 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.

- Read more about Weyl Geometry and the Nonlinear Mechanics of Distributed Point Defects
- Log in or register to post comments
- 1680 reads

## Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics

Thu, 2012-01-05 09:34 - Arash_YavariWe 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.

- Read more about Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics
- Log in or register to post comments
- 2188 reads

## Riemann-Cartan Geometry of Nonlinear Disclination Mechanics

Fri, 2011-12-23 18:33 - Arash_YavariIn 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.

- Read more about Riemann-Cartan Geometry of Nonlinear Disclination Mechanics
- 5 comments
- Log in or register to post comments
- 4450 reads

## Covariantization of Nonlinear Elasticity

Thu, 2011-12-15 17:28 - Arash_YavariIn 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.

- Read more about Covariantization of Nonlinear Elasticity
- Log in or register to post comments
- 3444 reads

## Recent comments