# Arash_Yavari's blog

## PhD Position in Geometric Mechanics at Georgia Tech

I 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.

## 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. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids.

## 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. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field.

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

Compatibility 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. It turns out that there may be topological obstructions to compatibility and this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C.

## On superelastic bending of shape memory alloy beams

In this paper, a closed-form solution is presented for bending analysis of shape memory alloy (SMA) beams.

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

In 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. As opposed to some other approaches to the continuum definition of the Burgers vector, our definition is completely geometric, in the sense that it involves no ambiguous operations like the integration of a vector field--when we integrate a vector field, it is a vector field living in the tangent space at a given point in the manifold.

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

In 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.

## Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics

We 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. More importantly, in this geometric framework we are able to calculate the residual stress fields assuming that the nonlinear elastic body is incompressible.

## Riemann-Cartan Geometry of Nonlinear Disclination Mechanics

In 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. The problem then reduces to embed this manifold in Euclidean 3-space following the procedure of a classical nonlinear elastic problem.

## Covariantization of Nonlinear Elasticity

In 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.

## Mathematics and Mechanics of Solids

Dear Friends:

I would like to encourage you to consider submitting papers to Mathematics and Mechanics of Solids. The focus of this journal is on applications of mathematical techniques to solid mechanics problems. You can find more information in the following link: http://mms.sagepub.com/

Please feel free to contact me (arash.yavari@ce.gatech.edu) if you have any questions regarding this journal.

Regards,
Arash

## Ph.D. Position at The Georgia Institute of Technology

I am looking for a Ph.D. student to work on geometric mechanics of growing bodies (both surface and bulk growth). Candidates with strong math and mechanics backgrounds are encouraged to apply. Interested candidates should email me (arash.yavari@ce.gatech.edu) their CV along with the names of three references.

## Influence of Material Ductility and Crack Surface Roughness on Fracture Instability

This paper presents a stability analysis for fractal cracks. First, the Westergaard stress functions are proposed for semi-infinite and finite smooth cracks embedded in the stress fields associated with the corresponding self-affine fractal cracks. These new stress functions satisfy all the required boundary conditions and according to Wnuk and Yavari's embedded crack model they are used to derive the stress and displacement fields generated around a fractal crack. These results are then used in conjunction with the final stretch criterion to study the quasi-static stable crack extension, which in ductile materials precedes the global failure.

## Analysis of the Rate-Dependent Coupled Thermo-Mechanical Response of Shape Memory Alloy Bars and Wires in Tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied.It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate-dependency that has been known in the literature.

## The James Clerk Maxwell Young Writers Prize

Congratulations to Julian Rimoli (who's one of the moderators of iMechanica) for winning the 2010 James Clerk Maxwell Young Writers Prize!

## Two Faculty Positions in Structural Engineering, Mechanics and Materials at the Georgia Institute of Technology

The School of Civil and Environmental Engineering invites applications for two tenure-track faculty positions in structural engineering/mechanics/materials (SEMM). Candidates at all ranks are sought with expertise in one or more of the following areas: (1) computational/solid mechanics; (2) infrastructure materials. The expected starting date is August, 2011.

## Convergence Analysis of the Wolf Method for Coulombic Interactions

A rigorous proof for convergence of the Wolf method for calculating electrostatic energy of a periodic lattice is presented. In particular, we show that for an arbitrary lattice of unit cells, the lattice sum obtained via Wolf method converges to the one obtained via Ewald method.

## Call for abstracts for a minisymposium on the multiscale constitutive modeling of materials at the 11th US NATIONAL CONGRESS ON

The session organizers would like to invite researchers to participate in a minisymposium titled "Multiscale constitutive modeling of materials" at the 11th US NATIONAL CONGRESS ON COMPUTATIONAL MECHANICS (USNCCM 11) to be held in Minneapolis, Minnesota from July 25-29, 2011.

This session will focus on recent multiscale constitutive modeling techniques including variational methods and energy principles. The topics covered will include the formulation of energy based potentials, variational constitutive updates, model parameters identification, validation and applications. Material models spanning different length scales will be addressed.
Targeted themes:
- Discrete mechanics for crystalline solids with defects

## Effect of External Normal and Parallel Electric Fields on 180^o Ferroelectric Domain Walls in PbTiO3

We impose uniform electric fields both parallel and normal to 180^o ferroelectric domain walls in PbTiO3 and obtain the equilibrium structures using the method of anharmonic lattice statics. In addition to Ti-centered and Pb-centered perfect domain walls, we also consider Ti-centered domain walls with oxygen vacancies. We observe that electric field can increase the thickness of the domain wall considerably. We also observe that increasing the magnitude of electric field we reach a critical electric field E^c; for E > E^c there is no local equilibrium configuration. Therefore, E^c can be considered as a a lower bound for the threshold field E_h for domain wall motion. Our numerical results show that Oxygen vacancies decrease the value of E^c.

## Professor Jerrold E. Marsden

It is with the deepest sadness that I inform you that Professor Jerrold E. Marsden passed away on September 21.

## Joint Postdoctoral Fellow Position at KAUST and Georgia Institute of Technology

Joint Postdoctoral Fellow Position at KAUST
and Georgia Institute of Technology — Accepting applications until the
position is filled

We have one more opening and have no U.S. visa waiver requirement anymore.

## Atomic Structure of Steps on 180^o Ferroelectric Domain Walls in PbTiO3

Using the method of anharmonic lattice statics, we calculate the equilibrium structure of steps on 180^o ferroelectric domain

## Joint Postdoctoral Fellow Position at KAUST and Georgia Institute of Technology

Joint Postdoctoral Fellow Position at KAUST and Georgia Institute of Technology — Accepting applications until the position is filled

We have one more opening and have no U.S. visa waiver requirement anymore.