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 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.
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 MechanicsSubmitted by Arash_Yavari on Fri, 2012-09-14 17:47.
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.
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.
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.
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.
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.
Multiple Tenure Track Faculty Positions, School of Civil and Environmental Engineering, Georgia Institute of TechnologySubmitted by Arash_Yavari on Wed, 2011-12-14 20:42.
Please see the attached ad.
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 (email@example.com) if you have any questions regarding this journal.
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 (firstname.lastname@example.org) their CV along with the names of three references.
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 TensionSubmitted by Arash_Yavari on Tue, 2011-04-19 12:23.
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.
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 TechnologySubmitted by Arash_Yavari on Thu, 2011-02-17 19:27.
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.
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 ONSubmitted by Arash_Yavari on Sun, 2011-01-16 15:39.
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.
- Discrete mechanics for crystalline solids with defects
Effect of External Normal and Parallel Electric Fields on 180^o Ferroelectric Domain Walls in PbTiO3Submitted by Arash_Yavari on Fri, 2010-11-26 22:48.
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.
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 — Accepting applications until the
position is filled
We have one more opening and have no U.S. visa waiver requirement anymore.
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 — Accepting applications until the position is filled
We have one more opening and have no U.S. visa waiver requirement anymore.
In this letter we obtain the finite-temperature structure
of 180^o domain walls in PbTiO_3 using a quasi-harmonic
lattice dynamics approach. We obtain the temperature dependence of
the atomic structure of domain walls from 0 K up to room
temperature. We also show that both Pb-centered and Ti-centered
180^o domain walls are thicker at room temperature; domain
wall thickness at T=300 K is about three times larger than that of
T=0 K. Our calculations show that Ti-centered domain walls have a
lower free energy than Pb-centered domain walls and hence are more
likely to be seen at finite temperatures.
In this paper we first obtain the order of stress singularity for a dynamically propagating self-affine fractal crack. We then show that there is always an upper bound to roughness, i.e. a propagating fractal crack reaches a terminal roughness. We then study the phenomenon of reaching a terminal velocity. Assuming that propagation of a fractal crack is discrete, we predict its terminal velocity using an asymptotic energy balance argument. In particular, we show that the limiting crack speed is a material-dependent fraction of the corresponding Rayleigh wave speed.