Line and Point Defects in Nonlinear Anisotropic Solids

In this paper, we present some analytical solutions for the stress fields of nonlinear anisotropic solids with distributed line and point defects. In particular, we determine the stress fields of i) a parallel cylindrically-symmetric distribution of screw dislocations in infinite orthotropic and monoclinic media, ii) a cylindrically-symmetric distribution of parallel wedge disclinations in an infinite orthotropic medium, iii) a distribution of edge dislocations in an orthotropic medium, and iv) a spherically-symmetric distribution of point defects in a transversely isotropic spherical ball

Structure of Defective Crystals at Finite Temperatures: A Quasi-Harmonic Lattice Dynamics Approach

In this paper we extend the classical method of lattice dynamics to defective crystals with partial symmetries. We start by a nominal defect configuration and first relax it statically. Having the static equilibrium configuration, we use a quasiharmonic lattice dynamics approach to approximate the free energy. Finally, the defect structure at a finite temperature is obtained by minimizing the approximate Helmholtz free energy. For higher temperatures we take the relaxed configuration at a lower temperature as the reference configuration. This method can be used to semi-analytically study the structure of defects at low but non-zero temperatures, where molecular dynamics cannot be used. As an example, we obtain the finite temperature structure of two 180^o domain walls in a 2-D lattice of interacting dipoles. We dynamically relax both the position and polarization vectors. In particular, we show that increasing temperature the domain wall thicknesses increase

Estimating Terminal Velocity of Rough Cracks in the Framework of Discrete Fractal Fracture Mechanics

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

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. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance