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

Geometric Phases of Nonlinear Planar NN-Pendula via Cartan's Moving Frames

We study the geometric phases of nonlinear planar $N$-pendula with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold $\mathcal{B}$, and fibers $\mathcal{F}$ along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan's structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold. For a double pendulum the metric is pseudo-Riemannian if the total angular momentum $\mathsf{A}<0$, and the shape manifold is an expanding spacetime with the Robertson-Walker metric and positive curvature. For $\mathsf{A}>0$, the shape manifold is the hyperbolic plane $\mathbb{H}^2$ with negative curvature. We then generalize our results to free $N$-pendula. We show that the associated shape manifold~$\mathcal{B}$ is reducible to the product manifold of $(N-1)$ hyperbolic planes $\mathbb{H}^2$~($\mathsf{A}>0$), or Robertson-Walker $2$D spacetimes~($\mathsf{A}<0$). We then consider $N$-pendula subject to time-dependent self-equilibrated moments. The extended autonomous Hamiltonian system is considered. The associated shape manifold is a product space of $N$ hyperbolic planes $\mathbb{H}^2$~($\mathsf{A}>0$), or Robertson-Walker $2$D spacetimes~($\mathsf{A}<0$). In either case, the geometric phase follows by integrating a $2$-form given by the sum of the sectional curvature forms of $\mathcal{B}$. The Riemannian structure of the shape manifold provides an intrinsic measure of the closeness of one shape to another in terms of curvature, or induced geometric phase