Gradient plasticity in isotropic solids

We discuss a framework for the description of gradient plasticity in isotropic solids based on the Riemannian curvature derived from a metric induced by plastic deformation. This culminates in a flow rule in the form of a fourth-order partial differential equation for the plastic strain rate, in contrast to the second-order flow rules that have been proposed in alternative treatments of gradient plasticity in isotropic solids.

Universal deformations in inhomogeneous isotropic nonlinear elastic solids

Universal (controllable) deformations of an elastic solid are those deformations that can be maintained for all possible strain-energy density functions and suitable boundary tractions. Universal deformations have played a central role in nonlinear elasticity and anelasticity. However, their classification has been mostly established for homogeneous isotropic solids following the seminal works of Ericksen. In this article, we extend Ericksen’s analysis of universal deformations to inhomogeneous compressible and incompressible isotropic solids. We show that a necessary condition for the known universal deformations of homogeneous isotropic solids to be universal for inhomogeneous solids is that inhomogeneities respect the symmetries of the deformations. Symmetries of a deformation are encoded in the symmetries of its pulled-back metric (the right Cauchy–Green strain). We show that this necessary condition is sufficient as well for all the known families of universal deformations except for Family 5.

On Eshelby’s inclusion problem in nonlinear anisotropic elasticity

In this paper, the recent literature of finite eignestrains in nonlinear elastic solids is reviewed, and Eshelby’s inclusion problem at finite strains is revisited. The subtleties of the analysis of combinations of finite eigenstrains for the example of combined finite radial, azimuthal, axial and twist eigenstrains in a finite circular cylindrical bar are discussed. The stress field of a spherical inclusion with uniform pure dilatational eigenstrain in a radially-inhomogeneous spherical ball made of arbitrary incompressible isotropic solids is analyzed. The same problem for a finite circular cylindrical bar is revisited. The stress and deformation fields of an orthotropic incompressible solid circular cylinder with distributed eigentwists are analyzed.

Probing the in-plane liquid-like behavior of liquid crystal elastomers

When isotropic solids are unequally stretched in two orthogonal directions, the true stress (force per actual cross-sectional area) in the larger strain direction is typically higher than that in the smaller one. We show that thiol-acrylate liquid crystal elastomers with polydomain texture exhibit an unusual tendency: The true stresses in the two directions are always identical and governed only by the area change in the loading plane, independently of the combination of imposed strains in the two directions. This feature proves a previously unidentified state of matter that can vary its shape freely with no extra mechanical energy like liquids when deformed in the plane. The theory and simulation that explain the unique behavior are also provided. The in-plane liquid-like behavior opens doors for manifold applications, including wrinkle-free membranes and adaptable materials.

Solidity without inhomogeneity: perfectly homogeneous, weakly coupled, UV-complete solids

Solid-like behavior at low energies and long distances is usually associated with the spontaneous breaking of spatial translations at microscopic scales, as in the case of a lattice of atoms. We exhibit three quantum field theories that are renormalizable, Poincaré invariant, and weakly coupled, and that admit states that on the one hand are perfectly homogeneous down to arbitrarily short scales, and on the other hand have the same infrared dynamics as isotropic solids. All three examples presented here lead to the same peculiar solid at low energies, featuring very constrained interactions and transverse phonons that always propagate at the speed of light. In particular, they violate the well known $$ {c}_L^2>\frac{4}{3}{c}_T^2 $$ c L 2 > 4 3 c T 2 bound, thus showing that it is possible to have a healthy renormalizable theory that at low energies exhibits a negative bulk modulus (we discuss how the associated instabilities are absent in the presence of suitable boundary conditions). We do not know whether such peculiarities are unavoidable features of large scale solid-like behavior in the absence of short scale inhomogeneities, or whether they simply reflect the limits of our imagination.

Thermoelasticity if isotropic solids containing non-deformable thread-like inclusions

The paper derives integral equations of heat conduction and thermoelasticity of isotropic solids with non-deformable perfectly thermally conducting thread-like inclusions. It is observed that, in spite of the order of singularity, the integral equations obtained are hypersingular due to the symmetry of the kernels. Non-integral terms of these equations are derived. A boundary element method scheme for numerical solution of formulated problems is proposed. A numerical example is provided.

An Improved Technique for Elastodynamic Green's Function Computation for Transversely Isotropic Solids

Elastodynamic Green's function for anisotropic solids is required for wave propagation modeling in composites. Such modeling is needed for the interpretation of experimental results generated by ultrasonic excitation or mechanical vibration-based nondestructive evaluation tests of composite structures. For isotropic materials, the elastodynamic Green’s function can be obtained analytically. However, for anisotropic solids, numerical integration is required for the elastodynamic Green's function computation. It can be expressed as a summation of two integrals—a singular integral and a nonsingular (or regular) integral. The regular integral over the surface of a unit hemisphere needs to be evaluated numerically and is responsible for the majority of the computational time for the elastodynamic Green's function calculation. In this paper, it is shown that for transversely isotropic solids, which form a major portion of anisotropic materials, the integration domain of the regular part of the elastodynamic time-harmonic Green's function can be reduced from a hemisphere to a quarter-sphere. The analysis is performed in the frequency domain by considering time-harmonic Green's function. This improvement is then applied to a numerical example where it is shown that it nearly halves the computational time. This reduction in computational effort is important for a boundary element method and a distributed point source method whose computational efficiencies heavily depend on Green's function computational time.