Unstable Elastic Materials and the Viscoelastic Response of Bars in Tension

Some homogeneous elastic materials are capable of sustaining finite equilibrium deformations with discontinuous strains. For materials of this kind, the energetics of isothermal, quasi-static motions may differ from those conventionally associated with elastic behavior. When equilibrium states involving strain jumps occur during such motions, the rate of increase of stored energy in a portion of the body may no longer coincide with the rate of work of the external forces present. In general, energy balance now includes an additional effect due to the presence of moving strain discontinuities. As a consequence, the macroscopic response of the body may be dissipative. This fact makes it possible to model certain types of inelastic behavior in solids with the help of such "unstable" elastic materials; see, for example, Abeyaratne and Knowles (1987a,b,c)

A note on the driving traction acting on a propagating interface: Adiabatic and non-adiabatic processes of a continuum

An expression for the driving traction on an interface is derived for an arbitrary continuum undergoing an arbitrary thermomechanical process which may- or may not be adiabatic

An Elementary Model of Focal Adhesion Detachment and Reattachment During Cell Reorientation Using Ideas from the Kinetics of Wiggly Energies

A simple, transparent, two-dimensional, nonlinear model of cell reorientation is constructed in this paper. The cells are attached to a substrate by “focal adhesions” that transmit the deformation of the substrate to the “stress fibers” in the cell. When the substrate is subjected to a deformation, say an in-plane bi-axial deformation with stretches λ1 and λ2, the stress fibers deform with it and change their length and orientation. In addition, the focal adhesions can detach from the substrate and reattach to it at new nearby locations, and this process of detachment and reattachment can happen many times. In this scenario the (varying) fiber angle Θ in the reference configuration plays the role of an internal variable. In addition to the elastic energy of the stress fibers, the energy associated with the focal adhesions is accounted for by a wiggly energy ϵacos Θ / ϵ, 0 < ϵ≪ 1. Each local minimum of this energy corresponds to a particular configuration of the focal adhesions. The small amplitude ϵa indicates that the energy barrier between two neighboring configurations is relatively small, and the small distance 2 πϵ between the local minima indicates that a focal adhesion does not have to move very far before it reattaches. The evolution of this system is studied using a gradient flow kinetic law, which is homogenized for ϵ→ 0 using results from weak convergence. The results determine (a) a region of the λ1, λ2-plane in which the (referential) fiber orientation remains stuck at the angle Θ and does not evolve, and (b) the evolution of the orientation when the stretches move out of this region as the fibers seek to minimize energy

On the commutability of homogenization and linearization in finite elasticity

We study non-convex elastic energy functionals associated to (spatially) periodic, frame indifferent energy densities with a single non-degenerate energy well at SO(n). Under the assumption that the energy density admits a quadratic Taylor expansion at identity, we prove that the Gamma-limits associated to homogenization and linearization commute. Moreover, we show that the homogenized energy density, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the quadratic term associated to the linearization of the initial energy density

Beyond Kinetic Relations

We introduce the concept of kinetic equations representing a natural extension of the more conventional notion of a kinetic relation. Algebraic kinetic relations, widely used to model dynamics of dislocations, cracks and phase boundaries, link the instantaneous value of the velocity of a defect with an instantaneous value of the driving force. The new approach generalizes kinetic relations by implying a relation between the velocity and the driving force which is nonlocal in time. To make this relations explicit one needs to integrate the system of kinetic equations. We illustrate the difference between kinetic relation and kinetic equations by working out in full detail a prototypical model of an overdamped defect in a one-dimensional discrete lattice. We show that the minimal nonlocal kinetic description containing now an internal time scale is furnished by a system of two ordinary differential equations coupling the spatial location of defect with another internal parameter that describes configuration of the core region.Comment: Revised version, 33 pages, 9 figure

Traveling Waves for Conservation Laws with Cubic Nonlinearity and BBM Type Dispersion

Scalar conservation laws with non-convex fluxes have shock wave solutions that violate the Lax entropy condition. In this paper, such solutions are selected by showing that some of them have corresponding traveling waves for the equation supplemented with dissipative and dispersive higher-order terms. For a cubic flux, traveling waves can be calculated explicitly for linear dissipative and dispersive terms. Information about their existence can be used to solve the Riemann problem, in which we find solutions for some data that are different from the classical Lax-Oleinik construction. We consider dispersive terms of a BBM type and show that the calculation of traveling waves is somewhat more intricate than for a KdV-type dispersion. The explicit calculation is based upon the calculation of parabolic invariant manifolds for the associated ODE describing traveling waves. The results extend to the p-system of one-dimensional elasticity with a cubic stress-strain law.Comment: 17 pages, 5 figure