On selection criteria for problems with moving inhomogeneities

We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Studying simple examples for lattice motion we then compare the implications for both resting and moving inhomogeneities.Comment: revised version contains new introduction, numerical simulations of Riemann problems, and a more detailed discussion of the causality principle; 18 pages, several figure

Steady-State Cracks in Viscoelastic Lattice Models II

We present the analytic solution of the Mode III steady-state crack in a square lattice with piecewise linear springs and Kelvin viscosity. We show how the results simplify in the limit of large width. We relate our results to a model where the continuum limit is taken only along the crack direction. We present results for small velocity, and for large viscosity, and discuss the structure of the critical bifurcation for small velocity. We compute the size of the process zone wherein standard continuum elasticity theory breaks down.Comment: 17 pages, 3 figure

Does the continuum theory of dynamic fracture work?

We investigate the validity of the Linear Elastic Fracture Mechanics approach to dynamic fracture. We first test the predictions in a lattice simulation, using a formula of Eshelby for the time-dependent Stress Intensity Factor. Excellent agreement with the theory is found. We then use the same method to analyze the experiment of Sharon and Fineberg. The data here is not consistent with the theoretical expectation.Comment: 4 page

Oscillatory wave fronts in chains of coupled nonlinear oscillators

Wave front pinning and propagation in damped chains of coupled oscillators are studied. There are two important thresholds for an applied constant stress $F$: for $|F|<F_{cd}$ (dynamic Peierls stress), wave fronts fail to propagate, for $F_{cd} < |F| < F_{cs}$ stable static and moving wave fronts coexist, and for $|F| > F_{cs}$ (static Peierls stress) there are only stable moving wave fronts. For piecewise linear models, extending an exact method of Atkinson and Cabrera's to chains with damped dynamics corroborates this description. For smooth nonlinearities, an approximate analytical description is found by means of the active point theory. Generically for small or zero damping, stable wave front profiles are non-monotone and become wavy (oscillatory) in one of their tails.Comment: 18 pages, 21 figures, 2 column revtex. To appear in Phys. Rev.

Energy radiation of moving cracks

The energy radiated by moving cracks in a discrete background is analyzed. The energy flow through a given surface is expressed in terms of a generalized Poynting vector. The velocity of the crack is determined by the radiation by the crack tip. The radiation becomes more isotropic as the crack velocity approaches the instability threshold.Comment: 7 pages, embedded figure

Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators

We present a stability theory for kink propagation in chains of coupled oscillators and a new algorithm for the numerical study of kink dynamics. The numerical solutions are computed using an equivalent integral equation instead of a system of differential equations. This avoids uncertainty about the impact of artificial boundary conditions and discretization in time. Stability results also follow from the integral version. Stable kinks have a monotone leading edge and move with a velocity larger than a critical value which depends on the damping strength.Comment: 11 figure

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