We introduce a phenomenological continuum model for the mode III dynamic fracture that is based on the phase-field methodology used extensively to model interfacial pattern formation. We couple a scalar field, which distinguishes between “broken” and “unbroken” states of the system, to the displacement field in a way that consistently includes both macroscopic elasticity and a simple rotationally invariant short-scale description of breaking. We report two-dimensional simulations that yield steady-state crack motion in a strip geometry above the Griffith threshold.

The early stages of growth of highly strained ${\mathrm{In}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As on GaAs(100) have been investigated as a function of composition. The evolution of the film microstructure as determined by in situ STM and RHEED is from a two-dimensional rippled surface in the beginning stages of growth to a three-dimensional island morphology. A growth mode is proposed whereby strain relaxation is initially achieved through the kinetically limited evolution of surface morphology. In contrast to traditional critical-thickness theories, significant strain relief is accommodated by a coherent island morphology. This study represents a new view for both the growth mode and initial strain relaxation in thin films.

We present a mean-field theory for the evolution of RNA virus populations. The theory operates with a distribution of the population in a one-dimensional fitness space, and is valid for sufficiently smooth fitness landscapes. Our approach explains naturally the recent experimental observation [I. S. Novella et al., Proc. Natl. Acad. Sci. U.S.A. 92, 5841–5844 (1995)] of two distinct stages in the growth of virus fitness.

We introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion. The intended application is to systems for which competing stabilizing and destabilizing forces act on the phase boundary to produce irregular or patterned structures, such as those which occur in solidification. We discuss the kinematics of moving interfaces in two or more dimensions in terms of their intrinsic geometric properties. We formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems. Special solutions for dendritic crystal growth and their stability are analyzed in some detail.

A general class of models is introduced which relate the motion of a phase boundary to properties of the local interfacial geometry. These systems can undergo successive destabilizations as they grow, possibly giving rise to nonequilibrium spatial patterns. This formalism has applications to a wide variety of physical problems, especially including dendritic solidification.