Nucleation and Growth in Elastodynamics

Structures made of transforming materials exhibit a striking capacity to hysteretically recover significant deformation with a controllable amount of energy absorbed in the process. The unusual properties of these materials are due to the fact that large deformations and inelastic behavior are accomplished by coordinated migration of mobile phase or domain boundaries. Intensive research in recent years has led to well-defined static continuum theories for some of the transforming materials (see Pitteri and Zanzotto (1997) for a recent review). Within the context of these theories, the main unresolved issues include history and rate sensitivity in the constitutive structure. In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. The multiplicity of solutions at the continuum level can be viewed as arising from a constitutive deficiency in the theory, reflecting the need to specify additional pieces of constitutive information through some kind of phenomenological modeling (see, for instance, Truskinovsky, 1987; Abeyaratne and Knowles, 1991). Here we take a different point of view and interpret the nonuniqueness as an indicator of essential interaction between macro and micro scales. We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses; nonlinear elasticity with nonconvex energy is a theory of this type.