Continuum-mechanical models of phase-transforming solids and related issues
James K. Knowles (California Institute of Technology)

Lecture 1


This lecture presents the fundamentals of a one-dimensional, continuum-mechanical theory of stress-induced phase transitions in nonlinearly elastic solids. Although the emphasis will ultimately be on dynamic response in contexts such as impact, the lecture will begin with background material concerning equilibrium phase mixtures and quasi-static processes of the kind often arising in experiments on hysteresis in materials such as shape-memory alloys. This setting provides the natural motivation for the continuum-mechanical notion of driving force, and it exhibits the connection between this notion and the corresponding concept in materials science. Quasi-static processes also furnish the simplest framework in which to exhibit the ill-posed character of the basic problems of continuum mechanics when formulated for phase-transforming materials, making clear the need for a nucleation criterion and a kinetic relation. Next, we turn to the energetics of the fundamental dynamical system of partial differential equations of mixed type arising in this model, and the effect of inertia on the definition of driving force. We then formulate the impact problem for phase-transforming solids and discuss the global structure of solutions in the especially simple case of the two-phase trilinear material model. The roles of the nuclear criterion and the kinetic relation are discussed in detail.



Lecture 2


We begin by applying the theory of the impact problem developed in the first lecture to a series of experiments by D.J. Erskine and W.J. Nellis involving a phase transition from graphite to diamond induced by compressive impact. In particular, we describe

1. the determination of the kinetic relation from the experimental results,
2. the prediction by the model of the onset of the so-called överdrivenresponse observed in the experiments, and
3. the relationship of this phenomenon to ßupersonicphase boundaries for which kinetics cannot be prescribed.

We turn next to the incorporation of thermal effects into the theory by sketching the modifications needed to model phase-transforming thermoelastic solids, and we briefly describe the impact problem in this context. Finally, we discuss the sense in which the notions of driving force and kinetic relation may be mathematically applicable in a model for impact-induced tensile waves in a rubberlike material, for which the system of partial differential equations, while always hyperbolic, fails to be genuinely nonlinear.