Solid-solid Phase Transformations:Non-existence of One-dimensional Stress Problems, Model Equation and Uniqueness Conditions
Dr. Hui-Hui Dai
Many people have used pure one-dimensional theories to study boundary-valueand/or initial-value problems of phase-transforming materials. We show that forsuch materials there do not exist one-dimensional stress problems. Thus, these pureone-dimensional theories have some essential defects. To model these materialsphysically and mathematically, it is important to consider the influence from theother dimension(s). For a slender circular cylinder, by taking into account the effectsdue to the radial deformation, we establish the proper model equation, which showsthat the problem is a singular perturbation one. The model equations used in theliterature are only the leading order equations valid in the outer regions. The lackof uniqueness of solutions in the pure one-dimensional theories is well-known. In theliterature, the kinetic relation, which is regarded as an extra constitutive relation, isproposed to give an additional condition to obtain unique solutions. Here, by usingour model equation and matching its traveling wave solution to those in the outerregions, we obtain three relations for three unknowns, which provide the uniquenessconditions for solutions. Also, these three relations are given in terms of the stressfunction alone, independent of the notion of the kinetic relation. Our results seemto resolve the long outstanding issue of nonuniqueness of solutions in modelingdynamical problems of phase-transforming materials.