James K. Knowles

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.

Dynamics of propagating phase boundaries: Thermoelastic solids with heat conduction

This paper is concerned with the incorporation of thermal effects into the continuum modeling of dynamic solid-solid phase transitions. The medium is modeled as a one-dimensional thermoelastic solid characterized by a specific Helmholtz free-energy potential and a specific kinetic relation. Heat conduction and inertia are taken into account. An initial-value problem that gives rise to both shock waves and a propagating phase boundary is analyzed on the basis of this model.

On Stress-Strain Relations and Strain-Energy Expressions in the Theory of Thin Elastic Shells

The stress-strain relations of Flugge and Byrne for thin elastic shells are inverted to express strain quantities, and therewith the strain energy, in terms of stress resultants and couples. In this form, and upon omission of terms which are small of order h2 /R2 , the stress-strain relations and the strain-energy expression are shown to be simply related to corresponding results of Trefftz. The strain-energy formula of Trefftz is generalized to arbitrary orthogonal middle surface co-ordinates.