Gareth P. Parry

A complete list of invariants for defective crystals

The classical theory of continuous distributions of dislocations has traditionally focused on the Burgers’ vectors and the dislocation density tensor as descriptions of defectiveness. We prove that, generally, there is an infinite number of tensor densities with similarly descriptive properties, and that there is a functional basis for this list which strictly includes the Burgers’ vectors and dislocation density. Moreover the changes of state which preserve these densities turn out to represent slip in certain surfaces associated with crystal geometry, so that the basic mechanism of plasticity emerges naturally from abstract ideas which neither anticipate nor involve the kinematics of particular types of crystal defects.

On defect-preserving deformations in crystals

Abstract In the context of a continuum theory of crystals with defects, one can define a particular list of invariants so that elements of this list do not change when the crystal is deformed elastically. Here we characterize (most of) the deformations which leave these elements unchanged and find that these “defect-preserving” deformations.strictly include the elastic deformations. This characterization allows a judgement of the completeness of the list of invariants. Furthermore it turns out that the defect-preserving deformations which are not elastic generally involve some kind of rearrangement, or slip, of the crystal lattice, and that if one admits such deformations in variational principles determining equilibria of the lattice, then the crystal is necessarily weak, in some sense; for example, the crystal may be able to equilibrate only simple pressure over its boundary.

On diatomic crystals

Abstract Strain energy functions which will appear in phenomenological theories of the mechanical behaviour of diatomic crystals are constructed in accordance with the constraints imposed by the crystal symmetry.

On internal variable models of phase transitions

Some models of phase equilibria as local minima of a Lagrangian functional are shown to allow configurations where macroscopic strain is generally discontinuous, but internal variables (and derivatives) vary smoothly across the ‘phases’. The stability properties of ‘regularizations’ of such equilibria are also investigated.

Twinning in nonlinearly elastic monatomic crystals

Abstract Two-phase mixtures are admitted to the energy criterion of stability. This generality allows phase transitions to be interpreted as global instabilities, and so enables the prediction of the onset of these transitions. Stability is assessed in various loading environments, corresponding broadly to the well known hard and soft devices. The work is illustrated mainly by reference to deformation twinning.

A Model for Twinning

The modelling of twins in crystals with strain gradient theories provides interesting problems both in thermodynamics and in the calculus of variations. Here, Dunn and Serrins thermomechanical theory of interstitial working is used to obtain a variational principle that governs the equilibria of materials with non-convex Helmholtz free energy. In some geometries, this principle reduces to a novel calculus of variations problem; an example is described in which symmetry-related uniform equilibrium states can be connected by nonconstant extremals which realiselocal minima of the free energy. Certain implications of different definitions of local minima are also discussed.

On shear-induced phase transitions in perfect crystals

Abstract This is an analysis of structural phase transitions in b.c.t. crystals controlled by tensile loading along a crystallographic axis.

Uniform rearrangement in defective crystals

We show that an abstract notion of defectiveness yields the slip mechanism of classic plasticity theory by reasoning in purely kinematical terms, and we catalogue the canonical forms of slip that occur in this way. There appears, in consequence, a precise version of the idea of elastic-plastic decomposition.

On a Class of Invariant Functionals

A characterization of a class of functionals invariant under isochoric changes of domain is obtained. This class contains strictly the null lagrangians.

VARIATIONAL PROBLEMS FOR CRYSTALS WITH DEFECTS

Equilibrium configurations for crystals with defects are analyzed. Using the theory of compensated compactness the state functions are characterized in the case where minimizing sequences develop oscillations. A new class of variational problems involving bulk and surface energy terms is studied.