J. L. Ericksen

Equilibrium of bars

For elastic bars, we discuss some material instabilities.RésuméPour une barre élastique nous discutons quelques instabilités matérielles.

Homogenization and effective moduli of materials and media

Generalized Plate Models and Optimal Design.- The Effective Dielectric Coefficient of a Composite Medium: Rigorous Bounds From Analytic Properties.- Variational Bounds on Darcys Constant.- Micromodeling of Void Growth and Collapse.- On Bounding the Effective Conductivity of Anisotropic Composites.- Thin Plates with Rapidly Varying Thickness, and Their Relation to Structural Optimization.- Modelling the Properties of Composites by Laminates.- Waves in Bubbly Liquids.- Some Examples of Crinkles.- Microstructures and Physical Properties of Composites.- Remarks on Homogenization.- Variational Estimates for the Overall Response of an Inhomogeneous Nonlinear Dielectric.- Information About Other Volumes in this Program.

Constitutive theory for some constrained elastic crystals

Abstract Some of the observations of A-15 superconductors near cubic-tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints. Here we begin to develop a theory of this kind.

Special Topics in Elastostatics

Publisher Summary This chapter emphasizes the fact that, mathematically, elasticity theory has much in common with modern theories of thin elastic shells, plates, or rods; static theories of liquid crystals; and various other local theories associated with the most common types of variational principles. This chapter discusses that the interest in elasticity theory lies in improving the theory underlying the somewhat mystical process whereby definite forms of constitutive equations are selected. The domain of constitutive equations may include subdomains that are, in principle, inaccessible to the experimentist. The operationalist holds a contrary view. The chapter highlights that the elasticity theory can predict the effects that are not commonly thought of as being associated with the adjective “elastic.” In such cases, elasticity theory enters into free competition with other theories capable of describing the effect at hand.

The Cauchy and Born Hypotheses for Crystals.

Abstract : Commonly, hypotheses introduced by Cauchy or Born are used to relate macroscopic deformation to atomic motions, in molecular theories of elasticity. Our purpose is to discuss the applicability of these to crystal-crystal phase transformations and the ambiguities which are involved in estimating the deformation from observations of lattice vectors. (Author)

A thermo-kinetic view of elastic stability theory

Abstract It seems to be common to regard thermodynamic stability and mechanical stability as two distinct subjects. We here explore the possibility of combining the two in a conceptually clear manner, in a rather limited context. Primarily, we work within the contexts of nonlinear elasticity and thermoelasticity theories, exploring relations between energy criteria for stability and consequences of a kinetic definition of stability.

Nonlinear elasticity of diatomic crystals

Abstract We here explore symmetry considerations which seem to us relevant for simpler mechanical theories of ideal diatomic crystals, using elasticity theory for illustrative purposes. These considerations differ from those commonly employed in continuum mechanics, though the two approaches seem not to be incompatible. In more microscopic views of crystals, similar ideas are encountered in discussions of slip or twinning.

Weak Martensitic Transformations in Bravais Lattices

Nonlinear thermoelasticity theory is being used, with some success, to analyze phenomena associated with phase transitions in some crystals, involving a change in crystal symmetry, what are often called Martensitic transformations. Roughly, these are the crystals which are, or at least behave as if they were Bravais lattices. For such lattices, molecular theories of thermoelasticity imply that the Helmholtz free energy should be invariant under an infinite discrete group. However, workers often use constitutive equations which are invariant only under a finite subgroup, to analyze behavior near transitions, Physicists are likely to use polynomials of as low degree as is feasible, what is sometimes called “Landau Theory”, to treat second-order or “weak” first-order transitions. I don’t know how to give a precise meaning to the notion of a weak transition. By one rather pragmatic interpretation a transition is weak if behavior near the transition of interest can be analyzed, satisfactorily, with a free energy function which is invariant only under some finite subgroup. Using this idea, one can deduce some properties which transitions must have, to be considered weak. My purpose is to elaborate this, to try to get some better understanding of what limits the ranges of applicability of what are, really, two versions of thermoelasticity theory.

Stable Equilibrium Configurations of Elastic Crystals

Nonlinear thermoelasticity theory for crystals is generating some knotty questions in the calculus of variations and in the theory of related partial differential equations. This stems from invariance assumptions which are suggested by molecular theory, and seem to be needed to analyze some commonly observed phenomena, such as twinning. What might be regarded as the simplest problem is to characterize the most stable configurations of unloaded crystals. Even this is difficult, because these are not all unique, but form infinite sets. A number of these do seem to match configurations observed in real crystals. After elaborating this a bit, I will present methods for constructing solutions which are special, but resemble configurations which are observed.

Some Surface Defects in Unstressed Thermoelastic Solids

In the literature on crystals, “twinning” is a word used to describe a variety of phenomena involving different but symmetry-related configurations which coexist in crystals, meeting to form surfaces of discontinuity. As is discussed by Pitteri [1], there have been various attempts to formulate a more precise general definition of the word, as it applies to crystals. His discussion makes clear that some types of twinning are outside the scope of thermoelasticity theory. His definition excludes some phenomena which some experts on crystals call twins, like the “rotational twins” described by Barrett & Massalski [2, p. 406], things which seem to me more reminiscent of the multiple births we commonly describe by other words, like triplets or sextuplets. Whatever one calls them, they are of physical interest, as are other somewhat similar phenomena. My purpose is to present elements of thermoelasticity theory for things of this general kind.