J. M. Ball

Fine Phase Mixtures as Minimizers of Energy

Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

Proposed Experimental Tests of a Theory of Fine Microstructure and the Two-Well Problem

Predictions are made based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors. The crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries. The predictions concern primarily the two-well problem, that of determining all possible energy-minimizing deformations that can be obtained with two coherent and macroscopically unstressed variants of martensite. The set of possible macroscopic deformations obtained is completely determined by the lattice parameters of the material. For certain boundary conditions the total free energy does not attain a minimum , and the finer and finer oscillations of minimizing sequences are interpreted as corresponding to microstructure. The predictions are am enable to experimental tests. The proposed tests involve the comparison of the theoretical predictions with the mechanical response of properly oriented plates subject to simple shear.

Discontinuous Equilibrium Solutions and Cavitation in Nonlinear Elasticity

A study is made of a class of singular solutions to the equations of nonlinear elastostatics in which a spherical cavity forms at the centre of a ball of isotropic material placed in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the stored-energy function W for large strains and is consistent with strong ellipticity of W. Under appropriate hypotheses it is shown that a singular solution bifurcates from a trivial (homogeneous) solution at a critical value of the surface traction or displacement, at which the trivial solution becomes unstable. For incompressible materials both the singular solution and the critical surface traction are given explicitly, and the stability of all solutions with respect to radial motion is determined. For compressible materials the existence of singular solutions is proved for a class of strongly elliptic materials by means of the direct method of the calculus of variations, an important step in the analysis being to show that the only radial equilibrium solutions without cavities are homogeneous. Work of Gent & Lindley (1958) shows that the critical surface tractions obtained agree with those observed in the internal rupture of rubber.

W1,p-quasiconvexity and variational problems for multiple integrals

Abstract Variational problems for the multiple integral I Ω (u) = ∝ Ω g(▽u(x))dx , where Ω⊂R m and u:Ω→R n are studied. A new condition on g, called W1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of I Ω in W 1,p (Ω;R n ) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W 1,p (Ω;R n ) , p ⩽ n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.

Null Lagrangians, weak continuity, and variational problems of arbitrary order

A control system for a fabric drying apparatus comprising a resistive heating element in series with high limit and cycling thermostats and connected across the two power lines of a three wire alternating current power supply. The drive motor of the timer is electrically connected between the junction of two legs of the heating circuit and the neutral wire of the power supply. The impedance of the timer drive motor is high compared to the resistance of the heating element; therefore, when the heating element is energized for drying fabrics, the current flow to the timer drive motor is negligible and the timer will not advance. When the cycling thermostat in one of the legs opens at a predetermined temperature, the heating element will be deenergized and current will flow in the other leg and neutral line and the timer will advance until the thermostat closes to reenergize the heating element.

Controllability for Distributed Bilinear Systems

This paper studies controllability of systems of the form

Some Open Problems in Elasticity

{{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w

Global invertibility of Sobolev functions and the interpenetration of matter

where

Initial-boundary value problems for an extensible beam

\mathcal{A}

The Discrete Coagulation-Fragmentation Equations: Existence, Uniqueness, and Density Conservation

is the infinitesimal generator of a