Robert V. Kohn

The relaxation of a double-well energy

This paper studies coherent, energy-minimizing mixtures of two linearly elastic phases with identical elastic moduli. We derive a formula for the “relaxed” or “macroscopic” energy of the system, by identifying microstructures that minimize the total energy when the volume fractions and the average strain are fixed. If the stress-free strains of the two phases are incompatible then the relaxed energy is nonconvex, with “double-well structure”. An optimal microstructure always exists within the class of layered mixtures. The optimal microstructure is generally not unique, however; we show how to construct a large family of optimal, sequentially laminated microstructures in many circumstances. Our analysis provides a link between the work of Khachaturyan and Roitburd in the metallurgical literature and that of Ball, James, Pipkin, Lurie, and Cherkaev in the recent mathematical literature. We close by explaining why the corresponding problem for three or more phases is fundamentally more difficult.

Variational bounds on the effective moduli of anisotropic composites

Abstract The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ∗ of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C ∗ of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ∗, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C ∗ for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.

A new model for thin plates with rapidly varying thickness

Abstract We study the bending of a thin plate with rapidly varying thickness, for example one with rib-like stiffeners or perforated by small holes. We obtain a fourth-order equation for the midplanc displacement, using an asymptotic analysis based on three-dimensional linear elasticity. The coefficients of this equation represent the constitutive law relating bending moments to midplane curvature; they are explicitly determined by the plate geometry. Our analysis distinguishes between three different cases, in which the thickness varies on a length scale longer than, on the order of, or shorter than the mean thickness.

Branching of twins near an austenite—twinned-martensite interface

Abstract We study the role of surface energy in determining twin width near an austenite—twinned-martensite interface. We find that there are two distinct regimes, depending on the relative values of the elastic moduli, surface energy density and grain size. In the first regime the twin width w is constant, of order L 1/2, where L is the grain size. In the other regime the twin width is not constant; rather, it varies as l 2/3, where l is the distance to the austenite. Most prior analyses have captured only the first regime, and it seems to be the one most commonly observed. However, twin branching consistent with our second regime has been observed in some materials.

Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics

in a bounded domain Q c W, n > 2, with appropriate initial and boundary data. Our interest is in the limiting behavior of u = U’ as E + 0. Formal analysis suggests the following picture: ~8 separates Q into two regions, where U&Z +l and ~8% 1, respectively, and the interface between them moves with normal velocity equal to the sum of its principal curvatures. Our goal here is to present two rigorous results which tend to confirm this picture. The first is a compactness theorem: we show that as E + 0, the solutions of (1.1) are in a certain sense compact as functions of space-time (see Theorem 2.3 and Remark 2.5). Thus it makes sense to discuss the limiting behavior. Our second result is a verification of the picture for certain radial solutions: we prove that lim,,, U’ exists and has the expected form if Q is a ball, U’ is radial with one transition sphere, and the boundary condition is of Dirichlet type (see Theorem 3.1).

Symmetry, texture and the recoverable strain of shape-memory polycrystals

Shape-memory behavior is the ability of certain materials to recover, on heating, apparently plastic deformation sustained below a critical temperature. Some materials have good shape-memory behavior as single crystals but little or none as polycrystals, while others display good shape-memory behavior even as polycrystals. In this paper, we propose a theoretical explanation for this difference: we show that the recoverable strain in a polycrystal depends on the texture of the polycrystal, the transformation strain of the underlying martensitic transformation and especially on the change of symmetry during the underlying transformation. Roughly, we find that the greater the change in symmetry during transformation, the greater the recoverable strain. We include an extensive survey of the experimental literature and show that our results agree with these observations. We make recommendations for improved shape-memory effect in polycrystals.

Numerical implementation of a variational method for electrical impedance tomography

A variational method for computing electrical conductivity distributions from boundary measurements was proposed by Kohn and Vogelius (1987). The authors explore the numerical performance of that technique. A version of Newtons method is used for the minimisation, and synthetic data for the boundary measurements. The variational method is found to be generally stable and robust, reproducing the locations and shapes of conducting objects well, provided that smooth boundary data are used. Early termination appears to have a desirable smoothing effect upon the reconstruction. Contrary to the suggestion of Kohn and Vogelius, the method is not enhanced by allowing the conductivity to be anisotropic.

Dual spaces of stresses and strains, with applications to Hencky plasticity

\int \epsilon^{-1} (1-|\nabla u|^2)^2 + \epsilon |\nabla \nabla u|^2

Numerical study of a relaxed variational problem from optimal design

in two space dimensions. We introduce a new scheme for proving lower bounds and show the bounds are asymptotically sharp for certain domains and boundary conditions. Our results support the conjecture, due to Aviles and Giga, that folds are one-dimensional, i.e., \nabla u varies mainly in the direction transverse to the fold. We also consider related problems obtained when (1-|\nabla u|2)2 is replaced by (1-δ2 ux2 - uy2)2 or (1-|\nabla u|2)2γ .