VIGRE Summer School - Nonconvex Problems in the Calculus of Variations
Macroscopic Models for Phase Transforming Materials
1. Introduction: Mathematical Models, nonlinear elasticity theory, and the failure of the direct method in the calculus of variations
The importance of microstructures in materials has been known for a long time. The austenite-martensite transformation in single crystals is named in honor of the metallurgists Adolf Martens (1850-1914) and Sir William Chandler Roberts-Austen (1843-1902). The existence of different phases in the solid state has a significant impact on the elastic properties of the materials, for example the shape memory effect.
Mathematical models in the general framework of nonlinear elasticity theory lead to variational problems in which one tries to minimize the energy of the system in a class of deformations, typically a subset of a Sobolev space with suitable boundary conditions. The stored energy density W reflects the symmetries of the underlying crystalline lattice through special symmetries. As a consequence, the densities typically fail to satisfy the crucial convexity assumption in the direct methods in the calculus of variations (quasiconvexity in the sense of Morrey) and thus the questions of existence, uniqueness, and regularity of minimizers cannot be obtained from the general theory.
2. Quasiconvexity and weak lower semicontinuity
The pioneering work of Morrey established the crucial connection between lower semicontinuity of functionals on Sobolev spaces and quasiconvexity of the integrand. Intuitively, quasiconvexity implies global stability of affine deformations, i.e., if one minimizes the energy functional subject to affine boundary conditions, then the homogeneous affine deformation is a minimizer of the functional and the energy cannot be lowered by local perturbations. Despite its fundamental importance, a lot of questions related to quasiconvexity are still open, in particular there are only very few examples of functions that are quasiconvex. A necessary condition for quasiconvexity is rank-one convexity and a sufficient condition is polyconvexity.
3.Relaxation
The variational problems that describe phase transforming materials typically are not lower semicontinuous and cannot be analyzed by the direct method in the calculus of variations that is based on weak lower semicontinuity of the functional. One approach to overcome this difficulty is to analyze instead the largest lower semicontinuous functional J below the given functional. It turns out that this functional is again an integral functional in which the energy density W is replaced by its quasiconvex envelope. This envelope is the largest quasiconvex function less than or equal to W and it describes at the same time the macroscopic or effective energy of the system, i.e., the smallest energy per unit volume that is needed to deform an infinitesimal volume with given affine boundary conditions. Here the material is allowed to form any (including infinitesimally fine) microstructure that minimizes the energy. An example is the relaxation of the minimum of two quadratic energies with the same elastic moduli that was obtained by Kohn. It is a common feature that the relaxed energy contains different domains, in which the energy can be reduced by the formation of microstructure (typically laminates) and in which the original energy is already sufficiently convex to prevent any reduction of the energy.
4. Minimizers and Minimizing sequences
The approach presented so far leads naturally to the following problems. Suppose that the energy density is nonnegative and that its zero set K is not empty.
1) What can we say about minimizing sequences, say subject to affine boundary conditions, along which the energy tends to zero?
2) What can we say about minimizers of the energy? This requires that the gradient of the minimizer lies in K for a.e. x. The function u is also said to be a solution of a differential inclusion.
The right tool to analyze the first question is the so-called gradient Young measure (GYM) generated by the sequence of deformation gradients. The set of all affine boundary conditions for which there exists a minimizing sequence with infinitesimally small energy is therefore given by the set of centers of mass of GYM supported on K and is called the quasiconvex hull of K. It can also be defined as the set of all points that cannot be separated by quasiconvex functions from K. Based on the necessary and sufficient condition for quasiconvexity we define the rank-one convex and the polyconvex hull analogously. Finally we call the hull that is obtained by inductively adding rank-one segments to K the lamination convex hull of K. Explicit examples show that the inclusions between the various hulls can be strict, the lamination convex hull being the smallest, the polyconvex hull being the largest. An example with connections to austenite-martensite transformations is the two-well problem, where K is the union of two distinct energy wells SO(2)A and SO(2)B. The quasiconvex hull of K is equal to K if the two wells are not rank-one connected, and it is formed by laminates within laminates if each point in K is rank-one connected to two distinct points in K, the case of physical interest.
The second problem found a surprising answer in the work of Müller and Sverak: If the boundary conditions correspond to second laminates (points in the relative interior of the quasiconvex hull) then there exists a Lipschitz function that is a solution of the differential inclusion subject to the given affine boundary conditions. These solutions have an intrinsically complicated geometry: The length of the phase boundary is necessarily infinitely long.
5. Mathematical Models for Nematic Elastomers
This material system that combines the elastic properties of rubbers with the orientational instabilities of liquid crystals has attracted a lot of attention as a model system, not only because of potential applications as artificial muscles. Since the high temperature phase is isotropic, the zero set of the nematic low temperature phase depends only on the singular values of the deformation gradient. This invariance allows not only to give an explicit characterization of its quasiconvex hull, but also of the relaxed energy. It has three distinct regions: its zero set, in which the response of the material is ideally soft, an intermediate region, in which the macroscopic energy depends only on one singular value which leads to a ideally soft/solid response, and the region in which the original energy is already convex and the material behaves like a solid.
Literature
These lectures are mostly based on the following references:
A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Rational Mech. Anal. (2002), 181-204
G. Dolzmann, Variational methods for crystalline microstructure - analysis and computation, Springer Lecture Notes in Mathematics 1803, 2003
S. Müller, Variational methods for microstructure and phase transitions,
Springer Lecture Notes in Mathematics 1713, 1999
Georg K Dolzmann
Last modified: Sat May 28 14:30:02 EDT 2005