

Preprint 62/2007
The relaxation of two-well energies with possibly unequal moduli
Isaac Chenchiah and Kaushik Bhattacharya
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Submission date: 03. Jul. 2007
Pages: 75
published in: Archive for rational mechanics and analysis, 187 (2008) 3, p. 409-479
DOI number (of the published article): 10.1007/s00205-007-0075-3
Bibtex
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Abstract:
The elastic energy of a multiphase solid is a function of its microstructure. Determining
the infimum of the energy of such a solid and characterizing the associated
optimal microstructures is an important problem that arises in the modeling
of the shape memory effect, microstructure evolution and optimal design. Mathematically,
the problem is to determine the relaxation under fixed phase fraction of
a multiwell energy. This paper addresses two such problems in the geometrically
linear setting.
First, in two dimensions, we compute the relaxation under fixed phase fraction
for a two-well elastic energy with arbitrary elastic moduli and transformation
strains, and provide a characterization of the optimal microstructures and the associated strain.
Second, in three dimensions, we compute the relaxation under fixed phase
fraction for a two-well elastic energy when either (1) both elastic moduli are
isotropic, or (2) the elastic moduli are well-ordered and the smaller elastic modulus
is isotropic. In both cases we impose no restrictions on the transformation strains.
We provide a characterization of the optimal microstructures and the associated
strain.
We also compute a lower bound that is optimal except possibly in one regime
when either (1) both elastic moduli are cubic, or (2) the elastic moduli are wellordered
and the smaller elastic modulus is cubic; for moduli with arbitrary symmetry
we obtain a lower bound that is sometimes optimal. In all these cases we impose
no restrictions on the transformation strains and whenever the bound is optimal we
provide a characterization of the optimal microstructures and the associated strain.
In both two and three dimensions the quasiconvex envelope of the energy can
be obtained by minimizing over the phase fraction. We also characterize optimal
microstructures under applied stress.