Abstract
This paper describes some analytical results for the equilibrium configurations of a three (one austenitic and two martensitic) phase elastic isotropic material. Constitutive relations in each of the material phase are provided explicitly. Direct methods in the calculus of variations are employed to compute the relaxed energy. An analytical expression of the relaxed energy envelope is derived for a particular case where both the martensitic and austenitic phase share the same value of the shear modulus and the chemical energy of the austenitic phase becomes negative. The computed relaxed energy afterward is used in the energy minimization method for finding the equilibrium-state solution of the three phase elastic material. Deformations when computed using exact relaxed constitutive relations exhibit mesh independence and this property is also illustrated in a specific case by computing the deformation of a single crystal in tension and compression by using Finite Element Method.
Highlights
Relaxation methods have been employed in the energy minimization problems for the analysis of martensitic micro-structures in several different situations; see for instance the work of Ball et al.,1,2 Bartels et al.,3 Carstensen et al.,4,5 Gobbert et al.14,15 and Kohn.16 In general, research has shown that it is difficult to compute the exact relaxed envelope for a material with multi-well potential
Very few results can be found in literature on the explicit representation of the relaxed energy6–8,10,11,17–20,22 and most of them are for some special cases
An analytical expression for the relaxed energy is derived for a particular case where the shear modulus of all phase energies has same value and the chemical energy of the austenitic phase becomes negative
Summary
Relaxation methods have been employed in the energy minimization problems for the analysis of martensitic micro-structures in several different situations; see for instance the work of Ball et al., Bartels et al., Carstensen et al., Gobbert et al. and Kohn. In general, research has shown that it is difficult to compute the exact relaxed envelope ( known as quasi-convex envelope) for a material with multi-well potential. Very few results can be found in literature on the explicit representation of the relaxed energy and most of them are for some special cases Another idea is to compute the bi-conjugate function ( called second Legendre transform) of the given potential for three phase elastic material. They provide a more accurate constitutive description of the material They guarantee the existence of minimizers when employed in the energy minimization problems and enables the material to attain the equilibrium state in general and in particular with a unique minimizer in the case when the relaxed energy is exactly the convex envelope.
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