Fine phase mixtures as minimizers of energy
type: Item
instance of: scholarly article
title: Fine phase mixtures as minimizers of energy
zbMATH Open document ID: 0629.49020
DOI: 10.1007/BF00281246
published in: Archive for Rational Mechanics and Analysis
publication date: 1987-01-01T00:00:00Z
review text: Solid-solid phase transformations lead to certain characteristic microstructural features involving fine mixtures of phases. In this paper the martensitic transformation is investigated by theoretical methods. The main idea that the austenit/finely twinned martensite interface is modelled by certain minimizing sequences for the total free energy. It is assumed that the body prefers to be deformed in three states specified by three constant deformation gradients E, \(F^+\), \(F^-\). Here E is the unit tensor. The authors show that it is possible to arrange a very fine mixture of layers \(F^+/F^-/F^+/..\). on one side of an appropriately oriented interface so that the average deformation gradient of these layers does approximately satisfy conditions of compatibility with E. These configurations are minimizers of the total free energy. The finely twinned configurations of martensites are described as the approximations of generalized curves in Sobolev space. It is shown that the failure of lower semicontinuity of the free energy functional is a typical property for solids which change phase. The paper gives some different examples of fineness in energy minimizers. These examples are as follows: 1. Fine twins in minimization problems with no absolute minimum; 2. Strongly elliptic energy with minimizers having fine boundary wrinkles; 3. Minimizers of energy having a finer and finer mixture of phases as an interface is approached from one side.
Mathematics Subject Classification ID: 49S05
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 74A99
Mathematics Subject Classification ID: 74E30
zbMATH DE Number: 4024133
zbMATH Keywords: Solid-solid phase transformations
zbMATH Keywords: fine mixtures of phases
zbMATH Keywords: martensitic transformation
zbMATH Keywords: twinned martensite
zbMATH Keywords: free energy
zbMATH Keywords: energy minimizers
Wikidata QID: Q116037667
author: J. M. Ball
author: Richard D. James
reviewed by: István Ecsedi
MaRDI profile type: Publication
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: Strict convexity, strong ellipticity, and regularity in the calculus of variations
cites work: Null Lagrangians, weak continuity, and variational problems of arbitrary order
cites work: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity
cites work: Quasiconvexity at the boundary, positivity of the second variation and elastic stability
cites work: \(W^{1,p}\)-quasiconvexity and variational problems for multiple integrals
cites work: Lyapunov functions for thermomechanics with spatially varying boundary temperatures
cites work: Equilibrium configurations of crystals
cites work: Some phase transitions in crystals
cites work: Constitutive theory for some constrained elastic crystals
cites work: Some surface defects in unstressed thermoelastic solids
cites work: Q3802724
cites work: Variational methods for elastic crystals
cites work: Q3278168
cites work: Two-phase deformations of elastic solids
cites work: Finite deformation by mechanical twinning
cites work: Displacive phase transformations in solids
cites work: Q3977298
cites work: Q5666248
cites work: Explicit relaxation of a variational problem in optimal design
cites work: Regularization of optimal design problems for bars and plates. I
cites work: Absolute continuity on tracks and mappings of Sobolev spaces
cites work: Q3768459
cites work: Q3051526
cites work: Q3793816
cites work: Q3865070
cites work: On the stability of conformal mappings in multidimensional spaces
cites work: Liouville's theorem on conformal mappings for minimal regularity assumptions
cites work: On failure of the complementing condition and nonuniqueness in linear elastostatics
cites work: On the positivity of the second variation in finite elasticity
cites work: Q3320728
cites work: Q4435456
description: scientific article; zbMATH DE number 4024133