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**Fine phase mixtures as minimizers of energy.**
*(English)*
Zbl 0629.49020

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.

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.

Reviewer: I.Ecsedi

### MSC:

49S05 | Variational principles of physics |

49J20 | Existence theories for optimal control problems involving partial differential equations |

74A99 | Generalities, axiomatics, foundations of continuum mechanics of solids |

74E30 | Composite and mixture properties |

### Keywords:

Solid-solid phase transformations; fine mixtures of phases; martensitic transformation; twinned martensite; free energy; energy minimizers
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\textit{J. M. Ball} and \textit{R. D. James}, Arch. Ration. Mech. Anal. 100, No. 1, 13--52 (1987; Zbl 0629.49020)

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