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Restrictions on microstructure. (English) Zbl 0808.73063

Summary: We consider the following question: given a set of matrices \({\mathcal K}\) with no rank-one connections, does it support a nontrivial Young measure limit of gradients? Our main results are these: (a) a Young measure can be supported on four incompatible matrices; (b) in two space dimensions, a Young measure cannot be supported on finitely many incompatible elastic wells; (c) in three or more space dimensions, a Young measure can be supported on three incompatible elastic wells; and (d) if \({\mathcal K}\) supports a nontrivial Young measure with mean value 0, then the linear span of \({\mathcal K}\) must contain a matrix of rank one.

MSC:

74A60 Micromechanical theories
74M25 Micromechanics of solids
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
74A15 Thermodynamics in solid mechanics
49Q20 Variational problems in a geometric measure-theoretic setting
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