Restrictions on microstructure

Kaushik Bhattacharya, Nikan B. Firoozye, Richard D. James, Robert V. Kohn

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Abstract

We consider the following question: given a set of matrices Jf 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 k supports a nontrivial Young measure with mean value 0, then the linear span of Jf must contain a matrix of rank one.

Original languageEnglish (US)
Pages (from-to)843-878
Number of pages36
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume124
Issue number5
DOIs
StatePublished - 1994

Bibliographical note

Funding Information:
Section 2 draws from the Ph.D. thesis of Firoozye (New York University, 1990). The 'four-gradient example' was worked out while James was visiting the Mathematical Sciences Institute at Cornell University during the spring of 1988. Firoozye held a postdoctoral position at the Institute for Mathematics and its Applications in 1990-91, and a visiting position at Universitat Bonn in 1992-3. Firoozye and James were affiliated with the Mathematics Department of Heriot-Watt University during 1991-2. Bhattacharya was a graduate student at the University of Minnesota (Ph.D. 1991) then a Visiting Member at the Courant Institute (1991-3) during the execution of this research. Partial support from the following agencies is gratefully acknowledged: AFOSR (K.B., R.D.J., R.V.K.), ARO (K.B., R.D.J., R.V.K.), NSF (K.B., N.B.F., R.D.J., R.V.K.), ONR (R.D.J.), and SERC (N.B.F, R.D.J.).

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