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A complete characterization of invariant jointly rank-\(r\) convex quadratic forms and applications to composite materials. (English) Zbl 1107.74039
Summary: The theory of compensated compactness of F. Murat and L. Tartar [Cherkaev, Andrej (ed.) et al., Topics in the mathematical modelling of composite materials. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 31, 139–173 (1997; Zbl 0939.35019)] links the algebraic condition of rank-\(r\) convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-\(r\) convex forms arises. In the present paper, we define the concept of extremal 2-forms and characterize them in the rotationally invariant jointly rank-\(r\) convex case.
MSC:
74Q20 Bounds on effective properties in solid mechanics
74Q15 Effective constitutive equations in solid mechanics
74E30 Composite and mixture properties
15A90 Applications of matrix theory to physics (MSC2000)
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