Relaxed Energy for Transversely Isotropic Two-Phase Materials |
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Authors: | C Padovani M Šilhavý |
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Institution: | (1) Istituto di Scienza e Tecnologie dell' Informazione “Alessandro Faedo” ISTI–CNR, Via G. Moruzzi, 1, San Cataldo, 56100 Pisa, Italy;(2) Mathematical Institute of the AV ČR, Žitná 25, 115 67 Prague 1 Czech Republic |
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Abstract: | The paper gives a simple derivation of the relaxed energy W
qc
for the quadratic double-well material with equal elastic moduli and analyzes W
qc
in the transversely isotropic case. We observe that the energy W is a sum of a degenerate quadratic quasiconvex function and a function that depends on the strain only through a scalar variable.
For such a W, the relaxation reduces to a one-dimensional convexification. W
qc
depends on a constant g defined by a three-dimensional maximum problem. It is shown that in the transversely isotropic case the problem reduces to
a maximization of a fraction of two quadratic polynomials over 0,1]. The maximization reveals several regimes and explicit
formulas are given in the case of a transversely isotropic, positive definite displacement of the wells.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | double-well materials transverse isotropy quasiconvexity |
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