On the convexity of transversely isotropic chain network models

The basic concern of the present work is the systematic derivation of a new constitutive framework for transversely isotropic materials with a particular application to soft-tissue biomechanics. The constitutive equations are motivated by micromechanical considerations based on the statistical mechanics of long-chain molecules. The effective assembly of a representative number of individual model chains defines the macroscopic response of the overall chain network. The resulting class of models is characterized through a limited number of micromechanically motivated parameters with a clear physical interpretation. The newly derived framework captures not only transversely isotropic chain network effects but also classical isotropic network effects and classical transverse isotropy as special cases. Moreover, to account for biomechanically induced remodelling, we allow the material's principal axes to align progressively with the axes of principal strain. The convexity properties of the newly developed model are elaborated systematically for the selected test cases of uniaxial tension, equibiaxial tension, pure and simple shear. †Dedicated to Professor Ahmed Benallal on his 65th birthday.

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