Contravariant tensor algebra for anisotropic hyperelasticity

Analysis of tensors in oblique Cartesian coordinate systems always requires the definition of a set of orthogonal covariant basis vectors called the Reciprocal basis. This increases the complexity of the analysis and hence makes the method cumbersome. In this work a novel method is presented to effectively carry out the various transformations of tensors to and between oblique coordinate system/s without the need to create the covariant reciprocal basis. This will simplify the procedure of transformations involving problems where tensors are required to be defined in the oblique coordinate system. This work also demonstrates how the analysis of contravariant tensors can be applied to hyperelasticity. Continuum material and damage models can integrate this approach to model anisotropy and non linearity using a much simpler approach. The accuracy of the models was illustrated by matching the predictions to experimental results. A finite element analysis of material and damage model based on contravariant tensors was also carried out on a simple geometry with a re-entrant corner.

Direct Eulerian formulation of anisotropic hyperelasticity

Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangean strain tensor referred to the initial configuration and Lagrangean structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. Abstract In the present note, we show that the described two-step procedure is not necessary and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is further applied to the particular case of the transverse isotropy for the sake of illustration.