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Biological tissues have been shown to behave isotropically at lower strain values, while at higher strains the fibers embedded in the tissue straighten and tend to take up the load. Thus, the anisotropy induced at higher loads can be mathematically modeled by incorporating the strains experienced by the fibers. From histological studies on soft tissues it is evident that for a wide range of tissues the fibers have an oblique mean orientation about the physiological loading directions. Thus, we require a mathematical framework of tensors defined in nonorthogonal basis to capture the direction-dependent response of fibers under high induced loads. In this work, we propose a novel approach to determine the fiber strains with the aid of the contravariant tensors defined in an oblique coordinate system. To determine the fiber strains, we introduce a fourth-order contravariant fiber orientation transformation tensor. The approach helps us successfully in determining the fiber strains, for a family of symmetrically and asymmetrically oriented fibers, with the aid of a single anisotropic invariant. The proposed model was fitted with the experimental data from literature to determine the corresponding material parameters.

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.