The Use of Contravariant Tensors to Model Anisotropic Soft Tissues

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.

Automated summary

Biological tissues exhibit isotropic behavior at lower strain values, but show anisotropic behavior at higher strains as fibers straighten and bear load. This anisotropy can be mathematically modeled by considering the strains experienced by fibers. It is found that fibers have an oblique mean orientation around physiological loading directions, requiring a mathematical framework with tensors defined in a nonorthogonal basis to capture the direction-dependent response.