A closed form solution for the uniaxial tension test of biological soft tissues

This paper concerns the modeling of biological soft tissues in the framework of anisotropic hyperelasticity. A closed form solution is proposed in the special case where uniaxial unconstrained tension loading is applied to a strip modeled by Holzapfel–Gasser–Ogden's (HGO) strain energy. The classical Cardano's formula is used to calculate the solution. This solution is investigated for different values of ββ which represents the angle between the collagen fibers and the circumferential direction. This study allows for the understanding of the reason why a one-to-one correspondence does not exist between the principal stretch and the fourth invariant of the right Cauchy–Green deformation tensor. It is demonstrated that the relationship becomes non-bijective when ββ is greater than a critical angle of 54.73∘54.73∘. The HGO model is implemented into an in-house finite element program and numerical results are in good agreement with theoretical ones.