A simple polyconvex strain energy density with new invariants for modeling four-fiber family biomaterials

Abstract

We introduce in this paper a new hyperelastic model for the prediction of nonlinear mechanical properties of anisotropic hyperelastic materials under biaxial stretching. The proposed strain energy function (SEF) can be applied for understanding the nature of behavior laws for materials with four-fiber family structures, which has a large potential of applications, particularly in biomechanics, surgical and interventional therapies for peripheral artery disease (PAD). This SEF is built with a recent and new invariant system based on the mathematical theory of invariant polynomials. By recombining them in an appropriate manner, we demonstrate that it is possible to build a polyconvex integrity basis of invariants. Accuracy and reliability of the corresponding numerical model were validated by a comparison with experimental and numerical results extracted from Kamenskiy et al. (2014).11We warmly thank Assistant Professor Kamenskiy to have kindly provided us the numerical data corresponding to the measurements included in Kamenskiy et al. (2014). These results concerned diseased superficial femoral (SFA), popliteal (PA) and tibial arteries (TA) from one patient under planar biaxial extension. For each kind of arteries tested with 5 combinations of different biaxial stretches, the predicted results of the proposed model and the experimental data are consistent. Our model includes 7 material parameters and their identification result in a single solution because of the linear form we have chosen for the SEF with respect to the material parameters.