Dynamic finite element implementation of nonlinear, anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.