Smooth hyperelastic potentials for 1D problems of bimodular materials

A family of one parametric infinitely differentiable hyperelastic potentials for three-dimensional infinitesimal problems of bimodular isotropic materials is constructed, yielding a set of uniform approximations to the discontinuous stepwise elastic modulus adopted in the original one-dimensional bimodular formulation. The introduced potentials enable either analytical solutions or construction of the explicit governing equations for a number of static and dynamic problems. Theorem of convergence to the discontinuous bimodular modulus is proved.