Smooth hyperelastic potentials for 1d problems of bimodular materials

Introduction. A large number of natural and artificial materials exhibit various mechanical properties under compression and tension. Such materials are called bimodular. It can also be noted that materials having the same modulus of elasticity under compression and tension during testing exhibit bimodular properties during their operation in building structures.

Materials and methods. The research described in the paper is related to the construction of a family of one-parameter smooth infinitely differentiable hyperelastic potentials for incompressible bimodular materials in one-dimensional motion; the derivation of closed-form solving equations, including the equation of motion, the Hadamard compatibility equation and the energy balance equation; performing a dynamic analysis of the propagation of harmonic vibrations in a semi-infinite rod. The developed method is based on a combined mechanical and thermodynamic approach combined with an energy-saving explicit numerical Lax – Wendroff scheme.

Results. A family of one-parameter infinitely differentiable hyperelastic potentials for three-dimensional infinitesimal problems on bimodular isotropic materials is constructed, which gives a set of homogeneous approximations to the discontinuous stepwise modulus of elasticity adopted in the initial one-dimensional bimodular formulation. The introduced dependencies make it possible either to obtain analytical solutions or to derive explicit solving equations for a number of static and dynamic problems. The theorem of convergence to a discontinuous module for bimodular materials is proved.

Conclusions. Shock wave fronts that appear in one-dimensional rods made of nonlinear materials modeled by a family of smooth hyperelastic potentials clearly demonstrate that their formation is not caused by a discontinuity in the stress-strain ratio corresponding to bimodular materials. Shock wave fronts occur in materials modeled by the considered smooth hyperelastic potentials both in the case of bimodular material and any other hyperelastic material. The propagation of shock wave fronts leads to the dissipation of mechanical energy, which implies a decrease in amplitudes with distance.

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