Transverse Impact of a Hyperelastic Stretched String

Abstract

Governing equations are derived for the plane motion of a stretched hyperelastic string subjected to a suddenly applied force at one end. These equations can be put in the form of a quasilinear system of first-order partial differential equations, which is totally hyperbolic for an admissible strain energy function. There are, in general, two wave speeds and two corresponding shock speeds. Special consideration is given to the jump relations across the shocks. Similarity solutions for a string moved at one end in loading or unloading are obtained for a general hyperelastic solid. The results are applicable to the familiar neo-Hookean or Mooney-Rivlin material, and the nature of the solution for another special hyperelastic material is discussed. These solutions are valid for a semi-infinite string, or until the first reflection occurs. It is shown that a special case of the similarity solution is valid for the normal impact of a stretched string by a constant speed, point application of load. Exact solution to the equations for the neo-Hookean model is derived in terms of elliptic integrals, and some numerical results are provided.