On complex-variable formulation for finite plane elastostatics of harmonic materials

Abstract

Although complex-variable formulation provides a powerful method for linear plane elastostatics, finite plane elastostatics does not admit an efficient complex-variable method. Here, to explore the potential of complex-variable methods to finite elastostatics, a new derivation is presented for a complex-variable formulation for plane-strain deformation of compressible hyperelastic harmonic materials, developed first by Varley and Cumberbatch [8]. The present derivation is remarkable in its mathematical conciseness, and has the potential to stimulate further interest in the complex-variable methods for finite plane elastostatics. To demonstrate the powerfulness of the complex-variable method, a complete solution is given for an interface crack in a special class of compressible harmonic materials, which defines a real parameter similar to the Dundurs's parameter in linear elasticity. In particular, this parameter becomes unity and all oscillatory singularities disappear when the asymptotic behavior of the harmonic materials obeys a constitutive restriction proposed by Knowles and Sternberg [10]. Finally, main features of finite deformation of the interface crack are discussed with a comparison to the well-known results of linear elasticity.