Constitutive equations and stiffness related properties for elastic and hyperelastic solid surfaces: Theories and finite element implementations

Abstract Surface stress and surface stiffnesses depend on the frame of reference. From the perspectives of theoretical analysis, experimental explanations, and finite element modeling, we perform a systematic investigation on the frame-dependent constitutive equations for surfaces, surface stress, and surface stiffness related properties. When interpreting experimental results and choosing finite element modeling parameters, extra attention needs to be paid to whether the surface stiffnesses are in the Eulerian or Lagrangian frame of reference. Area modulus, surface Young’s modulus, and surface Poisson’s ratio, the respective counterparts of bulk modulus, Young’s modulus, and Poisson’s ratio on surfaces are defined. The surface elasticities obtained from the biaxial and uniaxial tests for an incompressible gel from other researchers are well correlated by using the area modulus and the surface of the gel is explained as area-preserving, i.e. incompressible. A constitutive equation for hyperelastic incompressible surfaces is thus proposed. The finite element implementations of the constitutive equations for the elastic and hyperelastic surfaces are presented, which are based on the analogy of the equivalent shells with initial in-plane stresses. When the surface is incompressible and the deformation is large, the finite element computations by using the hyperelastic equivalent shell are more accurate and robust than those by using the elastic equivalent shell. The finite element results are verified with theoretical results from other researchers. The finite element methods in this work are easy and robust to be implemented since most conventional finite element codes and software can be directly used without any user-defined algorithm.