A comparison of second-order constitutive theories for hyperelastic materials

To develop a ®nite elastic constitutive equation for a real material, it is necessary to experimentally determine a number of material functions. This is much more dicult than determining the material constants needed for construction of a second-order constitutive equation for the same material. So it can be advantageous to use a second-order constitutive equation if it provides an adequate description of a materialʹs mechanical behavior over the deformation regimes to be considered. Of course, the second-order expansion of a constitutive relation always provides a good description of the mechanical response for suciently small strains and rotations. But by neglecting terms in these expansions which are higher than second order, we can construct a St Venant±Kirchho€-type material model which may be applied for any deformation. In this paper we investigate the circumstances under which such St Venant±Kirchho€-type second-order constitutive equations can successfully be used in lieu of the fully nonlinear elastic constitutive equation to solve boundary value problems. Two nonlinear elastic materials are considered: the generalized Blatz±Ko and the harmonic materials. For these materials we examine the performance of constitutive equations which are second order in the displacement gradient, the Biot strain, and the Green strain. A variety of boundary value problems are solved with these second-order constitutive equations, and the resulting solutions are compared to the solutions obtained with the corresponding fully nonlinear constitutive equation. We ®nd that both the nature of the material and the nature of the deformation have a large impact on the performance of these three second-order constitutive equations. Overall, the constitutive equation which is second order in the Biot strain provides the most accurate solutions over the largest range of strains and rotations