On isotropic tensors and elastic moduli

Abstract The use of even-order isotropic tensors in non-linear elasticity theory is discussed in this paper. A notation is adopted through which these tensors can be represented conveniently so that their interdependence is clearly shown. Information about the number of independent elastic constants required is then readily available for use in an expansion of the stress to various orders in the strain relative to the undistorted configuration of the elastic material in question. For an incompressible isotropic hyperelastic solid, it is shown that each principal component of the distortional part of the stress is expressible as a function only of the corresponding principal component of strain to the fourth order. Under certain conditions, which are not too restrictive, this result can be extended to higher orders.