Generalisation of Hooke’s law for finite strain to include the elastic range of strain-hardening materials

The representation theorem for isotropic tensor-valued functions of symmetric second-order tensors is considered in the context of two parameters based on the Lode and Fromm parameters. A geometrical representation is established using the concept of a characteristic representation intensity function. It is shown that this geometrical representation identifies the only admissible form of the representation intensity function to be piecewise linear and continuous. This conclusion imposes a restriction on how the representation theorem can be used to formulate constitutive equations. The representation theorem is used to formulate a generalisation of Hooke’s law for finite strain that is applicable to the initial elastic range of strain-hardening materials, including the elastic conditions at initial yield.