On the inversion of nonlinear constitutive relations for hyperelastic anisotropic materials

The polynomial elastic potentials represented by the power functions of their arguments are considered for hyperelastic anisotropic materials. The conditions for the elastic free energy $W(\varepsilon)$ and Gibbs potential $V(\mathbf{T})$ in isothermal processes are assigned so that the nonlinear constitutive relations can be inverted. For polynomial elastic potentials, whose coefficients are dependent on elastic constants of the second and third orders, a dependence between the coefficients of the potential $W(\varepsilon)$ (elasticity constants) and the coefficients of the potential $V(\mathbf{T})$ (elastic compliances) is obtained.
The relationships between the elastic constants and the coefficients of elastic compliance of the second and third orders for an isotropic material and for an anisotropic material corresponding to a cubic crystallographic system are found. For a copper crystal belonging to the cubic system, uniaxial loading along one of the anisotropy axes is considered. The stress-strain dependence obtained from direct and inverted relations coincides in the vicinity of zero.
The stress-strain dependence calculated using direct and inverted relations for copper crystals has made it possible to determine the strain range in which the results of calculations using direct and inverted relations differ by less than 5%.