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2.4.1 Isotropic Hyperelasticity
The hyperelastic constitutive equations discussed so far are unrestricted in their application. Isotropic material symmetry is defined by requiring the constitutive behavior to be independent of the material axis chosen and, consequently, must only be a function of the invariants of , where the invariants of C are defined here as, As a result of the isotropic restriction, the second Piola-Kirchhoff stress tensor can be written as, The second order tensors formed by the derivatives of the invariants with respect to C can be evaluated as follows: Introducing expressions (2.4.1-4) into equation (2.4.1-3) enables the second Piola-Kirchhoff stress to be evaluated as, where , , and .
The Cauchy stresses can now be obtained from the second Piola-Kirchhoff stresses by using (2.3.3-5): Note that in this equation , , and still involve derivatives with respect to the invariants of . However, since the invariants of are identical to those of , the quantities , and may also be considered to be the derivatives with respect to the invariants of .