In order to fully close the equations outlined in Finite Element Nonlinear Discretization . it is necessary to define a constitutive equation that defines the relationship between stresses and strains. For large displacement problems, it is assumed that a relationship exists between Green-Lagrange strains and the second Piola-Kirchhoff stresses. Thus, the first-order tensor form can be  defined as:

(16–26)

The vector-matrix form is:

(16–27)

The second Piola-Kirchhoff stress vector is defined as:

(16–28)

The Green-Lagrange strain vector is defined as:

(16–29)

The constitutive matrix C would result in a 6x6 matrix tensor. A full derivation of constitutive matrix for large strains is not covered here but we just state that compressible hyper-elastic neo-Hookean material is used as a default for Nonlinear Elasticity. The neo-Hookean model is used as first choice because of its generality and because it is a natural extension of the usual linear isotropic model (Hooke’s law) to large deformation.