Determination of the tangent stiffness tensor in materials modeling in case of large deformations by calculation of a directed strain perturbation

The Finite Element Method in the field of materials modeling is often relying to the tangent stiffness tensor of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness tensor can be derived analytically only with great effort, if at all. Numerical methods like the forward-difference, the central-difference, and the complex-step derivative approximation approach are widely used for its calculation. For each of these methods it is necessary to generate a specific strain perturbation. However, in large strain formulations it is not possible to perturb the strain directly but one can only modify the deformation gradient.We present our methods to generate a directed strain perturbation for the Green-Lagrange, Euler-Almansi and logarithmic strain measures as a function of the deformation gradient and compare them with other commonly used methods. An increase in accuracy and rate of convergence can be achieved with the proposed procedures.