Closed-form representation of matrix functions in the formulation of nonlinear material models

• Derivation of closed-form representation of 3×3 matrix functions.
• Closed form representation includes first and second derivatives.
• Solution to the ill-conditioning in the vicinity of multiple equal eigenvalues.
• Derivation based on symbolic description, automatic coding and automatic differentiation.
• Application includes finite strain Cam-Clay model.

The paper presents new approach to the evaluation of matrix functions operating over tensors that are essential part of formulation of complex nonlinear material models in mechanics of solids. A method is presented how to automatically derive numerically efficient closed-form representation of an arbitrary matrix function and its first and second derivatives for 3×3 matrices with real eigenvalues. The method offers an unique solution to the standard problem of ill-conditioning in the vicinity of multiple equal eigenvalues which is characteristic for all closed-form representations. A compiled library of subroutines with derived closed-form representation of most commonly used matrix functions along with their first and second derivatives has been created and is available for the use in general finite element environments. Consequently, the matrix functions can become as accurate, efficient and commonly available as their scalar counterparts, resulting in more common use of advanced strain and stress measures, such as Hencky strain measure which have so far been considered difficult for implementation. Accuracy and efficiency of the derived closed-form representations was compared with corresponding truncated series expansion and a speed up between 20 and 80 times has been observed depending on the matrix. The proposed methodology was tested on a set of selected nonlinear material models where matrix functions play an essential part in nonlinear finite element formulation.