Numerical differentiation for use in integrating unified constitutive equations

TLDR: In this paper, a numerical method for evaluating partial derivatives of the constitutive equations is developed, thus avoiding the cumbersome task of analytically re-deriving the partial derivatives when improvements to the non-linear constitutive model are evaluated.

Abstract: The NONSS method for numerically integrating time-dependent constitutive equations requires evaluation of the partial derivatives of the constitutive equations. A numerical method for evaluating these partial derivatives has been developed, thus avoiding the cumbersome task of analytically re-deriving the partial derivatives when improvements to the constitutive model are evaluated. This method is based upon the adaptive finite-difference algorithm published by Stepleman and Winarsky. Modifications of their algorithm are described that allow its efficient use within an extended version of NONSS, called NONSS-ND. Using the MATMOD constitutive equations, the capabilities of NONSS-ND are demonstrated by comparing the results of a wide variety of deformation simulations with those of NONSS, for which the derivatives are computed from analytical solutions. The numerical differentiation within NONSS-ND is shown to be sufficiently accurate for use in integrating time-dependent constitutive equations, but results in a 30–100 per cent increase in computation time. Multi-element structural calculations are therefore found to be uneconomical, but NONSS-ND is well suited for testing alternative constitutive models owing to its accuracy and its general applicability to highly non-linear systems of differential equations.