A direct finite element implementation of the gradient-dependent theory

The enhanced non-local gradient-dependent theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical theories and the classical (local) continuum. They are successful in explaining the size effects encountered at the micron scale and in preserving the well-posedeness of the (I)BVP governing the solution of material instability triggering strain localization. This is due to the incorporation of an intrinsic material length scale parameter in the constitutive description. However, the numerical implementation of these theories is not a direct task because of the higher order of the governing equations. In this paper a direct computational algorithm for the gradient approach is proposed. This algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches (Int. J. Solids Struct. 1988; 24:581–597; Int. J. Numer. Meth. Engng 1992; 35:521–539; Eng. Comput. 1993; 10:99–121; Dissertation, 1994; Int. J. Numer. Meth. Engng 1996; 39:2477–2505; Int. J. Numer. Meth. Engng 1996; 39:3731–3755; Comput. Meth. Appl. Mech. Eng. 1998; 163:11–32; Comput. Meth. Appl. Mech. Eng. 1998; 163:33–53; Euro. J. Mech. – A/Solids 1999; 18:939–962; Int. J. Solids Struct. 2000; 37:7481–7499). A predictor–corrector scheme is proposed for the solution of the non-linear algebraic problem from the FEM. The expressions of the continuum and consistent tangent matrices are provided. The method is validated by conducting various numerical tests. As a result, pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered