New strong formulation for material nonlinear problems based on the particle difference method

Abstract

In this paper, we propose a strong form method for analyzing material nonlinear problems. The method utilizes the Particle Difference Method (PDM) which is classified as a strong form meshfree method and discretizes the governing equations based on a complete nodal computation without any integral formulation. The conventional strong form meshfree methods could not explicitly handle the nonlinear material model since they are mostly discretized based on the Navier's equation where kinematic variables are unified into displacement. To explicitly treat the nonlinear constitutive equation in the framework of strong formulation, a double derivative approximation is devised, which removes the need for the use of second order derivative approximation. The momentum equation is directly discretized through the double derivative approximation and is linearized by Newton's method to yield an iterative procedure for finding a converged solution. Stresses and internal variables are updated by the return mapping algorithm and efficiency of the iterative procedure is dramatically improved by the algorithmic tangent modulus. The consistency of the double derivative approximation was shown by the reproducing test. The accuracy and robustness of the developed nonlinear procedure were then verified through various inelastic material problems in one and two-dimensions.