A nonlinear finite element plasticity formulation without matrix inversions

• A finite element formulation is proposed for plastic analysis of ductile materials.
• Local constitutive equations lead to the solution of one nonlinear scalar equation.
• The nonlinear scalar equation boils down to a single variable algebraic equation.
• The analytical solution of the algebraic equation is found in exact closed form.
• No iterative method is required for the solution of the local constitutive problem.
• An alternative expression of algorithmic plastic consistency condition is presented.
• The consistent tangent operator is derived without computing inverse matrices.
• The computational procedure shows a quadratic rate of asymptotic convergence.

In the present paper a robust computational procedure is proposed for the finite element solution of elasto-plastic boundary value problems in the structural analysis of ductile materials. An implicit numerical scheme is detailed based on a return mapping algorithm. In the present computational scheme the local constitutive equations lead to the solution of a nonlinear scalar equation which ultimately reduces to a single variable algebraic (polynomial) equation in the plastic rate parameter. The proposed integration scheme allows to find the analytical solution of the algebraic equation in closed form. Consequently, in the present algorithmic procedure no iterative method is required to solve the local constitutive problem. The algorithmic scheme presented herein is also characterized by an alternative formulation of the algorithmic counterpart of the plastic consistency condition. Furthermore, in the algorithmic strategy a suitable procedure is presented for the consistent linearization of elasto-plastic boundary value problems. In fact an alternative procedure is proposed herein for the consistent derivation of the tangent operator associated to the algorithmic scheme so that in the present approach the consistent tangent operator is determined without the necessity of computing matrix inversions. The presented computational approach combined with the proposed consistent tangent operator ensures a quadratic rate of asymptotic convergence when an iterative method is pursued for the global solution procedure of elasto-plastic boundary value problems. Numerical applications and illustrative examples are finally reported to show the robustness and effectiveness of the proposed computational procedure in the finite element analysis of elasto-plastic boundary value problems.