An Implementation of the Gurson-Tvergaard-Needleman Plasticity Model for ABAQUS Standard Using a Trust Region Method

Abstract

The Gurson-Tvergaard-Needleman (GTN) model is a material plasticity model in which the accumulation of ductile damage is represented by the nucleation, growth and coalescence of micro-voids. The model has been implemented in full for the ABAQUS finite element code. The model supports fully the nucleation, growth, and coalescence of voids, and differs from the porous plasticity model provided in ABAQUS/Standard. The GTN model is just one model from a particular class of pressure-dependent plasticity models in which the response is dependent on the development of the hydrostatic stress as well as the deviatoric stress tensor. A formal derivation of the constitutive equations is presented in this paper. It is shown that the model can be formally represented by a coupled system of four non-linear equations. A novel approach to solving the equations has been adopted based on a hybrid solution method and a trust region to ensure convergence. The solution of non-linear equations in more than one variable is usually attempted using iterative methods. For the calculation of a material’s response, the method adopted must be sufficiently robust to ensure that the correct result is obtained at each of the material points in the component or structure being modelled. Moreover, the solution method must be as efficient as possible for practical use. In this paper, we present an implementation of a trust region method that allows the solution of the GTN constitutive equations to be derived with confidence. The method utilises iterative corrections and a trust region surrounding the current estimated solution. In the early stages of the iteration, when the estimate may be far removed from the true solution, the steepest descent method is used to improve the solution, while at later stages Newton’s method, with its superior convergence, is used. A hybrid step (part steepest descent step, part Newton step) may also be taken using Powell’s dogleg method with the constraint that the corrections do not take the solution outside the current trust region. A measure of the quality of each step is used to shrink or expand the radius of the trust region during the iteration. The solution algorithm has been implemented in Fortran 90 as a user subroutine for ABAQUS/Standard. The method provides faster convergence than the porous plasticity model in ABAQUS and allows for the representation of void coalescence. Examples of application of the GTN model to study the response of axi-symmetric bars are provided and comparisons are made with the porous plasticity model where appropriate.