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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.

This paper presents the results of a numerical study undertaken to assess the influence of residual stresses on the ductile tearing behavior of a high strength low toughness aluminum alloy. The Gurson–Tvergaard model was calibrated against conventional fracture toughness data using parameters relating to void nucleation, growth, and coalescence. The calibrated model was used to predict the load versus ductile tearing behavior of a series of full-scale and quarter-scale wide-plate tests. These center-cracked tension tests included specimens that contained a self-balancing residual stress field that was tensile in the region of the through-wall crack. Analyses of the full-scale wide-plate tests indicated that the model provides a good prediction of the load versus the ductile tearing behavior up to approximately 3mm of stable tearing. The influence of residual stress on the load versus the crack growth behavior was accurately simulated. Predictions of the load versus the crack growth behavior of full-scale wide-plate tests for crack extensions greater than 3mm and of the quarter-scale tests were low in terms of predicted load at a given amount of tearing. This was considered to result from (i) the “valid” calibration range in terms of specimen thickness and crack extension, (ii) the development of shear lips, and (iii) the differences in the micromechanism of ductile void formation under plane strain and under plane stress conditions.

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