Semi-implicit finite strain constitutive integration of porous plasticity models P. Areias a,d,n,1 , T. Rabczuk b , J. César de Sá c a Department of Physics, University of Évora, Colégio Luís António Verney, Rua Romão Ramalho, 59, 7002-554 Évora, Portugal b Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstraße 15, 99423 Weimar, Germany c Mechanical Engineering Department, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal d ICIST, Instituto Superior Técnico, Lisboa, Portugal article info Article history: Received 11 December 2014 Received in revised form 16 May 2015 Accepted 19 May 2015 Keywords: Constitutive integration Porous plasticity Finite strains Semi-implicit Löwdin's method abstract Two porous plasticity models, Rousselier and Gurson–Tvergaard–Needleman (GTN), are integrated with a new semi-implicit integration algorithm for finite strain plasticity. It consists of using relative Green– Lagrange during the iteration process and incremental frame updating corresponding to a polar decomposition. Lowdin's method of orthogonalization is adopted to ensure incremental frame- invariance. In addition, a smooth replacement of the complementarity condition is used. Since porous models are known to be difficult to integrate due to the combined effect of void fraction growth, stress and effective plastic strain evolution, we perform a complete assessment of our semi-implicit algorithm. Semi-implicit algorithms take advantage of different evolution rates to enhance the robustness in difficult to converge problems. A detailed description of the constitutive algorithm is performed, with the key components comprehensively exposed. In addition to the fully detailed constitutive algorithms, we use mixed finite strain elements based on Arnold's MINI formulation. This formulation passes the inf– sup test and allows a direct application with porous models. Isoerror maps for two common initial stress states are shown. In addition, we extensively test the two models with established benchmarks. Specifically, the cylindrical tension test as well as the butterfly shear specimen are adopted for validation. A 3D tension test is used to investigate mesh dependence and the effect of a length scale. Results show remarkable robustness. & 2015 Elsevier B.V. All rights reserved. 1. Introduction For common metals at room temperature, void nucleation and particle debonding, void growth and coalescence are the mechanisms for crack growth from the process region [10]. Detailed multiple-scale analysis of these mechanisms require robust polycrystalline plasticity and representation of dislocation dynamics and grain boundary barriers (Roters et al. [30] present a comprehensive review). Therefore, the so-called phenomenological models (e.g. [18]) are considered a cost-effective tool for predicting damage and fracture in metals when overall quantities, like maximum load and dissipated energy, are of interest. The now well-established GTN model is based on the original yield surface by Gurson [16] and was subsequently modified by Tvergaard and Needleman [34,35]. The void growth part of the GTN model is based on the micromechanics of the ductile process. Phenomenological models formulated based on thermodynamical principles have also been proposed such as the Lemaitre model [18] and its subsequent modi fications. Among this class of constitutive models, the Rousselier model [31,4] is interesting from a computa- tional perspective as it is simpler and requires the speci fication of fewer parameters than the GTN model, while possessing the ability to closely match the predictions of the latter by determination of its parameters. It adheres to the idea that the occurrence of localization, crack initiation and propagation emerges as a direct consequence of strain softening due to void growth. Hence, unlike micro-mechanics models of porous metal plasticity, the Rousselier model does not directly model the mechanism of coalescence. However, two proper- ties are worth mentioning: 1. The Rousselier model predicts void fraction growth in pure shear. 2. The Rousselier yield surface has an isochoric vertex causing a non-zero plastic volumetric term component. As a more complete alternative to the Rousselier model, it was recently (see [22]) created an extended version of the GTN model, which includes the shear void fraction as an independent history variable. We here focus on variants of the GTN and Rousselier Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ finel Finite Elements in Analysis and Design http://dx.doi.org/10.1016/j.finel.2015.05.005 0168-874X/& 2015 Elsevier B.V. All rights reserved. n Corresponding author. 1 Researcher ID: A-8849-2013 http://www.researcherid.com/rid/A-8849-2013 Web-of-Science search: areias, p* Finite Elements in Analysis and Design 104 (2015) 41–55