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1 Computer Methods in Materials Science

1 2009

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Nonlocal integreal formulation for a plasticity-induced damage model

Andrade F. X. C, Pires F. M. A., de Sá J. M. A., Malcher L.

Computer Methods in Materials Science

2009

Vol. 9, No. 1

49-54

Numerical ductile failure prediction commonly involves the use of plasticity-induced damage constitutive models within the finite element method framework. Algorithms based on the so-called operator split concept, resulting in the standard elastic predictor/plastic corrector format, are particularly suitable for numerical integration of constitutive equations and are widely used in computational plasticity (Simo & Hughes, 1998). However, differences in the converged solution after the return mapping step may arise due to integration errors, leading to an undesirable sensitivity of the problem to some input parameters (e.g. the size of the global load-step). This is a direct consequence of the loss of accuracy of the integration algorithm, which dramatically reduces the precision of the prediction of ductile failure onset. This issue is investigated here with a simplified version of the Lemaitre?s damage model where only isotropic hardening is considered (de Souza Neto, 2002). The integration scheme for this material model has proved to be effective in situations where kinematic hardening can be disregarded, i.e., when load reversal is absent or negligible. Furthermore, its computational implementation is relatively straightforward and the resulting algorithm is very efficient. However, the accuracy of the algorithm, which can be assessed by means of isoerror maps, deteriorates as damage grows. One possible strategy to improve the accuracy of the solution is to reduce the global incremental load factor. This leads to the unwanted situation where a few critical Gauss points govern the global problem, yielding on loss of efficiency. In order to overcome this problem, we make use of a sub-stepping technique (Sloan, 1987; Pérez-Foguet et al., 2001). In this approach, we can subdivide the (pseudo) time-step, of a given critical Gauss point, into a finite number of sub-steps. As a result, it is possible to use larger steps at the global level while some critical points may demand sub-stepping. The impacts of using a sub-stepping approach with the simplified Lemaitre?s damage model are shown, in the present work, by means of iso-error maps at the Gauss point level. Furthermore, issues involved in the implementation of the sub-stepping technique within a finite element framework are addressed. It is well known that the introduction of new models into a nonlinear finite element problem with the Newton-Raphson method requires proper modifications both in the state update procedure and in the consistent tangent operator. Although the modification in the state update procedure is straightforward under infinitesimal strains, this is not the case under finite strains when considering a hyper-elastic based multiplicative elasto-plastic model. In addition, the derivation of the consistent tangent operator is also intricate. Here, we present some strategies to modify both procedures under infinitesimal and finite strains. Two simple examples are used to highlight the improvement of the numerical solution. The results show that the new strategy produces more accurate results together with less sensitivity to input parameters. Keywords: ductile failure, sub-stepping, damage, finite element method. References: DE SOUZA NETO, E.A. 2002. A fast, one-equation integration algorithm for the Lemaitre ductile damage model. Communications In Numerical Methods In Engineering, vol. 18, pp. 541?554. PÉREZ-FOGUET, A., RODRÍGUEZ-FERRAN, A., HUERTA, A. 2001. Consistent tangent matrices for substepping schemes. Computer Methods in Applied Mechanics and Engineering, vol. 190, pp. 4627-4647. SIMO, J.C., HUGHES, T.J.R. 1998. Computational Inelasticity. Springer, New York. SLOAN, S.W. 1987. Substepping scheme for the numerical integration of elastoplastic stress-strain relations. International Journal for Numerical Methods in Engineering, vol. 24, pp. 893-911.

Predykcja uszkodzenia indukowanego odkształceniem plastycznym powoduje zwykle konieczność wykorzystania metody elementów skończonych w zakresie sprężysto-plastycznym z modelem uszkodzeń. W miejscu gdzie tworzy się odkształcenia, cząstkowe równania różniczkowe tracą eliptyczności, co powoduje że rozwiązania numeryczne stają się zależne od siatki elementów. Jednym z rozwiązań tego problemu jest wprowadzenie wewnętrznej długości, która odpowiada zadanej wielkości mikrostrukturalnej materiału. W takim rozwiązaniu nielokalne sformułowania całkowe oraz rozszerzone sformułowania gradientowe są powszechnie wykorzystywane jako strategie regularyzacji. W niniejszej pracy sformułowanie elementów skończonych oparte o nielokalny schemat całkowania zaproponowany przez Strómberga & Ristinmaa rozszerzone zostało o uproszczoną wersję modelu uszkodzeń indukowanych plastycznie Lemaitre'a. Jako, że model ten ma wpływ jedynie na obliczenie sił wewnętrznych, jego implementacja w istniejących kodach MES jest stosunkowo prosta. Otrzymane wyniki z przykładowych obliczeń pokazują, iż implementowana strategia pozwala na pokonanie problemów związanych z lokalizacją odkształceń.