[Méthodes de pilotage indirect du chargement pour la description de réponses structurales instables induites par l’adoucissement en déformation : une analyse critique]
Des techniques de pilotage indirect du chargement pour décrire des réponses structurelles instables induites par l’adoucissement en déformation sont discutées. Après avoir rappelé les principaux ingrédients des formalismes introduits par Riks et Crisfield pour les méthodes de longueur d’arc pour le traitement de non-linéarités géométriques, un lien entre les deux façons (monolithique et partitionnée) de résoudre le problème d’équilibre augmenté qui en résulte est établi en se servant de la formule de Sherman–Morrison. L’utilisation de solveurs monolithiques repose sur l’hypothèse de différentiabilité des équations de pilotage par rapport aux inconnues du problème, le champ de déplacement et le facteur de chargement. Cependant, lorsqu’il s’agit de décrire des instabilités d’origines matérielles, des phénomènes locaux sont souvent responsables de la réponse instable obtenue au niveau global. Par conséquent, il est parfois préférable d’introduire des équations de pilotage permettant de contrôler le maximum d’un champ défini sur le domaine de calcul (e.g., la variation d’une mesure scalaire de déformation, le taux de variation d’une variable interne du modèle constitutif). Dans ce cas, la différentiabilité de l’équation de pilotage n’est plus garantie en raison de la présence de l’opérateur de maximum, ce qui rend utilisable uniquement la formulation partitionnée. Plusieurs exemples issus de la littérature scientifique pour les deux classes d’équations de pilotage sont présentés. Ensuite, on dérive les formulations d’éléments finis correspondantes et on décrit les avantages et inconvénients pour chacune d’elles. Une comparaison détaillée entre les différentes formulations est présentée sur la base d’un cas test bidimensionnel simple simulant la localisation de l’endommagement dans une poutre soumise à une sollicitation de traction. Un cas de test impliquant l’apparition de plusieurs snap-backs dans la courbe de réponse globale est enfin illustré pour montrer la robustesse des formulations considérées.
Path-following methods for describing unstable structural responses induced by strain-softening are discussed. The main ingredients of the formalisms introduced by Riks and Crisfield for arc-length methods for geometrical non-linearities are presented. A link between two ways (monolithic and partitioned) of solving the resulting augmented equilibrium problem is discussed based on the Sherman–Morrison formula. The original monolithic approach assumes that the path-following constraint equation is differentiable with respect to the unknown displacement field and load factor. However, when dealing with material non-linearities, it is often preferred to consider constraint equations controlling the maximum of a field defined on the computational domain (e.g., a scalar strain measure, the rate of variation of an internal variable of the constitutive model). In that case, differentiability cannot be guaranteed due to the presence of the maximum operator. This makes only the partitioned formulation usable. Several path-following constraint equations from the literature are presented, and the corresponding implementations in the finite element method are discussed. The different formulations are compared based on a simple two-dimensional test case of damage localization in a beam submitted to tension. A test case involving multiple snap-backs is illustrated, finally, to show the robustness of the considered formulations.
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DOI : 10.5802/crmeca.112
Mot clés : Méthodes de pilotage indirect du chargement, Formulations monolithique et partitionnée, Méthodes dissipatives, Méthodes non dissipatives, Non-linéarités matériaux
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@article{CRMECA_2022__350_G2_205_0,
author = {Giuseppe Rastiello and Hugo Luiz Oliveira and Alain Millard},
title = {Path-following methods for unstable structural responses induced by strain softening: a critical review},
journal = {Comptes Rendus. M\'ecanique},
pages = {205--236},
publisher = {Acad\'emie des sciences, Paris},
volume = {350},
year = {2022},
doi = {10.5802/crmeca.112},
zbl = {07526206},
language = {en},
}
TY - JOUR AU - Giuseppe Rastiello AU - Hugo Luiz Oliveira AU - Alain Millard TI - Path-following methods for unstable structural responses induced by strain softening: a critical review JO - Comptes Rendus. Mécanique PY - 2022 SP - 205 EP - 236 VL - 350 PB - Académie des sciences, Paris DO - 10.5802/crmeca.112 LA - en ID - CRMECA_2022__350_G2_205_0 ER -
%0 Journal Article %A Giuseppe Rastiello %A Hugo Luiz Oliveira %A Alain Millard %T Path-following methods for unstable structural responses induced by strain softening: a critical review %J Comptes Rendus. Mécanique %D 2022 %P 205-236 %V 350 %I Académie des sciences, Paris %R 10.5802/crmeca.112 %G en %F CRMECA_2022__350_G2_205_0
Giuseppe Rastiello; Hugo Luiz Oliveira; Alain Millard. Path-following methods for unstable structural responses induced by strain softening: a critical review. Comptes Rendus. Mécanique, Volume 350 (2022), pp. 205-236. doi : 10.5802/crmeca.112. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.112/
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