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Article Dans Une Revue Comptes Rendus. Mécanique Année : 2022

Path-following methods for unstable structural responses induced by strain softening: a critical review

Résumé

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.

Dates et versions

hal-04019577 , version 1 (08-03-2023)

Identifiants

Citer

Giuseppe Rastiello, Hugo Luiz Oliveira, Alain Millard. Path-following methods for unstable structural responses induced by strain softening: a critical review. Comptes Rendus. Mécanique, 2022, 350 (G2), pp.205-236. ⟨10.5802/crmeca.112⟩. ⟨hal-04019577⟩
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