Stress resultant plasticity for plate bending in the context of roll forming of sheet metal

Abstract

Modeling of roll forming process of sheet metal requires efficient treatment of plastic deformations of thin shells and plates. We suggest a new general approach toward constructing the governing equations of bending of an elastic-plastic Kirchhoff plate on the structural mechanics level. The generic function of the isotropic hardening law is formulated in terms of variables, which are meaningful for a material surface, namely resultant bending moments and dissipative work. This function is then identified by comparison of the exact solutions for a uni-axial or isotropic bending experiment within the structural model and with the continuum model of a through-the-thickness element. We validate the approach against both the fully three-dimensional simulations as well as the results of the traditional elastic-plastic plate analysis, the latter one featuring the continuum laws of plasticity treated in the integration points along the thickness direction. We considered benchmark problems from the literature as well as a prototype for the roll forming problem. Besides good agreement regarding the shape of plastic zones and force-deflection characteristics, experiments also demonstrate higher computational efficiency of the new stress resultant model.

An enhanced stress resultant plasticity model for shell structures with application in sheet metal roll forming

“…To close the formulation, we need to properly restate the governing equations of plasticity in terms of these stress resultants and a small number of internal plastic variables. In this respect, we follow the concept developed in [16]. It features the plastic dissipation work A p as internal variable that governs the evolution law for the effective yield stress.…”

“…This phenomenon of "structural hardening" is not to be confused with actual material hardening, which is neglected here owing to the elastic-ideal plastic material law. The model proposed in [16] is limited to the geometrically linearized framework of the plate bending problem, where N can be neglected and only the bending moments M need to be considered. This simplification is no longer feasible in the present context of the geometrically nonlinear shell theory owing to the inherent coupling of membrane and bending forces.…”

“…Consequently, so-called stress resultant plasticity models are developed, see [3,6,10,11,15,16,26], where the elastic-plastic constitutive laws are stated directly in the space of the stress resultants, thus rendering the through-thethickness integration obsolete. Many publications regarding stress resultant metal shell plasticity make use of some variants of the Ilyushin yield criterion [15], see e.g.…”

“…Another attempt to augment the Ilyushin yield criterion was carried out in [16] for geometrically linear plate bending, meaning only the bending moments enter the yield criterion. The evolution of the yield surface was described by means of an isotropic hardening function, which uses the dissipation work as an internal hardening variable.…”

“…To overcome certain limitations of the previously reported stress resultant plasticity models, we here modify the plate model of [16] in an effort to treat large deformation problems of Kirchhoff-Love shells. In essence, we arrive at an enhanced shell stress resultant model by appropriately pairing the isotropic hardening law of [16] with the augmented yield surface proposed by Crisfield [6]. The thus obtained yield criterion is still approximate, but in contrast to the plate-model of [16] features an additional account for small to moderate membrane forces.…”