Plastic Instability and Failure of Sheet Metals Subjected to Complex Stress States

Abstract

The rapid development of new emerging classes of steels has outpaced the methodologies and modelling strategies to exploit the superior mechanical properties in the design stage of structural automotive lightweight components. The conventional in-plane forming limit curve, employed to assess part feasibility, fails to account for the delay in plastic instability due to bending and tool contact pressure and can lead to overly conservative product designs, especially in materials with low inherent formability. This research focuses on the characterization and methodology development to account for process and boundary conditions in acute localization in combined loading of sheet metals. The ISO 12004-2 standard was followed for the baseline characterization of the stretching-dominated forming limits of the AA5182 and the DP980 AHSS. The non-linear strain path and higher limit strains in the Nakazima test, relative to the Marciniak test, are a natural consequence of the boundary conditions of the respective test method. A series of correction methods were applied to the Nakazima limit strains to linearize the strain path and to compensate for the triaxial stress state. The concept of phenomenological mapping under a constant in-plane stress ratio was shown to work reasonably well for the DP980 with its power law hardening characteristic but was problematic for the AA5182 with saturation-type hardening. Corrected limit strains for the AA5182 were sensitive to both the choice of the limit strain detection method and the hardening rate. Prediction of the plane stress forming limit curve (FLC) utilizing the phenomenological Modified Maximum Force Criterion (MMFC) correlated well with the Marciniak limit strains and the process-corrected Nakazima forming limits for the DP980. Key to avoiding artefacts associated with the shift of the FLC0 is a constrained calibration of the constitutive model to reflect the plastic uniform elongation obtained in a tensile test. The effect of a triaxial stress state on plastic instability was studied on a physical basis adopting the Hillier instability framework for the onset of instability. It was shown how commonly adopted instability criteria under plane stress loading are in fact special cases of the Hillier model under specific boundary conditions. Analytical solutions derived for triaxial loading revealed the explicit dependence of the boundary conditions upon plastic instability and highlighted the need to replace the traditional instability curve with a surface. Overall, a compressive normal stress caused an increase in limit strains for all studied boundary conditions. Highest formability gains were found for a contact pressure that proportionally evolves with the major in-plane stress. The formability increase in the presence of tool contact pressure between the phenomenological mapping criteria and the analytical solutions for plane strain tension only correlated for the proposed work-based mapping method and a hardening exponent of n = 0.1. An extension of the Hillier framework to diffuse localization was proposed in the Generalized Incremental Stability Criterion (GISC) that adopts the concept of neutral incremental stability to perform a quasi-stable transition of the stress state to plane strain tension associated with the formation of an acute neck. It was revealed that the MMFC is in fact physically motivated and represents one special case of the GISC under approximately proportional loading or proportional stressing with a prescribed minor load. The effect of out-of-plane loading with appreciable bending and tool contact pressure on the forming limits was characterized in an experimental test campaign comprising angular stretch-bend (ASB) tests and the VDA 238-100 tight radius V-Bend test. It was revealed that reliance upon the punch force for fracture detection in the V-Bend test may lead to inconclusive results since the punch force descends at large bend angles (>145°) as a consequence of the kinematic boundary conditions even in the absence of fracture. The proposed V-Bend stress metric mitigates inconclusive results by consideration of the geometric boundary conditions, punch force, and thinning of the cross-section. No new parameters were introduced such that the developed method is readily applicable to the VDA 238-100 test. A strain-rate based detection method was also considered as a local failure metric to detect lift-off of the specimen from the punch for both the 590R and the 270 Mild steel. Relative to the in-plane forming limit in Marciniak tests, forming limits for the 3rd Gen 1180 AHSS increased by a factor of approximately 2.6 in stretch-bending and by a factor of approximately five in tight-radius bending. To account for the delay in plastic instability in the presence of through-thickness stress-strain gradients, a control algorithm was developed to resolve the evolution of the stress-strain state over the sheet cross-section. The effect of a superimposed contact pressure on the stress state in plane strain stretch-bending was derived on continuum level and shown to effectively shift the strain path from plane strain tension to positive minor strains for material layers within the cross-section. The convex layer remains in a state of plane strain tension. To capture the mechanics, plasticity-related equations were derived from general bending mechanics and coupled with the GISC framework. It was revealed that an incremental multi-layer modelling approach is required to properly capture non-monotonic straining of the lower cross-section when the sheet wraps around the punch. Application to the 3rd Gen 1180 AHSS demonstrated that the developed instability framework is able to capture the overall trend of the experimental limit strains. A key outcome of this research is the understanding of the effect of a triaxial stress state and the importance of the boundary conditions on plastic instability. In particular, the analytical work to underline that the common assumption of a unique forming limit is fundamentally flawed and instead represents a forming limit surface are major accomplishments. Consideration of the combined effect of bending normal stresses and superimposed tool contact pressure in combined loading and implications upon acute necking limits constitute major novelties in sheet metal forming. The developed instability framework can be utilized by tool manufacturers to pursue a more aggressive product design as the part geometry can be tailored to the process window of the forming operation and costly tool re-cuts can be reduced.