Thomas B. Stoughton

A general forming limit criterion for sheet metal forming

The forming limit of sheet metal is defined to be the state at which a localized thinning of the sheet initiates during forming, ultimately leading to a split in the sheet. The forming limit is conventionally described as a curve in a plot of major strain vs. minor strain. This curve was originally proposed to characterize the general forming limit of sheet metal, but it has been subsequently observed that this criterion is valid only for the case of proportional loading. Nevertheless, due to the convenience of measuring strain and the lack of a better criterion, the strain- based forming limit curve continues to play a primary role in judging forming severity. In this paper it is shown that the forming limit for both proportional loading and non-proportional loading can be explained from a single criterion which is based on the state of stress rather than the state of strain. This proposed criteria is validated using data from several non-proportional loading paths previously reported in the literature for both aluminum and steel alloys. In addition to significantly improving the gauging of forming severity, the new stress-based criterion is as easy to use as the strain-based criterion in the validation of die designs by the finite element method. However, it presents a challenge to the experimentalist and the stamping plant because the state of stress cannot be directly measured. This paper will also discuss several methods to deal with this challenge so that the more general measure of forming severity, as determined by the state of stress, can be determined in the stamping plant.

A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming

Abstract Spitzig and Richmond [Acta Metall. 32 (1984) 457] proposed that plastic yielding of both polycrystalline and single crystals of steel and aluminum alloys shows a significant sensitivity to hydrostatic pressure. They further showed that under the associated flow rule, this pressure sensitivity leads to a plastic dilatancy, i.e. permanent volume change, that is at least an order of magnitude larger than observed. Indeed, the plastic dilatancy for most materials is on the order of the measurement error and must be zero in the absence of phase change and significant void nucleation during plastic deformation. A non-associated flow rule based on a pressure sensitive yield criterion with isotropic hardening is proposed in this paper that is consistent with the Spitzig and Richmond data and analysis. The significance of this work is that the model distorts the shape of the yield function in tension and compression, fully accounting for the strength differential effect (SDE). This capability is important because the SDE is sometimes described through kinematic hardening models using only pressure insensitive yield criteria.

A non-associated flow rule for sheet metal forming

Abstract An improved model of material behavior is proposed that shows good agreement with experimental data for both yield and plastic strain ratios in uniaxial, equi-biaxial, and plane-strain tension under proportional loading for steel, aluminum and possibly other alloys. This model is based on a non-associated flow rule in which the plastic potential and yield surface functions are defined by quadratic functions of the stress tensor. The plastic potential aspect of the model is identical to that proposed by Hill for a quadratic anisotropic plastic potential defined in terms of measured r values. The new model differs in that the yield surface, although also defined by a quadratic function of the stress tensor, is defined independently of the plastic potential in terms of measured yield stresses. The model is developed and implemented in an FEM code that is based on a convected coordinate system. Since the associated flow rule, which assumes equivalency between the plastic potential and yield functions, is commonly accepted as a valid law in the theory of plastic deformation of most metals, the arguments for the associated flow rule are also discussed.

Influence of transverse normal stress on sheet metal formability

Abstract Since the late 1960s, the strain space forming limit diagram, FLD ϵ , has been nearly universally employed as a method to help predict sheet metal failure for the open die stamping process. Traditional FLD ϵ s in use today are based upon a plane stress assumption. However, a plane stress assumption may not always be valid for many sheet metal forming operations. Through the employment of a strain-to-stress space mapping procedure, a new sheet metal formability model that takes into account the through-thickness normal stress ( σ 3 ) is proposed. Good agreement with a limited set of experimental data is found.

Modeling of Directional Hardening Based on Non‐Associated Flow for Sheet Forming

This work describes a material model for sheet metal forming that takes into account anisotropic hardening under conditions of proportional loading. Conventional isotropic and kinematic hardening models constrain the shape of the yield function to remain fixed throughout plastic deformation, which is not consistent with most test data from aluminum alloys obtained under proportional loading. Conventional hardening models are shown to introduce systemic errors in stresses in different loading conditions at low and high levels of strain that tend to amplify the effect of stress miscalculation on the prediction of springback. A new model is described in which four stress‐strain functions are explicitly integrated into the yield criterion in closed form solution. The model is based on non‐associated flow so that this integration does not affect the accuracy of the plastic strain components. The model is expected to lead to a significant improvement in stress prediction under conditions dominated by proportion...

Material characterizations for Benchmark 1 and Benchmark 2

This report summarizes material testing on three metals used in the Numisheet 2014 Benchmark Study, a DP 600 steel sheet, a TRIP 780 steel sheet, and an aluminum alloy 5182-O sheet. The tests include r value, yield stress, and hardening in uniaxial tension at 15 degree increments of the loading axis in the plane of the sheet, r value, yield stress, and hardening in equal biaxial tension, and forming limit curves for all three metals. In addition, cyclic tension-compression tests along the rolling direction are reported for the DP 600 and aluminum alloy.

Paradigm Change: Alternate Approaches to Constitutive and Necking Models for Sheet Metal Forming

This paper reviews recent work proposing paradigm changes for the currently popular approach to constitutive and failure modeling, focusing on the use of non-associated flow rules to enable greater flexibility to capture the anisotropic yield and flow behavior of metals using less complex functions than those needed under associated flow to achieve that same level of fidelity to experiment, and on the use of stress-based metrics to more reliably predict necking limits under complex conditions of non-linear forming. The paper discusses motivating factors and benefits in favor of both associated and non-associated flow models for metal forming, including experimental, theoretical, and practical aspects. This review is followed by a discussion of the topic of the forming limits, the limitations of strain analysis, the evidence in favor of stress analysis, the effects of curvature, bending/unbending cycles, triaxial stress conditions, and the motivation for the development of a new type of forming limit diagram based on the effective plastic strain or equivalent plastic work in combination with a directional parameter that accounts for the current stress condition.

Advances in characterization of sheet metal forming limits

This paper accounts for nonlinear strain path, sheet curvature, and sheet-tool contact pressure to explain the differences in measured forming limit curves (FLCs) obtained by Marciniak and Nakajima Tests. While many engineers working in the sheet metal forming industry use the raw data from one or the other of these tests without consideration that they reflect the convolution of material properties with the complex processing conditions involved in these two tests, the method described in this paper has the objective to obtain a single FLC for onset of necking for perfectly linear strain paths in the absence of through-thickness pressure and restricted to purely in-plane stretching conditions, which is proposed to reflect a true material property. The validity of the result is checked using a more severe test in which the magnitude of the nonlinearity, curvature, and pressure are doubled those involved in the Nakajima Test.

Two-surface plasticity Model and Its Application to Spring-back Simulation of Automotive Advanced High Strength Steel Sheets

A two-surface isotropic-kinematic hardening law was developed based on a two-surface plasticity model previously proposed by Lee et al., (2007, Int. J. Plast. 23, 1189-1212). In order to properly represent the Bauschinger and transient behaviors as well as permanent softening during reverse loading with various pre-strains, both the inner yield surface and the outer bounding surface expand (isotropic hardening) and translate (kinematic hardening) in this two-surface model. As for the permanent softening, both the isotropic hardening and the kinematic hardening evolution of the outer bounding surface were modified by introducing softening parameters. The numerical formulation was also developed based on the incremental plasticity theory and the developed constitutive law was implemented into the commercial finite element program, ABAQUS/Explicit and ABAQUS/Standard using the user-defined material subroutines. In this work, a dual phase (DP) steel was considered as an advanced high strength steel sheet and uni-axial tension tests and uni-axial tension-compression-tension tests were performed for the characterization of the material property. For a validation purpose, the developed two-surface plasticity model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference and successfully validated with experiments.

Die Face Morphing With Formability Assessment

A die face morphing concept was recently introduced for quick die design for evolutionary products from their prior generations. Based on this concept, this paper proposes a strain increment method for early formability assessment by predicting strain distribution directly from the part-to-part mapping process. This method consists of mapping the finite element mesh to the part geometry, solving a part-to-part mapping function with smoothness and strain gradient penalties, and extracting strain increment from geometric morphing. It is shown, through a case study, that the strain field estimated by the proposed strain increment method compares well with that from the direct finite element analysis. Since this method does not require the knowledge on new die surface, such formability assessment can serve as a tool for early manufacturing feasibility analysis on the new part design.