Marciniak-Kuczynski Approach Research Articles

An improved Marciniak-Kuczynski approach for predicting sheet metal formability

The Marciniak-Kuczynski (M-K) approach is the most renowned numerical method for forming limit strain curve (FLC) determination. In the commonly used M-K model, the thickness imperfection orientation (ψ) in the sheet plane is variable and the analysis is looped for different initial imperfection orientations (ψ0) for each strain route (ρ), which causes enormous computational waste. In addition, the predicted necking angle may obviously deviate from the theoretical one. This paper investigates the initial imperfection orientation and the necking angles (ψ*) of the AA5182-O sheet by both Nakazima experiments and M-K analysis. The effects of anisotropy on the necking angles and forming limit strains are investigated numerically. The results indicate that the necking orientation at equibiaxial stretching corresponds to the direction of the maximum r-value. The reason of necking angle deviation is well explained. Furthermore, the theoretical equation of necking angle is modified by an empirical formula, which is related to the ratio of r0 to rmax near the equibiaxial expansion. In light of the constant orientation of the imperfection deformation band during stretching, an improved M-K method is proposed based on the pre-assigned and non-evolving groove orientation. Compared with the modified M-K approach, this improved method achieves a higher calculation efficiency with a 97.7% reduction in computing time while maintaining the same level of accuracy.

Crystal plasticity finite element–Marciniak-Kuczynski approach with surface roughening effect in predicting formability of ultra-thin ferritic stainless steel sheets

Forming Limit Diagrams (FLDs) of 0.1 and 1 mm thick ultra-thin ferritic stainless steel (FSS) sheets are predicted using a combination of the crystal plasticity finite element (CPFE) approach and the Marciniak-Kuczynski (M-K) model. In a scale of few microns, especially for thin materials, surface roughness significantly affects sheet formability; therefore, the M-K model is further modified to address the surface roughening effect. The CPFE combined M-K model with the surface roughening effect can reproduce experimental FLDs of two FSS sheets, unlike the method without considering the surface roughening effect. The results highlight the importance of surface roughening, particularly in thinner sheets, where the portion of surface roughness is an order higher than that in the thicker sheets. The evolution of the deformation texture under various loading paths is further investigated and discussed. The findings coincide with reports of previous studies.

A Crystal Plasticity Formulation for Simulating the Formability of a Transformation Induced Plasticity Steel

Abstract The enhancement in formability of new advanced high strength steels (AHSS), such as duplex stainless steel, arises from increased hardening and ductility from complex deformation mechanisms, such as the transformation-induced plasticity (TRIP) effect. However, the interaction of dislocation and transformation mechanisms during deformation for various strain paths presents a challenge in evaluating formability. High fidelity simulation tools for evaluating formability need to capture these complex deformation mechanisms to allow manufacturers to realize their potential benefits. This work presents a rate-dependent crystal plasticity model with a micro-mechanics based transformation criteria to simulate the mechanical response of TRIP steel. A new stress-based transformation criterion, based on the micromechanics of fault band intersection on habits, was developed to initiate transformation. This model inherently captures the strain path effects of martensite transformation through the accumulated shear strain on slip systems. Simulations are calibrated and compared to experimental measurements of duplex stainless steel. Polycrystalline aggregate simulations show that although high Schmid factor habit planes were favorable for transformation, competition exists between the lower Schmid factor dislocation planes that generate higher elastic stress needed for transformation. The calibrated model is then used to predict the forming limit diagram using the Marciniak-Kuczynski approach. The mechanism of transforming from low strength austenite to high strength martensite showed enhanced formability by at least 20% compared to without transformation. This is achieved by the TRIP mechanism suppressing localization at critical moments during deformation. A parametric study of martensite transformation reveals the sensitivity of controlling the timing of martensite generation for improving formability.

Crystal Plasticity Based Transformation Induced Plasticity Formulation for Predictions of Forming Limit Diagrams

Significant effort has been invested into the development and research of Advanced High Strength Steel. In particular, transformation induced plasticity (TRIP) steel has shown promise as a candidate for lightweighting of vehicles because of their enhanced formability. However, predictive modeling of TRIP steel in formability is difficult due the combination of dislocation and transformation mechanisms occurring during deformation for various strain paths. This work presents a crystal plasticity constitutive model to simulate the behaviour of deformation induced γ → α’ transformation. A physics based transformation model is implemented into the crystal plasticity framework to fully capture the behavior of TRIP assisted steels. Simulations are calibrated and compared to experimental measurements of Duplex Stainless Steel. The calibrated model is then used to predict the forming limit diagram using the Marciniak-Kuczynski approach. The results highlight the importance of a physics based used in simulation to fully utilize TRIP assisted steels.

Investigation on the forming limits of 5754-O aluminum alloy sheet with the numerical Marciniak–Kuczynski approach

The aim of this work was to study the formability of 5754-O aluminum alloy sheet through utilizing the finite element M–K approach. The uniaxial tensile test and the Nakajima test were performed respectively for determining the mechanical properties and the forming limit strain pairs. The finite element M–K approach considering the Swift model, the Yld2000-2d yield function and the groove orientation was developed through utilizing software Abaqus. The predicted forming limit curve (FLC) agreed well with the Nakajima test results. Furthermore, the limit strains with the consideration of pre-strain under uniaxial tensile condition were experimentally determined, and numerically predicted by the original M–K approach. The numerical limit strains under different pre-strain paths and pre-major strains showed that the M–K approach could only predict partial FLC when the effective strain caused by the pre-strain was larger than the one at plane strain state. This phenomenon was interpreted by the evolution of the stress state in the groove of the M–K theory, i.e., the change of the strain path in loading process resulted in the stress state rotation, and consequently made the ratio of major strain increment in the groove to that out the groove increase when the stress state in the groove approached the stress state at plane strain state.

Forming limits of dual phase steels using crystal plasticity in conjunction with MK approach

A viscoplastic self-consistent crystal plasticity model was used to describe the mechanical behavior of two dual phase steel samples DP780 and DP1000. Mechanical anisotropy of these alloys was observed in uniaxial tests conducted along various directions. Additionally, X-ray diffraction was conducted to obtain the averaged crystallographic texture of ferrite and martensite. The hardening parameters were identified by fitting with the flow stress-strain curve obtained from bulge tests. The model-predicted and experimental flow-stress curves and R-values were compared in order to estimate the adequacy of the crystal plasticity model to describe the anisotropic behavior of the dual-phase steel samples. Furthermore, the crystal plasticity model, in conjunction with the Marciniak-Kuczynski approach, was used to predict forming limit diagram. The predictive accuracy was estimated by comparing with the experimental forming limit strains obtained through Nakazima tests. It turned out that several assumptions made in the current study led to somewhat poorer predictive accuracy in comparison with the previous reports, which implies that certain improvements in the model application are required for successful and valid model applications.

Numerical investigation of the combined effects of curvature and normal stress on sheet metal formability

A number of parts and components involved in the automotive industry are made of thin bent sheets, which are subjected to out-of-plane compressive stresses in addition to traditional in-plane stresses. Unfortunately, the classical predictions based on the conventional concept of Forming Limit Diagram (FLD) are no longer relevant when the strain distribution is heterogonous over the sheet thickness. Therefore, this conventional FLD concept is not capable of accounting for the effect of out-of-plane stresses on the onset of localized necking. The aim of the present contribution is to propose an extension of the well-known Marciniak–Kuczynski approach to simultaneously account for the effect of curvature and normal stress on formability prediction. The new developed tool allows predicting the limit strains for the whole range of strain paths. The mechanical behavior of the studied sheets follows the rigid–plastic flow theory. Through numerical results, it is shown that both curvature and normal stress tend to increase the formability limit of the sheet metal.

Prediction of Forming Limit Diagrams under combined Bending-Stretching loadings

A number of parts and components involved in the automotive industry are made of thin bent sheets. Unfortunately, the classical predictions based on traditional Forming Limit Diagrams are not relevant when the strain distribution is heterogonous over the thickness, as is the case for bent sheet metals. The aim of the present contribution is to propose an extension of the well-known Marciniak–Kuczynski approach to account for the effect of bending on formability prediction. The new developed tool allows predicting the limit strains for the whole range of strain paths. The mechanical behavior of the studied sheets follows the rigid–plastic flow theory. Through numerical results, it is shown that bending tends to decrease the formability limit of the sheet metal.

Effect of nonlinear strain paths on forming limits under isotropic and anisotropic hardening

The path-dependence of the conventional Forming Limit Diagram (FLD) is an important issue for its applications in industry. Great efforts have been made to understand the nature of the path-dependence with both experimental and theoretical approaches, many of them attempting to find a path-independent way for the application of forming limits. In this paper, we focus on the nonlinear strain path effect on forming limit predictions using both isotropic and anisotropic hardening models. The Forming Limit Diagram (FLD), Forming Limit Stress Diagram (FLSD) and Forming Limit Effective Strain Diagram (epFLD) of sheet metals subject to linear and nonlinear strain paths are analyzed and compared using the Marciniak–Kuczynski approach. An anisotropic hardening model based on Yoshida and Uemori development is adopted in this study, and it is coupled with the traditional Hill’48 yield surface. This model is capable of describing the complex Bauschinger phenomenon after the material undergoes the reverse loading process such as the early re-yielding, work-hardening stagnation and permanent softening. Two different scenarios for the change of strain paths are also investigated. In the first scenario, the sheet material is initially loaded with a fixed strain increment ratio, unloaded to the free stress state, and then reloaded with a different strain increment ratio until the forming limit is reached. In the second scenario, the material does not undergo elastic unloading. Instead, the strain path is abruptly changed to a different strain increment ratio and the material undergoes continuous loading until the forming limit is reached. It is found that the work-hardening behavior after the pre-straining and the loading scenario plays an important role in the path dependent behavior of forming limits. Detailed analysis reveals that the M–K approach may have contributed to the significance of path-dependence observed in this study, especially at high pre-strain levels.

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

Abstract An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.

Predictions of forming limit diagrams using crystal plasticity models

Crystal plasticity models are used in conjunction with the Marciniak-Kuczynski approach to predict forming limit diagrams (FLDs) of different annealed aluminium alloy sheet (AA6xxx series) with various initial crystallographic textures. The numerically determined FLDs are compared with experimental data. Two distinct models are used for FLDs predictions. The first method involves the calculation of yield surfaces from a rigid-plastic rate insensitive crystal plasticity model without recourse to the use of analytical yield functions. In the second method, the response of the material is based on an elastic-viscoplastic crystal plasticity model. Both methods take into account texture evolution during deformation. By taking into consideration the elastic part of the deformation, the elastic-plastic model can predict the shift of the forming limit curve minimum towards positive minor strains. It is found that the Goss orientation 011 〈 100〉 present in the initial texture as well as the microstructure influence significantly the formability.

On crystal plasticity FLD analysis

This paper is concerned with the computation of forming limit diagrams (FLDs) using a rate-sensitive polycrystal plasticity model together with the Marciniak–Kuczynski approach. Sheet necking is initiated from an initial imperfection in terms of a narrow band. The deformations inside and outside the band are assumed to be homogeneous and conditions of compatibility and equilibrium are enforced across the band interfaces. Thus, the polycrystal model needs only to be applied to two polycrystalline aggregates, one inside and one outside the band. Each grain is modelled as an FCC crystal with 12 distinct slip systems. The response of an aggregate comprised of many grains is based on an elastic–viscoplastic Taylor‐type polycrystal model developed by Asaro and Needleman (1985). The effects of initial imperfection intensity and orientation, initial distribution of grain orientations, crystal elasticity, strain rate sensitivity, single slip hardening and latent hardening on the FLD are discussed in detail. The predicted FLD is compared with experimental data for an aluminium alloy sheet.

Effects of plastic anisotropy and yield surface shape on sheet metal stretchability

The influence of plastic anisotropy and the shape of the yield surface on localized necking of thin metal sheets is examined. Forming limit curves (FLCs) of strain hardening, rate-sensitive sheets including Ti alloys, Al alloys, and steels are calculated on the basis of the Marciniak-Kuczynski approach using the quadratic Hill or the Drucker yield function in conjunction with either the flow or deformation theory of plasticity. The roles of the R-value and the yield surface shape in biaxial stretchability of sheet metals are delineated and discussed in relation to the plasticity theories and yield functions. It is concluded that the limit strains decrease with increasing R-value in the e2 > 0 region of an FLC but increase with the R-value in the e2 0 region, and are independent of the R-value at plane strain conditions. These mixed, strain-path dependent effects are explained in terms of the shape of the yield surface and a recently proposed critical thickness strain criterion.