Stress based prediction of formability and failure in incremental sheet forming

A strain-based forming limit criterion is widely used in sheet-metal forming industry to predict necking. However, this criterion is usually valid when the strain path is linear throughout the deformation process [1]. Strain path in incremental sheet forming is often found to be severely nonlinear throughout the deformation history. Therefore, the practice of using a strain-based forming limit criterion often leads to erroneous assessments of formability and failure prediction. On the other hands, stress-based forming limit is insensitive against any changes in the strain path and hence it is first used to model the necking limit in incremental sheet forming. The stress-based forming limit is also combined with the fracture limit based on maximum shear stress criterion to show necking and fracture together. A derivation for a general mapping method from strain-based FLC to stress-based FLC using a non-quadratic yield function has been made. Simulation model is evaluated for a single point incremental forming using AA 6022-T43, and checked the accuracy against experiments. By using the path-independent necking and fracture limits, it is able to explain the deformation mechanism successfully in incremental sheet forming. The proposed model has given a good scientific basis for the development of ISF under nonlinear strain path and its usability over conventional sheet forming process as well.     

Investigation on the strain-path dependency of stress-based forming limit curves

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain ( $ \bar{\varepsilon } \geqslant 0.35 $ for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.     

On the use of effective limit strains to evaluate the forming severity of sheet metal parts after nonlinear loading

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ε _e ?≥?0.37 for AISI-1012 steel and ε _e ?≥?0.25 for AA-2008-T4 aluminum).     

Influence of out-of-plane compression stress on limit strains in sheet metals

The prediction of the forming limits of sheet metals typically assumes plane stress conditions that are really only valid for open die stamping or processes with negligible out-of-plane stresses. In fact, many industrial sheet metal forming processes lead to significant compressive stresses at the sheet surface, and therefore the effects of the through-thickness stress on the formability of sheet metals cannot be ignored. Moreover, predictions of forming limit curves (FLC) that assume plane stress conditions may not be valid when the forming process involves non-negligible out-of-plane stresses. For this reason a new model was developed to predict FLC for general, three-dimensional stress states. Marciniak and Kuczynski (Int J Mech Sci 9:609-620, 1967) first proposed an analytical method to predict the FLC in 1967, known as the MK method, and this approach has been used for decades to accurately predict FLC for plane stress sheet forming applications. In this work, the conventional MK analysis was extended to include the through-thickness principal stress component (σ ₃), and its effect on the formability of different grades of sheet metal was investigated in terms of the ratio of the third to the first principal stress components (). The FLC was predicted for plane stress conditions (β = 0) as well as cases with different compressive through-thickness stress values (β ≠ 0) in order to study the influence of β on the FLC in three-dimensional stress conditions. An analysis was also carried out to determine how the sensitivity of the FLC prediction to the through-thickness stress component changes with variations in the strain hardening coefficient, in the strain rate sensitivity, in plastic anisotropy, in grain size and in sheet thickness. It was found that the out-of-plane stress always has an effect on the position of the FLC in principal strain space. However, the analysis also showed that among the factors considered in this paper, the strain hardening coefficient has the most significant effect on the dependency of FLC to the through-thickness stress, while the strain rate sensitivity coefficient has the least influence on this sensitivity.     

Investigations of the quadrature phase impact on the behavior of the Liu & Mahadevan’s criterion for fully reversed bending–torsion experiments

The present paper focuses on stress-based criterion of Liu & Mahadevan. A computation of this criterion from an implementation of a numerical procedure based on stress solutions leading to unit criterion is proposed. This approach is supported by an analytic analysis for brittle steel materials for validating the numerical study. A singular behavior of the criterion is observed for in-quadrature phase of stresses, when fully reversed bending–torsion experiments are considered. A comparison of results with some experimental data provided by literature is performed.     

Semi-analytical model based on generalized plane strain assumption for unidirectional composites with matrix micro cracks

Generalized plane strain state is assumed and stress-based finite strip method is formulated for analysis of unidirectional laminates with matrix microcracks. Total complementary potential energy is minimized and fourth-order Euler Lagrange governing equations are presented. This stress-based generalized plane strain approach analyzes general layup and loading conditions. It provides flexibility to control the number of finite strip nodal lines within each lamina; hence, stress behavior can be predicted across each lamina at the desired location of the structure. Along with all of the capabilities which are common with finite strip methodology based on plane strain assumption, this current work has extended the analysis of the cracked laminate. For example, by incorporating behavior of the out of the plane shear stress, boundary conditions including natural boundary conditions are imposed appropriately to solve governing Euler's equations. Results are compared with previously developed displacement-based formulation in the literature for cracked laminates. It has already been shown that a stress-based plane strain approach enhances variation-based cracked laminate analysis where only the case of cross-ply laminate under tension is considered. This current work applies generalized plane strain-based finite strip methodology to carry out analysis under different loading conditions.     

Formability prediction of high-strength steel sheet using experimental,analytical and theoretical analysis based on strain and stress forming limit curves and its application to automotive forming parts

In this work, the high-strength steel (HSS) sheet dual-phase 440 (DP440) were conducted to establish the forming limit curve (FLC) and analytical forming limit stress curve (FLSC) obtained from experimental forming limit curve. First, the Nakajima stretch forming examination was carried out to obtain forming limit curve of investigated sheet. Afterwards, the theoretical Marciniak–Kuczinsky (M–K) model was developed and calculated to evaluate localized necking limits both in strain and stress spaces combination with anisotropic yield criteria. Then, the analytical forming limit stress curves were plastically calculated by using experimental forming limit curve data combination with Swift hardening model and anisotropic yield criteria namely, Hill’48 and Yld2000-2d for representing anisotropic plastic deformation behavior on examined steel sheet. Finally, automotive stamping parts were performed in order to verify an applicability of all developed curves. It was observed that the analytical forming limit stress curves could more precisely predict the formability of automotive parts better than the forming limit curve based on strain. Particularly, the one based on Yld2000-2d predict better than the one based on Hill’48. Simultaneously, the experimental forming limit curve and analytical forming limit stress curve were also evaluated comparing with the theoretical calculated forming limit curve and forming limit stress curve using the Marciniak–Kuczinsky model. It should be noted again that the experimental forming limit curve and analytical forming limit stress curve are the best one. Therefore, the Yld2000-2d yield function better represented the anisotropic behavior of the high-strength steel sheet dual-phase 440 than Hill′ 48 yield function, and can suitable be used for the analysis prediction and design of bumper automotive parts under forming processes.     

利用最小厚度准则预测高强热轧钢板的FLC曲线

为更便捷地获得成形极限图(FLD)中的成形极限曲线(FLC),用最小厚度准则,通过少量成形极限试验结合数值模拟来预测FLC.采用Barlat1989屈服准则对QStE340TM、SAPH370、ZStE260P三种高强度热轧钢板进行成形极限模拟,并以最小厚度准则作为极限判据,根据数值模拟结果绘制FLC图.计算结果表明,采用平面应变路径下的成形极限实验数据作为最小厚度准则的已知参数时,数值预测结果与实验结果能较好吻合.故采用平面应变路径下的成形极限实验数据,结合最小厚度准则和数值模拟,即可得到材料完整的FLC曲线.     

A different approach for considering the effect of non-proportional loading path on the forming limit diagram of AA5083

In this paper, the effects of strain path change on the forming limit diagram (FLD) of AA5083 sheet were investigated. The aim is to predict the forming limit curve (FLC) with non-proportional loading path by ductile fracture criteria. For this purpose, some square blanks were pre-strained by uniaxial tension in rolling direction (RD) and transverse direction (TD), and some others were pre-strained by biaxial stretching over a hemispherical punch. Then, the FLD test specimens were prepared by trimming the pre-strained blanks with the longitudinal axis in the RD and TD directions. The out-of-plane formability test was used for obtaining the FLD. The commercial finite element software ABAQUSE 6.9 was used for simulation in accordance with the experimental procedure. For trimming in the simulation environment, a program was written in MATLAB 7.6 that could determine the elements and introduce their properties to the new simulation model. Ductile fracture criteria were used for predicting the failure, and the Hill’79 criterion was used for applying the anisotropic coefficients. The results show that pre-straining in biaxial tension generally reduces the FLC and shifts it to the right-hand side of the FLD, whereas pre-straining in uniaxial tension raises the FLC and shifts it to the left-hand side. The numerical results were compared with the experimental findings, and relatively good agreement was achieved.     

Numerical study of the effect of martensite plasticity on the forming limits of a dual-phase steel sheet

The formability prediction of dual-phase steel sheets is highly important in the present automotive industry. In this study, the forming-limit curve (FLC) of a DP-780 steel sheet is predicted based on the well-known Marciniak and Kuczynski (MK) theory using a Visco-Plastic Self-Consistent (VPSC) crystal-plasticity scheme. To calibrate the polycrystal model, the stress–strain curves of the ferritic and martensitic phases are inferred by accounting for three martensitic plastic behaviors. Thus, the effect of martensitic plasticity on the FLC simulation can be analyzed. In addition, two different hardening laws – namely saturation and Voce models – are considered in order to study the effects of the extrapolated hardening behavior on the shape of the predicted FLCs. The best agreement with experimentation is found when the FLCs are calculated using the saturation hardening law and when the martensite deformation is either not allowed or retarded to occur after the point of necking. An analysis of the ferritic/martensitic slip system activity inside and outside the MK instability band suggests that, within the MK-VPSC framework, localization occurs much faster in the ferritic than in the martensitic phase. In addition, it is found that, unlike uniaxial tension, after plane-strain deformation and equi-biaxial stretching there is a strong correlation between the orientation of the ferritic grain and the strain that it accommodates. The predictive capability of the model is also confirmed by comparing the measured and simulated crystallographic textures close to necking.     

管道失效判据简析

在外部载荷超过管道所能承受范围时管道即发生失效。通过研究管道在外力作用下的力学性能,有助于确定管道失效时的应力或应变的临界值,根据是取应力还是取应变作为衡量管道失效时的指标,分别有基于应力的失效判据和基于应变的失效判据。合理选用管道失效判据,可以节约管道投资、延长管道使用时间。简介材料应力—应变曲线的一般特征,分析管道基于应力的失效判据和基于应变的失效判据的方法。     

Finite element modelling of α particle size on the stress strain curve of near beta Ti alloy

In the present study the stress–strain curve of low cost near β titanium alloy (Ti–10V–4.5Fe–3Al) at 16% volume fraction of α was measured. Also, it was simulated to be based on the stress–strain curves of individual phases both α and β by using Finite Element Method (FEM), then measured result compared with the calculated curve. The average error between the predicted and measured one is 6.22% and coefficient of correlation (R) is 0.98, these results showed that the simulated results closely fit the measured one, which proves that this FEM model used in this work is appropriate. The distributions of stress and strain are also discussed with the help of FEM model. Additionally, an attempt was made to study the effect of α particle size on the stress–strain curve for the same alloy (16% volume fraction of α). As a result the α particle size increases, then the stress–strain curve is shifted down. The distributions of stress and strain are also discussed for different α particle sizes.     

Triaxial compression tests of QH-E lunar soil simulant under constant mean principal stress path using discrete element method simulations

In this paper, discrete element method (DEM) simulations are applied to investigate the triaxial compression tests of QH-E lunar soil simulant (developed by Tsinghua University, China) under constant mean principal (P) stress path. The P stress path is achieved by controlling the loading speed and direction of the axial stress and confining stress in the DEM simulations, the strain softening and dilatancy characteristics of QH-E lunar soil simulant both at low and conventional P stress levels are discussed. The results show that the deviatoric stress–strain curve is divided into hardening, softening descending, and residual strain stages, and the shear strength and residual strength increase with the increase of P stress, which are similar to the conventional triaxial compression tests. The volumetric strain versus shear strain curve shows a good linear relationship both at the linear shear dilatancy and residual shear dilatancy stages, but the slope of the linear dilatancy stage is larger than that of the residual dilatancy stage. Furthermore, it is found that the variations of shear dilatancy characteristic parameters with regard to P stress also show a good linear relationship, and the values of these parameters increase with the increase of P stress.     

Generalization of the moment of inertia method to estimate equivalent amplitudes for simplifying the analysis of arbitrary non-proportional multiaxial stress or strain histories

Several models have been proposed in the literature to account for fatigue damage under multiaxial load histories. Most of them require some measure of an equivalent stress or strain amplitude, in the sense of causing the same damage as the original history, which may be difficult to obtain for generic non-proportional multiaxial variable amplitude load histories. To identify individual load cycles, a multiaxial rainflow-like algorithm must be employed. For each rainflow-counted cycle, the equivalent stress or strain amplitude along its path is often computed using the so-called convex enclosure methods, which find minimum spheres, ellipsoids, or rectangular prisms that contain the load path in a deviatoric stress or strain space. However, such procedure involves information loss, in special if the path shape is very different from the shape of the enclosing convex surface, resulting in poor estimates of equivalent stress or strain amplitudes. To overcome this problem, the moment of inertia (MOI) method has been proposed in Meggiolaro and Castro (Int J Fatigue 42:217–226, 2012) to calculate equivalent amplitudes and mean components of two-dimensional stress or strain paths, generated, e.g., by tension–torsion or biaxial histories. In this work, the MOI method is extended to deal with generic 6D stress or strain paths, which include all normal and shear components. To accomplish that, the load history path is first represented in a 5D deviatoric stress or strain space and then assumed to be a homogeneous wire with unit mass, whose perimeter centroid is used to estimate the location of the path mean component. Then, the polar moment of inertia (PMOI) of such a hypothetical wire with respect to its mean component is calculated. The PMOI represents the distribution of the path about a single point, the perimeter centroid, giving a measure of how much the path stretches away from its mean component, which is used in the calculation of the equivalent amplitudes. Experimental results for 13 different multiaxial load histories prove the effectiveness of the proposed method to predict equivalent amplitudes and multiaxial fatigue lives.     

A strain-based Dugdale model for cracks under generally yielding conditions

To develop an analytical method for quantifying the growth behaviour of short cracks embedded in notch plastic zones, a critical assessment of the Dugdale model is first made by comparison against finite element analysis for an edge-cracked plate subjected to an applied strain varying linearly along the crack path. It is shown that the conventional stress-based Dugdale model provides accurate estimates for the crack-tip opening displacement and the plastic zone size provided that the applied strain does not exceed one third of the yield strain. These estimates become significantly inaccurate at higher strain levels. To overcome this limitation of the conventional model, a strain-based implementation of the Dugdale model is proposed in which the conventional equilibrium equation is replaced by strain compatibility. Comparison with finite element results shows that this strain-based model provides accurate values for both the crack-tip-opening displacement and the plastic zone size for applied strains up to four times the yield strain and with no evidence of decreasing accuracy with increasing strain. Furthermore, it is shown that the relevant plastic constraint factor to be used for plane strain is that appropriate for the notch plastic zone in the absence of a crack, rather than the more usual choice which is appropriate only for small-scale yielding conditions. This provides a practical and physically plausible approach for extending the scope of current predictive software for fatigue crack growth based on the Dugdale model to include conditions of large-scale yielding.     

A critical evaluation of superposition methods for cracks in grossly plastic gradient fields

This paper presents a critical assessment of three analytical methods in determining the crack-tip plastic deformation under large scale and gross-section yielding conditions. These theoretical approaches include two stress-based cohesive zone models and a strain-intensity factor approach, all developed on the basis of the principle of superposition. In the two stress-based cohesive zone models, the prospective stresses along the crack path are taken to be the elastic-plastic solution and the nominal elastic solution, respectively. The problem of an edge crack subjected to grossly plastic strain fields with a constant strain gradient is analysed using the finite element method. The plastic zone size and the crack-tip opening displacement determined by the finite element analysis are then compared with the analytical predictions, showing that these theoretical approaches are unable to capture the influence of gross plasticity on the plastic deformation at the crack tip. The present results highlight the deficiencies in existing stress-based superposition approaches.     

Constitutive modeling and the effects of strain-rate and temperature on the formability of Ti–6Al–4V alloy sheet

The constitutive model considering the strain-rate and temperature effects was presented by fitting the true stress–strain curves of Ti–6Al–4V alloy over a wide range of strain-rates (0.0005–0.05 s⁻¹) and temperatures (923–1023 K). The Forming Limit Curve (FLC) of Ti–6Al–4V alloy at 973 K was measured by conducting the hemispherical dome test with specimens of different widths. The forming limit prediction model of Ti–6Al–4V alloy, which takes strain-rate and temperature sensitivity into account, was predicted based on Marciniak and Kuczynski (M–K) theory along with Von Mises yield criterion. The comparison shows that the limit strain decreases with temperature lowering but strain-rate increasing. The comparison between theoretical analysis and experiment of FLC verifies the accuracy and reliability of the proposed methodology, which considers the strain-rate and temperature effects, to predict limit strains in the positive minor strain region of Forming Limit Diagram (FLD).     

The mode-amplitude technique and hierarchical stress elements—a simplified and natural approach

This paper reviews the derivation procedure of conventional stress-based elements and then shows how the procedure can be simplified by using degrees-of-freedom which are amplitudes of the boundary loading modes (‘mode-amplitude technique’). This gives one class of element based on stress assumptions and uses only one virtual principle. The natural formulation of hierarchical stress elements is shown.     

A New Curve Fitting Method for Forming Limit Experimental Data

The forming limit curve (FLC) can be obtained by. means of curve fitting the limit strain points of different strain paths. The theory of percent regression analysis is applied to the curve fitting of forming limit experimental data. Forecast intervals of FLC percentiles can be calculated. Thus reliability and confidence level can be considered. The theoretical method to get the limits of limit strain points distributing region is presented, and the FLC position can be adjusted according to practical requirement. Method for establishing FLC with high reliability using small samples is presented at the same time. This method can make full use of the current experimental data and the previous data. Compared with the traditional method that can only use current experimental data, fewer specimens are required in the present method to obtain the same precision and the result is more accurate with the same number of specimens.