Necking In Sheet Metals Research Articles

The bending dependency of forming limit diagrams

The forming limit diagram (FLD) is commonly used in the forming industry to predict localized necking in sheet metals. However, experimental and mathematical determination of forming limit curves is based on in-plane deformation without taking a bending component into account. This paper addresses the question whether a forming limit diagram predicts localized necking too conservatively or too liberally for different amounts of superimposed bending. Furthermore a methodology is proposed on how to determine the influence of bending quantitatively and how to incorporate the results into a FLD. Finite element analysis (FEA) is used to model the angular stretch bend test (ASBT), where a strip of sheet metal is locked at both ends and a tool with a radius stretches and bends the center of the strip until failure. By using different radii, the sample is exposed to different amounts of superimposed bending during the stretching. The FEA model is verified by experimental work using a dual phase steel (DP600). Applying a strain rate dependent identification method for determining the onset of localized necking, the FEA model is utilized to access the forming limit strains for different stretch bending conditions. Thereby the Keeler-Brazier FLD is extended by a third dimension, the superimposed bending dimension, to assess the effect of bending on the forming limit of sheet metal parts. The more severe the superimposed bending, the higher the formability of the steel.

Theoretical and numerical investigation of the impact of out-of-plane compressive stress on sheet metal formability

In modern sheet metal forming processes, such as hydroforming and single point incremental forming, sheet metals are often subjected to out-of-plane compressive stresses in addition to traditional in-plane stresses. However, the effect of these out-of-plane stresses on the onset of plastic strain localization is not considered when classic necking criteria are used, as the latter are generally formulated based on the plane stress assumption. The main objective of the present investigation is to overcome this limitation by developing numerical tools and analytical relations that allow considering the influence of these compressive stresses on the prediction of localized necking. In the different tools developed, and for comparison purposes, finite strain versions of both the deformation theory of plasticity and the rigid-plastic flow theory are used to describe the mechanical behavior of the metal sheet. Furthermore, both the bifurcation theory and the initial imperfection approach are employed to predict the onset of strain localization. Various numerical predictions are reported to illustrate the effect of normal stress on the occurrence of localized necking in sheet metals. From these different results, it is clearly demonstrated that out-of-plane stresses may notably enhance sheet metal formability and, therefore, this property can be effectively used to avoid the initiation of early strain localization.

On Complete Solutions for the Problem of Diffuse Necking in Sheet Metal

Abstract The paper focuses on the assessment of so-called complete solutions for the problem of diffuse necking in sheet metal. The assessment is based on a virtual tensile test generated with the aid of a finite element model using a reference strain hardening behavior. The resulting displacement fields within the diffuse neck are then used to numerically deform a speckle pattern. Subsequently, the deformed images are post-processed using DIC software to acquire displacement fields and strain fields. The identification of the strain hardening behavior is based on the virtual force-elongation curve, the displacement fields and associated strain fields within the diffuse neck.

Extended evaluation of Erichsen cupping test results by means of laser speckle

Erichsen cupping test has usually been used to determine the height of the cup at the moment of sheet metal fracture, defined as the Erichsen Number IE. The paper presents a proposal of using Erichsen cupping test to determine the additional parameter – the height of the cup LN-IE at the moment of localized necking in sheet metal, which usually precedes fracture of sheet metal. For this purpose, a method of localized necking detection was developed. It is based on the phenomenon of laser speckle. The results of height measurements were presented for three different grades of sheet metals used in the automotive industry: 5754 H22 aluminum alloy, DC04 deep drawing steel, and Dogal 800DP high strength steel. The differences in cup heights at the moments of localized necking (LN-IE) and fracture (IE) of the tested sheet metals were in the range of 0–8%.

Recent Approaches for the Determination of Forming Limits by Necking and Fracture in Sheet Metal Forming

Forming limit diagrams (FLD's) are used to evaluate the workability of metal sheets. FLD's provide the failure locus at which plastic instability occurs and localized necking develops (commonly designated as the forming limit curve - FLC), and the failure loci at the onset of fracture by tension (FFL) or by in-plane shear (SFFL). The interest of metal formers in controlling localized necking is understandable because the consequence of plastic instability is an undesirable surface blemish in components. However, because under certain loading conditions fracture can precede necking in sheet metal forming processes, there is a growing interest in characterizing the forming limits by necking and fracture in the FLD's. This paper gathers together a number of recently developed methodologies for detecting the onset of local necking and fracture by in-plane tension or in-plane shear, and discusses their applicability to determine experimentally the FLC's, FFL's and SFFL's.

Marciniak–Kuczynski type modelling of the effect of Through-Thickness Shear on the forming limits of sheet metal

The Marciniak–Kuczynski (MK) forming limit model is extended in order to predict localized necking in sheet metal forming operations in which Through-Thickness Shear (TTS), also known as out-of-plane shear, occurs. An example of such a forming operation is Single Point Incremental Forming. The Forming Limit Diagram (FLD) of a purely plastic, isotropic hardening material with von Mises yield locus is discussed, for monotonic deformation paths that include TTS. If TTS is present in the plane containing the critical groove direction in the MK model, it is seen that formability is increased for all in-plane strain modes, except equibiaxial stretching. The increase in formability due to TTS is explained through a detailed study of some selected deformation modes. The underlying mechanism is a change of the stress mode in the groove that results in a delay of the onset of localized necking.

Prediction of Necking in Tubular Hydroforming Using an Extended Stress-Based Forming Limit Curve

This paper presents an extended stress-based forming limit curve (XSFLC) that can be used to predict the onset of necking in sheet metal loaded under non-proportional load paths, as well as under three-dimensional stress states. The conventional strain-based ϵFLC is transformed into the stress-based FLC advanced by Stoughton (1999, Int. J. Mech. Sci., 42, pp. 1–27). This, in turn, is converted into the XSFLC, which is characterized by the two invariants, mean stress and equivalent stress. Assuming that the stress states at the onset of necking under plane stress loading are equivalent to those under three-dimensional loading, the XSFLC is used in conjunction with finite element computations to predict the onset of necking during tubular hydroforming. Hydroforming of straight and pre-bent tubes of EN-AW 5018 aluminum alloy and DP 600 steel are considered. Experiments carried out with these geometries and alloys are described and modeled using finite element computations. These computations, in conjunction with the XSFLC, allow quantitative predictions of necking pressures; and these predictions are found to agree to within 10% of the experimentally obtained necking pressures. The computations also provide a prediction of final failure location with remarkable accuracy. In some cases, the predictions using the XSFLC show some discrepancies when compared with the experimental results, and this paper addresses potential causes for these discrepancies. Potential improvements to the framework of the XSFLC are also discussed.

Forming limits of AL 6022 sheets with material damage consideration—theory and experimental validation

Characteristics of localized necking in sheet metals are examined with anisotropic plasticity as well as anisotropic damage developed progressively after load application. The vertex theory is employed to describe basic mechanisms of localized necking. Anisotropic plasticity is accounted for by Hill's quadratic yield criterion. An anisotropic damage model based on Continuum Damage Mechanics is reviewed and expanded. The anisotropic damage model is combined with a modified vertex theory to generate damage-coupled localized necking criteria on both sides of forming limit diagram (FLD). The criteria lead to explicit expressions of critical hardening modulus on both sides of FLD. It is shown that the damage-coupled FLD model can be readily reduced and used to predict the forming limit strains of damage-free materials satisfying power hardening law given by other researchers (Hill, J. Mech. Phys. Solids. (1952)1,19; Zhu et al., ASME J. Eng. Mater. Tech. (2001) 123, 329). Critical damage value at localized necking can be computed from the damage-coupled localized necking criteria as a function of stress/strain states and strain paths. Tests on the formability and material properties of Al 6022, such as hardening and damage law, anisotropic plasticity parameter, have been performed. The measured FLD of the material are compared with the predictions based on the damage-coupled localized necking criteria for validation of the proposed FLD model. Material damage is observed to have a definite effect on the forming limits of Al 6022, thus providing a more accurate prediction than that of the conventional models.

金属板の局所くびれ解析に見られる解析構造の矛盾点

金属板の局所くびれ解析に見られる解析構造の矛盾点

Analytical and experimental studies of necking in sheet metal forming processes

Abstract Two approaches are detailed for the determination of forming limit diagrams (FLD). The first one is experimental: a correlation technique has been developed for the displacement field measurement on a sheet. The second approach is theoretical: a necking criterion based on the load-instability and plane strain localization assumptions is proposed. It includes the possibility to introduce the modified Gurson–Tvergaard damage model taking account Hill's or Barlat and Lian's anisotropy. The criterion is written in an intrinsic form and is applicable to linear strain paths as well as to non-linear strain paths. Finally, the FLD for three aluminum sheets and a mild-steel sheet obtained by experimental and theoretical ways are compared.

A Consistent Criterion for Diffuse Necking in Sheet Metals Using Hill’s 1979 Yield Surface

A review of some existing criteria for diffuse necking in sheet metals is given and their limitations are discussed. The introduction into production of new sheet materials whose plastic deformation is impossible to be modeled using Hill’s 1948 anisotropic yield function necessitates improvements of these existing criteria to accurately describe their necking behavior. In this paper, a generalization of the existing diffuse necking criteria for materials describable by Case IV of Hill’s 1979 anisotropic yield function is presented. The proposed criterion is consistent with the previous criteria. It predicts a significant effect of Hill’s 1979 yield surface shape factor on the critical principal strain in the range of negative minor strains while in the range of biaxial tension this influence is small.

Localized Necking of Sheet at Negative Minor Strains

Localized necking in sheet metal has been examined for strain paths between uniaxial tension and plane strain (i.e., the negative minor strain region of a forming limit diagram). The behavior of sheet with preexisting imperfections has been analyzed and is contrasted to that free of imperfections. In particular, it is shown that the size and orientation of an imperfection is critical in determining whether or not localized necking is initiated along the imperfection. The influence of strain hardening, strain rate hardening, and plastic anisotropy on localized necking of an imperfect sheet is also examined. One of the most significant conclusions obtained from present analysis and from a reexamination of Hill’s theory is the prediction of a critical thickness strain criterion for the onset of localized necking at negative minor strains, regardless of whether or not an imperfection is present. The critical thickness strain criterion is observed in Ti alloys, Al alloys, steels, and brass.

The use of a shear instability criterion to predict local necking in sheet metal deformation

A shear instability criterion can provide a consistent approach to the onset of local necking in sheet metal forming under biaxial stretching. The neck is anticipated to initiate in the direction of pure shear when the shear stress attains some critical value. The yield function proposed by Hill is employed and the material is assumed to display only normal anisotropy. Empirically it is found that the additional material parameter required by this yield function is simply related to R, the coefficient of normal anisotropy, for a number of materials. This allows the limit strain to be predicted in terms of three well established plastic properties, viz. work hardening coefficient, coefficient of normal anisotropy and initial pre-strain. The influence of these on the limit strain curve is analysed and the coefficient of work hardening shown to play the most important role. Data available in the literature are employed in a comparison of the present theory with that due to Marciniak. In general the predicted limit strains are in reasonable agreement with the trend of experimental results for a wide range of materials. In the case of isotropic materials with work hardening coefficients in the range 0·2-0·6 predictions from the present theory are almost identical with those from that presented by Stören and Rice. The theory presented here exhibits a good correlation with experimental limit strains for materials with high work hardening coefficients, of approx. 0·4 or more. Generally, for low work hardening materials, with coefficients of 0·25 or less, the shear instability theory tends to an underestimate of limit strains and a Marciniak type of analysis may be more appropriate. However, bearing in mind the scatter of the experimental data the present theory constitutes a safe lower bound on limit strains and, in addition, has the advantage of simplicity in the mathematical calculation required.

Conditions for shear localization in the ductile fracture of void-containing materials

This paper investigates the possibility that ductile fracture occurs by the McClintock-Berg mechanism of localization of deformation within a narrow shear band, owing to the progressive softening of the material by increasing porosity due to void growth. The ductility predicted for a macroscopically homogeneous sample of a voided material is shown to be unrealistically large and hence an initial inhomogeneity of properties is considered, in the sense of an analysis by Marciniak and Kuczynski in the related problem of local necking in sheet metals. General conditions for a localization bifurcation with an initial inhomogeneity (imperfection), concentrating deformation to allow localization within it, are derived. The initial imperfection is taken in the form of a void-containing, thin slice of a material and is assumed to have a void volume fraction slightly larger than the outside of the imperfection. Elastic-plastic constitutive rate relations for void-containing materials proposed by Gurson are adopted to the conditions for the localization bifurcation. The critical conditions are analyzed numerically to discuss the sensitivity of localization conditions to an initial imperfection, in consideration of the implications for the theory of ductile fracture. The results suggest that the existence of an initial imperfection makes it possible for localization to occur at a reasonable strain, and the predictions from this analysis seem broadly consistent with reported experimental observations.

Failure in thin sheets stretched over rigid punches

Criteria for diffuse and local necking in sheet metals stretched over a hemispherical punch have been developed. During stretching, a peak develops in the distribution of strain across the dome. This peak is a ring of thinned material around the pole which moves outward radially with continued deformation. The present approach essentially focuses on the loading rate of the region bounded by this ring of maximum thinning referred to here as thecrown. While diffuse instability is identified with a vanishing rate of crown loading, local necking is associated with the onset of a rapid rate of thinning under falling crown load. The principal surface strains were measured from incremental stretching of several different materials and time dependent constitutive relations were used to predict the instability strains. Such predictions agree with experimentally observed forming limits defined by the onset of visible necking and provide an understanding of the failure of biaxially stretched sheets deformed by a rigid punch. Since no satisfactory plasticity analyses have been developed to predict the strain distribution during punch stretching, the empirical input to the present instability criterion is required.