COMPARISON OF BIFURCATION ANALYSIS AND MAXIMUM FORCE CRITERIA IN THE PREDICTION OF NECKING IN STRETCHED METAL SHEETS

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1 Comparison of bifuration analysis and maximum fore riteria in the predition of neking in strethed metal sheets XIII International Conferene on Computational Plastiity. Fundamentals and Appliations COMPLAS XIII E. Oñate, D.R.J. Owen, D. Peri and M. Chiumenti (Eds) COMPARISON OF BIFURCATION ANALYSIS AND MAXIMUM FORCE CRITERIA IN THE PREDICTION OF NECKING IN STRETCHED METAL SHEETS F. ABED-MERAIM *, R.H.J. PEERLINGS AND M.G.D. GEERS * Laboratoire d Étude des Mirostrutures et de Méanique des Matériaux LEM, UMR CNRS 79, Arts et Métiers ParisTeh 4 rue Augustin Fresnel, 5778 Metz Cedex, Frane farid.abed-meraim@ensam.eu, Department of Mehanial Engineering Eindhoven University of Tehnology P.O. Box 5, 56 MB Eindhoven, The Netherlands R.H.J.Peerlings@tue.nl - M.G.D.Geers@tue.nl Key words: Diffuse Neking, Strethed Metal Sheets, Bifuration Analysis, Maximum Fore Criteria, Critial Neking Strains, Formability Limits. Abstrat. In the present work, diffuse neking is investigated for strethed metal sheets using two different approahes, namely bifuration theory and maximum fore priniple. The ontribution inludes a ritial analysis and a systemati omparison of their respetive ability to predit neking. In partiular it is shown that, in ontrast to bifuration theory, whih is of quite general appliability, some restritions are assoiated with the appliation of maximum fore onditions. It is noteworthy that the well-known Swift diffuse neking riterion is reovered through bifuration analysis. Reall that Swift s riterion has long been attributed in the literature to the maximum fore priniple, while it is shown here to rather originate from the bifuration analysis, whih provides it with a sound theoretial justifiation. INTRODUCTION In the literature dealing with plasti instabilities, in general, and espeially those related to material (loal) instabilities in relation to sheet metal forming analysis, a large number of neking and formability riteria have been proposed. However, a thorough and rigorous omparison of their theoretial foundations and underlying assumptions is still laking. An exhaustive list of those riteria is diffiult to be given, onsidering the multitude of variants derived from some approahes. A short review reveals, however, that those riteria ould be lassified into at least four distint ategories, depending on their fundamental basis as well as their theoretial or physial bakground. Early instability riteria were based on the maximum fore priniple, originating from Considère [] and its two-dimensional extension by Swift [] for appliation to metal sheets. These riteria, in their original form, were known to predit diffuse neking. Later, these maximum-fore-based riteria were extended by Hora et al. [] in order to predit loalized 8

2 neking. Conurrently, Hill s zero-extension riterion [4] was developed to predit loalized neking on the left-hand side of the forming limit diagram (FLD). Another approah, postulating a pre-existing defet in the material sheet, was proposed by Mariniak and Kuzyński [5]. In its original version, the Mariniak Kuzyński (M K) model an be regarded, in a sense, as a omplementary approah to Hill s zero-extension riterion, whih is only appliable to left-hand side of FLDs as no zero-extension diretion exists for positive biaxial strething. However, sine loalized neking in biaxial strething is observed in pratie, a geometri imperfetion has to be introdued in the M K model to apture this phenomenon, whih may provide some justifiation to this imperfetion theory. This model was subsequently extended by Huthinson and Neale [6] to the predition of the left-hand side of the FLD by allowing the imperfetion band to rotate until a loalized nek is deteted. Druker s [7] and Hill s [8] theory of loss of material stability, also referred to as the general bifuration riterion, represents another lass of approahes for neking predition. Belonging to the same lass, limit-point bifuration appeared later [9], and it has been shown that for assoiative elasto-plastiity, limit-point bifuration oinides with general bifuration. For loalized neking, Rudniki and Rie [,] established a bifuration riterion based on loss of elliptiity (i.e., singularity of the aousti tensor), also known as the disontinuous bifuration riterion. In the same way, some authors suggested the use of loss of strong elliptiity [,], whih was shown to oinide with Rie s riterion for assoiative elastoplasti models. A final signifiant lass of riteria onerns those based on stability theory. Within this approah, neking and loalization phenomena are takled by stability analysis of the loal equilibrium equations. The starting point is the mathematial onept of stability, introdued by Lyapunov [4] and ommonly applied to strutural instability problems (see, e.g., [5,6]). The assoiated tehnique of linear perturbation was extended to material instability problems by Molinari and Clifton [7]. To investigate the rate of growth of the perturbation, its governing equations are linearized, and the resulting eigenvalue problem will haraterize stable and unstable modes. For strain-rate dependent material models, this approah ould be regarded as an interesting alternative to bifuration theory; the latter is known not to apply for strain-rate sensitive materials. From the above overview of the various approahes for neking and loalization predition, an interesting observation an be made. Indeed, while the M K analysis and Maximum Fore Criteria (MFC) have been widely used in the literature, few appliations of bifuration theory to sheet metal formability have been published, whih were initially mostly restrited to simple onstitutive models (see, e.g., Doghri and Billardon [8]). Reently, Rie s bifuration riterion has been used to investigate formability limits of metalli materials (see, [9,], using phenomenologial onstitutive modeling, and [], using miromehanial approahes). Besides its sound theoretial basis, this bifuration riterion has also been shown to provide a useful tool to investigate the impat of mirostrutural mehanisms on the formability limit of multiphase polyrystalline materials [,]. In the present ontribution, attention is restrited to diffuse neking preditions applied to the in-plane strething of metal sheets. Although the investigation of diffuse neking, as an approah to formability limits, may be seen as onservative when ompared to loalized neking preditions, there is yet a need for the former analysis. Besides its evident aademi interest, expliit expressions for the ritial hardening moduli allow loading paths, whih are 8

3 less favorable to neking, to be seleted, and an therefore be used in experiments speifially designed for the material parameter identifiation relying on mehanial tests with homogeneous deformation. To this end, two approahes are thoroughly investigated, namely the maximum fore priniple and the bifuration approah. Their respetive ability to predit this type of geometri instability phenomena are systematially ompared for different workhardening models. BOUDARY VALUE PROBLEM The problem statement is given here by desribing the assoiated boundary value problem. Let us onsider a metal sheet as illustrated in Fig., whih is subjeted to in-plane biaxial loading onditions. The problem geometry is defined by its urrent parameters l, l, l, and their initial values L, L, L, whih denote the dimensions along the major strain diretion, the minor strain diretion and the thikness diretion, respetively. Figure : Illustration of a metal sheet subjeted to in-plane biaxial loading ( ) The sheet is strethed by applying an in-plane biaxial loading F ( ), F ( ) λ λ on its lateral edges, where λ () t is a monotonously inreasing funtion ating as a load ontrol parameter, while the upper and lower surfaes (i.e., normal to the sheet thikness) remain tration free. Only isotropi material models are onsidered in this work so that the prinipal diretions of stresses and strains remain parallel to the loading diretions (, ). FUNDAMENTAL EQUILIBRIUM PATH Subjeting the metal sheet shown in Fig. to in-plane biaxial strething results in an initially homogeneous stress and strain state, referred to as the fundamental path, to whih bifuration analysis and maximum fore priniple will be subsequently applied. The investigation will be onduted under plane-stress onditions and rigid plasti onstitutive modeling. The strain and stress tensors and σ, assoiated with this fundamental solution, are given by their orresponding matries and σ with respet to the Cartesian basis as 84

4 σ =, σ= σ ( + ). () Let us first onsider a general form of elasto-plasti onstitutive equations with inompressible elastiity and isotropi hardening. This onstitutive framework, despite its simpliity, an inlude a large variety of work-hardening models as typially used in sheet metal forming. Using the additive deomposition of the strain rate tensor into its elasti and e p plasti parts ɺ and ɺ, respetively, the elastiity law reads tr ( ) e p ɺ = ɺ ɺ = σ ɺ E σ ɺ E I, () where E is the Young s modulus, tr ( ) stands for the trae operator, and I denotes the seond-order identity tensor. The plasti strain rate tensor is given by the usual assoiative flow rule ɺ p F = pɺ σ, () where pɺ denotes the plasti multiplier and F the yield surfae, here given by eq p ( ) F = σ Y, (4) in whih σ = σ : σ is the von Mises equivalent stress, funtion of the deviatori stress eq tensor σ. Y is the yield stress desribing isotropi hardening, whih is a funtion of the p equivalent plasti strain, with ɺ p p p = ɺ : ɺ. Making use of the onsisteny ondition, the plasti multiplier an be determined along with the elasto-plasti tangent modulus in the following partiular tensorial form: ( ) E σ ɺ tr σ ɺ I = ɺ ( σ σ ):, ɺ (5) E E ( E+ h) σ eq p in whih h = dy d is the salar hardening modulus. The ase of rigid-plastiity, whih is of interest here, an be reovered from the previous equations in the limit of E +. The plasti strain rate, thus equal to the total strain rate, is given by the same flow rule (), with p the plasti multiplier given by pɺ = ɺ = ɺ. Therefore, the rigid-plastiity onstitutive equations are simply dedued from Eq. (5), as a speial ase, under the tensorial form or in terms of stress and strain omponents ɺ = ( σ σ ):, ɺ (6) σ eq 85

5 + = + ( ɺ + ɺ ) = σ ( σɺ + σɺ ) σ eq ( ɺ ɺ ) σ ( σɺ σɺ ) σ eq By setting β = ɺ ɺ, a onstant parameter that haraterizes the proportional in-plane loading path ( β [, ], e.g., β = for uniaxial tension (UT), β = for plane-strain tension (PT), β = for equibiaxial expansion (EBE)), it an be shown that ɺ σ ɺ σ = α, with the following relationship between β and α : α + β β = α = α + β (7). (8) Making use of these proportionality fators, the rigid plasti onstitutive equations an be rewritten in a more ompat form h ɺ σ = ( + β) ɺ h ɺ σ = ( β) + ɺ Integrating the above equations, the fundamental quasi-stati equilibrium solution, whih is indiated by supersript, an be obtained as follows: ( λ) = ( λ) β, σ ( λ) = σ ( λ) α ( + β ) Y in whih = λ( t) ats as a loading ontrol parameter (with λ( t), for σ σ (9), () = < ), and λ() t Y σ = σ + ( + β) hdλ, () Y with σ = σ y α α +, and σ y the initial yield stress. For a linear hardening model, for whih the hardening modulus h is onstant, a losedform solution an be easily obtained, while nonlinear hardening models require numerial integration. In all ases, the following onvenient relationship an be derived: β + β + =. () 86

6 In fat, Eqs. (,) represent a one-parameter family of fundamental paths, and for eah value of parameter α or β orresponding to a partiular loading path, we an investigate the bifuration or maximum fore riteria. 4 MAXIMUM FORCE PRINCIPLE This lassial approah was oined by Considère [] who observed that neking ours in a rounded bar under uniaxial tension when the applied load reahes a maximum. This earlier one-dimensional neking riterion has subsequently been extended to biaxial loading onditions by Swift []. Note that both the Considère and Swift 5 riteria are known to predit diffuse neking, whih generally ours prior to loalized neking in the ontext of sheet metal forming. For the predition of loalized neking, Hora et al. [] extended Considère s riterion by taking into aount the strain-path evolution after diffuse neking towards a plane-strain tensile state. It is worth noting that this lass of riteria, referred to as the MFC, is notieably popular in the literature and has been widely applied to determine FLDs at diffuse or loalized neking. In this setion, attention is restrited to diffuse neking by reonsidering the basi equations on whih the Considère and Swift 5 riteria are based. 4. General onsiderations on the appliability of the MFC For the above-desribed sheet under strething (see Fig. ), the quasi-stati equilibrium equations in terms of the prinipal Cauhy stress omponents ( σ, σ ) read F F = σ ll = σ ll For bifuration analysis or appliation of MFC, the rate form of the equilibrium equations is usually more onvenient. The latter, making use of the inompressibility ondition, reads Fɺ ll Fɺ = ɺ σ σɺ = ɺ σ σɺ ll In the one-dimensional ase, Eq. (4 ) leads to the well-known expression of Considère s riterion (i.e., ɺ σ ɺ = dσ d = σ ), whih expresses that neking starts when the uniaxial hardening modulus h= dσ d drops to the value of the stress. For biaxial loading, the extension by Swift [] suggests the appliation of the MFC at a simultaneous maximum of the fores. However, the simultaneous ourrene of maximum fores is only possible for some trivial loading paths. This has been demonstrated through experiments (see, e.g., Habbad [4]), but an also be shown by further analyzing Eqs. (4). Presribing a linear loading path haraterized by a onstant parameter β = ɺ ɺ, and the orresponding stress ratio α, as defined by Eq. (8), it is straightforward from Eqs. (4) and (8) that a simultaneous maximum of the fores is only possible for α = or β =. The first () (4) 87

7 ase, α =, orresponds to the uniaxial tensile (UT) test, and the seond, β =, to equibiaxial expansion (EBE). Note that for both ases, the simultaneous maximum amounts to the ondition of a maximum of the fore along the major strain diretion. Indeed, in UT the seond ondition ɺ σ σɺ =, derived from Eq. (4 ), is obviously always satisfied, sine F = σ =. For EBE, the symmetry of the problem reveals that ondition (4 ) simply redues to ondition (4 ). To summarize, it has been shown that the ondition of a simultaneous maximum of the fores ( ɺ σ= σɺ and ɺ σ = σɺ ) only ours for two partiular loading paths, whereas the ondition of maximum fore along the major strain diretion ( ɺ σ= σɺ ) may be possible for the whole range of loading paths that make up an FLD, and ould therefore represent an alternative riterion. On the other hand, the ondition of maximum fore along the minor strain diretion ( ɺ σ = σɺ ) is shown not to hold for some loading paths; therefore, this latter ondition will no longer be investigated in the sequel. 4. Appliation of the MFC along the major strain diretion Considering the limitations shown above as to the validity of the ondition of simultaneous maximum of the fores, whih restrit the range of appliability of the MFC, fous is onfined to the maximum fore ondition along the major strain diretion. Aordingly, for a linear loading path haraterized by parameter β, the ombination of the onstitutive equations (9) and the maximum fore ondition along the major strain diretion (i.e., ɺ σ= σɺ ) gives h σ = + h σ = + ( β ) ( β ) The determination of the ritial hardening modulus in Eq. (5) is ahieved by solving the following equation given by the yield funtion F = : ( ) (5) σ α α + = Y, (6) in whih the ritial stress σ is replaed by its expression in terms of Finally, the ritial strains are given by = = β β + β + h dy d =. For omparison purposes, Table below gives the ritial stress and strain states for three partiular loading paths, namely UT, PT, and EBE. (7) 88

8 Table : Critial states given by the MFC for three typial loading paths Loading path Critial state UT PT EBE β = β = β = σ h 4h σ h h h 5 BIFURCATION ANALYSIS ( ) As disussed previously, the appliation of a biaxial loading state F( λ), F( λ ), whih is haraterized by the loading ontrol parameter λ ( t), results in a quasi-stati response indiated by supersript. This response ( λ), σ ( λ ), referred to as the fundamental ( i i ) equilibrium path, may exhibit bifuration when the loading reahes a ritial value. Theoretially, this means that the equilibrium equations may lose uniqueness for some ritial values of the loading. The bifurated solution, whih intersets the fundamental equilibrium path, is haraterized by the ritial load and the assoiated bifuration mode. For pratial appliations, the analysis of suh instabilities amounts to solving an eigenvalue problem, in whih one seeks the first eigenvalue, orresponding to the lowest ritial load. 5. General onsiderations on bifuration analysis The bifuration equations are lassially obtained by first assuming that there exist two different solutions for the rate equilibrium equations (i.e., the fundamental path and the bifurated solution). Then, the orresponding governing equations (i.e., the rate equilibrium equations assoiated with these two solutions) are subtrated from eah other when evaluated at the first bifuration point. At this bifuration point, where the two solutions interset, there may be only loss of uniqueness for some rate variables; the non-inremental quantities themselves are equal at this point of onset of bifuration. Applying this proedure to Eq. (4), whih represents the rate equilibrium equations for the above sheet homogeneously deformed under strething, the bifuration equations are derived as follows: ɺ σ σ ɺ = ɺ σ σ ɺ = where A ɺ = A ɺ A ɺ denotes the differene between any rate variable Aɺ evaluated on the bifurated path and on the fundamental equilibrium path, respetively. (8) 89

9 To proeed further with the bifuration analysis, the onstitutive equations are required, so that the orresponding eigenvalue problem is ompletely defined and an therefore be solved. This will be done in the next setion, where the supersript will be omitted for oniseness. 5. Appliation of the bifuration analysis The bifuration analysis in this ontext of rigid-plastiity involves two main equations. The first equation is obtained by ombining the two equations in (7), whih gives ( ) ( ) σ σ ɺ + σ σ ɺ =. (9) The seond bifuration equation is derived starting from the onsisteny ondition Fɺ = ɺ σeq Yɺ( ) =. By replaing the equivalent strain rate by the expression for the plasti multiplier pɺ = σ : ɺ σ eq, and making use of Eqs. (8), whih are derived from the rate form of the equilibrium equations, the onsisteny ondition leads to ( h) ( h) σ σ σ ɺ + σ σ σ ɺ =. () The two-equation system assoiated with this bifuration problem (i.e., Eqs. (9,)) reads ɺ σ σ σ σ K, with K = = σ ( σ σ h) σ( σ σ h. () ɺ ) The above linear algebrai system results in an eigenvalue problem, in whih the bifuration ondition neessarily involves the singularity of matrix K, leading to 4h ( α α ) ( + α)( 4α 7α + 4) + σ = σ = ασ The orresponding ritial strains (, ) an be obtained by onsidering Eqs. (6) and (7), where the ritial equivalent strain needs to be first determined by solving the generally nonlinear Eq. (6). Then, the assoiated ritial hardening modulus h is simply obtained by taking the derivative h = dy d. It is remarkable that Eq. (), in onjuntion with Eq. (6), is exatly the expression given by the Swift 5 diffuse neking riterion, whih an be rewritten as ( + )( ) / 4( + ) dy α α α = Yd α α (). () This riterion has been frequently used in the literature to onstrut forming limit diagrams at diffuse neking for metal sheets. Table below gives the ritial stress and strain states for the three partiular loading paths previously investigated. 9

10 Table : Critial bifuration states for three typial loading paths Loading path Critial state UT PT EBE β = β = β = σ h 4h σ h h h The results of Table, whih are inidentally idential to those of Table for these three speifi loading paths, depend on the partiular hardening law adopted. For loading paths other than those reported in Tables and, it is on the ontrary shown that the ritial neking states predited by bifuration theory are different from those obtained by MFC.... Major strain MFC Bifuration Linear hardening law, σ k =. y Major strain MFC Bifuration Hollomon hardening law, n = Minor strain Minor strain..5.. Major strain MFC Bifuration Swift hardening law, n =.989 =.98x 4 Major strain.5..5 MFC Bifuration Voe hardening law, =.78 σ Q =.7 y Minor strain Minor strain Figure : FLDs at diffuse neking given by bifuration and MFC for four ommonly used hardening laws 9

11 For the full range of loading paths investigated, Fig. ompares FLDs at diffuse neking obtained with the bifuration approah and the MFC for a set of hardening models. These FLDs apply to opper foils modeled by rigid plasti onstitutive equations. The material parameters orresponding to four hardening models (i.e., linear, Hollomon, Swift, and Voe laws) have been identified using a uniaxial tensile experiment provided in [5], and the relevant values of these parameters are reported in Fig.. Moreover, Fig. reveals the impat of the seleted material model on neking preditions. Both the shape and the level of the FLDs are strongly affeted by the material model. As an be expeted, a linear hardening model leads to unrealistially high limit strains, as opposed to the Voe model, whih is a rapidly saturating hardening law providing the lowest FLDs. The intermediate hardening models of Hollomon and Swift, whih give similar results, orrespond to more ommonly observed FLDs, and this is also supported by the fat that they fit the experimental UT data better. 6 CONCLUSIONS In this paper, the basi equations underlying the bifuration analysis and those relating to the maximum fore priniple have been first speified. Some restritions to the maximum fore onditions have been disussed. In partiular, the simultaneous ourrene of the maximum of the fores is shown to be limited to some speifi loading paths. Throughout the analysis, the differene between the onepts of bifuration and maximum fore priniple has been learly evidened. Moreover, these two distint riteria have been systematially ompared for rigid plasti onstitutive equations involving a variety of isotropi work-hardening models. One of the main results of the present ontribution is the fat that it was demonstrated that the Swift 5 neking riterion is founded on bifuration theory rather than any maximum fore ondition. This provides a better justifiation for its wide appliation in the ontext of sheet metal forming. Indeed, the well-known expression given by the Swift 5 diffuse neking approah has been shown to be a natural outome of the bifuration analysis. In the same way, Considère s riterion in lassial elasto-plastiity has been given full justifiation within the bifuration theory, whereby its proper extension to multiaxial loading onditions should be undertaken within the same theory of bifuration. Indeed, this theoretial approah not only provides a sound foundation to the MFC for some partiular onstitutive equations, but also proves to be appliable and reliable in more general situations. REFERENCES [] Considère, A. Mémoire sur l emploi du fer et de l aier dans les onstrutions. Annales des Ponts et Chaussées (885) 9: [] Swift, H.W. Plasti instability under plane stress. J. Meh. Physis Solids (95) :-8. [] Hora, P., Tong, L. and Reissner, J. A predition method of dutile sheet metal failure in FE simulation. Pro. of Numisheet 996, Dearborn, Mihigan, USA, (996) pp [4] Hill, R. On disontinuous plasti states, with speial referene to loalized neking in thin sheets. Journal of the Mehanis and Physis of Solids (95) :9-. [5] Mariniak, Z. and Kuzyński, K. Limit Strains in the Proesses of Streth-Forming Sheet Metal. International Journal of Mehanial Sienes (967) 9:6-6. 9

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