On the prediction of side-wall wrinkling in sheet metal forming processes

Transcription

1 International Journal of Mechanical Sciences 42 (2000) 2369}2394 On the prediction of side-wall wrinkling in sheet metal forming processes Xi Wang, Jian Cao* Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA Received 6 January 1999; accepted 30 June 1999 Abstract Prediction and prevention of side-wall wrinkling are extremely important in the design of tooling and process parameters in sheet metal forming processes. The prediction methods can be broadly divided into two categories: an analytical approach and a numerical simulation using "nite element method (FEM). In this paper, a modi"ed energy approach utilizing energy equality and the e!ective dimensions of the region undergoing circumferential compression is proposed based on simpli"ed #at or curved sheet models with approximate boundary conditions. The analytical model calculates the critical buckling stress as a function of material properties, geometry parameters and current in-plane stress ratio. Meanwhile, the sensitivities of various input parameters and integration methods of FEM models on the prediction of wrinkling phenomena are investigated. To validate our proposed method and to illustrate the sensitivity issue in the FEM simulation, comparisons with experimental results of the Yoshida buckling test, aluminum square cup forming and aluminum conical cup forming are presented. The results demonstrate excellent agreements between the proposed method and experiments. Our model provides a reliable and e!ective predictor for the onset of side-wall wrinkling in sheet metal forming processes Elsevier Science Ltd. All rights reserved. Keywords: Plastic buckling; Wrinkling; Sheet metal forming; Analytical solution; Energy method 1. Introduction Wrinkling is usually undesired in "nal sheet metal parts for aesthetic or functional reasons. It is unacceptable in the outer skin panels where the "nal part appearance is crucial. Wrinkling on the * Corresponding author. Tel.: ; fax: address: jcao@nwu.edu (J. Cao) /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 2370 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 Nomenclature ¹ external work done by membrane forces ; bending energy b width of the plate a length of the plate t thickness of the plate w normal de#ection x, y, x coordinates, i"1, 2, 3 E elastic modulus ν Poisson's ratio σ stress components, α, β"1, 2 ε strain components s stress deviator, i, j"1, 2 δ Kronecker delta E, ν secant modulus and equivalent Poisson's ratio instantaneous moduli M instantaneous moduli for plane stress condition E stretching strains κ bending strains N membrane stress resultants M bending moments b curvature tensor of the middle surface u displacements in the in-plane directions σ initial yield stress σ, σ stress components σ e!ective stress ε e!ective strain α span angle along the hoop direction β inclination angle r, r radii of the top and bottom in a tapered curved sheet r, r, θ cylindrical coordinates σ, σ, σ stress components in cylindrical coordinates σ critical buckling stress σ applied compressive hoop stress σ absolute value of the calculated compressive stress K material strength coe$cient n strain-hardening exponent A, B, C parameters in the Voce's law e nominal strain in the Yoshida test e critical nominal strain a, b e!ective length and width

3 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} w de#ection amplitude m, n wave number in the hoop direction and lateral/radial direction m critical wave number in the hoop direction r, r radii of the top and bottom at the e!ective compressive area γ, R, q parameters I, I, I, A parameters (i"1, 2, 3, 4) Fig. 1. Schematic of sheet metal forming process. mating surfaces can adversely a!ect the part assembly and part functions, such as sealing and welding. In addition, severe wrinkles may damage or even destroy dies. Therefore, the prediction and prevention of wrinkling are extremely important in sheet metal forming. During the deep drawing process shown in Fig. 1, the sheet under the blank holder is drawn into the deformation zone by the punch. As a result, compressive hoop stress and thus wrinkling can be developed in the sheet metal under the holder (#ange wrinkling) as well as those in the side-wall, as wrinkling is a phenomenon of compressive instability. The magnitude of the compressive stress necessary to initiate the side-wall wrinkling is usually smaller than that for the #ange wrinkling since the wall is relatively unsupported. Hence, the formation of side-wall wrinkles is relatively easier especially when the ratio of the unsupported dimension to sheet thickness is large. In addition, the trim line of the part is usually located a little inside the die radius, and only the wrinkling in the frustum region appears in the "nal part. Hence, side-wall wrinkling is the problem of greater industrial importance and interest. The prediction on the initiation of #ange wrinkling has been addressed analytically and numerically in a number of previous works [1}4]. A detailed review can be found in Tomita [5] and Esche et al. [6]. Research e!orts on the prediction of wrinkling have been made in the past 50 years. The analytical solution can provide a global view in terms of the general tendency and the e!ect of individual parameters on the onset of wrinkling and can be achieved in an almost negligible computational time. However, past analytical work has been concentrated on some relatively simple problems such as a column under axial loading, circular ring under inward tension, and annular plate under bending with a conical punch at the center, etc. Plastic bifurcation analysis is one of the most widely used analytical approaches to predict the onset of wrinkling. Hutchinson and Neale [7] and Neale and Tug( cu [8] studied bifurcation phenomenon of doubly curved sheet metal by adopting Donnell}Mushtari}Vlasov (DMV) shell approximations. The investigation was

4 2372 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 applicable to the regions of the sheet which are free of any surface contact. Tug( cu [9,10] extended their approach to the wrinkling of a #at plate with in"nite curvatures. Wang et al. [11] used a similar approach to study wall wrinkling for an anisotropic shell and applied the criterion to axisymmetric shrink #anging. However, all the above analyses are limited to long wavelength shallow mode and the boundary conditions or continuity condition along the edge of the region being examined for wrinkling are neglected. Triantafyllidis [12] numerically studied the puckering instability problem in the hemispherical cup test based on a proposed bifurcation criterion using phenomenological corner theory instead of J2 theory. The e!ect of geometry and material properties on the onset of non-axisymmetric plastic instability was also investigated. Fatnassi et al. [13] carried out theoretical investigations to predict the non-axisymmetric buckling in the throat of circular elastic}plastic tubes subjected a nosing operation along a frictionless conical die. The buckling point and associated modes are determined by Hill's bifurcation theory in conjunction with a non-axisymmetric buckling mode. Other than bifurcation analysis, Zhang et al. [14] used a modi"ed adaptive dynamic relaxation approach to investigate the plastic wrinkling in the conical cup using an axisymmetric model. This method allowed for the complete analysis, including pre-failure deformation, the prediction of wrinkling and the post-wrinkling deformation. Nevertheless, it is hard to apply this theoretical analysis in the 3-D sheet metal forming with complicated geometry and boundary conditions. Energy method has been another approach to analytically investigate the buckling problem such as #ange wrinkling in Senior [15], Yu and Johnson [16], Yossifon and Tirosh [17,18], Cao and Boyce [1], Wang and Cao [3] and Cao and Wang [4], etc. To our knowledge, there is still no attempt to use this energy method in studying the side-wall wrinkling. Other than the analytical approach, experiments and numerical simulations have been conducted to determine wrinkle formation tendencies in sheet metal forming. Cup forming tests with various geometry are the common experiments to investigate side-wall wrinkling phenomenon. Yoshida et al. [19] developed a simple test (Yoshida buckling test) to provide a reference of the wrinkling-resistant properties for various sheet metals. It involves the stretching of a square sheet along one of its diagonals. Numerical and experimental investigations have been conducted to correlate Yoshida Test results with the material properties (Szacinski and Thomson [20], etc.). However, most of these results were focused on the wrinkling height while few were related to the onset of the wrinkling. The onset and growth of wrinkles and the e!ects of material properties in the Yoshida test were studied analytically and numerically in Tomita and Shindo [21]. The wrinkling point was found by using Hill's bifurcation theory and Mindlin type plate theories in conjunction with the "nite element approximation. Wang and Lee [22] employed a thin shell element to study the wrinkling behavior of the Yoshida test. However, no extensive comparisons with experimental results were given in either study. Numerical simulation using "nite element method (FEM) with either an implicit or explicit integration method has become a prime tool to predict buckling behavior for the sheet metal operation involving complicated geometry and boundary conditions including friction. Using an implicit method to predict wrinkling is essentially an eigenvalue approach, and it is hard to initiate wrinkles without initial imperfections, for example, a speci"c mode shape and/or material imperfection, built into the original mesh. Unlike the implicit solver, the explicit method as a dynamic approach can automatically generate deformed shapes with wrinkles due to the accumulation of numerical error. However, the onset and growth of the buckling obtained from the explicit code

5 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} is sensitive to the input parameters in the FEM model, such as element type, mesh density, simulation speed, etc. Generally, three types of elements are employed in the sheet metal forming simulation, i.e. membrane element, continuum element and shell element. Membrane elements have been widely used to model the forming processes, due to its simplicity and lower computation time, especially in the inverse and optimization analysis where many iterations of forming are required. However, it does not include bending sti!ness, therefore, it may not be appropriate in modeling the processes where the buckling phenomena is important unless some special treatment (such as post-processing) is given [2]. In general, the bending-dominant processes are simulated by the continuum or shell elements. In continuum analysis, the bending e!ect can be taken into account by having multiple layers through the thickness. However, this leads to extremely large computation time especially for three-dimensional problems. Shell elements may be considered as the compromise between the continuum and membrane elements. It is possible to take into account the e!ect of bending with much less computation time than continuum analysis although integration in the thickness section is still needed. Therefore, using shell elements in explicit code becomes a possible approach to study the side-wall wrinkling in deep drawing processes. Nevertheless, as we will discuss in this paper, the reliability of such an approach still must be veri"ed due to its sensitivity to the FEM model parameters. This motivates us to develop a stress-based wrinkling predictor such that we can take advantage of the reliability of FEM static (implicit) analysis for the complicated forming simulations and also overcome the numerical di$culty and sensitivities of the FEM buckling analysis in both implicit and explicit methods. In the present paper, a modi"ed energy approach is presented to provide a stress-based criterion for the side-wall wrinkling during deep drawing processes. Energy method is adopted for the development of the wrinkling predictor due to its simplicity and the previous success in predicting #ange wrinkling [3,4]. The e!ective dimensions over the region undergoing compressive hoop stress are introduced as dimension parameters. Energy equalities for #at plate and curved sheet models are established and a stress predictor is proposed in Section 2. To verify its predictive capability, in Sections 3}5, the present approach is applied to the case studies of the Yoshida buckling test, aluminum square cup forming, and aluminum conical cup forming. In each section, the corresponding model with appropriate boundary conditions is established; then the critical buckling stress is obtained as a function of material properties, geometrical dimensions, and stress ratio; "nally, numerical predictions and comparisons with experimental results are reported. In Section 5, the investigation of wrinkling prediction for aluminum conical cup using various FEM simulation methods is also given for comparison with our proposed approach in terms of the reliability of the prediction. 2. Wrinkling criterion This work aims to investigate the onset of wrinkling in the side-wall area of the forming part. Sheet metal in that region is free of any surface contact with the die/punch, other than the edge displacement restriction imposed by the tooling. For a forming part with complicated geometry, the side-wall can be divided into several relatively uniform sections, which can always be classi"ed into two categories: #at plate and curved sheet. Therefore, in this paper, we will establish the general formulations of wrinkling criterion and apply them for #at plate and curved sheet with

6 2374 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 uniform curvature in the hoop (compressive) direction using a modi"ed energy approach in the case studies. In the following analysis, the pre-buckling stress state in the side-wall over the region examined for wrinkling is assumed at membrane state, and thus, the shear strains and stresses are ignored. All the formulations are developed within the context of thin plate and shell theory, therefore, the thickness of the sheet and all the stress states through the thickness are assumed to be uniform before buckling. Strains are expected to be small and the characteristic wavelength is large compard to sheet thickness and yet small compared to the radii of the curvature of the sheet such that the strain measures given by Donnell}Mushtari}Vlasov (DMV) approximations can be adopted. Deformation theory is employed in the analysis since proportional loading before buckling is assumed. Another limitation is that this approach cannot predict non-symmetric or anti-symmetric wrinkling. Energy method has been extensively employed in Timoshenko [23] to study the elastic buckling of thin plates and shells with various boundary conditions. In his energy approach, a de#ected form may be assumed for the plate and the critical buckling condition can be assessed by equating the internal energy of the buckled plate, ;, and the work done by the in-plane membrane forces, ¹. If the internal energy for every possible assumed de#ection is larger than the work produced by membrane forces, the sheet is under a stable equilibrium condition. Hence, the stability condition is expressed as ¹) ;. (1) To obtain the bending energy for every possible assumed de#ection, the formulation for general doubly curved sheet is employed as detailed in Hutchinson and Neale [7] and Neale and Tug( cu [8]. At the instant of buckling, the in-plane components of Lagrangian strain tensor ε at a distance x from the middle surface of the curved sheet can be approximated by ε "E #x κ, (2) where the Greek indices range from 1 to 2, E and κ represent the stretching and bending strain which are given by E " (u #u )#b w, κ "!w, (3) where a comma denotes covariant di!erentiation with respect to in-plane coordinates (x, x ), u and w are the displacements in the in-plane direction (x, x ) and the buckling de#ection normal to the middle surface of the sheet, b is the curvature tensor of the middle surface in the prebuckling state. If a 3-D constitutive law with the form of σ " M ε is adopted, where the instantaneous moduli M are de"ned in the Appendix. The relationships for the membrane stress resultants are given by N " σ dx "t M EQ (4) and the bending moments are given by M " σ x dx " 12 M t K. (5)

7 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} The strain energy under the assumption of DMV thin plate and shell theory can be obtained as ;" M dκ #N de ds, (6) where S is the region of the sheet middle surface over which the wrinkles occur. By assuming the virtual displacements u "0 and using deformation theory, Eq. (6) can be simpli"ed as ;" 24 t M w w ds#t M 2 b b w ds. (7) The external work done by the membrane forces acting in the middle plane is represented as ¹" 1 2 (N w #N w )ds. (8) For a thin sheet, the boundary condition or continuity condition along the edges of the region being examined for wrinkling strongly a!ects the critical buckling condition since the admissible de#ection mode will be di!erent. By appropriately choosing the de#ection form to re#ect the boundary restriction and equating the energy ;" ¹, the critical buckling stress can be calculated analytically as a function of in-plane stress ratio, material properties, and geometry parameters. This wrinkling stress limit can be implemented into FEM simulation of complicated 3D forming processes to predict the onset of wrinkling in the sheet region free of any surface contact. A #ow chart of the proposed method is shown in Fig. 2. During the FEM simulation, the principal stresses at every material integration point in the region being examined for wrinkling are recorded. The critical buckling stress, σ, is calculated based on the boundary conditions and the in-plane stress ratio following the procedure presented above. The critical buckling stress, σ, is then compared to the actual compressive stress in the sheet, σ.ifσ exceeds σ, wrinkling occurs. Generally, the energy equality is considered over the entire region being examined and the stress "eld before wrinkling is assumed to be uniform over the entire region. However, from the FEM simulation, it is found that the stress distribution is not uniform and the hoop stress is even not completely compressive in the frustum region. Considering that wrinkling is a phenomenon of compressive instability in the presence of excessive in-plane compression, here in this work, the actual dimensions of the compressive area are employed to determine the critical buckling condition. In other words, the actual dimensions of the compressive area, instead of the dimensions of the entire frustum region, will be implemented in the above energy integration in Eqs. (7) and (8). The importance of using this e!ective dimension of the compressive area will be demonstrated in the case studies. In an e!ort to verify the predictive capability of this criterion, Yoshida Bucking test, an aluminum square cup forming, and an aluminum conical cup forming will be studied following the procedure illustrated in Fig. 2. In each case, the problem setup, the formulation of critical buckling stress and the prediction results compared with experimental data will be given. For these simulations, the commercial "nite element code ABAQUS with implicit integration solver is employed unless otherwise speci"ed. The material is characterized as an elastic}plastic material

8 2376 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 Fig. 2. Flow chart of detecting the onset of wrinkling in FEM simulation using the proposed stress predictor. following a hardening law σ "Kε or Voce's law σ "A!B exp(!cε ), and Hill's (1948) yield criterion is employed to describe normal anisotropic behavior of the material. 3. Case study 1: Yoshida buckling test 3.1. Problem setup In the Yoshida buckling test, a square sheet is stretched along a diagonal. As shown in Fig. 3, a standard test piece of 100 mm square and 0.7 mm thickness is speci"ed with a gripping width (CC and DD) of 40 mm at the corners and a gauge length (GG) of 75 mm. The nominal strain e over GG at the onset of the wrinkling is de"ned as the critical nominal strain, e. The yield stress of the material, mild steel, is 207 MPa, the strain-hardening exponent n is 0.22, and the material strength coe$cient K is 812 MPa Buckling condition In the Yoshida test, a combination of tension in the y-direction and an uneven blank geometry results in a compressive stress in the x-direction. Fig. 4 shows the contour of σ in a quarter of the deformed plate. Notice that the actual region under compression with e!ective length a and e!ective width b is less than the total dimension of the plate. This problem is thus simpli"ed as the

9 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} Fig. 3. Thin square plate subjected to tension in a diagonal direction (Yoshida buckling test). Fig. 4. Contour of hoop stress in Yoshida buckling test and the de"nition of the e!ective dimensions in the analytical model (unit : Pa).

10 2378 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 Fig. 5. Thin plate under in-plane biaxial loading. buckling of a rectangular #at plate under plane stress condition (shown in Fig. 5) with an e!ective length a, width b and thickness t. The plate is compressed by the force uniformly distributed along the sides x"0 and a, and is stretched by the force uniformly distributed along the sides y"0 and b. Considering the possible failure modes, the boundary conditions of the rectangular plate model are simply supported at the sides x"0, a, and clamped at the sides y"0, b. Therefore, the de#ection surface of the buckled plate can be represented by the double sine wave as w" w sin 2 mπx a 1!cos 2nπy b, n, m"1, 2, 3, 2, (9) where w is a constant representing the amplitude of the de#ection, m is the wave number in the compressive x-direction, and n is the wave number in the lateral y-direction. From Eq. (7), where b "0 for the #at plate, the strain energy of the plate is obtained as ;" 24 t M w # 2 M w x x # M w y w y #4 M w xy dx dy. (10) For such a plate undergoing the uniform stress "eld of a compressive stress, σ, in the x-direction and a tensile stress, σ, in the lateral y-direction, the stress resultants are expressed as N "!tσ, N "!tσ, and the work done by the membrane forces acting in the middle plane is given by ¹"! 2 t σ w x #σ w y dx dy. (11) By equating the energies calculated from Eqs. (10) and (11), ;" ¹, the critical value of the compressive stress at a given σ becomes σ "!σ " πt 12 M m a # 1 3 M a b m 2n # 2 3 ( M #2 M ) 2n b #1 3 a 2n m b σ. (12)

11 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} The critical buckling stress, σ, is the smallest value among all the σ obtained from Eq. (12) when m varies and n equals to 1. The wave number m corresponding to σ is the critical wave number. It can be seen that M are functions of stress components, which indicates that Eq. (12) is an implicit function of the critical buckling stress, σ, and numerical iteration may be needed to assure that the calculated hoop stress, σ, coincides with the input hoop stress, σ, used for obtaining the instantaneous moduli M. We found that σ decreases monotonically with an increasing σ. As illustrated in Fig. 2 and Section 2, wrinkling occurs if σ is less than the actual compressive stress in the sheet, σ. Let σ "σ, if we have σ 'σ, that is, σ 'σ 'σ, then σ 'σ must hold, and vice verse. Consequently, in the implementation of the procedure shown in Fig. 2, we use σ (σ as the criterion to detect the occurrence of wrinkles to simplify the computation. No numerical iteration is, therefore, needed Prediction and comparison with experimental results Due to the symmetries, a quarter of the plate is simulated in FEM using four-node reduced integration shell elements (ABAQUS S4R) (see Fig. 3). The truncated edges D!D and C!C, where the stretching forces are applied, are considered as clamped without lateral displacement or rotation along the x and y axes. For every material integration point within the area undergoing compressive hoop stress, the average stresses of the "ve integration points through the thickness section are calculated, and the critical wrinkling stress is obtained from Eq. (12) using the e!ective dimensions and the average tensile stress of the integration point. Following the approach introduced in Section 2, we could examine the occurrence of side-wall wrinkling. As shown in Fig. 6, at the nominal strain of , the actual compressive stress in one integration point within the examined region is equivalent or slightly greater than the calculated σ, and thus the onset of wrinkling is predicted to occur at this critical nominal strain. The experimental results in Gibson and Hobbs [24] where the wrinkling heights were recorded with respect to the Fig. 6. Comparison of wrinkling point in Yoshida buckling test with experimental results in [24].

12 2380 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 nominal strain are also shown in Fig. 6 for comparison. The straight line representing their experimental data was obtained from the curve "tting of several experimental measurements. Though the onset of wrinkling was not explicitly measured in the experiments, it is evident that our prediction of the onset of wrinkling compares favorably with the tendency of the wrinkling height curve. From Eq. (12), it can be seen that the critical buckling stress strongly depends on material properties and sheet thickness. It is evident that a thinner sheet provides a lower buckling resistance. The e!ects of material properties are illustrated by their in#uence on instantaneous moduli. Higher material strength component K or lower hardening exponent n yields higher wrinkling limit. This is consistent with the observation from the previous work in Ni and Jhita [25]. 4. Case study 2: square cup forming From case study 1, it is indicated that the present approach provides a reasonably accurate prediction of the onset of plate wrinkling under uneven stretching. The e!ects of material properties and sheet thickness have been quanti"ed. The following two more general case studies (Sections 4 and 5) will be presented to further demonstrate the capability of the proposed approach in dealing with two types of common cup forming tests Problem setup In a practical forming process, a sheet of material is plastically deformed into a desired shape as shown in Fig. 1. Besides the e!ects of material properties and sheet thickness on wrinkling as addressed in the Yoshida test, the forming process involves more complicated issues such as friction e!ect, tooling e!ect and process e!ect. In particular, the frictional #at binder and/or the drawbead are designed to provide adequate restraining force to prevent excessive metal draw-in and consequently to eliminate the occurrence of wrinkling. Here, a square cup forming with a frictional binder is studied to examine the capability of our stress-based predictor in capturing the onset of wrinkling. The forming geometry of the square cup is shown in Fig. 7. The circular blank of AL 2008-T4 with an initial diameter of mm and a thickness of 1.0 mm was formed, where the radii of punch, binder and die pro"les are 6.35 mm. The cross dimension of the punch is mm with the corner radii of mm, and the cross dimension of die is mm with the corner radii of 50.8 mm. The material properties used in the simulations are the initial yield stress σ "125 MPa, the material strength coe$cient K"515 MPa, the strain hardening exponent n"0.26, and the anisotropy parameter R" Buckling condition As shown in Fig. 7, the straight side-wall is constrained between the punch and the die and its deformation should be compatible with that of the corner section. Fig. 8 shows the contour of hoop stress in a quarter of the deformed cup. Accordingly, considering the actual compressive area in the

13 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} Fig. 7. Schematic of side-wall model in a square cup forming. Fig. 8. Contour of hoop stress in square cup forming and the de"nition of the e!ective dimensions in the analytical model (unit : Pa). straight side-wall, the e!ective examined region can be simpli"ed as a plate clamped at four sides of a b (note that a "a). Here, the e!ective dimension b refers to the maximum width where compressive circumferential stress develops in the side-wall area. Hence, the de#ection satis"es the following boundary conditions: w"0, w x "0 at x"0, a w"0, w y "0 at y"0, b (13)

14 2382 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 and it is assumed to be of the form w" w 4 1!cos 2mπx a 1!cos 2nπy b, n, m"1, 2, 3, 2 (14) Similarly, for the assumed uniform stress "eld, by substituting the above admissible de#ection form into Eqs. (10) and (11), the critical buckling stress can be obtained by letting ;" ¹: σ "!σ " πt 12 M 2m a # M a 2m 2n b # 2 3 ( M #2 M ) 2n b # a m 4.3. Comparisons and discussions n b σ. (15) By taking advantage of symmetric geometry and boundary conditions, one-eighth of the entire square cup is simulated. Again, four-node shell elements with reduced integration (ABAQUS S4R) are used to model the blank. The punch, die and binder are modeled as rigid surfaces with frictional interface, where the friction coe$cient between the punch and the sheet is 0.30, and 0.17 for that between the binder/die and the sheet [26]. The critical stress calculated from Eq. (15) is used to predict the onset of side-wall wrinkling based on the simulation results. A detailed procedure can be found in Fig. 2 and Section 3. For a deep drawing process, the blank holder force plays an important role in the formability of the blank. Fig. 9 displays the comparison of the critical forming height at various binder forces before wrinkling occurs between our predictions and experimental results. It shows that the critical forming height increases with the binder force. The analytical predictions from the present approach are in excellent agreement with experimental results in Bakkenstuen [27]. Experimental results showed that when the binder force was beyond 115 kn, another type of failure, tearing, would occur before the initiation of the wrinkling due to excessive stretching. Fig. 10 shows the deformed shape of one buckled cup obtained from the experiment, which shows a full wave in the draw-in direction and two full waves in the circumferential direction, compared with one full wave and three full waves, respectively, from our prediction. In addition to material properties and sheet thickness, the tensile stress resulting from binder design and friction also e!ects the wrinkling limit of the plate under biaxial loading. The critical buckling stress curve with respect to the tensile stress is plotted in Fig. 11 for t"1 mm, a/t"48, and b/t"30. It is shown that the critical compressive stress decreases while the tensile stress increases. This trend is consistent with the observation in Cao and Wang [4] where the tension in the transverse direction reduces the wrinkling limit for the plate under normal constraint. Note that the present approach is not applied to the analysis of cup forming where the length of the straight side is small compared with the corner radius. In that case, the e!ect of the transition between the corner section and the straight wall is signi"cant so that the assumed boundary condition will be inappropriate.

15 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} Fig. 9. Comparison of critical forming heights in the square cup forming. Fig. 10. Deformed shape of a buckled square cup [27]. Fig. 11. Calculated critical buckling stress σ as a function of applied tensile stress σ for a given geometry.

16 2384 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 Fig. 12. E!ective compressive dimension b versus forming height at various binder force cases in the square cup forming. Fig. 13. E!ect of dimension b on the calculated critical buckling stress σ. From Fig. 9, it can be seen that the restraining force due to the blank holder force and friction can improve the wrinkling height. Apparently, the restraining force increases the tensile stress component in the draw-in direction. Based on Fig. 11, it seems that a controversial conclusion was drawn, compared with the aforementioned prediction and experimental observation. However, from the FEM analysis, it is noticed that by increasing the restraining force, the e!ective width b is decreased for the square pan forming as shown in Fig. 12. The e!ect of the geometrical parameter b on the critical buckling stress is illustrated in Fig. 13. The critical buckling stress increases dramatically when b decreases. Therefore, it indicates that the decreased compressive width, not the increased tensile stress, accounts for the improvement of buckling resistance. The discovery leads to the basis for using the e!ective width undergoing compressive stress, b, rather than the total forming width in governing the onset of wrinkling. 5. Case study 3: conical cup forming From the above case studies, it is shown that the introduction of e!ective dimensions of compressive dimension make it possible to take advantage of energy method and yield some simple

17 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} Fig. 14. Schematic of curved wall model in a conical cup forming. formulations of critical buckling stress. Furthermore, the predictions obtained from this modi"ed energy approach match experiments well. In this section, we will apply the theory to another common type of geometry in sheet metal forming, i.e. conical cup or side-wall with curved section. In addition, FEM simulations using various integration methods and element types are investigated for their predictability in assessing the side-wall wrinkling Problem setup In an attempt to examine the capability of this stress-based predictor to capture wrinkling of a curved sheet, a symmetric conical cup forming is investigated here. The circular blank of AL5032-T4 with an initial diameter of 280 mm and a thickness of 1.0 mm is formed. The forming geometry is shown in Fig. 14, where the radii of punch, binder and die pro"les are 5 mm. The diameters of the punch and die are 100 and 200 mm, respectively. The material constants used in the simulations are R"0.92, and A" MPa, B"293.6 MPa, and C"7.112 in the Voce's law Buckling condition In the conical cup forming, the curved wall (as shown in Fig. 14) is also restricted by the tooling at the bottom and top. Therefore, it is simpli"ed as a clamped curved sheet. Fig. 15 displays the contour of hoop stress in the conical cup forming simulation. Considering possible buckling mode and the actual compressive area in the side-wall, the e!ective region investigated can be considered as the curved sheet with the width b as shown in Fig. 15. Similarly, the e!ective dimension b refers to the maximum width where compressive circumferential stress develops in the side-wall area. Let r and r denote the radii of the top and the bottom at the e!ective compressive area corresponding to b, and the de#ection satis"es the following boundary conditions: w"0, at r"r, r, w r "0 or w r "0, at r"r, r, (16)

18 2386 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 where r"r/sin β, and β is the inclination angle. In this analysis, only axisymmetric wrinkling mode is considered. Thus, the de#ection of the plate is assumed to be of the form w" w cos(mθ)(1!cos γ), (17) 2 where γ"2π(r!r )/R with R "r!r, r is the radius of any material point in the examined region, and m is the wave number along the circumferential direction. The expressions for the curved sheet are more complicated. Considering the curved sheet with an in"nite curvature in one direction as shown in Fig. 16, a cylindrical coordinate system is adopted. With b "1/r and b "0, the general formulation for the strain energy of the curved sheet in Fig. 15. Contour of hoop stress in conical cup forming and the de"nition of the e!ective dimensions in the analytical model (unit : Pa). Fig. 16. Schematic of a curved sheet model.

19 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} terms of an admissible normal de#ection w(r, θ) and w(r, θ) is given by M ;" 24 t w r # M #2 M 1 w rr # 1 w r θ w r 1 w r r # 1 w r θ #4 M 1 w r rθ! 1 w r θ # 12 t M w r rsin β dr dθ M " t 24 sinβ π w r # M 1 w r r # 1 w r θ #2 M #4 M w r 1 w r r # 1 w r θ rdr dθ, (18) 1 w r rθ! 1 w r θ # 12 tsin β M w r where α is the span angle along the hoop direction (in this case α"2π). With the stress resultants N "!tσ and N "!tσ, the external work done by the membrane forces acting in the middle plane yields ¹"! t 2 sin β π σ w r #σ 1 w r θ r dr dθ. (19) Considering the axisymmetry and force equilibrium, we have the following relationship for the radial stress assuming constant thickness in the region: σ r"σ r, (20) where r is the radius of any examined material point in the side-wall and σ is the radial stress at the associated point. Assume that the circumferential compressive stress is uniform over the region examined for wrinkling. Substituting Eqs. (17) and (20) into Eqs. (18) and (19), the energy equality ¹" ; yields the solution for the critical compressive stress as σ "!σ " 4 mi θ σ r π I # t R 12 sin βi, (21) where I " sin γ dr, (22)

20 2388 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} I " (1!cos γ) dr, (23) r with I " ( M A # M A #2 M A #4 M A )r dr, (24) A " 2π cos γ R, A " π sin γ!m R r 2r (1!cos γ) # 3 t sin β A " π cos γ π sin γ!m R R r 2r (1!cos γ), A " m (1!cos γ)!mπ 2r R r sin γ. 1 (1!cos γ), r The critical buckling stress, σ, is the smallest value among all the σ obtained from Eq. (21) for various values of m where n is taken to be 1. The corresponding wave number m is the critical wave number. From the above formulations, it can be seen that the critical buckling stress for the curved sheet also depends on the local curvature over the region being examined for wrinkling in addition to material properties, sheet thickness, e!ective compressive dimension and stress ratio. (25) 5.3. Comparisons and discussions Similarly as described in the FEM modeling of square cup forming, four-note shell elements (ABAQUS S4R) are employed to model one-quarter of the entire conical cup. In the simulation, the friction coe$cient between the punch and the sheet is 0.30, and 0.15 for that between the binder/die and the sheet. Fig. 17 displays the comparison of the critical forming height at various binder forces for the conical cup forming. The experimental data were provided by ALCOA. Excellent agreement between the predictions from the present approach and experimental results is obtained. As brie#y discussed in the introduction, various "nite element models with di!erent integration methods and element types have been used to study the onset and growth of wrinkling. Here, we will establish some of the most widely used FEM models for simulating this conical cup forming process and examine their predictions. Using the implicit method combined with four-node shell elements (ABAQUS S4R), with or without mesh/material imperfection, wrinkling did not occur in the side-wall area for various binder forces, even when the punch displacement reaches 40 mm, which is much higher than the experimental failure heights. Using the explicit code with four-node shell elements (Belytschko-Tsai Shell) in a commercial FEM package LS-DYNA, wrinkling occurs naturally in the side-wall area from the very early

21 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} Fig. 17. Comparison of critical forming heights in the conical cup forming. Fig. 18. History of maximum wrinkle amplitude obtained from the explicit FEM simulations of the conical cup forming with various mesh densities at a binder force of 100 kn. state, and grows with the increasing forming height. The maximum wrinkling amplitude in the side-wall area as a function of the forming height (or punch displacement) is used to study the wrinkling behavior. To determine the onset of wrinkling, a certain threshold for the tolerable wrinkling amplitude has to be speci"ed. It is obvious that the forming failure height is very sensitive to the level of the speci"ed wrinkle amplitude as shown in Fig. 18. In the following analysis, this threshold is taken to be 0.05 mm. As evident in Fig. 18, the predicted failure heights are sensitive to the mesh density. In addition, Fig. 19 illustrates this dependency for various binder forces. Using the same threshold, the failure heights from a "ner mesh are higher than those obtained from a coarse mesh. The existence of this sensitivity on mesh density makes FEM model with explicit integration method and shell elements not very reliable in terms of predicting the wrinkling behavior. The same phenomenon does not exist in our proposed method where the developed

22 2390 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 Fig. 19. Critical forming heights in the conical cup forming predicted from explicit FEM simulations with various mesh densities (1}1618, 2}2424, 3}3236, 4}4848, 5}6472). Fig. 20. Critical forming heights in the conical cup forming predicted from the present approach (stress-based predictor # implicit FEM simulation) with various mesh densities (1}1618, 2}2424, 3}3236, 4}4848, 5}6472). stress-based predictor is implemented in the implicit FEM analysis. Fig. 20 demonstrates that the predicted failure heights converge at certain values as mesh density increases. In addition to mesh density, the punch velocity used in the explicit models is another parameter that needs to be treated carefully. Fig. 21 shows the dependence of wrinkle height on the punch velocity under a binder force of 100 kn. Notice that the punch velocity is increased arti"cially to 1}20 m/s from a typical velocity of 100 mm/s in the real forming process in order to let the simulation "nish in a reasonable time. As shown in Fig. 21, the predicted di!erence of failure height among various punch velocities can be up to 1.5}2.0 mm. The above illustration of excellent results obtained from our proposed model and the investigation on various FEM models demonstrated that the current stress-based criterion ensures the reliable and e$cient prediction of the onset of side-wall wrinkling.

23 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} Fig. 21. E!ect of punch velocity in the explicit FEM simulation on the history of maximum wrinkle amplitude in the conical cup forming. 6. Conclusions Accurate prediction of side-wall wrinkling in sheet metal forming processes has been a challenging topic. As known, a pure analytical solution is suitable only for solving some simple geometry problems. Our investigation in Section 5 on the capability of various "nite element models for predicting the side-wall wrinkling in a conical cup forming demonstrates the limitations of these models. It concludes that the implicit FEM model with shell elements may overwhelmingly overpredict the failure heights and the predictions from explicit FEM models are sensitive to the selected critical wrinkle heights, the mesh density, the punch velocity, etc. To overcome these di$culties, a method combining the analytical solution and "nite element modeling is proposed here. The side-wall of even a complicated geometry can be characterized into a combination of local #at sheets or curved sheets. The analytical model for the onset of the buckling of an elastic}plastic #at/curved sheet is developed in this paper using the energy method (Section 2). The model assumes uniform stress distribution in the sheet and therefore yields an upper-bound solution. Most importantly, the model introduces the concept of e!ective compressive dimensions, which are the actual areas under compression in the side-wall obtained from the FEM simulation. The signi"cance of this e!ective dimension on the calculated critical buckling stress is illustrated in Section 4. By de"ning the appropriate boundary conditions, critical buckling stresses can be obtained in terms of material properties, in-plane stress ratio, sheet thickness and geometry parameters. The onset of wrinkling can be assessed by comparing this calculated critical buckling stress and the actual applied compressive stress obtained from FEM simulation. The present approach and the analytical model have been examined for predicting the onset of the wrinkling in Yoshida buckling test (Section 3), aluminum square cup forming (Section 4), and aluminum conical cup forming (Section 5). Excellent agreements between our predictions and experimental results in all these cases are obtained (Figs. 6, 9 and 17). Those cases demonstrated that the present analysis provides a simple and e!ective way to predict the onset of side-wall wrinkling. This approach can be implemented easily for analyzing complicated 3D forming problems and takes the e!ect of friction and other process parameters into consideration implicitly

24 2392 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369}2394 by utilizing the stress states obtained from a complete FEM simulation. However, the model does not address the issue of anti-symmetric wrinkling and wrinkling during unloading, which will be our next research target. A similar approach, but a di!erent analytical model, for predicting the onset of #ange wrinkling (sheet wrinkling with normal constraint) was presented in Cao and Wang [4]. The combination of these two works provides a more complete picture for accurately and e!ectively predicting the onset of wrinkling in sheet metal forming processes. For example, no #ange wrinkling is detected in the conical cup forming (Section 5) using the analytical model, which is consistent with the experimental observation. The reliability of these criteria provides engineers a robust tool in designing/optimizing the tooling and forming parameters and therefore may eliminate the costly trial-and-error approach. Acknowledgements The "nancial support for this work from NSF grant No. DMI is greatly appreciated. We also would like to acknowledge the ALCOA research center for providing the experimental data and collaboration through the Alcoa Foundation award to JC. Appendix For a 3-D constitutive law with the form of σr given by " εr, the instantaneous moduli are " E 1#ν 1 2 (δ δ #δ δ )# ν δ δ! 1 1!2ν q s s, (A.1) where s is the stress deviator and δ denotes the Kronecker delta, the secant modulus E "σ /ε, and the equivalent Poisson's ratio ν is obtained from ν E " ν E #1 2 1 E! 1 E. (A.2) The parameter q is given as 2E q" (1#ν) 3(E!E ) #1 2 σ. (A.3) 3 The incremental moduli for plane stress condition become M!. (A.4) "

25 X. Wang, J. Cao / International Journal of Mechanical Sciences 42 (2000) 2369} And Hill's (1948) anisotropic plasticity is employed here; therefore the e!ective stress σ for plane stress problem can be expressed as σ " σ #σ #R(σ!σ ), 1#R where R is the ratio of plastic strain in width direction to plastic strain through thickness in a uniaxial tensile test. (A.5) References [1] Cao J, Boyce M. Wrinkle behavior of rectangular plates under lateral constraint. International Journal of Solids and Structure 1997;34(2):153}76. [2] Cao J, Kara"llis A, Ostrowski M. Prediction of #ange wrinkles in deep drawing. In: Predeleanu, Gilormini P, editors. Advanced methods in material processing defects. 1997, pp. 301}10. [3] Wang X, Cao J. An analytical model for predicting #ange wrinkling in deep drawing. Transactions of NAMRI SME 1998;XXVI:25}30. [4] Cao J, Wang X. An analytical model for plate wrinkling under tri-axial loading and its application. International Journal of Mechanical Sciences 1999;42(3):617}33. [5] Tomita Y. Simulations of plastic instabilities in solid mechanics. Applied Mechanical Reviews 1994;47:171}205. [6] Esche SK, Kinzel GL, Taylan A. Review of failure analysis in sheet metal forming simulations. In: Lee JK, Kinzel GL, Wagoner RH, editors. Numisheet' p. 270}9. [7] Hutchinson JW, Neale KW. Wrinkling of curved thin sheet metal. Plastic instability, Paris: Presses Ponts et ChausseH es, p. 71}8. [8] Neale KW, Tugcu P. A numerical analysis of wrinkle formation tendencies in sheet metals. International Journal for Numerical Methods in Engineering 1990;30:1595}608. [9] Tug( cu P. Plate buckling in the plastic range. International Journal of Mechanical Sciences 1991;33:1}11. [10] Tug( cu P. On plastic buckling prediction. International Journal of Mechanical Sciences 1991;33:529}39. [11] Wang CT, Kinzel G, Altan T. Wrinkling criterion for an anisotropic shell with compound curvatures in sheet forming. International Journal of Mechanical Sciences 1994;36:945}60. [12] Triantafyllidis N. Puckering instability phenomena in the hemispherical cup test. Journal of Mechanical Physics Solids 1985;33:117}39. [13] Fatnassi A, Tomita Y, Shindo A. Non-axisymmetric buckling behavior of elastic-plastic circular tubes subjected to a nosing operation. International Journal of Mechanical Sciences 1985;27:643}51. [14] Zhang LC, Yu TX, Wang R. Investigation of sheet metal forming by bending, Part II: plastic wrinkling of circular sheets pressed by cylindrical punches. International Journal of Mechanical Sciences 1989;31:301}8. [15] Senior BW. Flange wrinkling in deep-drawing operation. Journal of Mechanical Physics Solids 1956;4: 235}46. [16] Yu TX, Johnson W. The buckling of annular plates in relation to the deep-drawing process. International Journal of Mechanical Sciences 1982;24:175}88. [17] Yossifon S, Tirosh J. On suppression of plastic buckling in hydroforming process. International Journal of Mechanical Sciences 1984;26:389}402. [18] Yossifon S, Tirosh J. Buckling prevention by lateral #uid pressure in deep-drawing. International Journal of Mechanical Sciences 1984;27:177}85. [19] Yoshida K, Hayashi J, Hirata M. Yoshida buckling test, IDDRG, 1981, Kyoto, Japan. [20] Szacinski AM, Thomson PF. The e!ect of mechanical properties on the wrinkling behavior in the Yoshida test and in a cone forming test. Proceedings of 13th Congress IDDRG, Meclboune, Australia, p. 532}7. [21] Tomita Y, Shindo A. Onset and growth of wrinkles in thin square plates subjected to diagonal tension. International Journal of Mechanical Sciences 1988;30:921}31.