Bending effect in the failure of stretch-bend metal sheets

BENDING EFFECT IN THE FAILURE OF STRETCH-BEND METAL SHEETS D. Morales*, A. Martinez, C. Vallellano, F.J. Garcia-Lomas Department Mechanical and Materials Engineering. University of Sevilla ABSTRACT: A study of the strain/stress gradient effect in the formability of stretch-bend metal sheets is presented. Depending on the severity of these gradients, two independent kinds of failure can be defined: Necking- and Fracturecontrolled Failure. In a previous work the authors assumed that the former failure mode takes place when the inner fibre necks, and the latter mode arises when the outer fibre of the sheet fractures. Obviously, this assumption could seem a priori too much restrictive. Both types of failure are remodelled in the present work, but now it assumes that failure is controlled by the evolution of the strains and stresses in a certain material volume at the inner and the outer surface of the sheet. The size of this critical volume and their effects in the failure predictions are discussed. KEYWORDS: Sheet-metal forming, Bending effect. 1 INTRODUCTION Traditionally, sheet metal failure criteria have been formulated assuming a uniform strain/stress field through the thickness of the sheet. Thus, the influence of a strain gradient is usually considered in the analysis by assuming that the mid-plane strain characterizes the formability of the sheet. This simple approximation is found to yield consistent and effective results, although it is sometimes too conservative [1]. The present paper analyzes the effect of bending in the formability of metal sheets. The failure conditions by necking and fracture under different combinations of bending moment and axial force under plane strain conditions are analyzed and discussed in detail. 2 STRAIN GRADIENT EFFECTS The sheet failure is mainly controlled by the initiation of necking or the initiation of ductile fracture in some fibres on thickness. Given that necking is a matter of plastic instability, it is reasonable to assume that the instability of a sheet is achieved only if all fibres in the sheet thickness are already necked. This means that the sheet failure is controlled by the capability of the less strained fibres to neck. In a stretch-bending process, a simple approach is to assume that sheet necking occurs when inner fibres reach the necking strain. This idea is consistent with work reported by Tharrett and Stougthon [1]. In fact, they pointed out that necking is observed on the sheet when the strain on the concave side of the sheet reaches the in-plane forming limit strain. They referred to this failure criterion as the Concave-Side Rule. As bending becomes more important the fibres on the concave side of the sheet may undergo compression and thicken. This means that necking will not govern the sheet failure any longer, giving way to ductile fracture as the dominant failure mechanism. The ductile fracture initiates with the appearance of a crack at the more stretched side (outer fibres) in the sheet. Therefore, two independent types of failures can be expected in a sheet under a strain gradient: (1) a Necking-controlled Failure, which takes place when all layers in the sheet thickness neck, and (2) a Fracturecontrolled Failure, which arises when outer fibres of the sheet fracture. 3 STRETCH-BENDING ANALYSIS The mathematical formulation presented here is based on previous work by El-Domiaty et al. [2]. The analysis of the deformation process has been simplified using the following assumptions: (1) Plane-strain conditions. (2) Radius of curvature greater than four or five times the thickness of the sheet, so the radial stress over the thickness is neglected. (3) The elastic behaviour of the sheet material is given by an isotropic Hook’s law, and the strain hardening is given by Swift’s law: σ eq = K (ε 0 + ε eq ) n , where σ eq and ε eq are the equivalent stress and strain. (4) The yielding behaviour is represented by the anisotropic non-quadratic yield criterion proposed by Hosford [3]: σ eq ⎛ r90 σ 1 a + r0 σ 2 a + r0 r90 σ 1 − σ 2 =⎜ ⎜ r90 (1 + r0 ) ⎝ 1 a ⎞a ⎟ ≡σ Y ⎟ ⎠ ____________________ * Camino de los Descubrimientos s/n, 41092 - Sevilla, Spain. Phone: +34 954 481 355. Fax: +34 954 460 475. [email protected] (1) where r0 and r90 are the Lankford coefficients along the rolling (0º) and the transversal (90º) directions, respectively, σ Y represents the elastic limit, and the exponent a is usually greater than 2. The suggested values are a = 6 for BCC metals and a = 8 for FCC materials. (5) Large plastic deformations, so the material is assumed to be nearly incompressible. These assumptions are used to develop a set of equations relating the process parameters. Let us assume the stretch-bending process of a sheet of initial thickness t 0 depicted in Figure 1. Because the material is assumed to be incompressible, rn t 0 = rm t , where rn and rm are, respectively, the radius of the neutral and the middle axis. A fibre of initial position λt 0 ( 0 ≤ λ ≤ 1 ) has a 2 current radius of curvature r = rin2 + λ (rout − rin2 ) , where rin and rout are the radius of curvature of the inner and the outer fibre, respectively. The strain distribution can be expressed as: ε 1 = ln ( ) r 1 = ln (1 − κ m / 2) 2 + 2λκ m − lnτ rn 2 (2) where κ m = t / rm and τ = t / t 0 are non-dimensional process parameters. Thus, the strain of the inner and the outer fibre are defined as ε 1,in = ε 1 (λ = 0) and Due to various combinations of N and M , the distribution of σ 1 over a cross-section may take one of the following three possible patterns: (i) An entirely elastic stress distribution (pattern E), for which no fibre in the sheet yields. (ii) A primary plastic stress distribution (pattern PI), for which partial yield on one side of the sheet occurs. (iii) A secondary plastic stress distribution (pattern PII), for which yielding occurs on both sides of the sheet (pattern PII− if yielding is of opposite sign on both sides of the neutral axis, and pattern PII+ if yielding is positive on the entire sheet). When the deformation of the sheet is sufficiently high the non-dimensional axial force and bending moment will reach a maximum and decrease due to the severe reduction in the sheet thickness. This global instability can be overcome by defining the variables: ν= nr τ μ= m τ2 (4) Figure 2 shows the evolution of the different strain/stress patterns in terms of ν and μ . This diagram corresponds with the M-N diagram previously presented by El–Domiaty et al [3] under the assumption of no thickness change, that is, setting τ = 1 in the present formulation (ν = n r and μ = m ). ε 1,out = ε 1 (λ = 1) . Via the constitutive relationship of the material, the stress distribution σ 1 can also be obtained as a function of the process parameters κ m and τ across the sheet thickness, λ . Then, the non-dimensional axial force and bending moment can be calculated as: nr = 1 Ne rout ∫ σ 1dr rin m= 1 Me rout ∫ σ (r − r )dr 1 m (3) rin where N e = σ 1,Y t and M e = σ 1,Y t 2 / 6 are the maximum elastic axial force and bending moment per unit of length, being σ 1,Y the elastic limit in direction 1. Figure 2: M-N diagram 3.1 FAILURE ON A SINGLE FIBRE Two independent types of failures are considered [3]: (1) a Necking-controlled Failure, which takes place when the inner fibre reaches the necking strain ratio, ε 1,in = ε 1,n (curve a-b in Figure 2), and (2) a Fracture-controlled Failure, which arises when the major strain at the outer fibre reaches a critical fracture strain, ε 1,out = ε 1, f (curve Figure 1: Variables in stretch-bend analysis d-c). As can be seen in Figure 2, the two curves intersect at point ‘o’. A piece-wise failure curve can then be defined (curve a-o-d). The stretch-bendability limit curve (curve a-o-d-e) is formed by the failure curve and eventually by the segment d-e, which corresponds to the maximum curvature of the sheet, that is, an inner radius equal to zero ( rin = 0 ). 3.2 FAILURE IN A MATERIAL VOLUME A more realistic hypothesis is to consider that failure takes place when not a single fibre but a certain material volume is damaged. Thus, as a consequence of the strain gradient across the sheet thickness, the strains acting on that material volume will be different to the strain at the external or internal fibre, modifying the failure criteria towards better predictions. Let’s consider that sheet failure occurs in the material volumes filled in Figure 3. Thus, sheet fracture occurs when all fibres in t f have reached the fracture limit strain, and necking occurs when all fibres above t n have necked. This means that both types of failure depends on what occurs in the less deformed fibre, λ f and λn , respectively. Thus, the necking- and fracture-controlled failure criteria can be expressed respectively as ε 1 (λn ) = ε 1,n and ε 1 (λ f ) = ε 1, f . Figure 3: Material volumes defined on the sheet Figure 4 shows the effect of the size of the damaged material on the failure curves. It can be seen that fracture curves move to the right as the size of the fractured volume increases ( t f ), increasing the pair μ-ν needed to cause fracture. On the other hand, necking curves move up as necked material volume ( t0 − tn ) decreases ( tn increasing), modifying the pair μ-ν required to trigger the local necking. 4 EXPERIMENTAL APPLICATION In order to explore the above ideas, the experimental results reported by Tharrett and Stoughton [1] have been analyzed. Briefly, a series of stretch-bend tests in plane strain conditions were carried out. Three different sheet thicknesses of 1008 AK steel were tested (Table 1). Figure 5 shows the experimental results. It shows the major strains measured on the convex side of the sheet for different t / R relationships, where R is the punch radius. The cases where a visible neck was observed at the concave side are represented as solid circles. Otherwise, open circles are depicted. Figure 5 also shows the failure criteria for different damaged material sizes. The in-plain necking strain is assumed to be ε 1,n ≈ FLD0 (see Table 1). As can be seen, if the necking at the concave side ( t n = 0 ) is considered, the experimental results are reasonably well reproduced for intermediate t / R values. However, the predictions differ somehow at low and high t / R values. Figure 4: Modified M-N diagram This trend can be improved by using the concept of damaged material volume. As can be seen as tn increases, that is, as unnecked material volume increases, the range of low t / R values is better predicted. In particular, a value of tn around 100-200 μm reproduces reasonably well the experimental results in the neckingcontrolled failure zone. Fracture predictions are also depicted in Figure 5. The fracture strain ε 1, f in the absence of bending is estimated through the Cockcroft and Latham’s classical ductile fracture criterion [5], along with the experimental result of Ayres et al. [6] (see [3] for details). The values of ε 1, f in plane-strain are given in Table 1. It can be seen that the fracture-controlled failure criterion at the outer fibre underestimates the trends of the experimental data for high values of t / R . However, a clear improvement is observed in the predictions assuming a damaged material volume around 100-200 μm. In particular, the agreement is remarkably good the thickest sheets. Finally, and for the sake of completeness, an arbitrary non-dimensional M-N diagram for 1008 AK steel sheets of 0.69 mm thickness is presented in Figure 6. Table 1: Mechanical properties and material constants for 1008 AK steel sheets t0 (mm) 1008 AK steel sheets 0.690 0.920 1.040 K (MPa) 549.1 552.0 556.8 n 0.230 0.230 0.240 r0 1.985 1.660 1.740 r90 1.860 2.120 1.800 a 6 6 6 FLD0 0.278 0.293 0.358 ε 1, f (plane-strain) 0.488 0.504 0.500 Figure 6: M-N diagram for 1008 AK steel sheets Two independent types of failures can be expected depending on the severity of the strain gradient: (1) a Necking-controlled Failure, which takes place when all layers in the sheet thickness become instable and neck, and (2) a Fracture-controlled Failure, which arises when the outer fibres of the sheet in the process zone fracture. It assumed that failure occurs in a certain material volume. This description results in a renewed M-N diagram. Experimental stretch-bend data reported by Tharrett and Stoughton [1] has been successfully analyzed. In the author opinion the critical size of the material volume to be considered might be related with microstructural size of the material. However, further work needs to be done to support this idea. ACKNOWLEDGEMENT The authors wish to express their gratitude to Spain's DGICyT for funding this research within the framework of the project DPI2006-06921. REFERENCES Figure 5: Influence of punch curvature on the stretching limits of 1008 AK steel 5 CONCLUSIONS The effect of strain gradient on the formability of stretch-bend metal sheets has been analysed. The mathematical description based on the previous work by El-Domiaty et al. [2] has been used and extended to stretch-bending operations in plane-strain conditions. View publication stats [1] Tharrett M. R., Stoughton T. B.: Stretch-bend forming limits of 1008 AK steel. SAE paper 200301-1157, 2003. [2] El-Domiaty A., Shabara M. A. N., Al-Ansary M. D.: Determination of stretch-bendability of sheetmetals. Int. Journal of Mach. Tools Manufact., 36(5):635-650, 1995. [3] Vallellano C., Morales D., García-Lomas F. J.: On the study of the effect of bending in the formability of metal sheets. In: Numisheet, 85-90, 2008. [4] Graf A. F., Hosford W. F.: Calculations of forming limit diagrams. Metallurgical and Materials Transactions A, 21(1):87-94, 1990. [5] Cockcroft M. G., Latham D. J.: Ductility and the workability of metals. Journal of the Institute of Metals, 96:33-39, 1968. [6] Ayres R. A.: Evaluating hardening laws at large tensile strains in sheet specimens. Metallurgical and Materials Transactions A, 14(11):2269-2275, 1983.