Sheet Metal Forming: A Review

William F. Hosford and John L. Duncan

Developments in the numerical modeling of stamping processes and experimental measurements now make it possible to design stamping processes using sound engineering principles. This article shows how experimental and theoretical contributions have led to the concept of a forming window in strain space that identifies the strains that can be developed safely in a sheet element. It is bounded by failure limits corresponding to localized necking, shear fracture, and wrinkling. A robust stamping process is one in which the strains in the part lie well within the forming window. The nature of the window and the influence of material behavior on its shape can be predicted.

SHEET FORMING

In stamping, drawing, or pressing, a sheet is clamped around the edge and formed into a cavity by a punch. The metal is stretched by membrane forces so that it conforms to the shape of the tools. The membrane stresses in the sheet far exceed the contact stresses between the tools and the sheet, and the through-thickness stresses may be neglected except at small tool radii. Figure 1 shows a stamping die with a lower counter-punch or bottoming die, but contact with the sheet at the bottom of the stroke will be on one side only, between the sheet and the punch or between the die and the sheet. The edge or flange is not usually held rigidly, but is allowed to move inward in a controlled fashion. The tension must be sufficient to prevent wrinkling, but not enough to cause splitting.

The limits of deformation, or the window for stamping, are shown in Figure 2. It is assumed that the failure limits are a property of the sheet. This assumption is reasonable if through-thickness stresses are negligible, and if each element follows a simple, linear path represented by a straight line radiating from the origin.

The path in stampings is described by the ratio of the membrane strains

b=e₂/e₁

which vary from equal biaxial stretching (b = 1) to uniaxial compression (b = -2.) Figure 3 shows the strain paths along two lines in a rectangular pressing. Such diagrams are strain signatures of the part. Unequal biaxial stretching (b 1) will occur in the middle, A. In the sidewall, C, plane strain is most likely. If the side of the stamping is long and straight, plane strain will exist also at D. Over the rounded corner of the punch at F, the strain is biaxial. From H to J, strains are in the tension-compression quadrant. The concept of the forming limit curve is that all possible strain signatures are bounded by an envelope that is a characteristic only of the material. The origins of this failure map were reviewed earlier,¹ and more recent developments are described here.

TENSILE INSTABILITY

Lankford, Gensamer, and Low² studied the tearing of aluminum alloys deformed along different strain paths. Their results for one aluminum alloy are replotted in Figure 4. Failure in the tension-compression quadrant (as in Figures 2 and Figure 4) occurs by local necking along a direction of zero extension. In this region, experiments agree well with the model of Hill,³ which is equivalent to a maximum tension criterion. Tension is the force per length along a section in a sheet, and is the product of the stress and the thickness, T = st. For a material with an effective stress-strain curve

This predicts failure at a constant thickness, t = t_oexp(-n), as the limit line on the left side of Figures 2 and Figure 4. This line intercepts the major strain axis at e₁ = n. The maximum tension criterion is a necessary condition for local necking, but in biaxial tension there is no direction of zero extension.

Keeler and Backofen⁴ measured failure strains in biaxial stretching (Figure 5). As the strain path becomes more biaxial, the measured failure strains increase, exceeding the strain at maximum tension. They introduced the term "forming-limit curve" to describe the plot of conditions that cause local necking.

Although these observations remained unexplained, they were confirmed and extended by additional work (e.g., References 5-8). Coupled with the development of circle-grid analysis, this formed a powerful method of diagnosing stamping failures. The forming-limit curves could be obtained by measurements in the press shop or laboratory. Figure 6 is a typical-forming limit curve for low-carbon steel.

DEFECT ANALYSIS

A pre-existing defect in the sheet, such as a local reduction in either thickness or strength, can have a large effect on the strain at failure. As an example, a tensile test specimen may have a defect region B that has a slightly lower load-carrying capacity than elsewhere (region A). The initial defect can be a region that is thinner or that has a lower flow stress because of variations in grain size, orientations, or composition. In any case, it can be characterized for mathematical analysis as though it were thinner.

f = tBo/tAo

(1)

where t_Bo and t_Ao are the initial thicknesses in the two regions. With this defect, the maximum load in the test-piece is reached when the strain in the imperfection reaches the value, n. The uniform strain (e_A = e_u) is

eAnexp(-eA) = fnnexp(-n)

(2)

An approximate solution⁹ for Equation 2 is

(3)

The values of the uniform strain given by Equations 2 and 3 are compared in Figure 7. It is interesting to note that templates frequently used for machining sheet tensile specimens often have a built-in defect of about 1 - f = 0.001, which, if n = 0.2, results in a uniform elongation of e_u = 0.18 instead of 0.2 (i.e., an imperfection of one part in one thousand reduces the maximum uniform strain by 10%).

Marciniak and Kuczynski^10,11 proposed that during biaxial stretching, local necks also form by the growth of very small defects. If a trough-like region perpendicular to the largest principal stress is weaker or thinner than the bulk of the material (Figure 8), the strain will localize there. Equilibrium requires that the tensile force perpendicular to the groove is the same in each region.

s1AtA = s1BtB

(4)

and compatibility requires that the strains parallel to the groove are equal,

e2A = e2B

(5)

If the uniform region deforms in a linear strain path, b_A, the strain in the groove accelerates along a nonlinear path with b_B changing from b_A to 0 (Figure 9). When the groove reaches plane strain, deformation in the uniform region ceases, and the forming limit for that value of b_A is given by the strains e₁* and e₂* in the uniform region, A.

The maximum tension criterion that leads to e₁* + e₂* = n appears to be a necessary, but insufficient condition for local necking. The other condition is de_2A /de_1A = 0 in the groove. In the tension-compression region of the diagram (e₂ £ 0), deformation in region A ceases immediately once the tension reaches a maximum, while the neck forms along a line of zero extension. In the tension-tension quadrant (e₂ > 0), straining continues beyond the tension maximum until plane strain is reached in the neck (unless fracture intervenes.)

The Marciniak-Kuczynski model provides a method for calculating forming-limit curves and the effects of material properties. However, calculations in a number of papers (e.g., References 12-14) predicted limits that deviated significantly from those observed experimentally.

YIELD CRITERIA

Figure 10a illustrates how the stresses in the groove, B, differ from those in the uniform region, while satisfying the equilibrium condition

s1b = (tA/tB)s1A

(6)

The forming limit is reached when region B reaches plane strain (Figure 10b). Sowerby and Duncan¹⁵ pointed out that this depends on the yield surface, which is influenced by the R-value, measured in the tensile test. Using Hill's anisotropic yield criterion,¹⁶ for plane stress

s12 + s22 + R(s1 - s2)2 =

(7)

Parmar and Mellor¹² predicted a large dependence of forming limits on anisotropy (Figure 11). However, experimental forming-limit diagrams show no appreciable effect of anisotropy.

A yield criterion with a high-exponent was proposed^17,18 to approximate upper-bound yield locus calculations based on crystallographic slip, that is,

s1a + s2a + R(s1 - s2)a = 2a

(8)

Exponents of a = 6 and a = 8 were proposed for body-centered cubic and face-centered cubic metals, respectively. This locus lies between the Tresca criterion and Hill's quadratic form as shown in Figure 12. Forming limits calculated using this high-exponent criterion¹⁹ predict almost no effect of anisotropy (Figure 13), because in the high-exponent criterion, anisotropy has little effect on the stress ratio needed for plane strain. Figure 14 from Barlat²⁰ shows the effect of the yield-criterion exponent on the forming limits. The yield locus for an exponent of is the Tresca locus. For such a material, the yield locus has a vertex at which the strain vector can have any orientation. Thus, when the tension is a maximum, plane strain can occur in the neck with immediate failure. The forming limit in the tension-tension quadrant does not coincide with the line

e₁* + e₂* = n

At one time, the discrepancy between calculated and experimental forming limits generated interest in vertex models of yield loci and the application of the deformation theory of plasticity, but these concepts have not been supported by experiments. Other factors may affect the yield surface. It is clear that in the biaxial stretching region, the material in the groove, B, follows a curved straining path. Effects similar to the Bauschinger effect, such as kinematic hardening, could diminish strain hardening in the groove and accelerate failure.

NATURE OF DEFECTS

Wilson and coworkers^21,22 have studied local variations in grain orientation and grain size that may be considered Marciniak defects. Local composition variations may also be of importance in some materials.

In calculating forming limits, a value for the initial imperfection, f, in the range 0.985 £ f £ 0.995 is usually chosen to create a fit with experimental findings. McCarron et al.²³ intentionally machined defects into steel sheets before subjecting them to biaxial stretching and found that the artificial defects needed to be in the range of 0.990 £ f £ 0.992 to localize the failure. The inability of smaller defects to initiate local necking suggests that defects in the range given were already present.

Industrial sheet probably has a characteristic spatial and size distribution of defects. In many stamping operations, the area of sheet subject to critical straining may be so small that it does not contain a large sample of defects in the sheet. The scatter of limit strains suggested that the limit is a band rather than a single line.²⁴ It was suggested²⁵ that the mean forming limit and scatter decreases as the physical scale of the part is increased and as strain gradients became more gradual.

STAIN HARDENING

The forming limit in plane strain is approximately equal to the strain-hardening exponent, n. If n is reduced (e.g., by cold work), the window in biaxial tension becomes very small (Figure 15). In fully cold-worked sheet, n ~ 0, the only processes that are possible without tearing are equal biaxial tension, as in stretching over a domed punch, and constant thickness deformation, e₁ = -e₂, as in deep drawing. These possibilities are exploited in forming the two-piece aluminum beverage can.

The critical events at necking occur at large strains so it is important in calculations to use the values of strain hardening at high strains. In the tensile test, information can only be obtained up to an equivalent strain of n. The loci of equivalent strains are shown in Figure 16; clearly limit strains are greater than these. The shape of the stress-strain curve at higher strains can be obtained from other tests, such as the hydrostatic-bulge test. Bird and Duncan²⁶ showed that materials with similar strain hardening in the tensile test range may diverge at higher strains (Figure 17), and this influences the limit strains.

STAIN-RATE SENSITIVITY

The strain-rate sensitivity of a material can be approximated by

(9)

For engineering metals at room temperature, the m-values are in the range of -0.005 to 0.015. Even these low values of m can be significant. In a typical tensile specimen, the strain rate is v/L if the length of the parallel section is L and the extension rate is v. Typically the width of the test-piece is about L/8, and the thickness is L/100. In a diffuse neck, deformation occurs within a length approximately equal to the width, so the strain rate increases by a factor of 8. In a local neck, deformation concentrates in a length about equal to the sheet thickness, which increases the strain rate 100-fold. Even a little strain-rate sensitivity increases the strength of the material in the neck, causing the neck to develop more slowly, allowing additional deformation outside the neck and increasing the limit strain. The effect of m on the forming limit curve is similar to the effect of the strain-hardening exponent, n.

FRACTURE LIMITS

Tearing usually occurs by a shear fracture in the root of the localized neck. The strain at fracture, e_f, is a material property like the reduction in area measured in round-bar tensile tests. If the fracture strain for the material exceeds the strain at which the local neck reaches plane strain (Figure 9), it does not influence the limit strains in the uniform region, A. If, however, the fracture occurs before plane strain-necking, it reduces the forming limit. This is illustrated in Figure 18 from LeRoy and Embury,²⁷ which shows that in biaxial tension, the limit strain in an aluminum alloy is governed by fracture rather than by local necking.

CHANGING STRAIN PATHS

In the experiments used to determine forming limits, the strain paths are almost linear. In some stampings, however, the strain paths are not linear. This problem can be handled theoretically²⁸ by simply imposing a changing strain path in region A. Experiments with single, abrupt, path changes tend to confirm this approach.²⁹

Figure 19 shows the effect of path changes in aluminum alloy 2008-T6. The change in forming limits depends on the preloading path. For a bilinear path, the effect of the first loading is to displace the minimum, either increasing or decreasing the limits.

THICKNESS EFFECTS

Higher forming limits are found for thicker sheets.³⁰ Figure 20 is a simplified representation showing how measured forming limits for plane strain increase with both thickness and strain-hardening exponent for steel. This measured effect is due partly to the fact that the same grid size is used regardless of the sheet thickness. The cross-sectional shape of necks is likely to be geometrically similar in thick and thin sheets. In this case, the regions sampled in thick sheets tend to lie closer to the centers of the neck and, therefore, have a higher average strain.
This explanation is supported by experiments³¹ in which both the grid size and thickness were changed (Table I).

TABLE I
The Dependence of Forming Limits at Plane Strain on Sheet Thickness and Grid Size31
	Engineering Strain
Sheet Thickness (mm)	2.54 mm Grid	5.08 mm Grid
0.76	0.38	0.34
1.52	0.47	0.41

Only a small change in the forming limit was observed when both the thickness and the grid size were doubled. The two sheets, 1.52 mm and 0.76 mm thick, had nearly the same composition, 0.049 wt.% C and 0.038 wt.% C, and n-values, 0.249 and 0.248, respectively. The measured forming limits in these results increase with sheet thickness, but the increase is small when the grid size increase is proportional to the sheet thickness. This is true for similar materials, but neck shape depends on the rate sensitivity of the material.

Necks are more gradual in materials having a higher m. Aluminum alloys, with very low m, have much sharper necks than low-carbon steel. With the sharper neck, the thickness effect is likely to be less significant. This is consistent with the much lower thickness effect in aluminum alloys found by Smith and Lee³² (Figure 21). It has been suggested^26,27 that if the absolute size of the defects remains constant, when the sheet thickness increases the inhomogeneity factor, f, would decrease, thereby raising the forming limits.

WRINKLING

Wrinkling may occur when the minor stress in the sheet is compressive. Wrinkling of the flange areas (e.g., Figure 3) can be suppressed by the blankholder. However, wrinkling may also occur in unsupported regions or regions in contact with only one tool. Figure 22 shows the forming of a shell with a conical wall. A compressive hoop stress may arise in the unsupported wall at C if too much material is allowed to be drawn into the cavity.³³ The usual remedy is to increase the blankholder force, B, that increases the radial stress, s₁, and strain, e₁. The lateral hoop contraction accompanying this radial stretching helps alleviate the hoop compression. How much stretching is required depends on the R-value of the material, which is the ratio of width-to-thickness strain in the tensile test. Less radial stretching is required with a high R-value, decreasing the chance of tearing failure. The R-value of drawing quality steel is typically >1 and that of aluminum sheet <1 so the wrinkling problem is more severe in aluminum. The wrinkling tendency is also affected by elastic modulus, sheet thickness, and tooling so there is no single wrinkling limit for a material, and the inclusion of this line in the general diagram (Figure 2) is not strictly valid.

Whitely³⁴ showed that increased R permits drawing of deeper cups, but his analysis does not explain why high R-value sheets are advantageous in forming shallow parts, such as car-body outer panels. Their use is due to their increased wrinkling resistance.

Acknowledgements

The authors acknowledge helpful discussions with many colleagues, including D. Lee, A.F. Graf , J.D. Embury, and B.S. Levy.

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William F. Hosford is a professor in the Department of Materials Science and Engineering at the University of Michigan. John L. Duncan is professor emeritus at the Department of Mechanical Engineering at the University of Auckland.

For more information, contact W.F. Hosford, University of Michigan, Department of Materials Science and Engineering, 2300 Hayward Street, Ann Arbor, Michigan 48109-2136; (734) 764-3371; fax (734) 763-4788; e-mail whosford@umich.edu.