1.1 Introduction

The High Velocity Forming Methods have a very peculiar history. They offer certain distinct advantages over conventional metal forming processes. These advantages led to the rapid development of these techniques for about fifteen years. Some of these techniques were even integrated into the production units of several companies. And then, suddenly, research on these processes were just abandoned for no apparent reason. However, several applications of these processes remained in the industry.

The activity in High Rate Forming Techniques over time, as measured by the literature published, has been traced in figure 1.1. A large portion of the work seems to have been done using the explosive forming technique. The literature during this period documents all the advantages of high rate techniques over conventional methods - increase in formability, the ability to obtain very close tolerances due to reduced springback, reduction in wrinkling, the ability to combine forming and assembly operations, reduced tool making costs and many others. Yet, very few systematic studies have gone into identifying and understanding the factors behind these. The bulk of the literature concentrates on applying these techniques to specific problems.
 

Figure 1.1 : Activity in High Rate Forming over time

There have been a few studies on springback in high rate sheet forming and none on wrinkling in high velocity sheet forming. Several analytical studies have gone into determining wrinkling in dynamic ring compression, mainly by the engineering mechanics community. However, there have not been many experimental studies on this. Springback in high rate ring compression is another area that is still completely unexplored. In the subsequent pages of this chapter, literature published on these topics has been reviewed. The literature on formability at high rates has already been well reviewed by V. Balanethiram[4] and M. Altynova[5].

1.2 Springback in Sheet Metal Forming

1.2.1 Fundamentals of Springback in Sheet Bending

The origin of springback lies in the elastic recovery of metals after bending. When a sheet metal is bent, it strains both plastically and elastically. When the deforming forces are removed, the elastic part of the strain is recovered causing a change in the shape of the metal. This process of springback also leaves behind residual stresses in the metal. The most simple case of springback is that of sheets under pure bending moment. An analysis of this case, developed by Hosford and Caddell [6] has been presented here.

Figure 1.2 : Schematic of a bent sheet metal [6]

The bent sheet metal is shown in figure 1.2. Let r be the radius of curvature measured to the mid-plane and z be the distance of any element from the mid-plane. The engineering strain at z can be derived by considering arc lengths, L measured parallel to the x axis. The arc length at the mid-plane, L₀ does not change during bending and may be expressed as L₀=rq , where q is the bend angle. At z, the arc length is L = (r+z)q . Before bending, both lengths were the same. So the engineering strain is e_x = (L-L₀)/L₀ = zq /rq = z/r. The true strain is

(1.1)

But, often the strains are small enough and this can be approximated to

e _x = z/r (1.2)

With sheet material, w >> t. So width changes are negligible. Therefore, bending can be considered as approaching a plane strain operation, where e_y = 0, e_z=-e_x. The value of e_x varies linearly from -t/2r at the inside (z=-t/2) to zero at the mid-plane (z=0) to +t/2r at the outside (z=t/2) from equation (1.2) and figure 1.3 (a). Knowing the strain distribution, the internal stress distribution can be found if the slope of the stress-strain curve is known. Assume a material that is elastic - ideally plastic as shown in figure 1.3 (b). If the tensile yield stress is Y, the flow stress in plane strain will be s ₀ = Y. The figure 1.3 (c) is a plot of the stress distribution through the sheet. The entire section will be at a stress, s_x = ±s ₀, except for an elastic core near the mid-plane, which will shrink as the bend radius decreases. For most bends, this elastic core can be neglected.

(a) (b) (c) (d) Figure 1.3 : Strain and stress distribution across sheet thickness. Bending strain (a) varies linearly across the section. For the non-work-hardening stress-strain relation (b), the bending moment causes the stress distribution in (c). Elastic unloading after removal of the loads results in the residual stresses shown in (d) [6]

To calculate the bending moment, M needed to produce this bend, it is assumed that there is no net external force in the x direction (S F_x = 0). However, the initial force, dF_x acting on any incremental element of cross section, wdz is dF_x = s_xwdz. The contribution of this element to the bending moment is the product of the force and lever arm, z. So dM = zdF_x = zs _xwdz. The total bending moment is found from

(1.3)

For the ideally plastic material with a negligible elastic core, s_x = s₀, and

(1.4)

The external moment applied by the tools and the internal moment resisting bending must be equal. So equation (1.4) applies to both. When the external moment is released, the internal moment must also vanish. As the material unbends ( springsback) elastically, the internal stress distribution results in a zero bending moment. Since the unloading is elastic,

D s_x = EíDe _x (1.5)

where, because of plane strain, Eí = E/(1 - n²). The change in strain is given by

(1.6)

where rí is the radius of curvature after springback. This causes a change in the bending moment, D M of

(1.7)

or  (1.8)

Since M - D M = 0 after springback, from equations (1.4) and (1.8)

(1.9)

or  (1.10)

The resulting residual stress, s_xí = s_x - Ds _x = s ₀ - EíD e_x = s₀ - Eíz(1/r - 1/rí) = s₀ -Eíz(3s₀/tEí),

(1.11)

This is plotted in figure 1.3 (d). Note that on the outside surface where z = t/2, the residual stress, s_xí is compressive and equal to -s_o/2. On the inside surface, where z = - t/2, s_xí is tensile and equal to +s₀/2.
 
 

1.2.2 Factors Affecting Springback

Several material and process parameters affect springback in conventional forming methods. The material parameters include the yield strength, ultimate tensile strength, thickness of the sheet and temper of the material. Process parameters include the hold-down force on the sheet, lubrication between the die and the sheet metal and the compressive stress on the sheet metal in the thickness direction. In high rate techniques, in addition to these factors, the energy used to deform the material, the stand-off distance and vacuum level also have an effect on springback.

Since springback is caused by the elastic portion of the strain, any factor that decreases the ratio of elastic to plastic strain reduces springback. In conventional methods, lower material yield strength, higher tensile strength, higher hold-down force on the sheet, absence of lubrication between the die and the sheet metal and higher compressive stresses on the sheet in the thickness direction cause a reduction in springback. The temper of the material indirectly affects springback by altering the yield stress and the ultimate tensile stress. Springback is also found to be less in thicker sheets.

1.2.3 Springback in High Rate Sheet Forming

A very extensive literature search has brought to light only three studies on springback in High Rate Sheet Forming. Two of these studies have been conducted using explosives and one using impulsive hydraulic pressure to form the metal. These have been discussed chronologically.

The first of these studies was by H.G. Baron and R.H. Henn [7] in 1964. Disks of two aluminum alloys, DTD.687 (5.5% Zn Mg Cu) and HS.10 (1% Si Mg) in the fully heat-treated condition were formed into a shallow spheroidal section cavity in a steel die. RDX/TNT charges were used to supply the energy for forming and water was used as the medium to transmit the shock pulse. A schematic of the die with the sample in position is shown in figure 1.4.

Figure 1.4 : Steel die, holding ring and test piece [7]

The sheet metal was formed into the die cavity by detonating a pre-determined amount of the charge at a fixed stand-off distance. A series of holes was drilled across a diameter of the dish formed. The dish was then placed back into the die and a dial gage fitted with a needle probe was used to measure the distance between the inner (concave) surface of the dish and the die surface at each of these holes. The thickness of the dish at each point was subtracted from these measurements to arrive at the springback values.

Baron and Henn have carried out two sets of experiments : (1) using two stand-off distances - 4 in. and 16 in. - for a 0.047 in. thick DTD.687 sheet (2) using HS.10 sheets of two different thicknesses - 0.034 in. and 0.126 in. - for a stand-off distance of 4 in. In each set of experiments, the weight of the explosive used was varied from the minimum required for the sheet to completely contact the die to that at which extensive failure of the sheet occurs.

(a) (b)  Figure 1.5 : Effect of charge weight on the springback of 0.047 in. DTD.687 after forming in water. (a) Stand-off distance 4 in. (b) Stand-off distance 16 in. [7]

The springback values measured from these experiments with DTD.687 are plotted in figure 1.5. Comparison of figures 1.5 (a) and 1.5 (b) shows that the stand-off distance has little effect on springback. At each distance, the smallest charge which produced complete contact between the die and the sheet resulted in a springback of about 0.1 in. at the pole. Increased charge weight reduced the springback, but the maxima in the curves moved away from the pole. Surface smoothness was found to deteriorate, with the development of faint marks in concentric circles and a small raised dimple was formed at the center. Further increase in charge weight led to the formation of a hole at the center and a concentric ring of cracks about 3 in. in diameter (in the region of maximum springback). A spring-forward effect was also noticed for the largest charges.
 

(a) (b) Figure 1.6 : Effect of charge weight on the springback of alloy HS.10 after forming in water with a 4 in. stand-off. (a) Sheet 0.034 in. thick. (b) Sheet 0.126 in. thick. [7]

The springback values measured for the experiments with HS.10 are shown in figure 1.6. HS.10 is reported to have shown lesser springback than DTD.687. This could probably be because HS.10 has a lower yield strength than DTD.687. Large charges produced a series of pronounced ripples in the thinner HS.10 alloy and a 2 oz charge gave a remarkable eruption at the center. There were no concentric ripples in the thicker alloy. The spring-forward effect was again observed, being particularly noticeable in the thicker alloy. As can be seen from figures 1.6 (a) and (b), the largest charges of 2 oz caused a large springback again at the center, suggesting the presence of rebound effects.

Baron and Henn attribute the presence of concentric circles in the thicker alloys and ripples in the thinner alloy to the movement of metal towards the center during the deformation process. The progressive thickening of the metal at the center is manifested as concentric circles in the thicker alloys. In the thinner alloys, this movement of the metal towards the pole is said to cause buckling which ends up as ripples. The spring-forward effect is also explained by a large dimple caused due to accumulated metal at the center. Thickness measurements along the diameter of the dishes seem to support the theory of movement of mass towards the pole.
 

Figure 1.7 : Experimental Set-Up of Behera et. al. [8]

In 1977, T. Behera, S. Misra and J. Banerjee [8] used cordtex explosives to form brass and copper blanks into parabolic dies. The figure 1.7 is a schematic of their experimental set-up. Water was used as the medium for transmitting the shock waves. Parabolic reflectors were used to improve the efficiency of energy transfer. They have studied the effect of the following parameters on springback : (1) size of explosive charge (measured by its length in mm), (2) die material, (3) blank holding force, (4) blank thickness, (5) energy transfer medium and (6) the use of a blanket.

Figure 1.8 : Effect of charge on springback [8]

The variation of springback with the size of the explosive charge is shown in figure 1.8. The sheet metal did not fully contact the die for charges of length 25 mm and 37 mm and hence the curves corresponding to these cannot be considered as springback curves. The 50 mm long charge caused complete contact between the die and the sheet and this caused the minimum springback. Subsequent increases in charge lengths progressively increased the springback. This suggests the presence of rebound effects.

Figure 1.9 : Comparison of springback for sheets formed into mild steel (m.s) die and a wooden die [8]

The figure 1.9 is a comparison of the springback values obtained using mild steel and wooden dies. The samples formed with the wooden die show lesser springback. This is attributed to the lower coefficient of restitution between the die and the sheet when wooden dies are used. Wood absorbs the kinetic energy of the moving blank more than a metallic die. This again points to the presence of rebound effects.

Figure 1.10 : Effect of blank hold-down force on springback [8]

From the figure 1.10, it can be seen that an increase in blank holding force reduces springback. This is explained by the increased stretching of the sample (since drawing is prevented) which increases the tensile stress on the blank. This could also cause the neutral axis in the analysis by Hosford and Cadwell to move out of the sample.

Figure 1.11 : Effect of blank thickness on springback [8]

Figure 1.12 : Effect of energy transfer medium on springback [8]

It can be seen from figure 1.11 that the springback is lesser for thicker samples. The effect of energy transfer medium on springback is shown in figure 1.12. The springback is lesser with water compared to an air cell. It is suggested that the second shock wave in the water medium opposes the rebound of the sample and that such an effect is not possible for an air medium.

Figure 1.13 : Effect of blanket on springback [8]

Finally, the effect of plasticine and rubber blankets is shown in figure 1.13. Use of a blanket seems to reduce springback, with a rubber blanket being more effective than a plasticine blanket. Behera et. al. feel that the blank and the blanket behave like a single piece of material of higher thickness which causes a reduced springback.

In 1981, Yamada et. al. [9] published a study on mechanics of springback in high speed sheet forming. They studied the deformation of rectangular titanium plates into a spherical section steel die both at low rates and high rates. In the static case, the plate was pressed between the dies and high velocity forming was done using impulsive hydraulic pressure. The main purpose of their study was to determine the effect of compressive stress in the thickness direction on springback.

Yamada et. al. have used a single parameter to quantify springback in contrast with the previous studies described. If "r₀" is the radius of curvature of the die and "r" is the radius of curvature of the deformed plate, then springback is defined as

(1.12)

After the plate is deformed, the coordinates of the die-side surface of the plate were measured at five points (both ends, center point and the mid-points between the center and the ends). The radius of curvature of the entire sample was taken as that defined by the end points and the center point. Three local radii of curvature were defined using the three sets of three adjacent coordinates. The corresponding springback values were calculated.

The high rate experiments were done by impinging a high speed plastic projectile on water, which transmits the shock wave to the sheet metal. The velocity of the projectile was measured using two laser beams. The collision speed of the sheet metal with the die surface was also measured using the electrical pin-contact method. This was done by repeating experiments (which were done with a steel die) using an identical dental stone die. Three electrical contact-pins embedded at the bottom of the dental stone die were used to obtain the velocity measurements.

Figure 1.14 : Effect of punch load on reduction of springback [9]

The figure 1.14 is a plot of the springback for low rate experiments. Springback is found to decrease monotonically with an increase in the punch load. But, for the range of punch loads used, springback does not reduce to zero.

Yamada et. al. feel that the compressive stresses acting along the thickness direction are responsible for reduction in springback with increasing punch load in low rate forming. The additional compressive stress caused by an increased punch load acts to make the stress distribution along the thickness more uniform which reduces the springback.

The shape of the high rate samples after deformation is shown in the figure 1.15. The figure 1.16 is a plot of the springback in the high rate samples. Springback is found to decrease with increasing collision speed and for collision speeds in excess of 66 m/s, springback is found to be negative, i.e. the radius of curvature of the sample is smaller than that of the die. For a collision speed of 47 m/s, the springback is very close to zero. This suggests that springback could be eliminated or made minimal by using an optimum collision speed.

Figure 1.15 : Specimen shape after bending [9]

Figure 1.16 : Collision speed dependence of springback [9]

The explanation for reduction in springback with increasing collision speed in high rate forming is based on a comparison with the low rate forming results. The figure 1.17 is a plot of springback as a function of compressive stress for both the low rate and high rate experiments. For the low rate experiments, the compressive stress was evaluated by dividing the punch load by the specimen area projected on a plane normal to the direction of load application. For the high speed experiments, the compressive stress was calculated from impedance mismatch using the following equation.

(1.13)

where P is the compressive stress produced, V is the velocity of the sheet and r₁U₁ and r₂U₂ are the shock impedance of the specimen and the die respectively. Since Hugoniot of the material was not available in the low stress region, evaluation was made upon an acoustic approximation.

Figure 1.17 : Compressive stress dependence of springback in static and high speed bending [9]

It can be seen from figure 1.17 that the springback variation with compressive stress is very similar for both the high speed and low speed samples. This suggests that the same mechanism controls springback in both cases. Yamada et. al. further argue that the shock impedance of steel and dental stone and hence the compressive stresses produced in them are very different. Yet, the compressive stress dependence of springback is the same for both materials and this observation is said to confirm that compressive stresses control springback. The author of this thesis refutes the validity of the latter argument. The compressive stress is computed from the collision speed of the sample. Since the impedance value is constant for a material, the compressive stresses for steel and dental stone will obviously show a similar variation with collision speed, even though the exact values may be different. Thus, the compressive stress and springback are two quantities that depend on the collision speed. There is no basis for relating the two by the latter argument put forth by Yamada et. al.

However, the fact that compressive stresses are responsible for reduced springback at low rates and that the variation of springback with compressive stress at high rates is similar to that at low rates suggest that compressive stresses could indeed be the most important factor affecting springback at high rates.

Yamada et. al. have also attributed the final shape of the high rate samples to the process of bending. Using high speed photography, they have captured snapshots showing the shape of the samples at different stages of deformation. The deformation sequence for the low speed and high speed samples is shown in figure 1.18. When the applied pressure is low, the deforming speed of the specimen is also low. Therefore, the plastic bending waves meet at the center of the sample and the sample collides with the die subsequently. The entire sample has a "V" shape before colliding with the die, as shown in figure 1.18 (a). Samples deforming by this process end up with a positive springback. On the other hand, when the applied pressure is high the deforming speed is also high. Consequently, as shown in figure 1.18 (b), the specimen collides with the die right from the beginning, starting at the ends and moving towards the center. This process of deformation is said to cause a negative springback.

(a) (b) Figure 1.18 : Deformation sequences of a sheet metal subjected to impulsive hydraulic pressure at (a) low rates (b) high rates [9]

Thus, all the three studies seem to indicate that springback goes through a minimum as forming energy or velocity is increased.

1.3 Wrinkling in Sheet Metal Forming

Wrinkling is caused due to the presence of excess material in the die during a forming operation. Consider a sheet being formed into a female die in the shape of a cone section using a punch. Three directions can be imagined.

(a) the direction of motion of the punch - the z direction

(b) the thickness direction of the sheet - the r direction and

(c) the circumferential direction - the q direction.

As the sheet moves into the die, there is excess material in the q direction. As the punch moves into the die, the sheet metal both stretches in the direction of punch movement and also draws in. These have opposing effects as far as material in the q direction is concerned. When the material stretches in the z direction, it causes a contraction of the sheet in the q direction. The drawing causes more sheet to come into the die and hence increases the amount of material in the q direction. If the contraction in the q direction due to stretching in the z direction does not compensate for the excess material brought in by draw-in, then the excess material results as wrinkles in the final component.

In conventional forming, one of the ways of eliminating wrinkling is to increase the extent of stretching and reduce the draw-in. This can be done by applying a hold down pressure on the sheet. The negative aspect of this is that the sheet is forced to stretch to fill the die and might tear if the forming limit is exceeded.

Wrinkling also strongly depends on the "r" value of the material. The "r" value is defined as the ratio of the width strain to the thickness strain when the sheet is stretched along its length, i.e.

(1.14)

Thus, if the "r" value is high, the width strain is high and the tendency of the material to wrinkle is lesser.

So far, no systematic procedure has been evolved to quantify or predict wrinkling. The solution to wrinkling problems is often to modify the tooling by trial and error till wrinkling is eliminated.

There has been no fundamental study on wrinkling in high rate forming.

1.4 Springback in Ring Compression

The springback in ring compression is important because of its effect on clamping stress. The electromagnetic process is widely used to compress rings or tubes onto other tubes or mandrels, i.e. the forming and the assembly processes are combined. Often, the requirement in such cases is that the ring or tube should hold on to the mandrel tightly. In other words, the clamping stress should be high. A reduced springback indicates an increase in the clamping stress.

The factors affecting springback in axisymmetric compression include the yield strength of the material, ultimate tensile strength of the material, thickness of the ring, height of the ring, temper of the material and energy used to compress the ring.

No literature was found on springback studies in ring compression.

1.5 Wrinkling in Ring Compression

Wrinkling in ring compression is caused by the non-uniform compression of the ring. Unless the reduction is extremely small, it is not possible to compress a ring uniformly under static conditions [10]. Below a critical wall thickness to diameter ratio, the ring will buckle under the action of external pressure. Using dynamic compression, larger reductions could be obtained before wrinkling occurs.

In dynamic compression, wrinkling is caused mainly due to small differences in the radial velocities of the ring at different points along its circumference. To the knowledge of the author, there have been no experimental studies on the compression of rings or short cylindrical shells. However, there have been several theoretical studies on buckling of rings and short cylindrical shells during dynamic compression [11, 12, 13]. In these studies, the shell is first assumed to deform uniformly and the effect of slight variations in radial velocity around the circumference are investigated. Representing these perturbations as a Fourier series, analysis shows that a particular term becomes amplified and the number of waves in the final buckled shape can be predicted. No attempt has been made to review the theoretical models proposed as these are beyond the scope of this thesis.

There has been one experimental study of wrinkling of long tubes under high rate compression by J.L. Duncan, W. Johnson and J. Miller [10]. The study has compared the effectiveness of different high rate techniques - electrohydraulic forming, explosive forming, explosive gas forming and water hammer technique - in uniformly reducing a thin-walled tube onto a mandrel. The experimental set up is shown in figure 1.19. The tube to be compressed and the mandrel were placed inside a thick steel cylinder. The tube, mandrel and the cylinder were concentric. The tube was sealed at both ends and the space between the tube and the mandrel was evacuated. The size of the mandrel was varied to obtain different reductions. Separate heads were attached to the tube for the different forming processes. The experiments were conducted on aluminum, mild steel and stainless steel tubes. The tubes were 12 inches long.
 
 

Figure 1.19 : Schematic of the forming equipment showing the four detachable heads [10]

Duncan et. al. observed a characteristic pattern in the geometry of the tubes deformed by explosive forming and electrohydraulic forming. In these processes, the intensity of the pressure pulse might be expected to diminish as the wave passes along the length of the tube. The figure 1.20 is a diagrammatic representation of a typical deformed tube. If the energy released was sufficient, a length "A" of the tube closest to the source was compressed uniformly onto the mandrel. Beyond this region "A", there was frequently a length "B" in which the deformed tube had a number of small wrinkles. Following the region of multiple wrinkling, there was a portion of the tube "C" in which a single wrinkle appeared as is characteristic of static reduction. Since the length of the tubes did not change, it was assumed that the reduction took place under plane strain conditions.

Figure 1.20 : Diagram of a typical formed tube showing the different modes of deformation [10]

The following observations were also made for the tubes formed with the electrohydraulic and explosive forming processes.

1. For the same amount of energy released, the length of tube reduced uniformly, i.e. length "A", decreased rapidly with increasing reduction.
2. For a constant reduction, the length "A" increased approximately linearly with the energy released.
3. The explosive forming process was found to be more efficient than the electrohydraulic process.

Figure 1.21 : Approximate forms of pressure waves in the different processes [10]

The difference in the effectiveness between the different process has been explained by the differences in the rise time and the intensity of the pressure pulse for the different processes. The approximate forms of the pressure waves in the different processes is shown in figure 1.21. The electrohydraulic, explosive and explosive gas forming processes have a very short rise time and hence the tube could be successfully reduced by these. The rise time for the water hammer technique is very long and hence the inability to reduce the tubes without wrinkling.

Thus, from the studies of Duncan et. al., it could be concluded that the rise time of the pressure pulse and the energy used to compress the ring or tube are main factors in controlling wrinkling. Also, three modes of deformation can be identified depending on the energy - uniform compression, a large number of small wrinkles and one or two large wrinkles.

References for Chapter 1 :

*1.1. Balaís thesis

*1.2. Marinaís thesis

*1.3. Hosford and Cadwell.

1.4. H.G. Baron and R.H. Henn, "Spring-back and metal flow in forming shallow dishes by explosives", International Journal of Mechanical Sciences, Vol. 6, pp. 435-444.

1.5. T. Behera, S. Misra and J.Banerjee, "Explosive Forming with Parabolic Dies", Indian Journal of Technology, Vol. 15, Sept. 1997, pp. 365-368.

*1.6. T. Yamada, K. Kani, K. Sakuma and A. Yubisui, "Experimental Study on the Mechanics of Springback in High Speed Sheet Metal Forming", pp. 306-314.

1.7. J.L. Duncan, W. Johnson and J. Miller, "Reducing thin walled tube by electrohydraulic and other processes", Conference on electrical methods of machining forming and coating, 1975, pp. 217.-228

1.8. A. Florence and H. Vaughan, "Dynamic plastic flow buckling of short cylindrical shells", Int. J. Solids and Structures, Vol. 4, 1968, pp. 741-756.

1.9. W. Stuiver, "On the buckling of rings subject to impulsive pressures", Trans. ASME, J. App. Mech., Vol. 87, Sept. 1965, pp. 511-818, 1965.

1.10. G.R. Abrahamson and J.N. Goodier, "Dynamic plastic flow buckling of a cylindrical shell", ASME, 4th Cong. App. Mech., Vol. 2, pp. 939-950.