ASM Metals HandBook Vol. 14 - Forming and Forging

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Fig. 1 Major elements of a computer-aided sheet metal formability prediction system. Source: Ref 10.

Plasticity and Constitutive Relations. Assuming an anisotropic, inelastic material dependent on deformation rate

under a general state of stress, the inelastic strain rate,

, may be expressed in terms of the flow potential, or the yield

function f and the scalar multiplier λ(Ref 11, 12):

(Eq 1)

where f is a generalized yield function such as that given in Ref 13, and the scalar multiplier is equal to (3/2) ( / )

where and represent equivalent strain rate and equivalent stress, respectively. For example, the yield function f can be

written in terms of the distortion matrix M

, which describes the variation of the flow stress with orientation, or:

f = M

ij i j

(Eq 2)

where σ

and σ

denote the stress vectors corresponding to the appropriate tensor counterparts, σ

. Therefore, it follows

that the equivalent stress, , can be expressed as = (M

)

1/2

. For the case of a plane stress loading condition (

= 0)

and materials with planar isotropy (M

= M

), it can be shown that the well-known anisotropy parameter R is related to

by R = (2/M

) - 1.

In order to describe the strain hardening and strain rate hardening behavior of materials, the equivalent strain rate, , may

be expressed in terms of the equivalent stress, , or:

(Eq 3)

where m is known as the strain rate sensitivity of flow stress; the effective flow stress k and the reference strain rate

are

defined below. The effect of temperature can be readily incorporated into Eq 3 by including an exponential function as an

Arrhenius-type expression (Ref 12).

The effective flow stress k and its dependence on can be determined from a uniaxial tension test at a reference strain rate

. It has been demonstrated that a number of experimental data could be represented most accurately by the well-known

Swift-type equation (Ref 14):

k = k

(

+ )

(Eq 4)

where k

and

are material constants and n is the well-known strain hardening exponent. Equations 3 and 4 can now be

combined to express the equivalent stress in terms of variations in and (Ref 12, 15, 16).

Prediction of Forming Limit Diagrams (FLDs). The detailed method of computing the limiting and elongation

FLDs has been recently outlined (Ref 8, 9). Specifically, there are several methods that can be used to compute the FLDs.

One method is to treat the onset of failure as the condition that leads to plastic instability (Ref 14, 17, 18); the second

approach incorporates J

deformation theory into a classical bifurcation analysis (Ref 4, 7, 19). Finally, the approach

based on the idea that necking develops from local regions of initial nonuniformity has been widely used in recent years

to establish FLDs (Ref 3, 6, 7, 8, 9).

Extending the idea of initial nonuniformity to start the neck growth process, it is possible to formulate an analytical model

that predicts the development of nonuniform flow in sheet metals under various plane-stress loading conditions (Ref 8, 9).

Specifically, the specimen is assumed to have an initial geometric nonuniformity in the form of a band of reduced section.

When the initial minimum thickness at the reduced section and the thickness at the uniform section are designated by

and , respectively, the nonuniformity index, η, is given by η= 1 - / ]. A uniform strain rate state is then

imposed on the specimen containing the initial nonuniformity. The growth of initial nonuniformity is described by

simultaneously integrating a series of equations at different but prescribed locations in the neck. A rate-dependent flow

theory of plasticity was used, and the computation of the detailed neck growth was repeated over a wide range of

proportional loading conditions to establish the limiting strain FLDs. In addition to η, the input material parameters that

are required for the calculation of FLD are m, n, ε

, and M

A computed limiting strain FLD and experimental data (Ref 15, 20, 21, 22) are compared for aluminum-killed steel in

Fig. 2. The η value of 0.008 was obtained from the measurement of sheet thickness variations. Because the limiting strain

refers to the ultimate level of strain in the uniform section while the neck growth has taken place, it is safe to conclude

that the material has failed when the state of limiting strain is reached. An additional lower bound is also shown in Fig. 2

by the dashed lines which separate safe from marginal regions, taken from an appliance business practice (Ref 23). An

added advantage of specifying such a lower bound is that it incorporates the empirical relationship that accounts for the

effects associated with sheet thickness and material properties at the onset of necking (Ref 24).

Fig. 2 Comparison of calculated limiting strain forming limit diagrams (FLDs) for aluminum-

killed steel with

experimental data. Experimental data are distinguished between those tested under the in-

plane (IP) and the

out-of-plane (OP) loading conditions.

Integrated Systems Approach. A method of processing and controlling the flow of information is outlined in Fig. 3

in the form of a simplified flow diagram. All the critical decision points are also indicated in the diagram. Briefly, the

finite element analysis program obtains the necessary input material data, geometry of the part, and boundary conditions

of loading, and thereby computes the stress and strain distributions on the formed part. Branching to the right side of the

diagram, the necessary material parameters are read again from the material data base, and the theoretical limiting FLD is

computed for the particular material. Because necking may occur below the limiting strain region, a boundary depicting

the safe and marginal regions obtained from the shop experience is also specified on the computed FLD. Finally, the

computed major and minor strains in the formed part are plotted directly on the composite FLD. If all the computed

strains in the formed part fall in the safe zone, the program terminates with a sign that indicates that the particular part

could be formed without any difficulty. On the other hand, if the computed strain levels are in the failure zone, the

program requests that the material, geometry, or the details of loading condition be changed. The entire routine may be

repeated until a satisfactory or safe condition is achieved.

Fig. 3 Flow diagram illustrating va

rious points of input, analysis, output, and decision for the formability

analysis.

Because one of the main objectives of computer-aided analysis is to identify critical design and processing parameters

while being cost effective, the use of a full-scale finite-element analysis method must be exercised with caution. For

example, a typical three-dimensional finite-element mesh generation, analysis, and post-processing of the output for a

structural analysis may easily require many days of effort.

References cited in this section

3. J.W. Hutchinson and K.W. Neale, Sheet Necking-III. Strain-Rate Effects, in Mechanics of Sheet Metal Forming,

D.P.

Koistinen and N. Wang, Ed., Plenum Press, 1977, p 269-285

4. A. Needleman and J.R. Rice, Limits to Ductility Set by Plastic Flow Localization, in

Mechanics of Sheet Metal

Forming, D.P. Koistinen and N. Wang, Ed., Plenum Press, 1977, p 237-267

6. Z. Marciniak and K. Kuczynski, Limit Strains in the Processes of Stretch Forming Sheet Metal, Int. J. Mech. Sci.,

Vol

9, 1967, p 609-620

7. J.W. Hutchinson and K.W. Neale, Sheet Necking-II. Time Independent Behavior, in

Mechanics of Sheet Metal

Forming, D.P. Koistinen and N. Wang, Ed., Plenum Press, 1977, p 127-153

8. D. Lee and F. Zaverl, Jr., Neck Growth and Forming Limits in Sheet Metals, Int. J. Mech. Sci., Vol 24, 1982, p 157-

173

9. Z.H. Lu and D. Lee, Prediction of History-Dependent Forming Limits by Applying Different Hardening Models,

Int.

J. Mech. Sci., Vol 29, 1987, p 123-137

10.

D. Lee, Computer-Aided Control of Sheet Metal Forming Processes, J. Met., Vol 34, 1982, p 20-29

11.

A.S. Argon, Physical Basis of Constitutive Equations for Inelastic Deformation, in

Constitutive Equations in

Plasticity, A.S. Argon, Ed., Massachusetts Institute of Technology Press, 1975, p 1-23

12.

D. Lee and F. Zaverl, Jr., Further Development of Generalized Constitutive Relations for Metal Deformation,

Metall.

Trans. A, Vol 11A, 1980, p 983-991

13.

C.F. Shih and D. Lee, Further Developments in Anisotropic Plasticity, J. Eng. Mater. Technol. (Trans. ASME),

Vol

100, 1978, p 294-302

14.

H.W. Swift, Plastic Instability Under Plane Stress, J. Mech. Phys. Solids, Vol 1, 1952, p 1-18

15.

A.K. Ghosh, A Numerical Analysis of the Tensile Test for Sheet Metals, Metall. Trans. A, Vol 8A, 1977, p 1221-1232

16.

Z. Marciniak, K. Kuczynski, and T. Pokora, Influence of the Plastic Properties of a Material on the Forming Limit

Diagram for Sheet Metal in Tension, Int. J. Mech. Sci., Vol 15, 1973, p 789-805

17.

R. Hill, On Discontinuous Plastic States, With Special Reference to Localized Necking in Thin Sheets, J. Mech.

Phys.

Solids, Vol 1, 1952, p 19-30

18.

S.P. Keeler and W.A. Backofen, Plastic Instability and Fracture in Sheets Stretched Over Rigid Punches, Trans. ASM,

Vol 56, 1963, p 25-48

19.

S. Stören and J.R. Rice, Localized Necking in Thin Sheets, J. Mech. Phys. Solids, Vol 23, 1975, p 421-441

20.

A.K. Ghosh and S.S. Hecker, Stretching Limits in Sheet Metals; In-Plane Versus Out-of-Plane Deformation,

Metall.

Trans. A, Vol 5, 1974, p 2161-2164

21.

S.S. Hecker, Simple Technique for Determining Forming Limit Curves, Sheet Met. Ind., Nov 1975, p 671-676

22.

H.J. Kleemola and J.O. Kumpulainen, Factors Influencing the Forming Limit Diagram; Part I--

The Experimental

Determination of the Forming Limits of Sheet Steel, J. Mech. Work. Technol., Vol 3, 1980, p 289-302

23.

W.E. Johnson and P.A. Stine, Application of Sheetmetal Research to High Production Fabrication Shops, in

Sheet

Metal Forming and Energy Conservation, American Society for Metals, 1976, p 235-244

24.

S.P. Keeler and W.G. Brazier, Relationship Between Laboratory Material Characterization and Press-

Shop

Formability, MicroAlloying '75, J. Krane, Ed., Union Carbide Corporation, 1977, p 21-31

Process Modeling and Simulation for Sheet Forming

D. Lee, S.A. Majlessi, and J. H. Vogel, Department of Mechanical Engineering, Rensselaer Polytechnic Institute

Analytical Methods

Two different types of analytical methods will be examined in this section, one dealing with simple and empirical

methods, and the other discussing more elaborate analytical methods for representative sheet metal forming processes.

Application of these methods should be based on trade-offs between computation time and accuracy. It should also be

pointed out that the following section does not include all the analytical methods that are available for sheet forming

processes.

Simple and Empirical Methods

Sheet shearing is a process of separating adjacent parts of a sheet through controlled fracture. This is accomplished by

placing the sheet between two edges of the shearing tools--in the case of blanking, a punch and a die. The quality of the

cut surface is largely influenced by the clearance between the two shearing edges. Based on experience, the clearance is

taken between 2 and 7% of the sheet thickness. The smaller clearance is used with a more ductile material. Because the

process is largely localized shear over a very narrow zone where strain hardening takes place, a good approximation of

shearing force, P

, is obtained from the following empirical equation (Ref 25):

= 0.7 (UTS) · h · l

(Eq 5)

where UTS stands for the ultimate stress, h is the sheet thickness, and l is the length of cut. When the shearing edges are

parallel, the entire length of the contour must be taken into consideration.

Shearing is practiced in a number of other processes. When shearing is accomplished between rotary blades, the process

is referred to as slitting; cutting along a straight line is simply shearing. A contoured part, such as a circular or more

complex shape, is cut between the punch and die in a press, and the process is called blanking. The economy of the

process depends on the proper physical arrangement of the parts to minimize scrap losses. Essentially, the same process is

also used to remove unwanted parts of a sheet, such as punching of a hole.

Sheet Bending. One of the main characteristics of the sheet bending process is stretching (tensile elongation) imposed

on the outer surface and compression of the inner surface (Fig. 4). There is only one line (the neutral line) that retains its

original length. For a given sheet thickness, the tensile strain increases with decreasing bend radius. A structural

weakening of the bent part occurs when elongation in the outer fiber exceeds the uniform elongation of the material,

, in

the tensile test. Referring to Fig. 4, the engineering strain e

is equal to:

= -y/

(Eq 6)

where y equals the sheet thickness, t, and is bend radius R + t. By substituting and rearranging,

(Eq 7)

The minimum permissible radius (or more generally, radius-to-thickness ratio) is related to the outer fiber strain, as shown

by Eq 7. For example, as the bend radius to sheet thickness ratio, R/t, increases, the strain in the outer surface decreases.

In order to develop a simple relationship between the geometry and material properties to achieve a maximum strain on

the outer surface, it is assumed that the true strain at cracking is equal to fracture strain

, that the material is isotropic

and homogeneous, and that the plane strain condition is valid, that is, that width is relatively large compared to thickness.

Fig. 4 A simple bending geometry.

The true strain at fracture in tension,

, is equal to:

= ln(A

)

(Eq 8)

where A

and A

are the initial and final areas. The reduction in area, r, is defined as l - (A

). Therefore,

= ln(1/(1 -

r)). Because the maximum surface strain,

, is equal to ln (e

+ 1), substituting the expression for e

, gives:

(Eq 9)

Combining the two equations above and setting

, the final expression is:

(Eq 10)

Failure may take place by either nonuniform deformation (thinning) or by splitting the outside surface layer, which

depends on

Sheet Drawing. Although cup drawing may appear to be a simple operation, a thorough analysis of the forming

process is rather difficult (Ref 26). A simple analysis method was, however, outlined by E. Siebel (Ref 27). From the

equilibrium condition in the radial direction, the required drawing load and its variations along the punch stroke were

determined analytically. The following equation for calculating the drawing load, F

, has been proposed (Ref 27):

(Eq 11)

where d

is the mean wall diameter, d is the instantaneous outside radius of the flange, S

is the blank thickness, r

is the

die radius,

f,m

is the mean value of flow stress of the blank material, is the coefficient of friction between flange and

die or blankholder, and P

is the blankholder pressure. In Eq 11, the first term represents the ideal deformation load

including the load increase due to friction at the die radius, the second term is the component produced by friction

between the flange and die or blankholder, and the last term is the load necessary for bending the sheet around the die

radius.

A typical load-stroke diagram for deep drawing, shown in Fig. 5, indicates that the flow stress increases continuously with

punch displacement as the flange diameter becomes smaller (Ref 26). It has been suggested that the maximum drawing

load is nearly independent of the workpiece material and the drawing ratio and occurs when d = 0.77d

(Ref 27), where d

is the initial blank diameter.

Fig. 5 Typical load-stroke diagram for the drawing process. The portion of the curve labeled B indi

cates the

load distribution for narrow clearances or ironing at the end of the draw. Maximum drawing load F

dmax

occurs

when diameter d 0.77d

, where d

is the initial blank diameter.

In general, buckling cannot occur as long as the blank is clamped rigidly between the blankholder and die. During cup

formation, the sheet thickness in the flange does not remain constant but may increase toward the outside edge; the center

portion near the die radius becomes thinner than the initial thickness, S

in the early stages of the draw. Because the

distance between the blankholder and die is determined by the largest flange thickness, there will be a gap between the

blankholder and sheet near the die radius. This gap may create an opportunity for the initiation of wrinkles. Thin sheets

> 25 to 40) are especially sensitive to wrinkle formation because their moment of inertia in buckling is relatively

small. The pressure necessary to avoid wrinkling depends on the material properties, the relative sheet thickness, and the

drawing ratio. Earlier work by E. Siebel and H. Beisswänger showed that the required blankholder pressure could be

estimated from the following empirical equation (Ref 27):

= 10

c[(β - 1)

+ 0.005 (d

)]S

(Eq 12)

where the factor c ranges from 2 to 3; β is the drawing ratio, which is defined as the ratio of the initial blank diameter d

to the inside diameter of the finished cup d

; and S

is the ultimate tensile strength of the material.

The flange-wrinkling behavior of sheet materials in deep-drawing operations was further analyzed by B.W. Senior (Ref

28) and J.M. Alexander (Ref 29). Their expression for the blankholder pressure is given as:

(Eq 13)

where the buckling modulus E

is equal to 4Eρ/(E

1/2

)

, E is the elastic modulus, ρ is the tangent modulus, t is the

sheet thickness, is the wrinkle amplitude, b is the flange width, n is the number of wrinkles, and a is the mean radius of

the flange. Equation 13 shows that blankholder pressure increases with the flange width in a complicated fashion.

Qualitatively, it is conceivable that proper control of the blankholder pressure may reduce fracture and wrinkles in simple

drawing operations while reducing the punch load at the same time. In fact, possible ranges of the variation of minimum

and maximum allowable blankholder pressure along the drawing depth were estimated by E. Doege and N. Sommer (Ref

30); the blankholder motion was also controlled by a cam mechanism to achieve greater drawing ratios by E.I. Odell (Ref

31).

Analytical Methods

Sheet forming operations may be roughly grouped into two types for the purpose of developing analytical models. One

class includes both drawing and stretch forming, where the variations in stress and strain through the thickness of the

sheet are usually neglected if any sort of closed-form solution is to be obtained. Within this class, drawing is one extreme,

in which the sheet experiences tension in one principal direction and compression in the other; the other extreme is biaxial

tension, which occurs in stretch forming. The other class of sheet forming operations is made up of those cases where

bending is dominant. In bending, through-the-thickness variations in stress and strain must be taken into account.

Often a given forming operation will involve both bending and stretching/drawing. To include both, it may be necessary

to divide the problem into parts, solving the bending and stretching/drawing problems separately and combining the

results. A general solution of the combined problem will usually require a numerical procedure such as the finite-element

method. In this section, the assumptions and equations used to develop a closed-form solution will be illustrated for both

types of operations, using a simple example of each, and some of the techniques developed for extending these solutions

to more difficult problems will be outlined.

The general approach to developing a model for a process involves several steps. The first step is to classify the problem

according to whether a bending or stretching/drawing type of solution is applicable, and then to express the appropriate

strains in terms of the geometry. This involves making simplifying assumptions about either the geometry or the strain

distribution; constancy of volume is also usually assumed.

The second step is to choose a constitutive model, relating the stresses in the material to the strains. A constitutive or

material model may consist of a definition of effective stress and effective strain and/or strain rate and the relationship

between them, plus a flow rule that further relates the individual components of stress to the components of strain.

The next step is to write an equation of equilibrium, usually a force balance obtained by equating the forces applied by the

tooling to the stresses in the material required to support these loads. In sheet forming, the effects of inertia and gravity

are usually neglected in this force balance. In order to solve the resulting equations and obtain a useful solution, it is often

necessary to refine the model, changing some of the assumptions or introducing some additional ones; therefore, the

whole process is usually somewhat iterative.

Bending Analysis. As a first example, consider the bending of a flat plate to a prescribed radius. As a bending problem

it is clear that the variations in both stress and strain through the thickness of the sheet are important. In order to relate the

strains in the sheet to the bending geometry, it is convenient to make several assumptions. First of all, plane strain

conditions are assumed to exist throughout the sheet, because the width of the sheet along the axis of the bend will not

change significantly. While this assumption is not valid near the edges of the sheet, it is a good assumption as long as the

width of the bend is much greater than the thickness of the sheet (w/t 1).

Second, the neutral axis is assumed to remain in the center of the sheet, meaning that the midplane of the sheet does not

change in length. The error introduced by this approximation is greater with tighter bends, but it is often reasonable for

practical bend radii. Another assumption is that planar cross sections of the sheet remain planar and do not warp during

the bend.

With the above assumptions in mind, the strains in the sheet may be related to the bend geometry. Referring to Fig. 6, let

, be the arc length at the midplane of the sheet, and let r

, be the radius of curvature of the midplane. Before bending, all

planes parallel to the surface of the sheet had the same length as the midplane (L

= r

θ), while the length of a plane in the

bent configuration is given by L = rθ. The engineering strain is:

(Eq 14)

where z = r - r

. This is not identical to the true strain, but for the strains encountered in most bends the two strains are

nearly equal. Because e

, has been assumed to be zero (that is, plane strain conditions exist throughout the sheet),

assuming volume constancy gives e

= -e .

Fig. 6 Geometry definitions for bending analysis.

In order to find the stress distribution, it is necessary to have a material model. For simplicity, assume that the material is

elastic-perfectly plastic (no work hardening), and that the stress-strain behavior is the same in tension or compression.

While no material is exactly described by this model, in many cases it will give good results if the plastic flow stress

chosen for the model is somewhere between the actual yield strength of the material and its ultimate strength.

With this model, the material toward the outside of the bend will be at some uniform value in tension, and the material at

the inner part of the bend will be at the same uniform value in compression; the central part of the sheet will be elastic

(see Fig. 7). Using both the von Mises yield criterion

( -

)

+ (

)

+ (

- )

= 2Y

(Eq 15)

where Y is the plastic flow stress, and the plastic flow rule

(Eq 16)

the plastic stress level may be determined. In the plastic flow equation, e refers to an increment of plastic strain, but in

bending, these relationships also hold for the total strain because the ratio of the strains does not change during the bend.

Using the results obtained for the strains, the stresses are computed to be:

= Y

= 0

(Eq 17)