Fundamentals of Metal Forming_wagoner Chenot

fundamentals of ffi[lHl fDHmlnG Hobert H. Wagoner The Ohio State l!niversity

Jean-loop Chenot Ecole Nationale Superieure des Mines de Paris

John lllileu &Sons. Inc. New York

Chichester Brjsbane

Toronto

Singapore

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rL

·



(2.29)

the set of equations can be written more compactly using matrix notation (and ntuch more compactly using indicial notation):

rAn

A11

!

~12

An

A13 A13

Ain A2n

.

LAn1

r1

fx2x'

'2 (2.29a)

=

An2 Ans

Ann lxn

'•

[AJ[x]=[r] Ai)xj;ri

It can be shown that such a set of-equations has a solution only if each one is independent of all the others. This condition may be checked by evaluating the determinant7 of the matrix, written jAj. lf the determinant is zero, the equations are not linearly independent> the matrix [A] is said to be singular~ and a unique solution cannot be found, The determinant is evaluated by summing a series of cofactors (the rule for a 3x3 matrix is shown below)" but the procedure is complex and cumbersome for large matrices:

11 A12 A13 21 A22 A13 = A31 A31 A33

~

AuCA22A33 -A32A23)-A12efficients relating hoo sets of axes; {T1j] is a physiali quantity that, for one given set of axes~ is represented by nine numbers.

[Rij]

(Tij] is tied to one set at a Ifme. One cannot speak, for instance, of tramiforming [Ri;]to aoother set of axes~ for it would not

straddlesf as it were, two sets ofaxetJ;

mean anything.13 This reply relies on the concept of a tensor as a physical reality, probably the clearest way to approach the question. (Mathematical tests of "tensomess" perform the same function but more obscurely, still relying on the user's kn~wledge of whether the tensor transformation rules can be utilized in a meaningful way.) If we can assign an appropri.. ate: meaning to

[Rij] thatambetransformedfrom oneooordinate system to another, then

in fact[Rij] could be considered a tensor. It is easily sfu:1wn that there are four closely related mal:ridx in the infinitesimal limit. With this continuum picture, which we shall mention again in the next section, the strain is defined at a point, exactly as the stress was defined at a point. Thus Eq. 4.7 becomes., in the non~homogeneous (i.e., general) case..

(4.8)

where dX1are understood to be the components of a vector of infinitesimal length dL before deformation. The corresponding length in the final configuration is dl.

4.2

GENERAL INTRODUCTION TO KINEMATICS Evolving from the physical picture presented in the previous section, there are many alternate formulations lor the geometry of deformation. In this section, we shall attempt to introduce the basic ideas without amplification. In subsequent sections, we shall-restrict our attention to the most important formulations and quantities used to represent metal deformation. Since metals usually undergo

4.2 General Introduction to li

4.5

T

(4.4-15)

RATE OF DEFORMATION In the foregoing, we used the basic Lagrangian method of following a material element from one time to another. This is the usual approach for solid mechanics when the material properties depend on the material history, and a steady state is not established. However, when considering rates of deforma* tion, it is usually more convenient to adopt the Eulerian approach, where a velocity field is considered the principal information available, Then~ using proper geometrical constructions, it is possible to compute the rat·es from the nearby velocity field. Consideramaterialvectordxattimet•. Ifv(x)isthematerialvelocityfie!dspeciiied in terms of spatial position, then the relative velocity of the head to the tail is (4.29)

or

136 Chapter 4

Stram (4.30)

where

(4.31)

and Lis called the velocity gradient or the spatial gradient of velocity.

Note: The connection with IT the deformation gradient, is clear if one considers two times separated infinitesimally~ say t and t + dt. Then~ du, I dt=dvi, anddJ I dt = L, as wngas the reference stale for J is chosen to be the state at t. This i's necessary because L 1·s based on the current spatial positions of a material element; that is~ in the usual E1llerian way. In an infinitesimal time, the relative displacements of material points are infinitesimal relative to their separation ([du[« [dx[) so that the small-strain formulation is exactly satisfied. Thus we follow the procedure of linear separation of L into symmetric parts, exactly corresponding to the deconvolution of {J] into [El and [e]: (L] = [D]+[WJ

where 2(D]=[L]+[L]

T

H

I( dv

dV·) H

2\o-1-+o-1UXj UXi

Dij =-1

(4.32)

rate of deformation tensor (4.33)

(4.34)

The role of [DJ and [W] can be seen by considering the small displacement counterparts [e] and [ro], which have already been discussed:ll [el= [D]dtor [e]=[D]

[ro] = [W]dt or 11

[ro] ~[W]

Some authors are careful to point out that[e.ij]"#.

(4.35)

(4.36)

[oiiJ because fsij] is based on material deriva»

tives while (Dill relies on spatial derivatives. In fact, there is no distinction when the displace-

ments are truly infinitesimal, because

xi ""'xi' In fact, thiS is a

denote i = D in ail i:ases «nd must-remember that: €. ?!' de/dt,

matter of convention: we

4.5 Rate of D~limnafion

137

(infinitesimal displacements only) Thus [DJ is shown to be the rate of change of [ e] (or [E] with the reference state at the curved state) Vlith time, whereas {WJ is the rate of change of fw) with time. It should be noted well that these correspondences are precise only for vanishingly small displacements. When the small-displacement quantities are considered to be approximations of the large-strain values; then these equivalences do not hold.

S~ow

how the Eulerian forms L and D are related to th.e rate of change of the Lagrangian quantities F and E.

The basic difference between the Eulerian quantities and the Lagrangian quantities lies in the current basis for the former and the original (undeformed) basis for the latter. For example, compare: (4.5·1)

F1i

(4.5-2)

where the definition for Fis taken from a time derivative of Eq. 4.15, and noting that the current velocity of a material particle currently at x and originally at X. The

v,

di only difference in the equations is apparent: the denominator of L uses a current spatial increment dx \vhereas Fis based on the variation 1vith respect to the original increment corresponding to this current increment. We can relate the tv"o siniply; dx=FdX

(4.5-3)

dv=("") a1 x =FdX

(4.5-4)

(4.5-5)

But

(45-6)

so

and, since

138

Chapter 4 Strain

(4.5-7)

the Lagrangian rate of strain,

E

1

may be found:

(4.5-8)

·r

T T

'

But, from above F = F L , and F = LF, so

(4.5-9)

and, since L = D + W where D is symmetric and W is anti-symmetric, (4.5-10)

Following the same procedure as in Exercise 4.4,. other useful equations relating various measures of deformation can be derived:

C=2il

(4.5-11) (4.5-12) (4.5-13)

where E* is the Eulerian strain tensor, referred to the current configuration. When Fis small (infinitesimal), all of the strain rates become the same: D = E= E* {infinitesimal deformation)

4.6

SOME PRACTICAL ASPECTS OF KINEMATIC FORMULATIONS In the foregoing sections, we have attempted to outline the basic kinematics nee assume that the origin of the cube does not translate (a rigid translation does not do internal work in any case), Then the internal power done by h1te cube of material is the sum

Chapter 4 Problems

149

TABLE4.1

Common Work-Conjugate Measures of Stress and Deformation Rate Strain Rate

Stress

D

(J

Type of Motion

Eulerian

rate of

Cauchy stress df = crda

deformation D=av ax

"(l)

D

volume= corrected Cauchy stress df = _e_cr( 1lda

rate of

Eulerian, deformation per original D=av material volume ax

Power

cr·D= power current volume

"(1).D= power original volume

Incremental Work Form

CT·dC

(de= Ddt)

crC1l. de (de=Ddt)

Po F

rC1)

mixed: Eulerian area, deformation Lagrangian gradient force 2 F=dF=a x dt axat rate of

first Piola = Kirchoff stress d(, = rC1lcta

.

1 .

rC2)

Eor c

second Piola = Kirchoff stress d(, = rC 2lcta

rate of Lagrangian

2

Lagrangian

rC1l. F= power original volume

rC 2l. E= power original volume

strain E=2.c:=FToF 2

Notes:

df =force vector acting on a current element of area da df., = df transformed back to the original configuration: df = Fdf da0 = the original area vector corresponding to the current area:

r

-Jr

p F da ::::-.£. p '

da 0

r(lldF

r( 2)dE or 2.rC2lctc 2

150

Chapter 4 Strain

of f • dv for the three moving faces of the cube. (The velocities are differential because the joint at the origin is stationary and the element is infinitesimal.) The velocities themselves are dv = Ldx, so the total power is the sum over the three (moving) faces: power=cr·L

(4.67)

However, because er is symmetric, only the symmetric part of L contributes to the inner product, so Eq. 4.67 reduces to power=cr·D

(4.68)

where Dis the symmetric part of L. Physically, it can easily be seen that the velocity of each plane caused by the shear-symmetric part of L ( oo) lies in the plane and therefore does no work in conjunction with the force acting on the plane. For the alternate stress measures, it can be readily shown that the workconjugate quantities are those shown in Table 4.1 on page 149. Any of the power-conjugate quantities can be viewed -in incremental work form by multiplying the rate term by dt.

CHAPTER4-PROBLEMS

A. Proficiency Problems

1.

Given: F

HG !]

a. Find EiJ and E w the components of the large- and small-strain tensors, respectively. b. Using E and£ directly, find the new length of the vectors OA and AB shown here. Note that the original vectors are of unit length.

B

A

0

c. 2.

Why are the deformed lengths of O'A' and O'B' different when calcu· lated using the two different measures of deformation?

Given this diagram for an assumed homogeneous deformation, write down the deformation gradient, F:

{6.6, 4.6}

(0,2) {6, 3}

10,0>

3.

(8.l, 3.Q)

{2, 0}

[0.1 Given: F = 0.3

0.2 0.4

~!]

0.7 0.8 0.9 Find: J, E, and e. 4.

In the following diagram, a point in a continuum (0) moves to anew point (O') as shown.

{4,2)

___ ,/,-~~~---->-~.

a. Find the new points A' and B', assuming homogeneous deformation for the following two cases:

J=[~ ~] b. For each deformation, find E, the large strain tensor.

152

Chapter 4 Strain 5.

Imagine that a line segment OP is embedded in a material that is deformed to a ne\v state. The line segment becomes O'P' after deformation, as shown:

~I P{2, l}

0

a,

Find the vector components ofO'P~ :

b. Find the length of O'P' ll

. [l 32] C= 3

c.

Find the components of OP if

J=[~ 6.

!]

A homogeneous deformation is imposed in the plane of the sheet. Two lines painted on the surface move as shown, with coordinates measured as shown: a.

Find F, the deformation gradient.

b. Starting with F, find C, E, I, and

E.

~t

y (1,

n

/

313.3J

(L8,2,l}

(3, 2)

(2:,0,1.5}

\

C

(4.1, 1.9)

Chapter 4 Problems 153 c. 7.

Find the principai strains and axes of E.

At time t, the position of a material particle initially at (X 1" Xv X-,) is

Obtain the unit elongation {i.e. change in length per unit initial length) of an element initially in the direction of X1 +X2.

S.

Take fixed

right~handed

gradient matrix,

following: a,

axes

Xv

x:z,

Xs· Write down the deformation

~for fue deformation

of a body from x to X for the

xi

Right-handed rotation of45° about x1,

x

b. Left-handed rotation of 45° about 2 , c. Stretch by a stretch ratio of 2 in the X3 direction. d. Stretch by a stretch ratio e.

of~

in the x2 direction.

Right-handed rotation of 90° about is.

Find the total deformation matrix for these motions carried out sequentially. Using this result, check the final volume ratio. 9.

From the following mapping, find C, U, and R:

Check whether this Is a permissible deforrMt!on in a continuous body.

154

Chapter 4 Problems

Chapter 4 Strain

10. Check the compatibility of the following strain components:

B. Depth Problems 11.

Consider the extension of an arbitrary small-line element AB. Start by examining how (AtB') 2 is related to (AB) 2 using the small extensional strain along that direction, en. Show that for small strains and displacements, rotations do not cause extension; that is, if extensions are zero, strains are zero.

12. In sheet forming, one often measures strains from a grid on the sheet surface. Then one can plot these strains as a function of the original length along an originally straight line:

Strain

Original Position

As shown in diagram (b) (for a simple forming operation), this originally straight line is curved and stretched. If the edges of the sheet do not move .(stretch boundary conditions), develop a rule that the measured strain distribution must follow. Consider that the original sheet length 10 be~ comes I at some later time. 13. Consider a 1-inch square of material deformed in the following ways: For each case:

Chapter 4 Problems

155

a. Find F. b. Find C and B.

c. Find the principal values and directions of E. d. Find the material principa1 directions after deformation. e. Which of these cases (a~d) are mechanically the same under isotropic conditions? Under general anisotropy?

14. Given

that[CJ=[~ ~l find Fwhen

a. the principal material axes do not rotate, and b. the principal material axes rotate by soa counterclockwise. Express your answers in the original coordinate system, 15. Derive a set of compatibility equations corresponding to Eq. 4.52 for the three~dimensional case.

C. Computational Problems 16. Consider a triangle of material defined by three points: A, B, and C, with coordinates before and after an in~planef homogeneous deformation as follows:

before:

156

Chapter 4 Strain

after: a.

Find F, E,

U~

R, and C in terms of the original and final coordinates.

b. Find the principal stretch ratios and true principal strains.

c. Assuming that strains are small (but displacements and rigid rotations may be large), find simplified expressions for principal strains and small-strain components.

d. Assume that all displacements are small, and find simplified expres-sions for principal strains and smaU....strain components. 17.

Repeat Problem 16 for the 3-D deformation where point A(A,, A,, A3) moves to a(a11 a2, a3),. and so on.

1

Standard mechanical Principles

Many alternate forms of equation can be derived to describe the condition of mechanical ,equilibrium, both for continuous bodies or for discrete assemblages of finite elements. In the latter case; it is possible to consider locat forms for isolated elements, and global forms describing the body that has been modeled discretely. Special mathematical tools are required for transforming one description of mechanical equilibrium to another. These are described briefly in this chapter and they are demonstrated by classically establishing the most important laws of deformable bodies. Integral forms are derived, and, finally, variational formulations are presented for the most standard classes of mechanical behavior: elasticity, plasticity, and viscoplasticity.

5.1

MATHEMATICAL THEOREMS Throughout this chapter, frequent use will be made of the Green theorem, or one of its various forms. The most simple expression is analyzed, and the mathematical proof is given in a simple case~ which can be generalized sufficiently to cover cases of practical interest for applications. As we axe not concerned \vith pure mathematical issues, we shall make the assumption that the domains of integrati6n {volume, surface, or curve) and the functions (scalar, vector~ or tensor) are regular enough for all !he integrals to be defined. 'A theorem forlhe necessary and sufficient condition for an integral to be null is given.

1

1

This chapter is more mathematical than th.e others, It develops principles essential for finite element analysis of furming operations, but-does not contain material needed to understand the remaining content of this book 0, then there exists a subdomain ro~ of n, containing x·, and verifying the conditions • \:f x e ro'

f(x);;,.! f(x') 2

(5.21)

By integration of both sides ofthe previous inequality (Eq. 5.21), we obtain

ff(x)dv0

(1 {f[xiX,t+1't),t+At)-f(x(X,t),l+AtJ}) 1

(5.25)

, I ) > llm le.t{f[x(X,t),t+At)-f(x(X,t),tj} At-->0

which gives the result

df(

dt ""

tl~of(x t)+IJlL(x t)ilx,(X,t)

at '

ax.' '

at

(5.26)

Finally, if the definition of the velocity of the material point with coordinate vector xis introduced into Eq. 5.26,, we obtain the desired result:

df df -(x,tl=-(x,t)+ di dt

2:

ill ;·-(x,t)vi(x,t)

(5.27)

oXi

This is often denoted by the more compact form: df

{If

dt

at

-~-+grad(f)·v

(5.28)

166

Chapter 5 Standard MechaniC12/ Principles

Material Derivative of a Vector or a Tensor Field The same approach can be carried out for each component separately, The case of a vector U(X, t) as a function of the Lagrange variables will give

du

au

t

dt

-d (X, t) = -:;-(X, t)

(5.29)

dU; (X t) = ilU; (X t) dt' at'

'(5.30)

with co1uponents

Wilh the Euler description, the vector field is denoted by u(x, t), and its :material derivative is

du

au

-=-+grad(u).v dt at

(5.31)

where the components are given by

(5.32)

An important material derivative, the acceleration,. is obtained from the velocity field:

av

dv a=-=-+grad(v).v di ilt

(5.33)

Note that the gradient of a vector is an operator with tl"/o indices. Three indices are necessary for the gradient of a tensor, the analogous ofEq. 5.32 being for the tensor cr:tl " Note that the-material: derivative of a tensor is notgeneraUy frame invariant-that is~ if the basU. function for the spatial coordinates is changed, the new material derivative o-f the same tensor tJ i.v:ilt be different.

5.2 The Material Derivative

167

(5.34a)

which can also be written dG

a(;

(5.34b)

-=-+grad(O')·v dt di Derivative of an Integral

This topic was already addressed in the previous section (see Eq. 5.11). Another useful form is obtained by first applying the Green theorem (Eq. 5.1) with f and g = vit and summing the contributions for i = l, 2, 3, which gives

(5.35)

The surface integral is eliminated between Eqs. 5.11 and 5.35, leaving

-d

dt

f,

fdV=

f, [ilf L af Lavil 0t

0t

-+ at

i

. -vi+f ax1

dV axi

i

{5.36)

J

Introducing the material derivative off, given by Eq. 5.27, and the diver~ gence of the velocity field,

we get

..Vith side 2Al,. and with center of coordinates (xi,

x21 x:J. The mass flux through face number 1 with outward normal is (5.M)

Similarly, face 1', with opposite normal, gives (5.1-2)

and '\\•e observe that by combining Eqs. 5.1-1and5.1-2, \ve get (5.1-3)

Two other expressions similar to Eq. S.1~3 can be written by considering the fluxes across faces 2 and 2>, and across faces 3 and 3', such that the totat mass increment is

(5.H)

We remark that the density change is linked to the mass transfer by lip "1' +Am rnO

(5.1-5)

The proof is completed by substituting Am between Eqs. Q.lw4 and 5.1-5~ dividing the resulting equation by At 6.P, and letting At tend to zero, which obtains

170

Chapter 5 Standard Mechanical Principles 1-> "1

X2

-X2 "22 _, "22 "2 _,, ''2

i3 -7

x3

0'33 ~ cr33

Cl3 __. 3

0"23 -? -0'23

0'4-> -04

0-13 ~ -O'i;3

"12 -C64 C15 :0 except

c1s--> - -c6s *

=0

c65 z:r.

0*

*These components 'Were already found to be zero by the X2 crystal rotation.

x

The final symmetry operation (about 1 ) is redundant, simply reproducing the information obtained by the first two operations. This result can easily be anticipated by performing the three rotations on the unit cell in Figure 6.1 and observing that the crystal is returned to its original position (the dot on one cotner can be used to distinguish the arbitrary orientation). We are left with the elastic constants for an orthogonal crystal, as expressed in a convCntionaJ coordinate system fixed along the symmetry axes:

rll

tie that results.

7.1 The Yield Surface

Elasti>::

221

Plastic

eeeeeeeepp

-.-..-..-o-+-•--+--+•

{)':!

0123466789

fol

(b)

Figure 7.2 Elastic¥plastic transition with constant c 2, We can continue this procedure for a different Value of cr2 ;;:: cr2 ° and identify another elastic~plastic transition within the accuracy of Ao1• If we repeat the process for many values, we can obtain a 2-D mapping of the discrete stress states that first cause plastic deformation (Figure 7.3a).

J.l:eeeeeeeeep p

-eeeeeeee

' •

w

p

p

m

Figure 7.3 Stress states for first observation of plastic deformation. When we interpolate between the last stress inducing purely elastic deformation and the first one inducing plastic deformation, we obtain a plot like the one shown in Figure 7.3b. Finally, by connecting the measured points with a smooth curve# we can arrive at a surface that represents, within the experimental limitations, the loci of all stress combinations that first .cause plastic deformation. This curve or surface is called the yield surface. There are several questions and criticisms that arise immediately from the procedure followed. There ls limited accuracy unless a great many experiments are dOne to reduce Aa1 and Acr2• Given the distinct points that are found, it is impossible to say confidently that the entire locus of such points is smooth. It could be that the pattern of yield stresses between points ls extremely rough, as shown in Figure 7.4. If such behavior were discovered, a great many additional experiments would be needed. As it is, the presence of strainhardenin~;requires a new, virgin specimen for ea 0 and w 2 > 0 fo:r a closed plastic path. The extension of these ideas to three dimensions is straightforward. The work done around a generat infinitesimal cycle starting from an elastic state requires consideration of all the stress terms and all of the resulting strains:

(7.26)

As in 1-0, we require that both w 1 and w 2 be greater than zero for an arbitrary path starting from an elastic: state in order to ensure that the material response is $table. If we start from the second-order work term (i.e., choose a A = a a for the path); we see that dcrip.eijP > 0 is required. Geometrically, it is easiest to think of dcr and condition that

d~.

as vectors (in 6-D or 9-D space). In such a case1 the

dcr·de>O

(7.27)

requires that the vectors dcr and d£ have a positive projection on one another (i.e., that the angle between the two be less than 90°). Furthermore, from the fundamental definition of flow theory plasticity, the strain increment direction can depend_ only on the stress G, not on the stress increment da.e Finally, we require dcr to have a component outward from the elastic region such that plastic deformation is induced.9 A 2~D representatiox1 of the situation is shown in Figure 7.11. 6

$

For example, consider general fluid fl.ow, whkh depends on total pressures, not on dp. The dire· Equation 7.45 lets us express the normality condition without arbitrary constants:

00

dc .. =-de IJ

{)crij

(7.46)

Equation 7.46 may be used directly to define the normality conditions without an arbitrary constant for Hill~s quadratic yield function, as illustrated in the following exercise.

Im

Find the nonarbitrary form of normality conditions for Hill's quadratic yield fu.nction directly from the yield function written in terms of effective stress. To derive the required result, we could rewrite Eq. 7.35 in order to find the desired form, a~il(roc. Roy Soc. London Ser A 4 108 (1925): 28.

l'- CJ.

8.3 Crystalline Deformation

285

sxT isxTI

d~=dSr·-­

(8.39) = -dy(m · T)Js x Tl

In the special case where _T lies in the s-m plane, then m · T = Is x T! =sin p, so that from Eq. 8.39, dP = -dysin2 p. Integration from some starting orientation P,,

yields (8.40)

When the loading is tensile, so that r is increasing, Eq. 8.40 shows that the sand Taxes approach one another and eventually coincide \vhen y -7 oo. For compressive deformation, so that y is negative, Eq. 8.40 indicates that T rotates away from sin Figure8.5a, along a path of increasing~- In fact, under compressive loading, Tis again rotating toward the direction of slip, which in this case is the minus-s direction in Figure 8.Sa. A more standard measure of strain during the test is Err, the tensile strain along the loading direction. The incremental strain, dc1T, along the current tensile axis equals the incremental change, dT = dT · T, in the length of the unit vector T. Using Eq. 8.37 and specializing to the case where m, s, and T

are coplanar, de,.,= -cot~ d~. This incremental relation may be integrated to give

Err= ln(sin~ 0 /sin~)

(8.41)

Figure 8.Sc shows how f3 depends on e..i-- Here, f3 decreases with En as for y. In the limit of large tensile strain,~ approaches zero, so that the tensile axis and slip direction coincide. The observation that f3 changes with strain is important for two reasons. First, the result suggests that any single active slip system will eventually rotate to decrease the resolved shear stress, and thus require higher applied stress to operate it. For the applied uniaxial stress along T considered here, the condition for yield Eq. 8.13 becomes

cr = wed plastic pclMbal for alum mum

Tresca

Shear

O Measured yIBld !Deus for o:. What aspect of compatibility is not satisfied here?

23. Outline the components of a computer program which will calculate the uniaxial stress~strain response of a £.c.c. single crystal loaded in tension. The tensile axis Tis a material onef similar to the case depicted in Fig. 8.5, and initially, Tis parallel lo Il 12 11). The sample is loaded from 0 lo 40MPa over a period of 300s; so that the stress rate? don/ dt is constant. Use a rate-dependent constitutive relation as described in Eqs. 8.73 and 8.74 with the following parameters:

312

Chapter 8 Crystal-Based Plasticity

'i~"l = 10-3 Is m=20

as the length scale approaches the important physical length scale (e.g., atomic spacing, dislocation spacing, or grain size) . .t\s shown in Figure 9.2, the actual contact area may be much less than the apparent contact area, perhaps only 5o/o to 10-0/o, so the true local stresses at local areas of contact may be much greater than the macroscopic ones that appear in Eq. 9.2. In standard friction terminologies, the high points of surface rough· ness--those most llkely to come in contact with the other surface-are called asp'erities. Adjacent regions that are not in contact may have virtually no true contact stress. However~ we shall restrict our attention to the apparent contact area and apparent stresses because these are· the ones that are used in phenomenological measurements and macroscopic simulations of metal forming. The microscopic pkture is more suitable for fundamental studies of friction.

316

Chapter 9 Fricfion

'" Figure 9..2 Macroscopic and microscopic views of contact areas. (b)

(•)

9.2

(d;

PARAMETERS AFFECTING FRICTION FORCES It is instructive to consider the kinds of variables that can affect friction forces;

even though ive will focus only on a phenomenological approach to friction in which friction forces file measured under conditions ciose to those in forming operations of interest. From our simple picture of friction from a physics textbook, it ls possible to identify immediately son1e of the effects.

° Contact pressure. Simple experiments shoi"¥ that the friction force in-

o

•

4

r.

o

°

creases as the contact pressure or normal force increases 1 at least at low pressures. Sliding speed. Most simple experiments show that the dynamic friction force (Le., the dissipative force opposing sliding motion) is less than the static friction force (i.e., the critical force to start sliding). Materials. The elastic and plastic properties of the materials in contact will certainly affect how difficult it is to slide the two materials across one another. At low true contact pressures, the asperities are more likely to interact elastically. Also, the materials may have different fracture characteristics that lead to changes in the surface topology and debris (from spalling) on the friction surface. Surface roughness. The shape and density of asperities may have a major effect on friction,. whether lubricated or dry. However,. even the measurement of sutface roughness is difficult and its quantification is not standard/ except for a few simple parameters. Lubrication and debris. The presence of other material at the conteu::t surfaces is important. The properties of the lubricant (rheology, compressibility, temperature sensitivity1 and so on) enter into consideration. The distribution of lubricant depends intimately on surface conditions, pressure, and sliding speed, .and knowledge of all of these depends on !he forming operation analysis. Temperature. The local temperature at contact points and in lubricants will depend on thermal -conductivity and plastic work dissipation. Concurrent defvnnation.. This effect is particularly germane for metal forming because the workpiece is usually deforming plastically while the friction forces are operatiflg. The deformatiDn changes the surface

9.4 Sticking Friction and Modified Sticking Friction

317

roughness,. eliminates most elastic effects in the workpiece, and opens up new surface by the action of dislocation slip.

With all of these variables influencing friction,. it is apparent why it is necessary to make measurements under conditions as close as possible to the real ones in the forming operations. For purposes of simulation,. it is necessary to adopt simple laws that allow measurement of a limited number of coefficients, These two principles, simplicity and similitude,. are sometimes contrary / but !hey guide !he design of mos! applied friction tests.

9.3

COULOMB'S LAW The most common form of friction law is known variously as Coulomb's law, or Amonton's law, Simply put, it states that the magnitude of the friction force is proportional to the magnitude of the normal force, or, equivalentlyf that the friction stress is proportional to the normal stress (or pressure}:

or

(9.5)

where1 as before, the friction force or friction stress are simply limiting values of the tangential force or stress that may be present because of frictional effects. µis called the friction coefficient, or Coulomb constantf with the basic idea that it is a constant with respect to contact pressure or stress for a given situation. Coulomb 1s Jawf Eq, 9.5, is conceptually correct at very small pressures.. in thatthe limiting tangential force approaches zero as the pressure tends to\vard zero. In fact, this law is usually found applicable at low contact pressures relative to material strength, before there is tl great deal of deformation. I-Iowever, at higher contact pressures, material properties should enter into consideration.

9.4

STICKING FRICTION AND MODIFIED STICKING FRICTION While Coulomb's law has the proper for1n suggested by silnple friction experiments for nondefonning bodies (which are by definition lightly loaded), it drastically overestimates friction at high contact pressures, A simple thought experhnent shows that friction at the interface cannot exceed the shear strength of the material.. because the material can slide at this shear strength even if the interface is bound tightly (welded). This extreme condition is called sticking friction, and may be represented generally as (9.6)

where i-max is a property of the softer material alone. Figure 9.3 illustrates conceptually the transition from very low contact pressure to pressures near the shear strength of the material.The term -emu in Eq. 9.6 is the shear strength of the

318

Chapter 9 Friction

Figure 9.3 Conceptual vie"'· of friction as a function of conta;;t pressure.

softer material. For a Tresca materiat the shear strength is equal to one~half of the niaterial strength in W\ia;.Uxorv"

u:i s &, "'r

""{T) ui

U11 >Q,t1 >"''t*

1'J

-•4

1

u~ Ol'

Vx

u"'s&t1 "'"*hmh 3~"' u.. > 'ti"""*

(b)

Figure 9.6 Regularization of friction forces by imposition of ever~slipping condition.

a See, for example, Y. Germain, K. Chung, and R.H. Wagonei:, Int. f, Mech. ScLr $1, 1989: 1-24. M. J. Saran and R.H. '\'\lagoner:ASME Trans. -].Appl. M£ch., 58, 1991:499~506.

i}

9. 7 The Rope Formula

323

As long as 3 is chosen sufficiently small relative to the average tangential displacements in each time step, the error introduced by these approximations· is not significant. Hoivever, care must be taken that errors are not introduced for small time steps, especially in programs that have automatic time step selection.

9.7

THE ROPE FORMULA Whereas the classic test defining ideas of friction is shown in Figures 9. la and b, the sliding block is not very similar to the situation in many forming operations. For example, the sliding block does not involve the macroscopic deformation inherent to forming.

,,,

'"

"'

Figure 9.7 Derivation of differential equation leading to the rope formula.

For generally curved contact and friction, like that shown in Figure 9.lc, different approaches are preferred. The basic tool for use with friction of thin deforming sheets is known as the rope formula because the conditions and results are characteristic of a flexible rope sliding over a fixed surface. The situation is shown in Figure 9.7a, where the following assumptions are made. • The sheet or ·rope has no significant bending strength; i.e., it is a membrane. • All points on the.rope or sheet are sliding relative to the fixed surface according to the Coulomb friction law. • The cross-sectiox1al area is assumed to remain constant.4 With these assumptions, we consider a differential element of the rope or sheet, Figure 9 .7b, and perform force balances in the normal N and tangential T directions as shown in Figure 9.7c: (9.13)

4

This assumption affects only the relationship between membrane stress and force, but the formula may be derived in terms of force without reference to cross-sectional area.

324

Chapter 9 Friction (9.14)

Solving Eqs. 9.13 and 9.14 obtains do

(9.15)

µfl=--

"

which may be integrated over the entire contact angle: (9.16)

and the standard rope formula is obtained: J Fi . µ=-ln~ 1 "' or µ=-ln-

p

"1

p Fi

(9.17)

The shape of the tool surface does not affect the calculation of friction coefficient, only the contact angle, also known as the wrap angle or angle of wrap.

Note: Equation 9.17 is often used in the reverse sense in sheet-forming arr.nlysis. That is, a standard friction coefficient w.11y be used in conjunction with a kn!YUJn geometry to estima.te t'he forces (and thus th.e stresses and strains via the constitutive equation) in tenns of the wrap angle: (9.18)

9.8

EXAMPLES OF SHEET FRICTION TESTS As mentioned earlier~ the basic principles guiding the selection of a friction test are simplicity {to separate frictional effects from others) and similitude (to assure that the many variables are similar to the actual operation of interest). In addition, there are many practical considerations involving tooling cost, complexity of the machinery needed,. size of the specimens,. and so on. In order to illustrate the trade-offs, we consider a few typical tests that have been proposed and used to determine the friction in sheet metal pressing or forming operations. Figure 9.8 shows several typical geometries for pinchwtype friction tests of sheet metal. These tests may be one-sided (Figure 9.8a) or two-sided• (Figures 9.Sb and c), and may have flat? cylindricat or inclined contact geometries. A back force may be applied to introduce a degree of tensile 5

Nakarnura~ M.Yoshida, A, Nishimoto,. Proc. 15lh 8ienniai Congr. lnternatfrmal Deep Drawing Research Group (May 1998): 77-83,

S,

9.8 Exa1nples of Sheet Friction Tests

(a)

(b)

3Z5

(c}

Figure 9.-S Pinch-type friction tests for metal sheets.

deformation that can be controlled independently from the normal Pinch-type devices have the advantage of simple interpretation:6

force~

fN.

(9.19)

Ho\vever, the sticking case is complicated by the unkno,vn contact area if curved dies are used. The disadvantages of such tests are serious. The geometry and deformation patterns ·mduced in a pinch-type apparatus are not similar to the tensile deformation and sliding in press dies~ where through-thickness compression is nearly always absent. Pinch-type tests are much more similar to conditions in rolling, drawing, or extrusion operations. Tests with the most _similitude to sheet forming are based on tensile deformation around a radiusr and these make use of the rope formula or direct measurement of normal forces, The device most immediately suggested from the rope formula involves drawing over a single radius~ as shown in Figure 9.9, In the simplest case_, a back force, F11 is applied~ and F2 is measured to move the sheet. In more complicated arrangements; the wr«p angle may be varied as shown in Figure 9,9b, and computer control may be used to vary F1, F2 and S throughout the test to Obtain combinations of stretch and wrap and their histories. In additionf the radius of the tool -can be changed and a roller may be substituted to investigate bending and other non-friction effects. Devices based on Figure 9 ,9 offer a great deal of flexib!Hty al !he expense of very complicated and specialized equipment. Because of the number of vari~ ables present, it is necessary to have a good characterization of the forming operation of interest. However, interpolatiOn is simple in viet\' of the direct measurement of Fl and F2, and use of the rope formula.

----,; The factor of! 2

1$ required be