Plastic deformation analysis of strain-rate sensitive materials under plane strain conditions

Int. J. Mech. Sci. Vol. 25, No. 4, pp. 251-263, 1983 Printed in Great Britain.

0020-7403[83/040251-13503.0010 Pergamon Press Ltd.

PLASTIC DEFORMATION ANALYSIS OF STRAIN-RATE SENSITIVE MATERIALS UNDER PLANE STRAIN CONDITIONS Y. S. LEE and A. T. MALE Research and Development Center, Westinghouse Electric Corporation, Pittsburgh, PA 15235, U.S.A. (Received 20 N o v e m b e r 1980; in revised [orm 11 M a y 1982)

Summary--The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stream functions of complex variables. Expressions for the stress, strain-rate and velocity are derived, assuming the stream function in the forms of both the summation and product of conjugate flow functions, for plastic flow in a nonlinear viscous (strain-rate sensitive)medium. The plastic states are also derived using a mixed mode solution expressed in terms of non-separable, independent conjugate complex variables. Application of the summation form solution is illustrated through the block indentation problem. Calculations are made on the effect of variation of the strain-rate sensitivity exponent on the contact stress. The predicted behavior of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation test for the rapid determination of the strain-rate sensitivity of real materials. By reducing the results of the indentation of a perfectly plastic material it is found that the contact stress is uniform and the external load is constant. The stress on the contact surface obtained using the present analysis is identical to that available from a slip line solution to the problem.

NOTATION Ai(i = 0 - 3) Ki(i = 0 - 3)

integral constants, Ai =- ai + ibi integral constants, Ki = ai - ibi 2~r0 (V3"~ '-'~

c a-Co-~-T/ I half width of the indentor m strain rate hardening exponent l+m /1 2 P applied load for unit thickness v punch velocity under applied load, P U, V horizontal and vertical velocity x, y horizontal and vertical axis z, ~ complex variable z = x + iy and conjugate of z m-I

13 #

~r0 ~r0 i0 ~0 sr rl

m

effective stress, ~ = ~0~m effective strain rate flow function stress tensor material constant strain rate tensor Arg(A0z+At) JAoz + All

0o Arlg A0 3~ ~b stream function

INTRODUCTION

The equilibrium equation for a plastic strain-rate hardening material derived earlier[l] is expressed in terms of a stream or stress function gradient with respect to the Cartesian coordinates. This equation was modified by introduction of the complex stream function in Ref. [2]. In the present paper, the solution technique is modified for application to the plastic strain-rate sensitive behavior exhibited by real materials during metal forming processes and is particularly appropriate for elevated temperature work. MS Vol. 25, No. 4 ~ C

25 1

252

Y.S. LEa and A. T. MALE

Three different a p p r o a c h e s are presented for solution of the plastic d e f o r m a t i o n problem. For the case of separation of variables, solutions using both f o r m s (summation and product) of conjugate flow functions are derived. W h e n the variables cannot be separated, a mixed m o d e solution e x p r e s s e d in t e r m s of an independent conjugate function is obtained. The s u m m a t i o n f o r m of the complex conjugate stream function can possibly be applied in the solution of plane strain contact problems with rate sensitive plastic material behavior, Solutions to the elastic contact problems are readily available [3, 4] and extensive literature exists on the slip-line solutions for rigid-perfectly plastic non-work hardening materials behavior[5, 6]. The indentation problem, b e c a u s e of its engineering application in various hardness tests, has been very popular in connection with these analyses [7, 8]. W o r k by Shaw and DeSalvo [9, 10] has s h o w n that for rigid indentation of a semi-finite block, the elastic-plastic interface significantly affects the constraint factor; h o w e v e r , the rigid-plastic behavior assumption could hold for the indentation of a finite block. To illustrate the solution procedure in this analysis, the plane strain contact problem of a semi-infinite rate sensitive plastic block being indented b y a rigid punch of rectangular cross section is considered. Stress and strain-rate on the surfaces are obtained for various values of the strain-rate sensitive exponent, subject to the assumption of zero friction at the contact interface and no w o r k hardening of the block. Thus the b o u n d a r y value problem illustrated in the analysis deals with the solution at the surface rather than that in the domain. ANALYSIS Nadai[12] considered that the shearing stress is assumed to be expressed in terms of the shear rate in the octahedral plane during the steady stage of creep (strain-rate hardening material) even though the stress components of rate dependent material may, in general, be expressed in terms of rate of deformation tensor. Nadai then established the relation between plastic stress and strain rates. The plastic strain rate components are given by a scalar multiplier 0() times the deviatoric stress components as in the Levy-Mises law without assuming rate-independent and without a yield condition. Hence, the applicability of the analysis is limited to small strains and a steady creep stage. The analysis is not valid for transient creep. A detailed explanation of the constitutive equation of steady creep material given by Nadai. For materials with stress-strain rate behavior of the type = cr0(~l~0)m

(1)

the governing equation (in the absence of work hardening) for plastic flow under plane strain conditions in accordance with the LevY-Mises constitutive relations[l] is derived from the two equilibrium equations. The constitutive equation, A(~rij+pS#)= ~s, the strain rate, e~j= (l/2)(u~.s + ui.i) and the incompressibility condition under plane strain condition is substituted into the two equilibrium equations. Using the stream function definition, u = (~9~/8y) and v = (-8q~Mx), the equilibrium equations are expressed in terms of the hydrostatic pressure gradient and the higher derivatives of the stream function with respect to x and y. Eliminating the hydrostatic pressure gradient from the two equilibrium equations gives,

82/2

8xsy aSTy, 8y2

(82, 82, /=0 8x2/t \ -0x /J

where ~ and ~ in equation (1) are the effective stress and strain rates, respectively. On transformation of the stream function ~b into the complex plane, this becomes

a: t(8:6~" 't82~x./

82 ~;8:6x"/8:~x ° '/

where n=(m+l/2) 0 < m < l and ~b is the stream function. Z = x + i y and 2 = x + i y . The detailed derivation is given in Appendix A. In terms of the complex flow.functions, the effective strain-rate in two-dimensional Cartesian coordinates and the scalar multiplier, ;t are given by

e = 4 d - l ( az6 a24~~'2 ~3\8Z2 - ~ - 1

;(

3~

where or0and e0 is a material constant.

3,i~"(4

I)t

~(aZ-OZ')

(4) (5)

Plastic deformation analysis of strain-rate sensitive materials

253

Note that for m = 1 or A = constant (perfectly viscous material [ 12]), equation (3) reduces to the familiar biharmonic form. Solution of the biharmonic equation subject to the boundary conditions of plane strain extrusion through a square-cornered die was illustrated in Ref. [21. Summation :orm solution Let ~(Z, ~z) = ~,(Z) + ~k2(Z)

(6)

denoting the second derivatives by 2

~2 and q_ _- d~-~

p =

(6a)

substitution of equation (6) and equation (3) yields

q

n d2p n-I

n d2q n i

=0

(7)

from which, on separating the variables, one obtains d2pn I - -

dZ 2

2

- Dp ~

n

1

dq . =Oand---~-~+Dq =0

(7a)

where D is a constant. Solutions for ~bt and ~b2 are given by ~bl(Z) = - 2(n - 1)(2n - 1) In (Z - Ao) + Ai Z + A2} for D = i. ~b2(2) = 2(n - 1)(2n - 1) in ( 2 - K0) + K , 2 + K2

(7b)

Using equation (7b) the deformation analysis of two viscous hollow cylinders buried in an infinitely large v i s c o u s m e d i u m d e s c r i b e d b y the p o w e r l a w w a s p e r f o r m e d a s s u m i n g an equal biaxial b o u n d a r y stress [ 13]. If D = 0 in equation (7a) the solution of equation (3) is given by

p = (A,,Z + A,) K,) '/I" '/(" "~ ')[ q = (K.2 +

(8)

where Ai = ai + ibi and K~ = ai - ibi From equations (6a) and (8), after integrating twice, one obtains cklT~(n-1)21)(AoZ+AO(2" I~'J - Ao2n(2n _

w("-t~+A,Z+A ~ (9)

- K,) ~b2(Z) = K o(n T ~-n- ~i )_2 1)(KoZ+ (2. ,,/,,,-,) + K 2 Z + K3. -

N o t i n g that A~ and K~ are conjugates, the stream function from equation (6) is given by 4' = 2 Re

{

(n-l)2 rat A,)'2" ":'" " + A 2 Z } n(2n 1)A0 2 ~ 0 - - + + A3 . -

(10)

This is a general expression for the stream function for 0.5 < n < 1 (0 < m < 1). For n = 0.5(m = 0), equations (9) are not valid, and hence the stream function is obtained directly from equation (8) as ~b = 2 re {~o2 In ~ } ,

n =0.5.

(11)

Hydrostatic pressure The hydrostatic pressure (negative mean stress) o-p can be obtained from the equilibrium equations written in terms of the stream function gradients[l]:

+±{! Ox

ay

02

a x \ A axOy]

ay~Xaxay/

aY[2~t\aY 2

/_o

ax2]J -

axt2-~\~-ax21J

Differentiating the first equation with respect to y, and the second with respect to x, and adding together yields -2

a2°'+ ( 0-~2+ axay

&2~[ 1 ( a 2 ~ - a 2 ~ ] \ay 2 ax2/t2,( \ a y 2 ~-x2/l = 0

(lla)

254

Y.S. LEE and A. T. MALE

or, in terms of the complex variables, from equations (5) and (8)

i{-02°'P + 0-'~-~ = 4°'~° (?)'-" m+ 11AoKo{(AoZ+ A,, '/l'' "+(KoZ+K02/1'' '~}. OZz

a~ 2 J

3 4om

(12)

m-

Hence, if the hydrostatic pressure is ~ = ~5,(Z) +

%2(23

one can obtain, as in the case of 4h and 4~2, 4~r0 (~.3)' " m - 1 K o ( a o Z + A,):,,,/.,, ~ ,~,.., o'p~(Z) = - 3d0---~ 2m Ao e 4~ro [X/3~' " m - l ao ,,,,,,, ,, ,~,/~, ~rp2(Z) = 3d0*" \ 4 ] 2m Koo ( K o Z + K O e ". Again, since A~ and K~ are conjugates, on addition, the negative mean stress component is given by 8~r0 /~/3~' "m-1 t,q-)

op

2,,,/,,,, ,,sin[ 2m r_20o]

[ q-l

(13)

]

where n = IAoZ+ Atl = {(a0x - boy + 002 + (aoy + box

+

bl)2} 1/2

[ aoy + box + bl ]

~"= arg ( A o Z + A0 = tan i I_aox - boy + a~ ' and 0o = arg (Ao)= tan ~ [b~].

From equation (10), the horizontal and vertical velocity components are given by u = ae, = i( o4, _ ocq

ay

\aZ

,~21

or U = - 2 [m-llI~--~--Y ~ ~ ..... ~'/'"' t, sin [~Z-~_ lrm+l ~'- Oo]+r: sin 0z} (14) V=

-

2 tm-~-i / m - I 70,7 1 (,..,,/(,. ,, cos I-m+l _0o]j + r~ cos 02/ [~--7-i_1 ¢ J

where rl = (ai2+ b2) ~12 and rt, r and 0~ are as defined in equation (13). Ai and Ki (i = 0, 1,2) are defined as Ai = r~exp 003 and Ki = r~exp (- i03. The strain-rate components are similarly given by

=i(a2 \aZ z

•

as a2~/-

_n (15)

2.0.~,,,,,,

[02q5 + 02~bX

The stress components are then obtained from the Levy-Mises equations with the scalar multiplier A given by equation (5) and the hydrostatic pressure % by equation (13): _ ~x

~x-~-~rp

= - 2 C ~ ''l"'' ~{ sin [ 2~" ]

L~-IJ

m-1 [ 2msr -20o]} 1 --m s i n t m _ l JJ

•

[ 2m• O'xY =

.~"

:

~ ZL"O

2m/(m

1)COS[ m2--~1]

200]} I

(16,

Plastic deformation analysis of strain-rate sensitive materials

255

where

2tro [V3'~ '-m 1 " (~0)"

c = T ~,Tj

The solution of equation (3) can be obtained by using the product and mixed mode form. Those solutions are given in Appendix B.

Boundary value problem The solution procedure described above can be applied to the problem of plane strain indentation of a semi-infinite block by a rigid rectangular punch of width 21, as shown in Fig. 1. The workpiece block is considered to be rate sensitive. It is assumed that no upward flow of material occurs at the edge of the punch. The summary of the boundary conditions associated with block indentation are given below: (1) 7xy = 0 o n y = 0

(2) cry=Oon y = O , l < i x I (3) 21~--@ydx = - P mlU (4) Velocities are continuous at x = I.

OV

(5) U = 0, ~-x = 0 at x = y = 0 (Symmetric Condition). (6) V = constant at y = 0, Ixl -< I. For the summation form of solution, the integral constants Ai, K~ in equation (14) can be manipulated to satisfy the above velocity boundary conditions. To apply the solution given in equation (16) to the present problem, z is written as z = x + l + iy and the conjugate of z is rewritten as ~ = x + l - iy. It is noted that x and I in the analysis are considered to be absolute values since the system is symmetric with respect to the y-axis and equations (14)-(16) have the branch points at x = ill. The plastic states at the branch points may or may not be continuous, but is not analytic. The requirements of the b r a n c h point.s are satisfied when the stress is discontinuous, and both the strain rates and velocities are continuous and not analytic at x = Ill, Y = o. These conditions are necessary for determining the plastic states in the non-contact boundary. For a frictionless contact interface (y = 0), the third of equation (16) implies that tan [m - 1] -~ = bj

(17)

a0

Note that ~ = tan -I y/(x + l ) + 00, (Ai = KI = 0 ) Denoting a = tan [m -

1] 4

and noting that equation (17) must be satisfied for all values of x, the constant must be related as

bo = aao.

(18)

Substitution of A0 from the above relations in the second of equations (16), yields the contact stress (vertical stress), o-y = - 2 C

1 cos

. . ~r

,2,,,/(,,,-i) ,

/ ( t + x)-'"""-'"'f"

In the case of plastic indentation from slip-line analysis, it has been observed that the vertical contact stress at the discontinuity (x = I, y = 0) is indeterminate; nevertheless, it has a finite value (as opposed to the elastic indentation case). The vertical stress at the indented surface peaks at the centerline and decreases

/

J

/- / Visc0usMedium / /

FIG. 1. Boundary value problem. Rigid rectangular Punch indenting a semi-infinite non-linear viscous medium under conditions of plane strain.

256

Y.S. LEE and A. T. MALE

gradually along the width to the edge of the indenter, where it decreases almost instantaneously to zero (neglecting elastic effects). Equation (19) does not show that the vertical stress is free on the non-contact surface. Hence, the stress representation on this surface is obtained separately. On the non-contact boundary, the stress at the point x = l, y = 0 is discontinuous. The relation between the hydrostatic pressure on the contact and non-contact boundary is considered as 6o-p, where 8 is an unknown constant and ~rp is hydrostatic pressure on the contact boundary, y = 0 and Ixl -< 1. It is noted that this relation is valid only for y = 0 and this is not valid in the domain since the stress solutions are analytic everywhere except the branch points and all solutions are unique values. The hydrostatic pressure is given in equation (13) and &rp is obtained by solving equation (1 la) multiplied by 6 or it is found simply by multiplying equation (13) by & Then the stress component in the vertical direction in the non-contact boundary (Ixl -> I, y = 0) is obtained from equation (16) by substituting 00 = ((m - 1)/4)1r and y = 0, . . . z,,,/i,._t)~f I + 8 m - 1} o'v = .zLVO) "

m

Ixl>-l.

at y = 0 ,

(20)

o ' . = - 2 C ( ' r l ) " " ' ' ~){1-6 m 2 1} The vertical stress on the non-contact boundary, y = 0 and x -> l vanish if the unknown constant 8 is m

6 - 1 - m"

(21)

The stresses in the boundary are shown in equation (16) and stresses are continuous and unique everywhere, except the branch point, Ix] = l, y = 0. The horizontal and vertical stresses on the non-contact boundary can be obtained by substituting 6 in equation (20), then those stresses are o- -

4C 2m(m-I) -

cry=O

~

] J

for

y = o , Ixl -

I.

(22)

Using equations (19) and the second expression of equation (22) the integral constant, a0 is obtained from the equilibrium condition,

-p=2fo,,,dx:2fo',,dx=

- 4 C 1 - m r_ C-.,/I,. ,,[1 -2m'~[ 1 ~-2m/[I .,),(l-3./I-m)r.(I-3rntl-rn)__ ][cosm_lzr~ , "t~\ ---T- /

1} (23)

(ao)(2,./,. i)

p (1-3m m \f m-I 24--C\-1---~ 1--2m) [ s e c T ~ r ]

y2"/~-")(1-3m)/(l-m) ~ - - 1 "

(24)

Substituting this constant into equation (19) after simplification, the nondimensionalized vertical stress becomes

o'~=-P [1-3m] 1 [ l ](2m/,-,,,) O'0 2 l o ' 0 \ 1 - m ] { 2 ( ' 3m/l-ra)--l}\/-~X/ "

(25)

The stress on the non-contact surface is given by equation (22). The horizontal stress is similarly obtained: crx= 1 -~Y 2m

(26)

on the contact surface and nondimensionalized ~rx on non-contact surface is o'x - P m(1-3m) ~ 1 ~ I ],2./,-,~) ~ro l~ro (1 - m)(1 - 2 m ) 12 (' 3,.)/(i ,.)_ lJ [/-~-xx] As seen from equations (24) and (27) the instability of the solution and equations continuous at the point Ix[ = I, y = 0. The shear stress at the contact interface zero friction. The strain-rate components equation (24) into equation (15) and setting

(27)

present approach is not valid when m*l 0.5, 1/3 due to the (26) for y = 0 and Ix[-< l and (27) for y = 0, I x [>- l are not and non-contact boundary is zero because of the assumption of on the surface are obtained by substituting the a 21(m-t~ from y = 0. Nondimensionalized strain rate gives,

ix _d_~_ V3[~/(3)mp l-3m 1 .¢,tm)[ 1 ~(2t,-.,) E~ = d o : - T ~ 2To/ (l-~n-)(T--2m)2 ° - 3 ~ / t - " ) - l ] [/-~xJ "

(28)

The velocity components may be obtained similarly from equations (14). The vertical velocity on the

Plastic deformation analysis of strain-rate sensitive materials

257

contact surface is the same as the velocity of the indenter, v. Hence the vertical velocity is given from equation (14). (note ~ = ( m + lira - I)00 in equation (14) for y = 0.) V lio

- - = - 2r2 cos 02 = vlldo.

(29)

The horizontal velocity at x = 0, y = 0 should be zero due to symmetry. From equation (14)

_ m- 1 1 F (a0)("*'/'-')(l)("+~/"-')q [ m-1 )(2:,. 1~ | -

U=-Zm-+-l~o[

L

2r

_i

tc°sT

2sin02 = 0

or -.- = lio

[m - I

{cosO41.r --'{ao}a/m

~)l(,.+,t,,_,~]llo+r2si n 0 2 = 0 .

(30)

From equations (29) and (30), r2 and 02 is r2 =

~(v 2 /m-l\:f 2m-1 X + ~ - + - - i ) [cos - - 4 -

]a/'"'ao,4/,_,)l,2(,.+,)/.,,,~{l ~ ]\li.o}

~rj

[ml m :"

2~-Ti/cos-T-0. = tan -~ "

{a,,} '2:" ,~l~..*... ,)

v

(31)

]

"

The velocity components shown in equation (14) are rewritten, to obtain the velocity field on the non-contact boundary. Ix]'-> l and y = 0. as

U-

--

2 m - I 1 r/(,.+l/m_l)sin[m+l r_00]+K ~%-ig

t ~ - - ~ -I

(14a) m+l V = - 2 7m, m + - ll lro -(m+'/"-')~"s[-~_l~'-0o]+K,,,.,,

K , = O.

The horizontal velocity in the non-contact boundary should be continuous at x = I and this implies that K in equation (14a) corresponds to -2r2 sin 02 in equation (30). The vertical velocity in the non-contact boundary is neither constant nor zero and the velocity, V, satisfies the velocity continuity condition at x = I, y = 0 and V = 0 as x approaches infinity. To obtain the vertical velocity in the non-contact boundary, the coordinates are transformed to new coordinates as shown in Fig. 1. The new coordinates with origin at x = I, y = 0 are denoted as £ and y. The relations between old and new coordinates are given as

$=l-x (32) ~] = y.

Substituting equation (32) into the vertical velocity component in the non-contact surface shown in equation (14a) and changing the arguments to ~:

l + m ~_00 z V = _ 2 (m-l),t/O+m/m_DCOS(~.~_l )

(33)

where

~ ~ = tan-' (/-~Yx ) - 0 o = tan ' Y + 0 o if J / > 0 and y = 0, then ~ : 0o / 0o = ~ - ~ if i < 0 a n d ~ = 0 , then ~ = ¢r+ 0o,J'

w.

(33a)

The non-dimensionalized vertical velocity on the non-contact surface can be found by substituting the second expression in equation (33a) into (33). V -ldo =

m - 1 1 7/(~+~a,._~) I • 1 + m is,n ~_--~ - 2 ~ - 1 ro

I 1

~] ~o

(33b)

on the non-contact boundary. Equation (33b) reduces to zero as x approaches infinity. The velocity continuity condition at x = l and y = 0 is then obtained from the condition, lira V(x)= x~l

lim V(x)

- x --* I + .

(33c)

258

Y.S. LEE and A. T. MALE

The l.h.s, of equation (33c) is the same as the punch velocity and substituting the integral constant is obtained in equation (24) into the r.h.s, of equation (33b), the relation between the external punch load and the punch velocity can be determined. The relation between the punch velocity and load is given as v (I-m)]'X/3 P m(l-3m) I /<'"~'~'¢:"/" '~ sin l + m ~rl 1~--~= m + l ( 2 ~01(1----~)(-1---2--m)2<' s ~ / ' - " ' - l J '" I 1-m I or ]

p 1 ,2,,~ . . . . o.(~1-%7~1 r

'"-°"~[l+m~"f,(l-")tl-2")t2" I\l-m] ( m(1-3m)

l)}

"

~""

-

• I+m

sln~"

m

l|

(~.)". ~teo/

l It is noted that if m ~ 1/3 or m = 0, the above expression is unstable and rate equation (33d) is not valid. The strain rates for Ix] -> I and y = 0 are obtained from equation (20) and stress-strain rate relation. They are equivalent to equation (28). The stream function, velocities, strain rates and stresses shown in equations (10), (14) (15) and (16), respectively, are valid within the boundary and they are unique except at the branch points, Ixl = l, y = 0• The summation form of the complex stream function associated with non-linear viscous material is applied for finding the solution on the boundary of the fiat indentation problem as mentioned in the introduction. The analysis of the indentation problem gives an approximate solution• The stress solution is modified by introducing Ixl instead of x in order to impose the symmetric stress distribution with respect to the y-axis. This implies that the equilibrium condition at x = 0 is not satisfied since Ixl is not analytic at the y-axis. Further analysis includes only the solution for y = 0.. Therefore the boundary value problem in the present analysis is incomplete and is an approximate solution based on complex variables. RESULTS AND DISCUSSION It can be seen from the expressions derived in the previous section that both the contact stress and strain-rate components are a function of the external load, P, on the punch. Variations of these components with respect to the strain-rate sensitivity exponent, m, of the indented medium are shown in Figs. 2-5. Figures 2 and 3 show the calculated variations in the vertical and horizontal stress components on the contact surface. For a perfectly plastic material (m = 0), the contact (vertical) stress is uniform over the entire surface. In a strain-rate sensitive material, however, this component peaks under the center of the punch and decreases smoothly towards the edge. The peak value increases with increasing value of the strain-rate sensitivity exponent m; however, the mean value for any given m remains equal to the uniform contact pressure corresponding to the perfectly plastic material• This suggests that for plastic material behavior, plane strain block indentation tests may be developed in which the ratio of the peak to mean contact pressure can be used as a measure of the strain-rate hardening characteristic of the material. The ratio of the peak to mean contact pressure as a function of the strain rate sensitivity exponent m is shown in Fig. 6. Frictional effects at the punch block interface have been shown to be negligible provided the indentation is not extensive[14], and would thus not provide complications for such a test. In the horizontal direction, the calculated horizontal stress, tr~ variation exhibits a similar pattern to that of the vertical stress, except that the mean value in this direction increases with increasing value of m. Also, for increasing values of m, the horizontal stress, tr~ is progressively higher than the vertical component. This is believed to be due to the assumption of no upward flow at the edges of the indenter and consequent material restraint. O.Q 1

q

I

i

I

i

I

i

I

f

I

i

I 1.O

0.8 0.7

~0•6~ E

0.4ooo:. 0.3 .'r 0.2 o•

l] i

i

~

I

0.2

i

I i I J L 0.4 0.6 0.8 Contact Surface, L

FIG. 2. Vertical contact stress variation for various values of strain-rate hardening sensitivity factor (m).

Plastic deformation analysis of strain-rate sensitive materials

259

As seen from the strain-rate expressions on the contact surface, the strain-rate is non-linearly dependent on the material constraints ~0 and m. as well as the load intensity P. No definite behavioral pattern for the strain-rates can therefore be observed as illustrated in Figs. 4 and 5. In a practical situation either the punch load or the punch velocity is the directly imposed variable. Therefore, the curves given in Fig. 7 have been calculated to depict the relationship of the punch pressure to the basic material flow stress as a function of the punch velocity (or vice versa) for various values of the material strain-rate hardening exponent. This analysis cannot be rigorously applied to the problem of indentation of metals, except possibly under conditions of high temperatures and low strain rates. However, the general forms of solution shown in Figs. 2 and 3 are not expected to change drastically by the introduction of work hardening behavior. If this is so, it is believed that this is the first work to postulate that vertical pressure variations can occur across the face of the indenting punch, which are caused by some inherent material behavior and not necessarily by interfacial friction between the workpiece surface and the punch face.

f

I

]

i

l

I

'

I

1.0 0.9

i 0.8

p 0.7 P,= .=~ 0.e

~=~ 0.5 = o¢

0.4-

-

0.30.20.1 J

I

I

I

0.2

¢

I

I

,

I

,

0.4 0.6 0.8 Contact Surface, £

I

1.O

FIG. 3. Horizontal contact stress variation for various values of strain-rate hardening sensitivity factor (m).

140

'

1

'

I

,

I

I

I

P---=5.0 too

120

--o 100

.w>,

.E N ~

6o

m= 0.05, 0.I0, O.

40--

~

,

@.

I

0.2

(very small) l

~

,

I

J

I

,

i

0.4 0.6 0.8 ContactSurface, t

,

--I

ql

1.0

FiG. 4. Vertical strain rate variation for various values of strain-rate hardening sensitivity factor (m).

260

Y.S. LEE and A. T. MALE

1500

i I I ~ I OJ I ~ I i I =

I0.0

A o

.¢~. 1000 _c i*

,o0

~:

0.2

0.4 0,6 0.8 Contact Surface,

1.0

FIG. 5. Vertical strain rate variations for various values of strain-rate hardening sensitivity facto~ (m).

1.4

i

I

L

I

~E 1.3

1.2

t~ 1.]

~

I

L

0.1 0.2 Strain-Rate Hardening Exponent(m)

0.3

FIG. 6. Relationship between normalized peak stress and strain-rate hardening exponent (m)

Plastic deformation analysis of strain-rate sensitive materials 20

T

261

P P* ~ I l + m Trl-rn ~ o o : ~--~-olSN 1 - m ]

I

18 ~

..q

O

4

--

V-ee

M: STRAIN RATE HARDENING EXPONENT

0.3 2

--

o

I 0

I

I I I I I1[

I

I

[

1 O0

I I I I I 1000

P U N C H V E L O C I T Y ( V / ~ ~ o)

FIG. 7. Relation between external load and punch velocity.

Although there is no published information concerning the experimental measurement of contact pressure variation across the face of an indenting punch, there are data available from somewhat analogous situations. Various research workers [16--19] have conducted experiments on pressure variations in the arc of contact during rolling operations. These works used essentially similar equipment, involving small pressure sensing pins incorporated into a rolling mill roll. They all made similar observations of a gradually increasing normal pressure variation reaching a maximum at the neutral point within the arc of contact. Similar observations have been made by other workers performing measurements of a like nature on the compression of cylindrical specimens being compressed between flat over hanging dies. Here the peak normal pressure is observed to occur at the specimen axis. such pressure variation has been explained in terms of the effect of interfacial friction. This current analysis offers an alternate explanation for such pressure variations. The present analysis emphasizes that the nonlinear creep material can be solved by introducing the complex stream function and it includes the partial solution of the flat indentation problem.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

REFERENCES A. H. SHAIBAIKand E. G. THOMSON, ASME, Series B 90, 343-352 (1968). Y. S. LEE and M. R. PATEL, ASME, Series B 99, 727-732 (1976). IAN N. SNEDDON, Fourier Transform, pp. 395--411. McGraw-Hill, New York. A. E. GREEN and W. ZERNA, Theoretical Elasticity pp. 274--275. Oxford at the Clarendon Press. W. JOHNSON, R. SOWERnY and J. B. HADt~3W, Plane-Strain Slip-line Fields: Theory and Bibliography. Elsevier, New York (1970). R. HILL, The Mathematical Theory of Plasticity. Oxford University Press (1950). A. E. ISHLINSKY,Prikladnaia Mathematics Mechhanika 8, 201 (1944). D. TABOR, The Hardness of Metals. Oxford University Press, London (1951). M. C. SHAWand G. J. DESALvO, ASME, Series B pp. 469-479 (1970). M. C. SHAWand G. J. DESALvO, ASIDE, Series B, pp. 480-494 (1970). R. T. SHIELD, Proc. Royal Soc. A2,33, 217 (1955). A. NADAI, Z App. Phy. 8, pp. 418--432 (1937). Y. S. LEE and L. C. SMITH, Trans. ASME., J. Appl. Mech. 48, pp. 486--492 (1981). A. T. MALE U.S. Air Force Materials Lab. Tech. Rep. AFML--TR-69-109, Part 1, Apr. 1969. L. M. KACHANOV, Foundation of the Theory of Plasticity, pp. 218-219. North-Holland, Amsterdam (1971). E. SI~BEL and W. LUEO Mitt K. W. Inst. Eisenf. 15, 1 (1933). E. OROWAN, Proc. Inst. Mech. Engng 150, 140 (1943). C. L. SMITH, F. H. ScoTt and W. SYLVESTROWICZJ. Iron Steel Inst., 170, 347 (1970). G. T. VAN ROOYENand W. A. BACKOFEN,J. Iron Steel Inst. 186, 235 (1957). APPENDIX A Appendix A is concerned with the derivation, of equation (3). The equilibrium equation is given as

OCrx+ L.~. = o o~ oy + ~ z = O

(A1)

262

Y.S. LEE and A. T. MALE

The constitutive equation for the creep material in the secondary creep region is considered to be

-=-o(:0)

(A2)

1.

where

i =3~ ~. Substituting equation (A2) to equation (A1)

Ox Ox

(A3)

. ~. .c9 ' ~ [l " i ~ + Za( 1 ' 4 ap

=°'

Using incompressibility condition under plane strain ix = - d. the definition of the stream function, u = (a~b/ay) and v = ( - a#lOx) and the linear strain rate, iii= (uu + ui.312, equation (A3) can be rewritten as

ap4 - a (1 a2* )+ *9 ( 1 (a2ck_O2~)]= O (A4) ay

0

ax

ay \,( axay/

eliminating hydrostatic pressure from equation (A4) the resulting equation is

fl \ ay 2

(AS)

ax2/J

The effective strain rate for an incompressible material under plane strain is given as _.

¢. = ~

2 ..2+i~y2),/2= 2 [ / a 2 6 ~ 2 , l[a2ck a24~'¢/'12 (~ ~7~ k a - ~ ] ~-4 k-~Y2 - ~Tx2] J "

(A6)

Since z = x + iy and r = x - iy o

,,

--ax = \ ~

a

=i(

o

_

a

), ~ ;- = ,~--~z2 - ~-~ ,j,..

(A7)

Substituting equation (A7) to equation (A6) yields

k az 2 ] k a~.2 ] J "

(A8)

Combining equations (AS) and (A2), the scalar multiplier, J( is obtained as = ~(~)1

m = .]~ 30mi [~074~ ~1

ra[c~2~)~k~Z2_~, ]02~)'~(I-ra/2)"

(g9)

Substituting equations (A9) and (A7) to (A5) and simplifying the resulting equation is ~2 f//92~x'ra

'/2)[1926x'm+'/2)~ ~2 ffO2,~(ra+l/2)[O2~(m-lt2'I

-

-

APPENDIX B

Product form solution Let ~b(Z, Z) = ~b,(Z)~b2(~r).

(B1)

Substituting equation (BI) into equation (3), the necessary condition to satisfy equation (3) is /A2~

\ (ra-1/2)

/A2.& \(m 112) a...+m)/u w2/ =/)2. ~'2 kd22 ]

(B2)

Plastic deformation Substituting 4, = obtained as

analysis of strain-rate sensitive materials

D,(A,Z+ A,)@and & = Q(K,Z+

263

K,)@ into equation (B2), the stream function 4(Z,,?) is

‘(” ’ =( #I(/3 -~)A,K,,)p( (A, + A,Z)(K, + K$))’ where D = (D1D2)“’ and /3 = (m - l/m). Strain rate, velocity, and stress components as in the case of the summation form of solution.

can now be obtained

Mixed mode solution The summation and product form of solutions described in the previous sections were associated with the method of separation of variables. The stream functions were decomposed into two holomorphic functions and the solution was obtained by using a semi-inverse method. In this section, a method of solution is presented when the stream function cannot be separated into functions of I+,(Z) and C+,(Z). The governing equation (3) is satisfied identically if the following general solutions are assumed

(~~“““(~)‘“““’ =*(A,z+

A, +

K,i+

K,)

(B4)

(a&J+““(~)‘,-“r’

=(A,Z+A,+K,.f+K,)

and the constant A is chosen so as to yield a unique function (e(Z, .?) from either of the equations above. Solving equation (B4) with respect to (a*+/&) and (#I#/@), the solutions are given by

a29=

a.?

where l= A,Z+ A, + K,.?!+K,. Integrating function is obtained to be 4

=

At~-d~mj

~(l+m/Zm)p/m)

I

twice the hrst of equation (BS) with respect to Z, the stream

m2 + A,Zf,(.$ + A&(i) A,, ( (l+ m)(l+2m) I ,J(‘+2mlm)

4

similarly, from the second of equation (BS) (#)

=

*(l+d2m)L K:

m2

(1 + m)(l + 2m)

I,p+*nJ”)+ K,2g,(z) + K3gQ).

(B7)

If the unknown constant A is chosen as

equations (B6) and (B7) become congruent and the stream function for the mixed mode solution is 4(z,

Z, = (2)“‘“‘(’ m2 )p+2dm) +A*zZ+ ADZ+ A$ KoAo(I+ m)(l + 2m)

Once the stream function is known, the velocity, strain rate and stress components shown previously in equations (14) and (16).

can be obtained as