n. aravas · 2005-09-29 · single crystal plasticity in this section, a new convenient description...

International Journal of Plasticity, Vol. 7, pp, 141-160, 1991 0749-6419/91 $3.00 + .00 Printed in the U.S.A. Copyright © 1991 Pergamon Press plc

O N T H E G E O M E T R Y O F S L I P A N D S P I N

I N F I N I T E P L A S T I C D E F O R M A T I O N

N. ARAVAS

University of Pennsylvania

E. C. AIFANTIS

Michigan Technological University

(Communicated by George Weng, Rutgers University)

A b s t r a c t - - A convenient description of the geometry of slip and the associated kinematics of finite deformation in an elastic-plastic crystal is presented. The physical meaning of the plastic spin is discussed in detail. The problem of uniaxial tension of a single crystal under multiple slip is analyzed, and a simple technique for the calculation of the rotation of the slip systems during plastic flow is presented. The paper closes with a relevant discussion of the kinematics of finite elastic-plastic deformations of a continuum.

I. INTRODUCTION

The mechanics of elastic plastic deformat ion of crystals has been well developed. The theory traces its origins to the work of TAYLOR [1938]. It was further advanced by HILL [1966], MANDEL [1971], RICE [1971], HILL and RICE [1972], ASARO and RICE [1977], HILL and HAVNER [1982], PIERCE, ASARO, and NEEDLEMAN [1982,1983], and AtlANTIS [1987]. A complete bibliography on the subject can be found in the review articles by HAVNER [1982] and ASARO [1983a,1983b].

A convenient description of the geometry of slip and the associated kinematics of finite deformations in an elastic-plastic crystal is presented in this paper with emphasis on the origin and interpretation of plastic spin. It is shown that, to within elastic strains, the plastic spin is the average angular velocity of the crystal relative to an observer who is spinning instantaneously with the angular velocity of the slip directions. The results of the analysis are applied to the problem of uniaxial tension of a single crystal under multiple slip, and a simple technique for the calculation of the rotation of the slip systems during plastic flow is presented. A description of the kinematics of finite plastic deformations of a continuum are also presented and the physical meaning of the plastic and relative spins of the continuum is discussed in detail.

The results of the present paper are based on a description of the geometry of slip given by MANDEL [1971] and AIFANTIS [1987] and give additional insight to the origin of plastic spin as discussed by DAFALIAS and AIFANTIS [1986,1990]. Moreover, they in- clude careful derivations of the expressions of the plastic and relative spins (ZBIB & AIFANTIS [1988a, 1988b]) and provide a detailed geometric interpretation of these spins for both crystal and continuum plasticity. In this connection, we point out that the present derivations hold true in the cases of both small and large elastic deformations,

141

142 N. ARAVAS and E. C. AIFANTIS

while in the discussion of AIFANTIS [1987], for example, attention was confined to rigid plastic deformations.

Standard notation is used throughout. Boldface symbols denote tensors, the order of which is indicated by the context. The following products are used in the text

and

(AB)ij = Aik Bkj ,

(Ab)i = Aikbk,

(bA)i = bkAki,

(a @ b)ij = aibj,

where the summation convention is used for repeated Latin indices.

il. SINGLE CRYSTAL PLASTICITY

In this section, a new convenient description of the geometry of slip and the associated kinematics of finite plastic deformation of elastic-plastic single crystals oriented for single or multiple slip is presented. The physical meaning and origin of the plastic spin is discussed in detail and a convenient geometric interpretation is given.

II. 1. Basic kinematics

We assume that crystallographic slip is the only mechanism of plastic deformation. It occurs via dislocation motion through the crystal lattice which, with the material embedded in it, undergoes elastic strains and rotations. I f F p is the purely plastic part of the deformation gradient, then the total deformation gradient F is given by (MAN- DEL [1971], RICE [1971], AIFANTIS [1987])

F = RUeF p, (1)

where R is a rotation tensor, and U e is the symmetric right elastic stretch tensor. Phys- ically, F p arises from irreversible slipping of crystal portions with respect to each other, U e arises from the reversible elastic lattice displacements, and R arises f rom the geometric constraints imposed by the kinematical boundary conditions.

The velocity gradient L is then given by

L = 1 7 F - I = RR r + R u e u e - I R r + RUeFPF p-I U e- I R r, (2)

where the superposed dot indicates material time differentiation, and the superscript T indicates the transpose of a second order tensor. The deformation rate D and the vorticity or spin W, defined as the symmetric and antisymmetric parts of L respectively, are now written as

D = D e + D p, (3)

The geometry of slip and spin 143

and

W -~- 0,1 "[- W e "[- W p, (4)

where

D e = R ( f t J e U e - l ) s RT, W e = R ( u e u e - I ) a R T , oJ = R R T, (5)

D p = R ( U e F P F p - I U e - I LR r, W p = R ( U e F P F p - I U e - I )aR T, (6)

where the subscripts s and a denote the symmetric and antisymmetric parts of the vari- ous second-order tensors respectively.

We consider first the case of single slip with one family of dislocations moving along their slip system. Figure 1 shows a schematic representation of the kinematics of single slip. We label the reference configuration as (Bo, the current configuration as (B, and the intermediate configuration defined by F p as (B1. In the undeformed configuration, the unit vector in the slip direction is p0 and the unit normal to the slip plane is no. In the deformed configuration and corresponding unit vectors are denoted by I, and n.

The plastic part of the deformation gradient, defined in the intermediate configuration (B~, is given by

F p = 1 + "yl'o @ no, (7)

where -y is the amount of slip, and 1 is the second order unit tensor. Physically, "r is related to the dislocation density and the corresponding distance (area) traveled (swept) by them. For straight-edge dislocations of average density p and Burgers vector b moving on the slip plane, one can write 7 = bpA where A denotes the average flight dislocation distance. In the intermediate configuration (BI we also define

D p = ( F P F p - I ) s = l ~ ( p o @no + no @Uo), (8)

and

W p = ( F P F P - I ) a = l # ( p o @no - no ® ~o). (9)

n d F p

I RU e

A~d _111 V /1 i lu~ i V

A I I I I V

Fig. 1. A schematic representation of the kinematics of single slip.


I f one identifies v and n with MANDEL'S [1971] director vectors, then the intermediate conf igura t ion 63L is the so-called "isoclinic conf igura t ion" o f MANDEL [1971].

The orientat ion o f the slip system in the current conf igura t ion 63 is given by

1 1 v = - - Fl'o = - - R U e v o , (10)

and

1 1 e n = - - no F-1 RU e- lno ,

where

~ = IF,,ol = [Ue'o[, and h e = Inol~-'l : IU ~ ' no [ . (12)

It should be noted that the slip direction p deforms with the cont inuum, whereas n re- mains normal to the slip plane and is, in general, along different material fibers at different times. The relat ionship (11) is simply a consequence o f Nanson ' s fo rmula (TRUESDELL & TOUPIN [1960], p. 249). Equat ion (10) makes it clear that both R and U e contr ibute to the rota t ion o f the slip system, unless J'0 is an eigenvector o f U e or the crystal is rigid plastic (U e = 1) in which cases R becomes the total rota t ion o f the slip direction in the current conf igura t ion 63. We ment ion again that the orientat ion o f the slip system during in the current conf igura t ion 63 is defined by the u n i t vectors u and n. A discussion o f the several possible choices for the description o f the evolution o f the slip direction, including the one just mentioned, can be found in ASARO and RICE [1977].

In view of eqn (6), the plastic deformation rate and the plastic spin in the current con- f iguration 63 are given by

D p = l ' ~ A e A e ( p (~ )n + n ~ ) p ) , (13)

and

W p 1 • e e = ~ A ~ A ~ ( v @ n - n @ v). (14)

We also record the following two relations which are easily obtained after elimina- t ion o f ~ f rom eqns (13) and (14)

W p = DP(n (~) n) -- (n (~ n)D p = - D P ( v ~) p) + (p (~) p)D p, (15)

and

D p : WP(n (~) n) - (n (~) n )W p = - W P ( I , (~) p) + (p (~) ¢)W p. (16)

In view of eqns (10), (11), (15), and (16) we can easily show that (Asngo & RICE [1977])

i, = [oJ + W e + D e ( v @ v) - (v @ p)De] v, (17)


and

!:! = [OO + W e - - D e ( n ® n) + (n ® n ) D e ] n . (18)

It is worth noting that the rate of rotat ion of the slip system in the current configuration (B consists of a rigid body spin ~0 and a spin caused by the elastic rate of distortion, the terms involving W e and D e. Furthermore, p and /~ are independent of F p which, as shown in Fig. 1, does not rotate the slip system. Similar expressions are also reported by AIVANTIS [1987] but for the most part they were valid for small elastic strains, a case which is discussed in detail in Section 11.3.

The above analysis is easily extended to the case of multiple slip as follows. In the intermediate configuration ( ~ we define

~'PF p-1 = ~ "~ ('~)ro ~a) (~) n(o ~), (19) t ~ = l

1 ~] .~=)(ro(,~) @ n(o =) + n(o =) Q ro(~)), Dg = 5~=, (20)

and

1 ~] ~,.~)(ro~,) ® n~o ~) _ n~o~ ® ro.~)). Wg = ~ = , (21)

where -yt~) and (vo ~) , n~o ~)) a = 1 . . . . . m are the amount of shearing and the orientation of the ~ slip system respectively, and m is the total number of active slip systems. It is interesting to note that, in multiple slip, F p is not generally a point function of the -y~) 's , but depends instead on their sequence of application (RICE [1971], ASARO & RICE [1977]). In the current configuration (B we have

m

(22)

1 m (23)

r(~) - 1 R u e vo(,~) ' n(,~) _ 1 R u e _ 1 n(o,~) ' (24) ,~<,~) A~ ~)

~(~) = [60 + W e + De(p (~) ~) v (=)) - (v (~) ~) V( 'x))oe]p (~), (25)

fi(~) = [ ~ + W ~ _ D e ( n (,~) ® n ('~)) + (n o~) Q n ( ~ ) ) D e ] n ('~), (26)

where Ape t~) and A,~ (~) are defined by equations similar to (12). It is pointed out that, in multiple slip, it is assumed that the crystal is not constrained on its boundary so that dislocation mot ion along one slip system causes material rotation or a rotation of the continuum but it leaves the relative orientation of the crystallographic directions unal- tered (RICE [1971]). This was taken into account in deriving eqn (24) which is valid for all slip systems, active or not.


II.2. A geometric interpretation o f the spin tensor to

Here, we provide a more direct interpretat ion o f the material spin to by employing a new intermediate conf igura t ion which enables us to absorb the last two terms o f (17) and (18) involving W e and D e and express the rate o f rotat ion o f the slip system, in the new intermediate configurat ion, solely in terms o f to. We rewrite the multiplicative de- composi t ion (1) as

F = VeRF p, (27)

where ve is the symmetric left elastic stretch tensor. We also consider a second intermediate conf igura t ion (B defined by the deformat ion gradient ~':

: RF p, (28)

so that

F = ¥ ~ ' . (29)

This conf igura t ion has been also used by LEE [1969], MANDEL [1971,1973a,b,1981], FARDSHISHEH and ONAT [1974] and DAFALIAS [1985a, 1987, 1988] in their studies o f con- t inuum plasticity, in which the elastic part o f the deformat ion gradient F e was assumed to be symmetric.

The corresponding velocity gradient, deformation rate, and vorticity at (B are given by

[~ = ~ F - I = RR r + R F p F p - l R r ' (30)

f J = D P and ~ W = t o + W P , (31)

where

I)P = R(FPFP-I )sRr = R D P R r and W p = R(FPFp-I ) ,~RT = R W ~ R r. (32)

The orientat ion o f any slip system in the intermediate conf igura t ion (B is given by

~ ) = Rv0 ~) and fi~) = Rnt0 ~), (33)

so that

~('~) = to~'~) and tic-) = tofi(,~). (34)

Equat ion (34) shows that to is the rate o f rotat ion o f all slip systems, active or not, in the intermediate conf igurat ion (B.

II .3. Small elastic strains

Motivated by the discussion o f the previous two sections and the relations (25), (26) and (33), (34) we consider the case o f small elastic deformat ions for which the config-


urations (~ and 63 differ by an infinitesimal amount , and the corresponding formulae simplify considerably due to the secondary role of elasticity. In this case we have

where

U ~ = 1 + 6U e, t l e = 6U ~, (35)

161<<1 , ~jeT=~_je, supllCrel) = 0 ( 1 ) ' (36)

and the equations for the kinematics of the crystal reduce to

D e = O(6) , W e ~--- O ( 6 2 ) , (37)

Dp = _1 ~ ~(~)(~,(~) ~) n(~) + n(~) ~) u(~) ) + O(6) = RDPR r + O(6) , (38) 2 ct=l

Wp = 1 ~, ~(~) ( ~ ) @ n(~) _ n(~) ~) ~(~)) + O ( e ) = R W ~ R T + O ( 6 ) ' (39) 2~=1

D = D p + O(6) , W = O~ + W p "F 0 ( 6 2 ) , (40)

v ~ ) = Ru0 (~) + O ( 6 ) , n ~ ) = Rn(0 ~) + O ( 6 ) , (41)

and

k (~) = ~ r (~) + O(e) , n(~) = ~on (~) + O(6) . (42)

It is interesting to note that, in the case of a rigid plastic crystal (E = 0), the configurations ~ and 63 coincide, and all slip systems, active or not, rotate by the same amount, specified by the rigid body rotation R.

II.4. A geometric interpretation o f the plastic spin W p

Here we provide a direct geometric interpretation of the plastic spin W p in terms of the change of the angles formed between the slip direction and the coordinate axes, for the case of small elastic strains.

First, we study the two dimensional deformation of the crystal. All active slip systems (i, ~) , n (~)) lie on the x i - x 2 plane and we can write

W = W ( - e l Q e2 + e2 @ e l ) , (43)

o~ = o~(-el ® e2 + e2 ® el), (44)

and

W p = WP(-el ® e2 + e2 ® el), (45)


where W, w, and W p are scalars, while el and e2 are unit vectors along the coordinate axes. Using (42) we can easily show that

~0 = 0 (~*) + O ( e ) , (46)

where 0 (4) is the rate o f rotat ion o f the slip direction v ~") in the current conf igurat ion 63 (see Fig. 2). A geometric interpretat ion o f the spin tensor W is given in the Appen- dix, where we show that

W = (0 ) , (47)

(0 ) being the average spin o f all material filaments a round a material point in the current conf igura t ion 63. Therefore, eqn (40b) implies that

W p = (0 ) - 0 ~" + 0 ( ~ ) . (48)

A similar interpretation exists in three dimensions. The different spin tensors are now written as

W -- W3( -e I (~) e2 + e2 @ el) + W l ( - e 2 (~) e3 + e3 (~) e2)

-F W 2 ( - e 3 (~) el + el (~) e3), (49)

W p : W P ( - e l (~) e2 + e2 ~) el) + Wi°( -e2 (~) e3 + e3 Q e2)

+ W P ( - e 3 Q el + el (~) e3), (50)

and

= w3(--el @ e2 + e2 @ el) + wz(--e2 @ e3 + e3 @ e2)

+ w2(--e3 @ el + el @ e3). (51)

X 2

Ca) //

> x 1

Fig. 2. Orientation of a slip direction in the current configuration.


Using (42) we can easily show that

0)1 .---- /)(or) -4- 0 ( ( ~ ) , (.O 2 ----" /)(or) + 0 ( ( ~ ) , a n d ¢o3 = 03(") + O ( e ) , (52)

where/)~"),/)2 ("), and/)3 ("), are the rates of rotation of the slip direction p(") in the current configuration (B about the Xl-, x2-, and xa-axes respectively (see Fig. 3). As shown in the Appendix, the components of W can be written as

W1 = </)l), WE = </)2), and W3 = </)3), (53)

where (/)1), (/)2), and (/)3) are the average rates of rotation of the material filament dx in the current configuration (B about the Xa-, x2 o, and x3-axes, respectively. Therefore, using eqn (40b) we find

w i ° = <0~> -0~ ") + o ( ~ ) , (54)

and

w £ = <02> -02 (") + 0 ( ~ ) ,

W P -~ (/)3) -O(°e) -1- O ( E ) .

(55)

(56)

Equations (48) and (54)-(56) show that, to within elastic strains, the plastic spin W p is the average spin of the continuum relative to an observer who is instantaneously spin-

X 5

I

X 1

~,(a)

>X 2

Fig. 3. Orientation of a slip direction in the current configuration.


ning with the angular velocity of the slip directions, or, put in other words, W p is, to within elastic strains, the spin of the continuum relative to the slip directions in the current configuration 63. It should be noted that the O(e) terms in (48) and (54)-(56) arise from the elastic rate of distortion, the terms involving W e and D e in (25) and (26).

Similarly, using the results of Section 2.2 we can readily show that

W p = (0,) -0('~), (57)

and

~o = (~02) --02 (cO, (58)

W~3 p = < 03 ) -03 ~), (59)

where all barred quantities are defined in the intermediate configuration (B. Equa- tions (57)-(59) show that, independently of the magnitude of the elastic strains, W p is the spin of the continuum relative to the slip directions in the intermediate configuration (B.

11.5. An example: uniaxial tension o f a single crystal

In this section, we demonstrate how the geometric boundary conditions can be used to define the rigid body spin ~0, and, therefore, the corresponding rotation tensor R.

We consider an elastic-plastic crystal that is deformed in tension with multiple slip. During the test, the crystal tends to rotate relative to the loading axis. This rotation is prevented by the action of the specimen grips which force all material fibers that are vertical in the undeformed configuration to remain vertical after deformation (see Fig. 4). This geometric constraint requires that

[ W + D ( e 2 (~ e2) - (e2 (~) e 2 ) D ] e 2 = 0. (60)

e2

T, e~

7°7 I

/~o (a)

8 ( a )

Fig. 4. Uniaxial tension of a crystal in single slip.


This follows directly f rom the general relation e = [W + D(e ® e) - (e ® e)D]e, which holds for any unit material filament e, by setting e = e 2 and e2 --" 0. In the case o f small elastic strains, (60) can be written as

[W p + ~0 + DP(e2 @ e2) - (ez (~) e2)D p + O(e)]e2 = 0. (61)

The spin ~o is written as

to = o~3(--e I (~) e 2 + e2 (~) el) + COl(--e2 Q e3 + e3 Q e2), (62)

where the rigid body spin about the x2-axis has no effect on the calculations and is arbitrarily set to zero. Using eqn (61) we can readily show that

~3 = DI°2 + W:2 + O(e) = ~,, "~(~)(e, .pt~))(n (~) .e2) + 0 ( 6 ) , ~=1

(63)

and

col = -D~'3 + W~3 + O(6) = - ~ - ] ~(~)(e3-v(~))(n(=).e 2) + 0 ( 6 ) . ct=l

(64)

The last two equations define ~0 in terms o f the kinematic solution -~ (~) and they can be used together with (42) to determine the change in or ientat ion o f the slip systems.

We consider next the case in which all active slip systems (~, (~), n t~) ) a = 1 . . . . . m lie on the Xl-X2 plane. Equat ions (63) and (64) now reduce to

603 = ~] 'Y (cOcOS20(~) + 0 ( 6 ) , (65) ~=l

and

wl = O. (66)

Using (42) we can easily find that in the current conf igura t ion 63 the rate o f ro ta t ion o f the slip directions, active or not , is given by

0(e~) = ~3 -F 0 ( 6 ) = ~ "y(3)COS20 (3) -F 0 ( 6 ) , /3=1

(67)

where 0 t") is the angle between the slip direction and the xl-axis in the current configura t ion 63 as shown in Fig. 4. Integrat ion o f the above equat ion determines the final or ientat ion o f the slip directions.

In the case o f single slip on the 0,(~),n ° ) ) slip system, the solution can be obtained in closed fo rm as follows. Equa t ion (67) becomes

0 ° ) = "y(1)COS20 (I) "{- 0 ( 6 ) , (68)

which leads to the well-known fo rmula (REID [1973], p. 124)

0 (l) = arctan(3, °) + tan0o ~1)) + O ( e ) , (69)


where 0o ") is the orientation of the slip system in the undeformed configuration 6] 0. Taking into account that, to within elastic strains, all slip directions rotate by the same amount , we can readily show that the orientation of all inactive slip directions in the current configuration 6~ is defined by

0 ¢~) = Oo ~) - Oo <1) + arctan(7 (1) + tanOo (1)) + 0 ( ~ ) . (70)

III . Cont inuum plasticity

The kinematics of finite elastic plastic deformation is best described by the multiplicative decomposition

F = FeF p, (71)

formally introduced in continuum mechanics by LEE and LIu [1967] and LEE [1969]. The intermediate unstressed configuration (B1, that is the configuration of the continuum after removal of F e, is not uniquely defined, since an arbitrary rigid body rotation can be superposed to it and still leave the configuration unstressed. Motivated by the kinematics of the single crystal and in order to remove the aforementioned ambiguity we use, instead, the decomposition (MANDEL [1971], AIFANTIS [1987], DAFALIAS [1987])

F = RUeF p = VeRF p, (72)

where R is the rigid rotation of certain characteristic directions in the continuum with respect to a global system, that is, the rotation of the characteristic directions in the configuration (B defined by F -- RFC It should be emphasized, however, that V ~ (or U ~) causes, in general, an additional rotation of the characteristic directions during the trans- formation f rom (~ to the current configuration (~, unless the material is rigid plastic, in which case, V e = U e = | . The intermediate configuration 631, determined by F p, is now uniquely defined in such a way that the orientation of the characteristic directions in (gl with respect to a global system is the same as the corresponding orientation in the reference configuration 6~0 (MANDEL [ 1971,1973a, 1973b, 1974], LORET [ 1983], NAGTEGAAL & WERTHEIMER [1984], DA~ALL~S [1987,1988]). The intermediate configuration 631 defined in such a way is the so-called "isoclinic configuration" (MANDEL [1971,1973a,1973b, 1974]). DAFALIAS [1987,1988] presented recently a thorough discussion on the subject and showed that the choice of the intermediate configuration can be made arbitrarily, provided, of course, that the corresponding kinematical quantities, such as the plastic part of the deformation rate and the plastic spin, are properly defined.

The aforementioned "characteristic directions" can be associated with the direction of certain material fibers and follow the deformation of the continuum, or they can be associated with the direction of the normal to material planes, the principal directions of second order tensors such as tensorial internal variables, and so forth, in which case their t ransformation is not necessarily identical to that of the continuum. The identifi- cation of the orthogonal part R of F ~ with the rigid body rotation of the characteristic directions was first made by MANDEL [1971] for the choice of the isoclinic configuration, and also by DAFALIAS [1987] where it was symbolized by R~.

In the current configuration 6~, the elastic and plastic parts of the deformation rate D and the spin tensor W are defined by equations (3)-(6) which are also listed here for convenience


D = D e + D p, W = 60 + W e + W p,

D e = R ( ( J e u e - l ) s RT , W e = R ( u e u e - I ) a R T , oo = RR r,

D p = R ( U e F P F p-I U e - I )sR r, W p = R ( U e F P F p-I U e - I )aRr.

In the intermediate configurations (B1 and (B we also define

D~ = ( F P F p - I ) s , W p = ( F P F p - I ) a ,

and

I ) = D P = RDPR r, ~W : 0O "1- W p ~--- 00 + RWPR r.

In the case of small elastic strains the above equations reduce to

D e = O(e) , W e = O ( e 2 ) ,

D p = D p + O(6), W p = W p + O ( 6 ) ,

and

D = D p + O ( e ) , W = o0 + W p + O ( e 2 ) .

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

ZBm and ArFANTIS [1988a, 1988b] introduced recently the concept of the "relative spin"

W s / o defined by

WS/D : W -- W S , (81)

where Ws is an antisymmetric second-order tensor related to the rate of rotation of the eigenvectors of the stress tensor in the case of hypoelastic materials. Equation (80b) can be written as

W p : W - oJ + 0 ( 6 2 ) , (82)

which shows that, to within elastic strains square, the plastic spin is the relative spin gen- erated by letting Ws = o~ = RR r. Below, we give two particular examples where we specify the "characteristic directions" of the material.

We consider first the case in which one of the characteristic directions at a material point is associated with the direction of a material fiber at that point. Let No be the unit vector along the characteristic direction in the undeformed configuration (Bo. The requirement that F p does not rotate the characteristic directions means that No is an eigenvector of F p so that

1 l q = R N o , N - - - RUeNo (83)

IUeSol

and

[Wg + D~(No @ No) - (No @ No)D~]No = 0, (84)


which also leads to

[WP + DP(/~ @ N) - (N @ N)DP]N = 0, (85)

where gl is the unit vector along the characteristic direction in the intermediate configuration (B, and N is the unit vector along the characteristic direction in the current configuration 63. The rotation of the characteristic direction in the current configuration consists of a rigid body rotation R and a rotation ue/pUeN01 caused by the elastic distortion. Equation (85) shows that lgl is the axial vector of the antisymmetric tensor W p + DP(N ® 1~) - (lq ® N)D p. In the case of plane motion on a plane that includes 1~ (85) also implies that

W p = (1~ @ N)D p - DP(N @ i~), (86)

which is a constitutive equation for W p. In the case of small elastic strains, (86) reduces to

W p = (N @ N)D - D(N @ N) + O(c) . (87)

We consider next the case in which the characteristic directions are associated with the principal directions of a second-order tensor A in the intermediate configuration (B. For simplicity, we assume that the eigenvalues of A are real and distinct and that the corresponding eigenvectors are normal to each other. The corresponding eigenvectors e~4, i = 1,2,3 are defined by

e~ = RAe/, (88)

where RA is a unique proper orthogonal second-order tensor, and ei denotes, as usual, a cartesian basis. The rate of change of the eigenvectors at (B is given by

e~ = WAe~, where W A = R ARA r. (89)

A note of caution is relevant at this point. The rate of change of the material fibers that coincide instantaneously with the eigenvectors e~ at (B is given by

(e t4) f iber = WAeA--i i (no sum on i), (90)

where

W--~ = W + D(e~ ® e~) - (e~ ® e~)D (no sum on i). (91)

It should be emphasized, however, that the eigenvectors e.~ coincide with different material fibers at different times and, therefore, the spin of the eigenvectors is, in general, different f rom the spin of the corresponding material fibers, that is,

i . m

W~ ~ WA. (92)

The rigid body spin o~ can be identified with any of the W~ 's or-any combination of them. In such a case, the rotation R is defined from the integration of ~, = RR r and


does not admit a direct kinematical interpretation since ~ is, in general, the spin (or combination of spins) of different material fibers at different times. Alternatively, ~ can be identified with WA, in which case R = RA. In this connection, we mention that eqn (2.6b) in ZBIB and AIFANTIS [1988a], namely W = RoR r where Ro is the rotation tensor that defines the orientation of the eigenvectors of D, is valid only for the special case in which the principal directions of D are aligned with the same material fibers at all times so that the spin of the eigenvectors and the fibers coincide. (Nevertheless, it turns out that their results hold true independently of this restriction.)

In the following, we use an averaging argument employed by DAFALIAS [1985b] (see also ZmB & AWANTIS [1988a, 1988b]) to obtain an expression for ~ which also leads to a constitutive equation for the plastic spin W p. We assume that ~, which is a measure of the spin of the eigenvectors of A at (B, is determined by the average value of the spins Wi, i = 1,2,3 defined by

Wi = W + 3~'Ai[D(e~ @ eh) - (e~ @ e~)D] (no sum on i) (93)

where ~" is a proportionality factor and Ai, i = 1,2,3 are the eigenvalues of A at (B. Then by using the definition

1 ~ Wi ' (94) 6 0 = 3 i = 1

we readily derive

~o = W - ~-(AD - DA). (95)

Finally, using (77b) we find

W p = ~'(A[~ - DA), (96)

which in the case of small elastic strains reduces to

W p = ~'(AD - DA) + O(¢) . (97)

It is interesting to note that in the case of rigid plastic materials (~ = 0) where the second order tensor A is identified with the back stress or, eqn (97) yields

W p = ~ ' (o tO p - - D P o t ) , (98)

which is exactly the relation proposed on phenomenological grounds independently by DAFALIAS [ 1983,1985b] and LORET [ 1983] and microscopically substantiated on the basis of single slip and a scale invariance argument by DAFALIAS and AIFANTIS [1986,1990] (see also AIrANTIS [1984,1987]).

We conclude this section by mentioning that in an isotropic material there are no pre- ferred directions and one could argue that the spin ~ can be identified with the average continuum spin in the intermediate configuration (B, that is ~ = W. Equation (77b) then becomes

W p = Wo v = 0, (99)


and eqn (75b) reduces to

W p = R(U~DPU ~-1 ),R r. 0oo)

Isotropy requires that Do p be coaxial with S e = det (Fe)F loF ~-r and, therefore, with U", so that eqn (100) reduces to

W p = O, (101)

a relation that has been also derived by DAFALIAS [1985a] using representation theorems for isotropic functions. In the above expressions, o is the true or Cauchy stress tensor defined at the current configuration 6~, and S e is the second Piola-Kirchhoff stress tensor defined at 6~.

IV. CLOSURE

The geometric interpretation of co and W p in continuum plasticity is similar to that discussed in Section 2.4 for the case of the single crystal. Using the results of the previous section we can readily show that ~0 is the spin of the characteristic directions in the intermediate configuration (B, and that WP is the spin of the continuum relative to the characteristic directions at (B. In the case of small elastic strains, the configurations (B and 6:3 differ by an inf ini tes imal amoun t , and the geometr ic in te rpre ta t ion o f W p is, to within elastic s trains, the same as that of W p.

Acknowledgemenls-The support of the National Science Foundation through Grants MSM-8657860 (NA) and CES-8800459 (ECA) is gratefully acknowledged. The support of the Mechanics of Microstructures Pro- gram during the visit of N. Aravas to Michigan Tech. in September 1988 and fruitful discussions with Prof. Y. F. Dafalias of the University of California at Davis are gratefully acknowledged.

1938 1953

1960

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1967

1969 1971

1971

1972

1973a

1973b

1973 1974

1974

1977

REFERENCES

TAYLOR, G.I., "Plastic Strain in Metals," J. Inst. Metals, 62, 307. Novozmeov, V. V., Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, NY, (translated from the first Russian ed., 1948). TRUESDELI_, C. and TOUHN, R. A., "The Classical Field Theories," in FtOccs~, S. (ed.), Handbuch der Physik, Vol. l I I / l , Springer-Verlag. HILL, R., "Generalized Constitutive Relations for Incremental Deformation of Metal Crystals by MuL tislip," J. Mech. Phys. Solids, 14, 95. LEg, E. H. and LIu, D. T., "Finite-Strain Elastic-Plastic Theory with Application to Plane Wave Anal- ysis," J. Appl. Phys., 38, 19. LEE, E. H., "Elastic-Plastic Deformations at Finite Strains," J. Appl. Mech., 36, 1. MANDeL, J., "Plasticit~ Classique et Viscoplasticit6," Courses and Lectures, No. 97, International Center for Mechanical Sciences, Udine, Wien-New York, Springer. RICE, J. R., "Inelastic Constitutive Relations for Solids: An Internal-Variable Theory and Its Ap- plication to Metal Plasticity," J. Mech. Phys. Solids, 19, 433. HIRE, R. and RICE, J. R., "Constitutive Analysis of Elastic-Plastic Crystals at Arbitrary Strain," J. Mech. Phys. Solids, 20, 401. MANDEt, J., "l~quations Constitutives et Directeurs darts les Milieux Plastiques et Viscoplastiques," Int. J. Solids Struct., 9, 725. MANDEL, J., "Relations de Comportement des Milieux 151astiques-Plastiques el t~lastiques- Viscoplastiques. Notion de Repere Directeur," in SAwczu~:, A. (ed.), Foundations of Plasticity (War- saw, 1972), Noordhoff, Leyden. REJD, C. N., Deformation Geometry for Materials Scientists, Pergamon Press, Oxford. FARDSmSHEJL F. and OYA'r, E.T., "Representation of Elastoplastic Behavior by Means of State Vari- ables," in SAWCZUK, A. (ed.), Problems of plasticity (Warsaw, 1972), Noordhoff , Leyden. MANDEL, J., "Director Vectors and Constitutive Equations for Plastic and Visco-Plastic Media," in SAWCZUK, A. (ed.), Problems of Plasticity (Warsaw 1972), Noordhoff, Leyden. ASARO, R. J. and RICE, J. R., "Strain Localization in Ductile Single Crystals," J. Mech. Phys. Sol- ids, 25, 309.


1980 1981

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1983

1983

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1986

1987 1987

1988

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1988b

1990

ERINGEN, A. C., Mechanics of Continua, Krieger, Huntington, NY. MANDEL, J., "Sur la D6finition de la Vitesse de D6formation l~lastique et sa Relation avec la Vitesse de Contrainte," Int. J. Solids Struct., 17, 873. HAVNER, K. S., "The Theory of Finite Plastic Deformation of Crystalline Solids," In HOPKINS, H. G. and SEWELL, M. J. (eds.), Mechanics of Solids, Pergamon Press, New York, pp. 265-302. H~L, R. and HAVNER, K. S., "Perspectives in the Mechanics of Elastoplastic Crystals," J. Mech. Phys. Solids, 30, 5. PEIRCE, D., ASARO, R. J., and NEEDLEMAN, A., "An Analysis of Nonuniform and Localized Defor- mation in Ductile Single Crystals," Acta Metall., 30, 1087. ASARO, R. J., "Micromechanics of Crystals and Polycrystals," Advances in Applied mechanics, 23, 1-115. ASARO, R. J., "Crystal Plasticity," J. Appl. Mech., 50, 921. DAFALIAS, Y. F., "Corotational Rates for Kinematic Hardening at Large Plastic Deformations," J. Appl. Mech., 50, 561. LEE, E. H., MALLETT, R. L., and WERTHEIMER, T. B., "Stress Analysis for Anisotropic Hardening in Finite-Deformation Plasticity," J. Appl. Mech., 50, 554. LORET, B., "On the Effects of Plastic Rotation in the Finite Deformation of Anisotropic Elastoplastic Materials," Mech. Materials, 2, 287. PEIRCE, D., ASARO, R. J., and NEEDLEMAN, A., "Material Rate Dependence and Localized Defor- mation in Crystalline Solids," Acta Metall., 31, 1951. AIFANTIS, PL. C., "On the Microstructural Origin of Certain Inelastic Models," J. Engng. Mat. Tech., 106, 286. NAGTEGAAL, J. C. and WERTHEIMER, T. B., "Constitutive Equations for Anisotropic Large Strain Plas- ticity," in WILLAM, K. J. (ed.), Constitutive Equations: Macro and Computational Aspects, ASME, New York, pp. 73-86. DAFALIAS, Y. F., "The Plastic Spin," J. Appl. Mechanics, 52, 865. DAFALIAS, Y. F., "A Missing Link in the Macroscopic Constitutive Formulation of Large Plastic Defor- mations," in SAWCZUK, k. and BIANCHI, G. (eds.), Plasticity Today, Elsevier, Appl. Sci., U.K., pp. 135-151. DAFALIAS, Y. F. and AIFANTIS, E. C., "Oil the Origin of Plastic Rotations and Spin," Appendix to the article of AIFANTIS, E. C. (1986) "On the Structure of Single Slip and Its Implications to Inelastic- ity" in: (eds.), GITTUS, J. et al., Physical Basis and Modelling of Finite Deformations of Aggregates, Elsevier Appl. Sci. Publ., Amsterdam, p. 283. AIFANTIS, E. C., "The Physics of Plastic Deformation," Int. J. Plasticity, 3, 211. DAFALIAS, Y. F., "Issues on the Constitutive Formulation at Large Elastoplastic Deformations, Part l: Kinematics," Acta Mechanica, 69, 119. DAFALIAS, Y. F. "Issues on the Constitutive Formulation at Large Elastoplastic Deformations, Part 2: Kinetics," Acta Mechanica, 73, 121. Zall3, H. M. and AIFANTIS, E. C., "On the Concept of Relative Spin and lts Implications to Large Deformation Theories. Part 1: Hypoelasticity and Vertex-Type Plasticity," Acta Mechanica, 74, 15. ZBIB, H. M. AND AIFANTIS, E. C., "On the Concept of Relative Spin and Its Implications to Large Deformation Theories. Part ll: Anisotropic Hardening Plasticity," Acta Mechanica, 74, 35. DAFALIAS, Y. F. and AIFANTIS, E. C., "On the Microscopic Origin of the Plastic Spin," Acta Mechan- ica, 82, 31.

Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, PA 19104, USA

Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University Houghton, MI 49931, USA

( Receive~t 2 July 1989; in final revised form 1 March 1990)


APPENDIX

It is a well-known result in continuum mechanics that the spin tensor W is the spin of the material fibers that coincide instantaneously with the principal directions of the deformation rate D (TRUESDELt ~ TOUPIN [1960], p. 355). Here, we show that W is also the average spin of all directions around a material point. A two-dimensional version of this interpretation has been given by Cauchy (see TRUESDELL & TOUPIN [1960], p. 355; NOVOZHILOV [1953], p. 27; ERINGEN [1980], p. 25), and has been also noticed by LEE et al. [1983] in the context of simple shear. For completeness, simple proofs for both the two- and the three-dimensional interpretations of W are given in the following and they are used in the geometric interpretation of the plastic spin W p.

A1. A geometric interpretation of the spin tensor in two dimensions

We consider the 2-dimensional motion of a continuum on the x~-x2 plane. The deformation rate and the spin tensors are given by

and

D = Dllel @ el + D22e2 @ e2 + D12(el @ e2 + e2 ® el),

W = W ( - e l @ e2 + e2 @ el),

(A.1)

(A.2)

where el and e2 are unit vectors along the coordinate axes.

Let

m = cos0el + sin0e2 (A.3)

be a unit vector attached to a material fiber of the continuum (see Fig. A1). The rate of change of m is given by

m = 0 ( - s in0e l + cos0e2), (A.4)

and

X 2

>X I

Fig. AI. Orientation of a unit vector in the current configuration.


m = [ W + D ( m ® m ) - ( m ® m ) D ] m .

Combining the above two equations we find that

0 = W + ½ (D22 - Dl l )s in20 + D12cos20.

We define the local mean rate of rotation (0) as

1 fo 2~ <0> = ~ 0d0.

Using eqn (A.6) we find that

<0) = w,

which shows that W is the average spin at a material point.

A2. A geometric interpretation o f the spin tensor in three dimensions

The deformation rate and spin tensors are now given by

D = D l l e l (~) ej + Dz2e2 @ e2 + D33e3 @ e3 + D lz ( e l @ e2 + e2 @ el)

4- D23(e 2 @ e3 + e3 @ e2) + D31(e3 @ el + el @ e3),

and

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

Using eqn (A.12) we find that

<03) = W3. (A.14)

W = W 3 ( - e I @ e 2 + e 2 @ ej) + W l ( - e 2 @ e3+e3 @ e2)

+ W2( -e3 @ el + el @ e3), (A.10)

where el, e, , and e3 are unit vectors along the coordinate axes. Let

m = c o s 03 COS 03ei + sin 03 cos 03e2 + sin 03e3 (A. 1 1)

be a unit vector attached to a material fiber of the continuum (see Fig. A2). The rate of change of 03 is easily found to be

03 = W3 + t an03( -W2s in03 + WI cos03) + ~ (022 -- Dij)sin203 + D12COS203

+ tan03( -D13 sin03 + 023 cos03). (A.12)

We define the local mean rate of rotation about the x3-axis by

- - 03 dO3 dO. (A. 13) (03) = 2 r 2 ,,'-7r/2


x3

:>,..

x I

: > x 2

Fig. A2. Orientation of a unit vector in the current configuration.

Similar ly , if we consider the ro ta t ion rates abou t the x~- and x2-axes, we f ind

<Ol> = Wj,

and

(A.15)

<62> = W2. (A.16)

Equa t ions (A .14) - (A .16) show that the spin tensor W is the average spin o f all direc- t ions a r o u n d a mate r ia l point .

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