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A dislocation theory of plasticity
Iat. J. E&rag
Sci,
1973, Vol. I I, pp. 1065- 1078.
Pergamon
A DISLOCATION
Press.
Printed
in Great
THEORY
H. T. HAHN* Air Force Materials Laboratory, Want-Patterson
Britain
OF PLASTICITY-l Air Force Base, Ohio, U.S.A
and
Department
W. JAUNZEMIS of Engineering Mechanics, The Pennsylvania State University, University Park, Pennsylvania, U.S.A.
Abstract-
Based on the.observation that plastic behavior is a macroscopic manifestation of the motion and interaction of dislocations, a plasticity theory is proposed that represents a connecting link between solid state physics and the conventional plasticity and viscopIasti~ity theories. The d~sio~tion picture is described-for a slip system a- by means of A; (immobile dislocation densities), ~4; (mobile dislocation densities), and the dislocation speeds Va. Suitable constitutive equations are introduced for V” and the timerate of change of A:, A’$ A two-dimensional extension of a rigid-viscoplastic body is analyzed to illustrate the nature of the proposed theory. 1. INTRODUCTION
IT HAS long been known that the basic mechanism of plastic flow is the motion of dislocations and that the a~angement of dislocations is characteristic of the thermomechanical state of crystalline sofids. Since the discovery of dislocations there have been continued attempts to reduce plastic properties to models of dislocation behavior. Some of this work, in particular what is called the continuum theory of dislocations, is really no more than a variant of elasticity theory. That is to say, if dislocations and their movement are considered as given, they can be incorporated into elasticity in the same way as distributed body forces. In a complete theory, the density and movement of dislocations should be derivable from a knowledge of initial conditions onfy. The progress towards a true dislocation theory of plasticity has been slow. In 1965 Mura[l, 2]$ gave a derivation of von Mises yield criterion and Prandtl-Reuss equations in terms of dislocation forces and a dislocation movement tensor. In 1966 an alternative approach to a continuum theory of dislocations in crystals was proposed by Fox[3]. This work was extended by Lardner[4,5], who accounted for the disIocation production neglected by Fox. Introducing a tensor which is an integral over time of Mura’s dislocation movement tensor, and also using dislocation forces, Eisenberg[6] interpreted the classical thermopl~ticity in terms of dislocation mechanics. Parallel to the development of continuum theories of dislocations much experimental evidence has been gathered particularly by Gilman and his coworkers[7], concerning the average behavior of dislocations and its relation to plastic properties. On the basis of this behavior, various constitutive equations have been proposed. These relat Presented at the 9th Annual Meeting of the Society of Engineering Science ( 197 1). SNRC Postdoctoral Associate: formerly Graduate Student, Department of Engineering The Pennsylvania State University. §Numbers in square brackets refer to References at end of paper. 1065
Mechanics,
H. T. HAHN
1066
and W. JAUNZEMIS
tions can, on the whole, explain satisfactorily inelastic phenomena like transient creep, propagation of plastic wave fronts etc. Summarizing, it seems reasonable to expect that any practicable dislocation theory of plasticity will contain the following special features: (1) a kinematical relationship between plastic flow and dislocation motion: (2) a selection of suitable dislocation densities (together with constitutive equations for the change of these densities); (3) constitutive equations for dislocation velocities; (4) a correlation of average dislocation properties with macroscopic thermomechanical properties (e.g. yield stress) [8]. In what follows we shall explore these features in some detail. as well as the theoretical and practical questions associated with them. 2. PLASTIC
FLOW
The gross shape deformation
AND
DlSLOCATlON
MOTION
of a body is described by x=x(X.t),
(1)
where x is the position at time t of that particle which in some reference configuration is located at X. The mapping x is assumed to possess a high degree of smoothness (continuity and differentiability). Let dX be a typical vector of an infinitesimal neighborhood N(X) of a particle X. Then, because of the differentiability of the mapping x, the entire neighborhood is mapped by a single second-rank tensor into its deformed shape: dx = F dX, dxi = Fu d Xj, (2) where Fij=Xi,j, J=detF
> 0,
(3)
is the tensor of deformation gradients. As a rule, a subscript ‘i’ indicates a derivative with respect to xi when preceded by a comma (,), and with respect to Xi when preceded by a period (.). Material time-rates will be denoted by a superposed dot. In elastoplastic behavior there exist, on a microscopic level, two distinct modes of deformation-a recoverable, thermoelastic deformation, and a permanent, plastic deformation caused, in effect, by relative slipping of adjacent layers. The classical approach to plasticity has been to decompose F into a (local) thermoelastic deformation @and a (local) plastic deformation P, thus [93 F=~~,g~det~>O,J~detF>O.
(4)
This is represented schematically in Fig. 1. The intermediate (local) configuration is called the plastic configuration hereafter; it may be obtained by a stress-temperature relaxation process. However, such distinctions must eventually be based on constitutive equations. Whereas F is always compatible, the same need not be true of @or F in general. *Here incompatibility means non-integrability of the differential forms associated with F and p (i.e. that F and F are not gradient fields). A plausible measure of incompatibility of a deformation @ is given by the dislocation density per unit area in the current configuration. (Yu= where
limk is
the alternator.
EimkFjjnEih,kt
(3
Because the density (Yis affected by elastic deformation,
1067
A dislocation theory of plasticity IntermedIate
Initial
Fig. 1. Decomposition
(local)
conflqurotion
Current
configuration
configuration
of deformation F into elastic deformation @and plastic deformation F.
one may, in order to avoid this, define a dislocation density tensor A per unit area in the plastic configuration: ..*
Ati = JFiilFj;* 1ak,,,. The time-rate of change of the gross shape is expressed gradient tensor L, L GEL+Jti=uij,
(6) by means of the velocity (7)
where v denotes partide velocity. Using the decomposition L can be decomposed into elastic and plastic parts: L = i+PLP-1,
(4), it is easily seen that (8)
where L and L are defined similarly to (7), i.e. 2, S
#p-1,
L
s
&l*
(9)
Turning now to the dislocational aspect of plastic flow, we first note that a group of dislocations in a sufficiently small neighborhood of a material particle is characterized by three variables: a common line direction k, a common direction ii~of Burgers vector, and a scalar density k. The vectors E and EI are all referred to the plastic configuration. Moreover, they are chosen to be of unit magnitude so that the density k is equivalent to the number of dislocations (per unit volume in the plastic configuration) of unit length and of unit Burgers vector. Thus, if cwdenotes a scalar density referred to the current configuration, it is related to 2 by OL= ]-‘A. When there are various groups of dislocations, specified by ka, iiib, a, b = 1,2, . . . , N, the corresponding scalar densities are denoted by AQb.The vectors P, iii” are like normalized vectors of a reference lattice: they show the possible directions in which dislocations can exist. For a single crystal the number of possible directions $“, EI~ is finite, but is in general infinite for a polycrystal. Moreover, because it is known that not all dislocations move when subjected to stress, we should distinguish mobile
1068
H. T. HAHN
and W. JAUNZEMIS
dislocations from immobile ones, thus jab
=
jab 1
+
Aub n-I f
(10)
where the subscripts Z and M denote the immobile and mobile parts, respectively. To relate dislocation motion to plastic flow we take a generic material element d(y) in the plastic configuration, which deforms plastically when dislocations pass through it. The net number of dislocations of density &?ij~, that cross the line d(y) per unit time is given by (%x V) - d(y)&, where v is the average velocity of the group &. Because each dislocation produces a relative displacement equal to its Burgers vector when it passes by, the amount of slip in d (y ) caused by the passage of all dislocations will be d(y) =fi[(iixV)
.d(y)]&,=Ed(y).
where the second equality follows from the macroscopic Therefore, from the above equation we have E = iii @ (E X
description z of plastic flow.
V)A,+f+ .il+j= iliiEjkJkVnJ&f*
(11)
where @ denotes the tensor product. If dislocation groups of various directions are present, the above equation becomes L=x
&@
(i;axpb)@$‘,
(12)
a,b
where ‘Vabdenotes the average velocity of the group A,. -ab This formula is the most important link between plastic flow and dislocation motion. [ 1,3]. The vectors Ea and Vab define a plane-with a unit normal vector i? x pb/Vnb - of dislocation motion of the group xgb. The quantity i? nb= p x pbl is the speed normal to Ea of the group Agb, because P is of unit magnitude. Such planes provide another alternative to the labelling of dislocation groups. Specifically, the summation in (12) can be rearranged as follows,
where the summation inside the bracket is taken over all dislocations that have Burgers vector in the iii” direction and are moving in a plane defined by @. Clearly, the density A$ of those dislocations with EF and I* is (13) Using the average speed Fb of the group Ahb, defined by ACbVCb
=
5
,+$hb,
a
the equation for E can finally be written as H,= 5: N*A$P, a
(14)
1069
A dislocation theory of plasticity
where each tensor N” is of the form
for some ~3 and P. At this point some comments regarding plastic incompressibility are appropriate. The time-rate of plastic volume change is given by tr L, and so plastic incompressibility results either from the motion of screw dislocations, for which
or from the glide motion of edge dislocations, characterized
by
or from a combination of both. The above equation implies that all three vectors K, 8, and IS lie in the same plane and hence the plane of motion coincides with the slip plane. A climb motion of edge dislocations is rare, except at high temperatures, and will be neglected in the following discussion. For an edge dislocation, this assumption defines a unique plane of motion except for its orientation because the slip plane is spanned by the vectors K and i% On the other hand, the slip plane of a pure screw dislocation is not predetermined; rather, it will in general depend on the applied stress. However, a real dislocation line is hardly straight, and a curved dislocation line can be thought of as composed of short straight pieces, some of which have edge components, and so any screw component will also have as its slip plane the plane in which the curved dislocation lies. Further, crystal structure permits screw dislocations to move more easily on certain planes. Therefore, it is assumed that each dislocation group A; belongs to a unique slip system Na. When this is the case, a term A&P represents the time-rate of shear caused by the moving dislocations A& in the slip system Na. This term is a basic variable in the slip theory of plasticity. (cf. [lo]) Thus, according to (15) a knowledge of the detailed arrangement of dislocations is not required-only the densities of dislocations according to their Burgers vector and slip plane must be specified.
3. PHYSICAL
PRINCIPLES
For later use the physical principles of classical thermomechanics will be summarized below. Conservation of mass requires that the mass density p = p (x, t) satisfy P+pdivv=O. In the absence of microstresses the form of Cauchy’s laws of motion
the balance of linear momentum divT+pb
= pv,
(17) is expressed
in (18)
where the symmetric tensor T = T(x, 1) is the Cauchy stress tensor, and b = b(x, t) denotes the body force density per unit mass. To state the first and second laws of thermodynamics the following additional variables must be introduced; the Helmholtz free energy density per unit mass +!J= $(x, t), a volume distribution of sources of heating per unit mass r = r (x, t) , a field of heat flux vectors q = q(x, t) , entropy density per unit mass r) = 71(x, I), and absolute
IJESVol.
1 I.No.
10-B
1070
H. T. HAHN and W. JAUNZEMIS
temperature
0 = 0(x. t). The balance of energy is then written as ~~=-divq+~+rr(TL)-~~e-~~~,
whereas the principle of non-negative Duhem inequality, becomes
(19)
entropy production,
referred to as the Clausius-
per=-P(~+se)-e-lq.g+tr(TL)
3 0,
m
where g s grad 8,
(21)
and y = y(x, t) is called the rate of entropy production density per unit mass. 4. CONSTITUTIVE
EQUATIONS
AND THERMODYNAMIC
RESTRICTIONS
With the introduction of kinematical variables suitable for describing the arrangement and motion of dislocations, the stage is now set for stating constitutive equations which, together with the physical principles, will yield a deterministic theory of thermoplastic behavior. The thermomechanical properties of crystalline solids depend, as a rule, upon the number, arrangement, and properties of defects (dislocations, gram boundaries, vacancies, interstitial atoms, and so on). Whereas the elastic properties of a solid are determined primarily by atomic forces, and crystal defects only introduce small deviations from the ideal behavior of a perfect crystal, the plastic properties largely depend upon the behavior of defects, especially dislocations. This is so because plastic deformation is the macroscopic manifestation of dislocation motion. The ease with which dislocations can move depends on the interaction of dislocations themselves with one another as well as on their interaction with other defects. This phenomenon results in strain-hardening of material when viewed on a larger scale. To describe mathematically those features of dislocation behavior which are significant in the macroscopic picture of plasticity, one needs first of all a set of suitable variables representing the number and a~angement of dislocations. These variables are here assumed to be provided by the dislocation densities A? and&, defined in section 2. Consequently, the list of variables for characterizing the state of an elastoplastic material will be:
Because an elastoplastic material behaves like an elastic material for deformations from a fixed plastic configuration, it is plausible to describe the thermoelastic behavior by the constitutive equations (22) where This set (22) is to be complemented by additional variables A;, A%, and V”. Production of dislocations during plastic flow include spontaneous nucleation of new dislocation pro&Jction by Frank-Read sources, and breeding is the most effective of them all).]71 The mobile
constitutive
equations for the plastic
involves various mechanisms. They loops under high stress, quasistatic through multipie cross glide (which fraction of total dislocations usually
1071
A dislocation theory of plasticity
decreases as plastic flow proceeds. The immobilization of mobile dislocations is primarily caused by interaction with other immobile dislocations. The average effects of these mechanisms under ordinary conditions are described, in a fairly adequate way, by (23) ky = C Bg*ALV*, k& = x B$‘ALV* b
b
where BF* = By*@, 8, A;, A;),
B$ = B$‘@, 8, A;, A;)
(24)
are ‘breeding coefficients’ [4]. The above equations also have the implication that dislocation densities change whenever dislocations move. Evidently the influence of temperature gradient on dislocation production has been neglected. To gain some idea about constitutive equations for Vu, (22), is substituted into (20) to yield (cf. [ 111, [ 121) p&y=tr[(T-pc3
~~‘)i]-~(a,~+s)e-pa,9.g-8-‘q.g+pecr
2 0,
(25)
where u is defined by o = $[P(TL)
-p
c (a,;$A;+aA:$A$)], a
T = p-l’&.
(26) (27)
Here tr (‘i;L) represents the plastic working by the stress T and the term z (aA,~A;+a,4,~& D denotes the time-rate of energy storage associated with dislocations. Experimental results show that this latent energy ranges from 0 to 10% of the plastic work done. Further, substituting (15) and (23) into (26) yields where the term
pt9u = x (Pa
T;)A;P,
(28)
Ta = tr(TNa)
(29)
is the resolved shear stress, caused by the Cauchy stress T, in the slip system Na. The term T;f = ; (Bf%,: I&+ B$,,
I$)
(30)
can be thought of as a latent stress serving to increase the free energy associated with dislocations; it will be the stress in the slip system induced by the presence of dislocations. Thus ( Ta - T;f ) is the net force acting on the dislocations of the slip system Na. Dislocations start moving when the resolved shear stress Ta exceeds the sum of T$ and a critical resolved shear stress T;. Moreover, they accelerate to steady state motion in a time very short compared with ordinary loading times, so that inertia effects can be neglected. Therefore, dislocations will move under the conditions Ta= T;:+T;+Ta,,
(31)
where T”,denotes a viscous stress resisting dislocation motion. The critical shear stress T; usually increases with plastic deformation, i.e. materials strain-harden. Note that strainhardening is also described in part by the latent stress
1072
H. T. HAHN
Tj) Because the effect of hydrostatic that
and W. JAUNZEMIS
pressure
T;I = T;(e,
For the viscous stress T; the constitutive
on Tg is negligible, it will be assumed
A:, A$).
(32)
equation is taken to be
T; = T;(C), A;, Aa,, IQ),
(33)
where _ais used to indicate that T; depends only on V” but not on P. b # U. When the direction of loading is reversed after having produced plastic deformation, (T;f + TE) usually decreases, i.e. there exists a Bauschinger effect. Because dislocation will move in the opposite direction upon reversal of loading direction, the original direction N of dislocation motion will change to -N. Thus the Bauschinger effect manifests itself in the form T;t + T; 2 T;-“’ + ,h?. Mechanisms for Bauschinger effect are not well understood. One possible mechanism may be the increase of mobile dislocations after the direction of loading is reversed. For instance, in the presence of an obstacle dislocations move under the applied stress until they become queued up at the obstacle, resulting in immobilization of mobile dislocations. If the direction of loading is changed, they will move again in the opposite direction. The ease with which dislocations can move will appear as a decrease of the stress (T;t + T;). It seems in general that equal hardening in both directions will require a special distribution of dislocations, which is unlikely to be produced by a unidirectional loading. When the viscous stress T; is negligible compared with other terms, which may be the case for quasistatic loading in certain materials, it is omitted in (3 1). The resulting theory .is then a rate-independent plasticity, whereas retaining T; leads to a ratedependent plasticity (viscoplasticity). (a) Rate-independent plasticity. For a fixed (A:, A;) the equation (3 l), T” = tr(TNa)
= Tff(e,T)
+ T;(e),
(34)
represents a hypersurface in a ten-dimensional stress-temperature space. It has been assumed here that T can replace i? in the domain of Tff. Because dislocations do not move when T” < T;E+ T& the region E(A;, A&) bounded by all hypersurfaces-each of which is described by an equation of the form (34)-is called the elastic range in a stress-temperature space corresponding to (A,“, A$). The boundary &?Z(Af, A$,) of the elastic range E(Ay, A$) is called the yield surface. When T;; is assumed independent of T,* the hypersurfaces at a fixed 8 become hyperplanes each of which is normal to N”. Moreover, if T;f can be written as T;t = tr (TEN”)
for some tensor T,, then (34) becomes tr[ (T--T,)Na]
The above equation is the generalized
= TE.
form of the kinematic hardening law of Prager
*This will be the case, e.g. when $ = Jr1(fi, 0) + &,(A;, A&), and By”, B$ are independent
ofk.
1073
A dislocation theory of plasticity
[ 131. In particular, if TR = @ for a scalar 5, and if T; = Tc, then the above equation reduces to an isopropic hardening law TC tr (TNa) = -.
A slip system Na is said to become potentially active when the resolved shear stress reaches a critical value, tr(TN”) = Tg+ Tl. (34) Further, tiont
dislocations
on a potentially
active slip system N” move subject to the condi-
tr(TNa) = Ti+
Tg
(35)
whenever tr[(Na’-
aTT;)?j
- a,( T;+
T;)d
> 0 (loading),
(36)
whereas they do not move either when tr [ (N”’ - aTT;f)?]
- a,( T;f + Tg)b = 0 (neutral loading),
(37)
or when tr [ (N”T - +T;t)@]
- a,( T;f + T,Z)b < 0 (unloading).
(38)
With the aid of (23), (35) can be written as tr [(N“’ - afTi)@]
- a,( Tg + TS)e = ): habALl/*, b
(39)
where ha* are defined by hab=C
[aA;(T;f+T~)BF*+a,;(T~+Tt4)B~].
(40)
Whereas c in (40) rang& over all slip systems, the summation in (39) is restricted to the activated systems only. Thus (39) provide the necessary number of equations forthe unknowns I/*. Therefore, if the matrix [ha*] is non-singular, (39) can be solved for AkV* as follows, A&I/*=x
(h-1)b”{tr[(N”T-dgT~)?T]-aag(T~+T~)8},
(41)
where ( h-l)ba denotes thl ba-component of the inverse of the matrix [ha*] .$ Consequently, the equation for L becomes E = z Nb(h-1)ba{tr[(N”‘-aaT;T;)f”]-&,(T;+T;)8}, *,a
(42)
where, again, the summation is taken over the activated systems only. Summarizing, the constitutive equations for plastic variables in the rate-independent plasticity are: (23), (24), (30), (32), and (42). (b) Rate-dependent plasticity. When the viscous stress T”, is of the same order of magnitude as the other stresses, the speed I/” is determined by solving (3 1): V” = V”( T” - Tff - T& 8, A;, A;) =OwhenT”-TT;f-Tg
when T"- Tff - T; > 0,
s 0.
tNote that fin = 0. SA discussion concerning special cases of [@I may be found in [14].
(43)
1074
H. T. HAHN and W. JAUNZEMIS
A restriction that should be imposed on I/” is that they be positive. monotonically increasing functions of (T”-- Ti- T”,), yet approaching certain limit values[7]. If the functions Vu can be replaced by a single function I/, then L becomes L = x Na&J’/( T” - Tj - Tg. 0, A:, A&).
(44)
Any admissible thermondynamic process is required by the postulate of nonnegative entropy production to obey the Clausius-Duhem inequality (20) at each particle X and time 1. As a consequence, the second law will impose some restrictions on the response functions formulated above. Specifically, following Coleman and Curtin [ 1.51,we can easily deduce (cf. [ 161) a,+ = 0,
(45)
T = ,@I/&‘,
(46)
7 =-a,*.
(47)
pey = p&r-- e-‘q * g 2 0,
(48)
and the general dissipation inequality,
where o is now called the internal dissipation. Because A;P 2 0 and Ta - T; 3 0, the internal dissipation u is non-negative. This is in agreement with the restriction on (T, Cr*0 imposed by (48) when (+ is independent
(49)
of g.
5. AN EXAMPLE
To illustrate how the theory developed in the preceding sections is applied to specific problems, we consider a simple extension of a rigid-viscoplastic body under isothermal conditions. A two-dimensional simple extension is described kinematically by x=X,y=(Y(f)U(t)Y,z=
U(t)Z,
(50)
where U(t) is the stretch in the z direction, and the undetermined coefficient a(t) describes a contraction in the y direction. The corresponding tensor F of deformation gradients is F=diag (l,(~u, U). (51) The body is assumed to be rigid-viscoplastic, and to have a single slip system which is characterized by the vectors n-r and fi as shown in Fig. 2. In the initial configuration they are given by rir = cos &ey + sin &e,, (52) ii = - sin &e, + cos &ez. The rigid-viscoplastic tion ti
behavior requires that the elastic deformation P=ff,j=
6 is a pure rota-
1,
and so the vectors m and n referred to the current configuration by
are related to M and ii
1075
A dislocation theory of plasticity
htial conflgurotlon
-e,
currentconflgurotlon
Fig. 2. Simple extension.
Moreover, deformation the x axis,
is taking place in the yz plane, so that fi will be a rotation about
k=
0 1 0 cos ($I-&) 0
Therefore,
sin(4-44
0 -sin(+--&) cos (4-4J
1.
(54)
m and n are given by m = cos 4e, + sin +e,, n = - sin #2e, + cos f$e,.
(55)
If the mobile dislocation density and the average speed are denoted by A,,, and V, respectively, the plastic deformation rate L becomes (cf. (15))
L = iii @i-i/l~~V. The plastic deformation
(56)
tensor F is then
F=e,~e,+~~P+1~fiI:,AVdr+a~a.
(57)
The elastic rotation k is not arbitrary because the condition (58) should be satisfied. Substituting (57) into (58) yields (59)
H. T. HAHN
1076
With the aid of (5 1), the components
and W. JAUNZEMIS
of (59) can be written as
F,, = a U = ~0s 4 cos 40 - cos 4 sin & ,: AwV dt + sin $ sin c$,,, F,,=
(60)
U=sin$sin&+sin+cos+,,~~&Vdt+cos+cos&,,
F,, = 0 = cos 4 sin CpO + cos $J cos do 1: A#
dt - sin 4 cos I&.
Eliminating cos c#I,,,: AMVdt from (60), and (60),, we find that A,Vdt=---
sin (p sin (PO cos Cp cos $0’
or, in terms of U, A#dt=’ Therefore,
U~-cos2Cpo -- sin+, cos 40’ cm 40
(61)
we have AuV=
For the dislocation production, are taken,
Uti cos cpovw - co9 $Jo. the following equations -suggested
(62) by Gilman[7] -
A = B~AMV $&)
(63)
= - B~B~(+)AMV,
where A is the total dislocation density, and B, and B2 are the breeding coefficients. The above equations are based on the following observations. Because new sources of dislocations are created mainly by multiple cross glide, the rate of change of A is proportional to A,,, and increases with the speed V. The rate of decrease of the mobile fraction AM/A is proportional to the rate of increase in A because this leads to more interactions, and to instantaneous mobile fraction itself because the fraction changes whenever dislocations move. Equations (63) can be integrated to yield
(64)
where A0 and AM0are the initial densities of the total and mobile dislocations, tively. When the applied stress is a simple tension,
respec-
T = Pe, 03 e, the resolved shear stress T becomes T = TUAi@ = Tunimj = P sin $J cos r$.
(65)
A dislocation theory of plasticity Assuming
1077
a linear relationship between dislocation speed and viscous stress Tv = kV, k = const.,
and further that TR = 0, Tc = const.,
the equation for V becomes T = Tc+kV.
Therefore,
(66)
the tensile stress P necessary to produce the prescribed deformation p=
Tc+kV
sin (P cos 4
(Tc+kV)U2
=
co!3 cp&J2
- cos2 &’
is given
(67)
where
v=
uu cm $du2
1
- co?dozy Xfu2 - cos” C#Jo ax
$0
Finally, the coefficient cxis easily seen to be 01=-.
1 U2
(68)
6. SUMMARY that plastic behavior
of crystalline solids is the macroscopic manifestation of the motion and interaction of dislocations, we have proposed a plasticity theory that represents a connecting link between solid state physics and the conventional plasticity and viscoplasticity theories. Macroscopic plastic deformation is represented by a second-rank tensor which is not necessarily a gradient field, whereas the dislocation picture is described by use of immobile and mobile dislocatioa densities together with average velocities. A kinematic relationship between plastic flow and dislocation motion is derived without the usual assumption of infinitesimal deformations. No climb motion of edge dislocations is allowed, however, resulting in the plastic incompressibility. Appropriate constitutive equations for the changes of those densities and velocities complete the formulation of the proposed theory. The theory, as expected, is much like the slip theory of plasticity. However, it is different in that plastic phenomena are explained in terms of more fundamental variables. Based
on the observation
Acknowledgemenrs-This AFOSR-1626-69.
work was supported by the Air Force Office of Scientific Research-Grant
REFERENCES [ll T. MURA, Phys. Smus. Solidi. 10,447 (1965). 121 T. MURA, Phys. Status. Solidi. 11,683 (1965). [31 N. FOX, J. Inst. Math. Applic. 2,285 (1966). [41 R. W. LARDNER, Z. angew. Math. Phys. u), 5 14 (1969). [5] R. W. LARDNER, Znt. J. Engng Sci. 7,4 17 (1969). [6] M. A. EISENBERG, Znt. J. Engng Sci. 8,26 1 (1970). [71 J. I. GILMAN, Micromechanics ofFlow in Solids. McGraw-Hill, (1969). [8] E. KROENER, J. Math. Phys. 42,27 (1963). [91 E. H. LEE, J. uppi. Mech. 36, l(l969).
1078
H. T. HAHN
and W. JAUNZEMIS
[IO]
T.H.LINandM.ITO,Inr.J.EngngSci.4,543(1966).
1111 1121 1131 [14]
J. KRATOCHVIL and 0. W. DILLON, Jr.,J. uppl. Phys. 40.3207 (1969). J. KRATOCHVIL and 0. W. DILLON. Jr.,J. appl. Phys. 41, 1470 (1970). W. PRAGER. Proc. Znstn mech. Engrs 169,41(1955). R. H1LL.J. mech. Phys. Solids 14.95 (1966).
[I51 B.
[ 161 A.
D. COLEMAN and M. E. GURTIN,J. E. GREEN and P. M. NAGHDI. Arch.
them. Phys. 47.597 (1967). ration. Mech. Analysis l&25
(Receiued Resume-Baste
mouvement
sur I’observation
13
1 (1965).
October 197 1)
que le comportement plastique est une manifestation une ttiorie de la plasticite est proposee
et de I’interaction de dislocations,
macroscopique qui represente
du un
maillon reliant la physique de l’etat solide avec les theories conventionnelles de la plasticite et de la viscoplasticite. L’image de la dislocation est d&rite-pour un glissement don& u-au moyen de A,a (densite de dislocation immobile)cl,” (densite de dislocation mobile) et la vitesse de dislocation Va. Des equations constitutives convenables sont introduites pour V” et la vitesse de variation dans le temps de A,” et A,,“. Un allongement bi-dimensionnel d’un corps viscoplastique rigide est analyse pour montrer la nature de la theorie proposee. Zussuumenfassung-
Begriindet auf der Beobachtung, dass plastisches Verhalten eine makroskopische Manifestierung der Bewegung und Wechselwirkung von Verlagerungen ist, wird eine Plastizitltstheorie vorgeschlagen. die ein Verbindungsglied zwischen Festkorperphysik und den konventionellen Theorien der Plastizit;it und Viskoplastizitat darstellt. Das Verlagerungsbild wird fur ein Gleitsystem u durch A,“(unbewegliche Verlagerungsdichten). A,v”(bewegliche Verlagerungsdichten) und die Verlagerungsgeschwindigkeiten Vu beschrieben. Geeignete Konstitutivgleichungen werden fiir V”-und die Zeitrate des Wechsels von A,“, A,,,a eingeftihrt. Eine zweidimensionale Ausdehnung eines Starr-viskoplastischen Korpers wird analysiert. urn die Natur der vorgeschlagenen Theorie zu erlautem.
In ase all’osservazione the il comportamento plastic0 e una manifestazione macroscopica del modo e dell’azione reciproca degli spostamenti. si propone una teoria di plasticita the rappresenta una maglia di collegamento fra la fisica di stato solid0 e le teorie di plasticit& convenzionale e viscoplasticita. Si descrive il quadro di spostamento nei riguardi di un sistema di slittamento a mediante A,” (densita di spostamento immobile), A,P(densita di spostamento mobile), e le velocita de spostamento V”. Si introducono opportune equazioni costitutive per Va e il ritmo di tempr di cambiamento de A, a, A,,,“. Si analizza un’estensione bidimensionale di un corpo rigido-viscoplastico per illustrare la natura della teoria proposta.
Sommario-
kiCXOn5i ki3 npenCTaB.BeHHB, ‘IT0 nnaCTHYeCKOe nOBeneHHe RBJIReTCIl MaKpOCKOnHYeCKHM IIOaBneHHeM nBHHteHHa B B3aHMOneuCTBuB nUCBOKaHHfi, npennOBOxeHa TeOpAR nnaCTHYHOCTH, KOTOpaa npeBCTaB.YBeT co6oP CBB3b MemAy @3HKOZi TBepBbIX TeJI ki 06bIYHbIMA TeOpHffMH IInaCTHYHOCTH H BR3KOnnaCTHYHOCTH. ,$IB CHCTeMbI CKOJIbXCeHHR(a) OnuCaHa KapTuHa AHCJtOKauHfi, HCnOJIb3yR A,” (TIBOTHOCTHHenOBBHXCHblXBHCJIOKauHti), AM’ (nnomocm nonBHmHbix BucnoKaqHir), ua (cK0p0c~H BBH)l(eHHRnuCJIOKaHH@. BBOABTC~~~H~O~H~I~KOHCTUT~TUBH~I~~~~BH~HURA~BU’~H~~~ CKOpOCTHH3MeHeHAfl A,“, A,” BO BpeMeHu. AH~BU~OM nByMepHOr0 pa3BHTHa )l(eCTKOrO BR3KOnBaCTuYeCKOrO TeJla HnJnocTpHpyeTcB xapaKTep npennoxennofi Teopuu. AtiCTpaKT
Sci,
1973, Vol. I I, pp. 1065- 1078.
Pergamon
A DISLOCATION
Press.
Printed
in Great
THEORY
H. T. HAHN* Air Force Materials Laboratory, Want-Patterson
Britain
OF PLASTICITY-l Air Force Base, Ohio, U.S.A
and
Department
W. JAUNZEMIS of Engineering Mechanics, The Pennsylvania State University, University Park, Pennsylvania, U.S.A.
Abstract-
Based on the.observation that plastic behavior is a macroscopic manifestation of the motion and interaction of dislocations, a plasticity theory is proposed that represents a connecting link between solid state physics and the conventional plasticity and viscopIasti~ity theories. The d~sio~tion picture is described-for a slip system a- by means of A; (immobile dislocation densities), ~4; (mobile dislocation densities), and the dislocation speeds Va. Suitable constitutive equations are introduced for V” and the timerate of change of A:, A’$ A two-dimensional extension of a rigid-viscoplastic body is analyzed to illustrate the nature of the proposed theory. 1. INTRODUCTION
IT HAS long been known that the basic mechanism of plastic flow is the motion of dislocations and that the a~angement of dislocations is characteristic of the thermomechanical state of crystalline sofids. Since the discovery of dislocations there have been continued attempts to reduce plastic properties to models of dislocation behavior. Some of this work, in particular what is called the continuum theory of dislocations, is really no more than a variant of elasticity theory. That is to say, if dislocations and their movement are considered as given, they can be incorporated into elasticity in the same way as distributed body forces. In a complete theory, the density and movement of dislocations should be derivable from a knowledge of initial conditions onfy. The progress towards a true dislocation theory of plasticity has been slow. In 1965 Mura[l, 2]$ gave a derivation of von Mises yield criterion and Prandtl-Reuss equations in terms of dislocation forces and a dislocation movement tensor. In 1966 an alternative approach to a continuum theory of dislocations in crystals was proposed by Fox[3]. This work was extended by Lardner[4,5], who accounted for the disIocation production neglected by Fox. Introducing a tensor which is an integral over time of Mura’s dislocation movement tensor, and also using dislocation forces, Eisenberg[6] interpreted the classical thermopl~ticity in terms of dislocation mechanics. Parallel to the development of continuum theories of dislocations much experimental evidence has been gathered particularly by Gilman and his coworkers[7], concerning the average behavior of dislocations and its relation to plastic properties. On the basis of this behavior, various constitutive equations have been proposed. These relat Presented at the 9th Annual Meeting of the Society of Engineering Science ( 197 1). SNRC Postdoctoral Associate: formerly Graduate Student, Department of Engineering The Pennsylvania State University. §Numbers in square brackets refer to References at end of paper. 1065
Mechanics,
H. T. HAHN
1066
and W. JAUNZEMIS
tions can, on the whole, explain satisfactorily inelastic phenomena like transient creep, propagation of plastic wave fronts etc. Summarizing, it seems reasonable to expect that any practicable dislocation theory of plasticity will contain the following special features: (1) a kinematical relationship between plastic flow and dislocation motion: (2) a selection of suitable dislocation densities (together with constitutive equations for the change of these densities); (3) constitutive equations for dislocation velocities; (4) a correlation of average dislocation properties with macroscopic thermomechanical properties (e.g. yield stress) [8]. In what follows we shall explore these features in some detail. as well as the theoretical and practical questions associated with them. 2. PLASTIC
FLOW
The gross shape deformation
AND
DlSLOCATlON
MOTION
of a body is described by x=x(X.t),
(1)
where x is the position at time t of that particle which in some reference configuration is located at X. The mapping x is assumed to possess a high degree of smoothness (continuity and differentiability). Let dX be a typical vector of an infinitesimal neighborhood N(X) of a particle X. Then, because of the differentiability of the mapping x, the entire neighborhood is mapped by a single second-rank tensor into its deformed shape: dx = F dX, dxi = Fu d Xj, (2) where Fij=Xi,j, J=detF
> 0,
(3)
is the tensor of deformation gradients. As a rule, a subscript ‘i’ indicates a derivative with respect to xi when preceded by a comma (,), and with respect to Xi when preceded by a period (.). Material time-rates will be denoted by a superposed dot. In elastoplastic behavior there exist, on a microscopic level, two distinct modes of deformation-a recoverable, thermoelastic deformation, and a permanent, plastic deformation caused, in effect, by relative slipping of adjacent layers. The classical approach to plasticity has been to decompose F into a (local) thermoelastic deformation @and a (local) plastic deformation P, thus [93 F=~~,g~det~>O,J~detF>O.
(4)
This is represented schematically in Fig. 1. The intermediate (local) configuration is called the plastic configuration hereafter; it may be obtained by a stress-temperature relaxation process. However, such distinctions must eventually be based on constitutive equations. Whereas F is always compatible, the same need not be true of @or F in general. *Here incompatibility means non-integrability of the differential forms associated with F and p (i.e. that F and F are not gradient fields). A plausible measure of incompatibility of a deformation @ is given by the dislocation density per unit area in the current configuration. (Yu= where
limk is
the alternator.
EimkFjjnEih,kt
(3
Because the density (Yis affected by elastic deformation,
1067
A dislocation theory of plasticity IntermedIate
Initial
Fig. 1. Decomposition
(local)
conflqurotion
Current
configuration
configuration
of deformation F into elastic deformation @and plastic deformation F.
one may, in order to avoid this, define a dislocation density tensor A per unit area in the plastic configuration: ..*
Ati = JFiilFj;* 1ak,,,. The time-rate of change of the gross shape is expressed gradient tensor L, L GEL+Jti=uij,
(6) by means of the velocity (7)
where v denotes partide velocity. Using the decomposition L can be decomposed into elastic and plastic parts: L = i+PLP-1,
(4), it is easily seen that (8)
where L and L are defined similarly to (7), i.e. 2, S
#p-1,
L
s
&l*
(9)
Turning now to the dislocational aspect of plastic flow, we first note that a group of dislocations in a sufficiently small neighborhood of a material particle is characterized by three variables: a common line direction k, a common direction ii~of Burgers vector, and a scalar density k. The vectors E and EI are all referred to the plastic configuration. Moreover, they are chosen to be of unit magnitude so that the density k is equivalent to the number of dislocations (per unit volume in the plastic configuration) of unit length and of unit Burgers vector. Thus, if cwdenotes a scalar density referred to the current configuration, it is related to 2 by OL= ]-‘A. When there are various groups of dislocations, specified by ka, iiib, a, b = 1,2, . . . , N, the corresponding scalar densities are denoted by AQb.The vectors P, iii” are like normalized vectors of a reference lattice: they show the possible directions in which dislocations can exist. For a single crystal the number of possible directions $“, EI~ is finite, but is in general infinite for a polycrystal. Moreover, because it is known that not all dislocations move when subjected to stress, we should distinguish mobile
1068
H. T. HAHN
and W. JAUNZEMIS
dislocations from immobile ones, thus jab
=
jab 1
+
Aub n-I f
(10)
where the subscripts Z and M denote the immobile and mobile parts, respectively. To relate dislocation motion to plastic flow we take a generic material element d(y) in the plastic configuration, which deforms plastically when dislocations pass through it. The net number of dislocations of density &?ij~, that cross the line d(y) per unit time is given by (%x V) - d(y)&, where v is the average velocity of the group &. Because each dislocation produces a relative displacement equal to its Burgers vector when it passes by, the amount of slip in d (y ) caused by the passage of all dislocations will be d(y) =fi[(iixV)
.d(y)]&,=Ed(y).
where the second equality follows from the macroscopic Therefore, from the above equation we have E = iii @ (E X
description z of plastic flow.
V)A,+f+ .il+j= iliiEjkJkVnJ&f*
(11)
where @ denotes the tensor product. If dislocation groups of various directions are present, the above equation becomes L=x
&@
(i;axpb)@$‘,
(12)
a,b
where ‘Vabdenotes the average velocity of the group A,. -ab This formula is the most important link between plastic flow and dislocation motion. [ 1,3]. The vectors Ea and Vab define a plane-with a unit normal vector i? x pb/Vnb - of dislocation motion of the group xgb. The quantity i? nb= p x pbl is the speed normal to Ea of the group Agb, because P is of unit magnitude. Such planes provide another alternative to the labelling of dislocation groups. Specifically, the summation in (12) can be rearranged as follows,
where the summation inside the bracket is taken over all dislocations that have Burgers vector in the iii” direction and are moving in a plane defined by @. Clearly, the density A$ of those dislocations with EF and I* is (13) Using the average speed Fb of the group Ahb, defined by ACbVCb
=
5
,+$hb,
a
the equation for E can finally be written as H,= 5: N*A$P, a
(14)
1069
A dislocation theory of plasticity
where each tensor N” is of the form
for some ~3 and P. At this point some comments regarding plastic incompressibility are appropriate. The time-rate of plastic volume change is given by tr L, and so plastic incompressibility results either from the motion of screw dislocations, for which
or from the glide motion of edge dislocations, characterized
by
or from a combination of both. The above equation implies that all three vectors K, 8, and IS lie in the same plane and hence the plane of motion coincides with the slip plane. A climb motion of edge dislocations is rare, except at high temperatures, and will be neglected in the following discussion. For an edge dislocation, this assumption defines a unique plane of motion except for its orientation because the slip plane is spanned by the vectors K and i% On the other hand, the slip plane of a pure screw dislocation is not predetermined; rather, it will in general depend on the applied stress. However, a real dislocation line is hardly straight, and a curved dislocation line can be thought of as composed of short straight pieces, some of which have edge components, and so any screw component will also have as its slip plane the plane in which the curved dislocation lies. Further, crystal structure permits screw dislocations to move more easily on certain planes. Therefore, it is assumed that each dislocation group A; belongs to a unique slip system Na. When this is the case, a term A&P represents the time-rate of shear caused by the moving dislocations A& in the slip system Na. This term is a basic variable in the slip theory of plasticity. (cf. [lo]) Thus, according to (15) a knowledge of the detailed arrangement of dislocations is not required-only the densities of dislocations according to their Burgers vector and slip plane must be specified.
3. PHYSICAL
PRINCIPLES
For later use the physical principles of classical thermomechanics will be summarized below. Conservation of mass requires that the mass density p = p (x, t) satisfy P+pdivv=O. In the absence of microstresses the form of Cauchy’s laws of motion
the balance of linear momentum divT+pb
= pv,
(17) is expressed
in (18)
where the symmetric tensor T = T(x, 1) is the Cauchy stress tensor, and b = b(x, t) denotes the body force density per unit mass. To state the first and second laws of thermodynamics the following additional variables must be introduced; the Helmholtz free energy density per unit mass +!J= $(x, t), a volume distribution of sources of heating per unit mass r = r (x, t) , a field of heat flux vectors q = q(x, t) , entropy density per unit mass r) = 71(x, I), and absolute
IJESVol.
1 I.No.
10-B
1070
H. T. HAHN and W. JAUNZEMIS
temperature
0 = 0(x. t). The balance of energy is then written as ~~=-divq+~+rr(TL)-~~e-~~~,
whereas the principle of non-negative Duhem inequality, becomes
(19)
entropy production,
referred to as the Clausius-
per=-P(~+se)-e-lq.g+tr(TL)
3 0,
m
where g s grad 8,
(21)
and y = y(x, t) is called the rate of entropy production density per unit mass. 4. CONSTITUTIVE
EQUATIONS
AND THERMODYNAMIC
RESTRICTIONS
With the introduction of kinematical variables suitable for describing the arrangement and motion of dislocations, the stage is now set for stating constitutive equations which, together with the physical principles, will yield a deterministic theory of thermoplastic behavior. The thermomechanical properties of crystalline solids depend, as a rule, upon the number, arrangement, and properties of defects (dislocations, gram boundaries, vacancies, interstitial atoms, and so on). Whereas the elastic properties of a solid are determined primarily by atomic forces, and crystal defects only introduce small deviations from the ideal behavior of a perfect crystal, the plastic properties largely depend upon the behavior of defects, especially dislocations. This is so because plastic deformation is the macroscopic manifestation of dislocation motion. The ease with which dislocations can move depends on the interaction of dislocations themselves with one another as well as on their interaction with other defects. This phenomenon results in strain-hardening of material when viewed on a larger scale. To describe mathematically those features of dislocation behavior which are significant in the macroscopic picture of plasticity, one needs first of all a set of suitable variables representing the number and a~angement of dislocations. These variables are here assumed to be provided by the dislocation densities A? and&, defined in section 2. Consequently, the list of variables for characterizing the state of an elastoplastic material will be:
Because an elastoplastic material behaves like an elastic material for deformations from a fixed plastic configuration, it is plausible to describe the thermoelastic behavior by the constitutive equations (22) where This set (22) is to be complemented by additional variables A;, A%, and V”. Production of dislocations during plastic flow include spontaneous nucleation of new dislocation pro&Jction by Frank-Read sources, and breeding is the most effective of them all).]71 The mobile
constitutive
equations for the plastic
involves various mechanisms. They loops under high stress, quasistatic through multipie cross glide (which fraction of total dislocations usually
1071
A dislocation theory of plasticity
decreases as plastic flow proceeds. The immobilization of mobile dislocations is primarily caused by interaction with other immobile dislocations. The average effects of these mechanisms under ordinary conditions are described, in a fairly adequate way, by (23) ky = C Bg*ALV*, k& = x B$‘ALV* b
b
where BF* = By*@, 8, A;, A;),
B$ = B$‘@, 8, A;, A;)
(24)
are ‘breeding coefficients’ [4]. The above equations also have the implication that dislocation densities change whenever dislocations move. Evidently the influence of temperature gradient on dislocation production has been neglected. To gain some idea about constitutive equations for Vu, (22), is substituted into (20) to yield (cf. [ 111, [ 121) p&y=tr[(T-pc3
~~‘)i]-~(a,~+s)e-pa,9.g-8-‘q.g+pecr
2 0,
(25)
where u is defined by o = $[P(TL)
-p
c (a,;$A;+aA:$A$)], a
T = p-l’&.
(26) (27)
Here tr (‘i;L) represents the plastic working by the stress T and the term z (aA,~A;+a,4,~& D denotes the time-rate of energy storage associated with dislocations. Experimental results show that this latent energy ranges from 0 to 10% of the plastic work done. Further, substituting (15) and (23) into (26) yields where the term
pt9u = x (Pa
T;)A;P,
(28)
Ta = tr(TNa)
(29)
is the resolved shear stress, caused by the Cauchy stress T, in the slip system Na. The term T;f = ; (Bf%,: I&+ B$,,
I$)
(30)
can be thought of as a latent stress serving to increase the free energy associated with dislocations; it will be the stress in the slip system induced by the presence of dislocations. Thus ( Ta - T;f ) is the net force acting on the dislocations of the slip system Na. Dislocations start moving when the resolved shear stress Ta exceeds the sum of T$ and a critical resolved shear stress T;. Moreover, they accelerate to steady state motion in a time very short compared with ordinary loading times, so that inertia effects can be neglected. Therefore, dislocations will move under the conditions Ta= T;:+T;+Ta,,
(31)
where T”,denotes a viscous stress resisting dislocation motion. The critical shear stress T; usually increases with plastic deformation, i.e. materials strain-harden. Note that strainhardening is also described in part by the latent stress
1072
H. T. HAHN
Tj) Because the effect of hydrostatic that
and W. JAUNZEMIS
pressure
T;I = T;(e,
For the viscous stress T; the constitutive
on Tg is negligible, it will be assumed
A:, A$).
(32)
equation is taken to be
T; = T;(C), A;, Aa,, IQ),
(33)
where _ais used to indicate that T; depends only on V” but not on P. b # U. When the direction of loading is reversed after having produced plastic deformation, (T;f + TE) usually decreases, i.e. there exists a Bauschinger effect. Because dislocation will move in the opposite direction upon reversal of loading direction, the original direction N of dislocation motion will change to -N. Thus the Bauschinger effect manifests itself in the form T;t + T; 2 T;-“’ + ,h?. Mechanisms for Bauschinger effect are not well understood. One possible mechanism may be the increase of mobile dislocations after the direction of loading is reversed. For instance, in the presence of an obstacle dislocations move under the applied stress until they become queued up at the obstacle, resulting in immobilization of mobile dislocations. If the direction of loading is changed, they will move again in the opposite direction. The ease with which dislocations can move will appear as a decrease of the stress (T;t + T;). It seems in general that equal hardening in both directions will require a special distribution of dislocations, which is unlikely to be produced by a unidirectional loading. When the viscous stress T; is negligible compared with other terms, which may be the case for quasistatic loading in certain materials, it is omitted in (3 1). The resulting theory .is then a rate-independent plasticity, whereas retaining T; leads to a ratedependent plasticity (viscoplasticity). (a) Rate-independent plasticity. For a fixed (A:, A;) the equation (3 l), T” = tr(TNa)
= Tff(e,T)
+ T;(e),
(34)
represents a hypersurface in a ten-dimensional stress-temperature space. It has been assumed here that T can replace i? in the domain of Tff. Because dislocations do not move when T” < T;E+ T& the region E(A;, A&) bounded by all hypersurfaces-each of which is described by an equation of the form (34)-is called the elastic range in a stress-temperature space corresponding to (A,“, A$). The boundary &?Z(Af, A$,) of the elastic range E(Ay, A$) is called the yield surface. When T;; is assumed independent of T,* the hypersurfaces at a fixed 8 become hyperplanes each of which is normal to N”. Moreover, if T;f can be written as T;t = tr (TEN”)
for some tensor T,, then (34) becomes tr[ (T--T,)Na]
The above equation is the generalized
= TE.
form of the kinematic hardening law of Prager
*This will be the case, e.g. when $ = Jr1(fi, 0) + &,(A;, A&), and By”, B$ are independent
ofk.
1073
A dislocation theory of plasticity
[ 131. In particular, if TR = @ for a scalar 5, and if T; = Tc, then the above equation reduces to an isopropic hardening law TC tr (TNa) = -.
A slip system Na is said to become potentially active when the resolved shear stress reaches a critical value, tr(TN”) = Tg+ Tl. (34) Further, tiont
dislocations
on a potentially
active slip system N” move subject to the condi-
tr(TNa) = Ti+
Tg
(35)
whenever tr[(Na’-
aTT;)?j
- a,( T;+
T;)d
> 0 (loading),
(36)
whereas they do not move either when tr [ (N”’ - aTT;f)?]
- a,( T;f + Tg)b = 0 (neutral loading),
(37)
or when tr [ (N”T - +T;t)@]
- a,( T;f + T,Z)b < 0 (unloading).
(38)
With the aid of (23), (35) can be written as tr [(N“’ - afTi)@]
- a,( Tg + TS)e = ): habALl/*, b
(39)
where ha* are defined by hab=C
[aA;(T;f+T~)BF*+a,;(T~+Tt4)B~].
(40)
Whereas c in (40) rang& over all slip systems, the summation in (39) is restricted to the activated systems only. Thus (39) provide the necessary number of equations forthe unknowns I/*. Therefore, if the matrix [ha*] is non-singular, (39) can be solved for AkV* as follows, A&I/*=x
(h-1)b”{tr[(N”T-dgT~)?T]-aag(T~+T~)8},
(41)
where ( h-l)ba denotes thl ba-component of the inverse of the matrix [ha*] .$ Consequently, the equation for L becomes E = z Nb(h-1)ba{tr[(N”‘-aaT;T;)f”]-&,(T;+T;)8}, *,a
(42)
where, again, the summation is taken over the activated systems only. Summarizing, the constitutive equations for plastic variables in the rate-independent plasticity are: (23), (24), (30), (32), and (42). (b) Rate-dependent plasticity. When the viscous stress T”, is of the same order of magnitude as the other stresses, the speed I/” is determined by solving (3 1): V” = V”( T” - Tff - T& 8, A;, A;) =OwhenT”-TT;f-Tg
when T"- Tff - T; > 0,
s 0.
tNote that fin = 0. SA discussion concerning special cases of [@I may be found in [14].
(43)
1074
H. T. HAHN and W. JAUNZEMIS
A restriction that should be imposed on I/” is that they be positive. monotonically increasing functions of (T”-- Ti- T”,), yet approaching certain limit values[7]. If the functions Vu can be replaced by a single function I/, then L becomes L = x Na&J’/( T” - Tj - Tg. 0, A:, A&).
(44)
Any admissible thermondynamic process is required by the postulate of nonnegative entropy production to obey the Clausius-Duhem inequality (20) at each particle X and time 1. As a consequence, the second law will impose some restrictions on the response functions formulated above. Specifically, following Coleman and Curtin [ 1.51,we can easily deduce (cf. [ 161) a,+ = 0,
(45)
T = ,@I/&‘,
(46)
7 =-a,*.
(47)
pey = p&r-- e-‘q * g 2 0,
(48)
and the general dissipation inequality,
where o is now called the internal dissipation. Because A;P 2 0 and Ta - T; 3 0, the internal dissipation u is non-negative. This is in agreement with the restriction on (T, Cr*0 imposed by (48) when (+ is independent
(49)
of g.
5. AN EXAMPLE
To illustrate how the theory developed in the preceding sections is applied to specific problems, we consider a simple extension of a rigid-viscoplastic body under isothermal conditions. A two-dimensional simple extension is described kinematically by x=X,y=(Y(f)U(t)Y,z=
U(t)Z,
(50)
where U(t) is the stretch in the z direction, and the undetermined coefficient a(t) describes a contraction in the y direction. The corresponding tensor F of deformation gradients is F=diag (l,(~u, U). (51) The body is assumed to be rigid-viscoplastic, and to have a single slip system which is characterized by the vectors n-r and fi as shown in Fig. 2. In the initial configuration they are given by rir = cos &ey + sin &e,, (52) ii = - sin &e, + cos &ez. The rigid-viscoplastic tion ti
behavior requires that the elastic deformation P=ff,j=
6 is a pure rota-
1,
and so the vectors m and n referred to the current configuration by
are related to M and ii
1075
A dislocation theory of plasticity
htial conflgurotlon
-e,
currentconflgurotlon
Fig. 2. Simple extension.
Moreover, deformation the x axis,
is taking place in the yz plane, so that fi will be a rotation about
k=
0 1 0 cos ($I-&) 0
Therefore,
sin(4-44
0 -sin(+--&) cos (4-4J
1.
(54)
m and n are given by m = cos 4e, + sin +e,, n = - sin #2e, + cos f$e,.
(55)
If the mobile dislocation density and the average speed are denoted by A,,, and V, respectively, the plastic deformation rate L becomes (cf. (15))
L = iii @i-i/l~~V. The plastic deformation
(56)
tensor F is then
F=e,~e,+~~P+1~fiI:,AVdr+a~a.
(57)
The elastic rotation k is not arbitrary because the condition (58) should be satisfied. Substituting (57) into (58) yields (59)
H. T. HAHN
1076
With the aid of (5 1), the components
and W. JAUNZEMIS
of (59) can be written as
F,, = a U = ~0s 4 cos 40 - cos 4 sin & ,: AwV dt + sin $ sin c$,,, F,,=
(60)
U=sin$sin&+sin+cos+,,~~&Vdt+cos+cos&,,
F,, = 0 = cos 4 sin CpO + cos $J cos do 1: A#
dt - sin 4 cos I&.
Eliminating cos c#I,,,: AMVdt from (60), and (60),, we find that A,Vdt=---
sin (p sin (PO cos Cp cos $0’
or, in terms of U, A#dt=’ Therefore,
U~-cos2Cpo -- sin+, cos 40’ cm 40
(61)
we have AuV=
For the dislocation production, are taken,
Uti cos cpovw - co9 $Jo. the following equations -suggested
(62) by Gilman[7] -
A = B~AMV $&)
(63)
= - B~B~(+)AMV,
where A is the total dislocation density, and B, and B2 are the breeding coefficients. The above equations are based on the following observations. Because new sources of dislocations are created mainly by multiple cross glide, the rate of change of A is proportional to A,,, and increases with the speed V. The rate of decrease of the mobile fraction AM/A is proportional to the rate of increase in A because this leads to more interactions, and to instantaneous mobile fraction itself because the fraction changes whenever dislocations move. Equations (63) can be integrated to yield
(64)
where A0 and AM0are the initial densities of the total and mobile dislocations, tively. When the applied stress is a simple tension,
respec-
T = Pe, 03 e, the resolved shear stress T becomes T = TUAi@ = Tunimj = P sin $J cos r$.
(65)
A dislocation theory of plasticity Assuming
1077
a linear relationship between dislocation speed and viscous stress Tv = kV, k = const.,
and further that TR = 0, Tc = const.,
the equation for V becomes T = Tc+kV.
Therefore,
(66)
the tensile stress P necessary to produce the prescribed deformation p=
Tc+kV
sin (P cos 4
(Tc+kV)U2
=
co!3 cp&J2
- cos2 &’
is given
(67)
where
v=
uu cm $du2
1
- co?dozy Xfu2 - cos” C#Jo ax
$0
Finally, the coefficient cxis easily seen to be 01=-.
1 U2
(68)
6. SUMMARY that plastic behavior
of crystalline solids is the macroscopic manifestation of the motion and interaction of dislocations, we have proposed a plasticity theory that represents a connecting link between solid state physics and the conventional plasticity and viscoplasticity theories. Macroscopic plastic deformation is represented by a second-rank tensor which is not necessarily a gradient field, whereas the dislocation picture is described by use of immobile and mobile dislocatioa densities together with average velocities. A kinematic relationship between plastic flow and dislocation motion is derived without the usual assumption of infinitesimal deformations. No climb motion of edge dislocations is allowed, however, resulting in the plastic incompressibility. Appropriate constitutive equations for the changes of those densities and velocities complete the formulation of the proposed theory. The theory, as expected, is much like the slip theory of plasticity. However, it is different in that plastic phenomena are explained in terms of more fundamental variables. Based
on the observation
Acknowledgemenrs-This AFOSR-1626-69.
work was supported by the Air Force Office of Scientific Research-Grant
REFERENCES [ll T. MURA, Phys. Smus. Solidi. 10,447 (1965). 121 T. MURA, Phys. Status. Solidi. 11,683 (1965). [31 N. FOX, J. Inst. Math. Applic. 2,285 (1966). [41 R. W. LARDNER, Z. angew. Math. Phys. u), 5 14 (1969). [5] R. W. LARDNER, Znt. J. Engng Sci. 7,4 17 (1969). [6] M. A. EISENBERG, Znt. J. Engng Sci. 8,26 1 (1970). [71 J. I. GILMAN, Micromechanics ofFlow in Solids. McGraw-Hill, (1969). [8] E. KROENER, J. Math. Phys. 42,27 (1963). [91 E. H. LEE, J. uppi. Mech. 36, l(l969).
1078
H. T. HAHN
and W. JAUNZEMIS
[IO]
T.H.LINandM.ITO,Inr.J.EngngSci.4,543(1966).
1111 1121 1131 [14]
J. KRATOCHVIL and 0. W. DILLON, Jr.,J. uppl. Phys. 40.3207 (1969). J. KRATOCHVIL and 0. W. DILLON. Jr.,J. appl. Phys. 41, 1470 (1970). W. PRAGER. Proc. Znstn mech. Engrs 169,41(1955). R. H1LL.J. mech. Phys. Solids 14.95 (1966).
[I51 B.
[ 161 A.
D. COLEMAN and M. E. GURTIN,J. E. GREEN and P. M. NAGHDI. Arch.
them. Phys. 47.597 (1967). ration. Mech. Analysis l&25
(Receiued Resume-Baste
mouvement
sur I’observation
13
1 (1965).
October 197 1)
que le comportement plastique est une manifestation une ttiorie de la plasticite est proposee
et de I’interaction de dislocations,
macroscopique qui represente
du un
maillon reliant la physique de l’etat solide avec les theories conventionnelles de la plasticite et de la viscoplasticite. L’image de la dislocation est d&rite-pour un glissement don& u-au moyen de A,a (densite de dislocation immobile)cl,” (densite de dislocation mobile) et la vitesse de dislocation Va. Des equations constitutives convenables sont introduites pour V” et la vitesse de variation dans le temps de A,” et A,,“. Un allongement bi-dimensionnel d’un corps viscoplastique rigide est analyse pour montrer la nature de la theorie proposee. Zussuumenfassung-
Begriindet auf der Beobachtung, dass plastisches Verhalten eine makroskopische Manifestierung der Bewegung und Wechselwirkung von Verlagerungen ist, wird eine Plastizitltstheorie vorgeschlagen. die ein Verbindungsglied zwischen Festkorperphysik und den konventionellen Theorien der Plastizit;it und Viskoplastizitat darstellt. Das Verlagerungsbild wird fur ein Gleitsystem u durch A,“(unbewegliche Verlagerungsdichten). A,v”(bewegliche Verlagerungsdichten) und die Verlagerungsgeschwindigkeiten Vu beschrieben. Geeignete Konstitutivgleichungen werden fiir V”-und die Zeitrate des Wechsels von A,“, A,,,a eingeftihrt. Eine zweidimensionale Ausdehnung eines Starr-viskoplastischen Korpers wird analysiert. urn die Natur der vorgeschlagenen Theorie zu erlautem.
In ase all’osservazione the il comportamento plastic0 e una manifestazione macroscopica del modo e dell’azione reciproca degli spostamenti. si propone una teoria di plasticita the rappresenta una maglia di collegamento fra la fisica di stato solid0 e le teorie di plasticit& convenzionale e viscoplasticita. Si descrive il quadro di spostamento nei riguardi di un sistema di slittamento a mediante A,” (densita di spostamento immobile), A,P(densita di spostamento mobile), e le velocita de spostamento V”. Si introducono opportune equazioni costitutive per Va e il ritmo di tempr di cambiamento de A, a, A,,,“. Si analizza un’estensione bidimensionale di un corpo rigido-viscoplastico per illustrare la natura della teoria proposta.
Sommario-
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