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Discussion of sufficient condition for plastic flow localization
0313-7944185 $3.00 + .OO Pergamon Press Ltd.
Engineering Fracture Mechanics Vol. 21, No. 4, pp. 761-179. 1985
Printedin the U.S.A.
DISCUSSION OF SUFFICIENT CONDITION PLASTIC FLOW LOCALIZATION
FOR
RYSZARD B. PECHERSKI Institute of Fundamental Technological Research, Warsaw, Poland? Abstract-The strong sensitivity of the predicted deformation instability initiation to the different finite-strain Jz theories commonly applied in studies of deformation instability exhibits the weakness of these theories and reveals the necessity for constitutivemodellingcorresponding to key physical mechanisms of large plastic strains, viz. formation of cell structure, fragmentation and development of terminated boundaries which produce large rotations of adjacent microvolumes. The aim of this paper is to incorporate these mechanisms into the continuum model of finite elastic-viscoplastic deformations of a single crystal and discuss a sufficient criterion for localization. The effect of the relative rotations of the crystal lattice is modelled by means of Somigliana dislocation, whereas development of terminated boundaries is described in terms of the movement of partial disclination dipoles. The existence of an “orientation” imperfection is considered a sufficient condition for localization. The orientation imperfection is represented by a critical value of the scalar density of partial disclination dipoles, which produce lattice misorientation and “geometrical softening” sufficient for the onset of localization.
1. INTRODUCTION THE
studies of plastic strain localization reviewed by Rice [I] are based mainly on Jz-type flow theory with smooth yield surface, deformation theory or theories taking into account such “destabilizing” effects as yield surface vertices, deviation from plastic “normality” and the dilatational plastic flow due to nucleation and growth of voids (cf. [21). This approach provides satisfactory qualitative predictions of plastic flow instabilities based on the necessary condition expressed as the loss of ellipticity of the governing incremental equations. However, quantitatively, the necessary condition leads to the upper bounds of the critical stress and strain or the lower bound of the strain hardening rate.S This could explain, at least in part, the discrepancy with experimental results discussed by Anand and Spitzig [4]. Furthermore, applications of different models of plastic flow for analyzing instability phenomena reveal strong sensitivity of the predicted deformation instability initiation to the constitutive relations adopted. This fact has been pointed out by Neale and Schrivastava [5]; it exhibits the weakness of existing finite-strain J2 theories commonly applied in studies of deformation instability and shows the importance of proper constitutive modelling for advanced strains. Adequate constitutive modelling for advanced strains and localization should correspond to the key physical mechanisms responsible for large inelastic strains and the onset of localization, viz. formation of dislocation cell structure with misorientation of neighboring cells, fragmentation of crystalline lattice and further development of bands propagating from the boundaries and terminating abruptly within the cell or fragment interior. The bands turn the crystal lattice by equal angles of opposite signs. The latter mechanism produces very high lattice misorientation. This can lead to achange of Schmid factors, in certain volumes of crystal, such that local “geometrical softening” occurs, triggering the onset of plastic flow instability. The aim of this paper is to identify and describe theoretically the most important properties of the above-mentioned mechanisms, incorporate them into the continuum model of finite elastic-viscoplastic deformations of a single crystal at advanced strains and formulate a pertinent criterion for plastic flow localization. The criterion is the sufficient one, for it derives from physical mechanisms originating at the onset of localization. The effect of the relative rotations of the crystal lattice in the co&se of the development of the cellular and fragmented ton leave at the Institut fiir Baumechanik und Numerische Mechanik, Universitlt Hannover, as a Felloi of the Alexander von Humboldt Foundation. $4 similar conclusion can be drawn from the results obtained by Pctryk [31, who has proved that the mentioned critenon for localization provides an upper bound to the onset of instability of the deformation process in the energy sense.
768
R. B. Pf$CHERSKI
structure is modelled by means of Somigliana dislocation, whereas the critical deformation instability effect of the formation of terminated bands is described in terms of the movement of partial disclination dipoles. The novel result arrived at is the incorporation of these new effects into the continuum theory of finite inelastic deformations. 2. AN OUTLINE OF PHYSICAL FOUNDATIONS Recently, there have become available a number of thorough review works in which a vast quantity of experimental results and data relating to the evolution of the microstructure in the course of large plastic strains of different metals is systematically analyzed, e.g. [6-81. Therefore, only the main conclusions necessary to the considerations of this paper will be discussed here. Our considerations are further limited to the behavior of metals at moderate temperatures where the mechanism of plastic glide prevails. One of the most important microscopic mechanisms responsible for the accommodation of large strains is the formation of a dislocation cell structure. At an earlier stage of deformation the average cellular misorientation is rather small, and the glide dislocations can cut through nearly all of the cell walls. At higher strains, however, some cell walls become impenetrable and form a new substructure. The network of impenetrable walls separates volumes of high lattice misorientation. The process of new subdivision is called the fragmentation of the crystalline lattice, and the volume elements are the fragments. It has been found that one of the main ways in which fragmentation develops is through the formation of band structures propagating from the cell or fragment boundary into the interior. Of great importance here, as noted by Vergazov et al. [9], is the fact that the neighboring boundaries of the bands terminate abruptly within the cell or fragment interior and turn the crystal lattice by equal angles of opposite signs. Similar phenomena have been analyzed by Kocks et al. [IO]. They have discussed pairs of subboundaries of opposite sign which produce large lattice misorientation and argued that the material volume containing such pairs of polarized subboundaries plays the role of small “nuclei” from which “shear bands” observed in free compression and in rolling might originate. Also Chandra et al. [l l] have found that unstable oriented crystals reveal fragmented bands of high o~entation. Torrealdea and Gil Sevillano 1121have analyzed similar structures of “double walled parallelogram-type cellular dislocation arrays forming banded structures,” called by them “microbands.” They have also noted that the formation of structures is assigned to localized unstable shear flow. The aforementioned mechanisms for the formation of terminated bands and polarized pairs of subboundaries and fragmented bands, as well as the “microbands,” are of similar character and can be interpreted by means of some cooperative forms of highly concentrated dislocation motions and the onset of collective degrees of freedom. This produces the local rotations of material volumes that eventually lead to the development of partial disclination dipoles and loops; cf. Rubtsov and Rybin [13], Rybin [14] and Orlov and Shitikova 1151, as well as Romanov and Vladimirov [8]. These authors consider the formation of partial disclination dipoles and loops as a critical state, very characteristic of the onset of plastic flow localization. On the other hand, the importance of the inhomogeneous lattice rotations for the onset of necking and shear band formation in single crystals have been recognized by Saimoto et al. [16], Chang and Asaro [17] and Peirce et al. [18] as well as by Lisiecki et al. 1191. Summing up the above discussion, we can note that the formation of partial disclination dipoles and loops produces very high lattice misorientation, which in certain volumes of crystal can lead to such a change of Schmid factors that local “geometrical softening” of the predominant slip systems operating in these volumes occurs (cf. 1191). 3. CONTINUUM MODEL OF FINITE INELASTIC DEFORMATIONS LATTICE MISORIENTATION
WITH HIGH
Let us consider a crystalline material as a continuous body. By the material point X of this body we understand a certain minimum lattice volume V which is sufficient for a valid continuum mechanics description of gross elastic-viscoplastic behavior (cf. 120, 211). Furthermore, as considered earlier by Mandel [22], it is assumed that the dominant lattice orientation in the
Sufficient condition for plastic flow localization
169
volume V is represented by the triad of director vectors ui, i = 1, 2, 3, attached to the material point x. Let us assume that the triad ui is orthonormal and remains the same throughout the body in the reference configuration K. Then it transforms into the piecewise continuous vector field of elastically deformed and misoriented director vectors e;, i = 1, 2, 3, determined on the current configuration + of the body. A comprehensive review of the main ideas concerning the continuum theory of finite inelastic deformations of crystalline materials with pertinent references was recently provided by Hill and Havner [23] and Asaro [24] (cf. also [21]). It is commonly assumed that dislocations traversing a volume element produce a change in its shape but not in its lattice orientation. This means that the material moves with respect to its underlying crystal lattice, whereas the lattice itself only undergoes elastic deformation. In our case the situation is different because advanced deformations produce, locally, high lattice misorientation because of the fragmentation process. The concept of Somigliana dislocation will be used to describe theoretically the fragmented structure of a strongly deformed single crystal near the onset of localization. The importance of Somigliana dislocation for the theoretical modelling of different kinds of interfaces within a crystal have been recognized by Eshelby [25] and Asaro [261, as well as by Volkov et al. [271. A Somigliana dislocation can be constructed as follows (cf. [25]): make a cut over a surface S and give the two faces of the cut an arbitrarily small relative displacement, removing material where there would be interpenetration, till in the gaps and weld the material together again. Let B(x) be the relative displacement at the point x of the cut. Then we have a Somigliana dislocation over S specified by the displacement jump B(x). Let us assume that, in the current configuration 4 of the continuous body, the vector fields ei, i = 1, 2, 3, suffer discontinuity on certain surfaces. These surfaces can be interpreted by means of Somigliana dislocations and they describe theoretically the formation of the fragmented structure. Furthermore, three groups of translational dislocations within the body are assumed (cf. [28, 291): those which leave it or are annihilated in it, those which remain continuously distributed in it and those which remain on the surface of the discontinuity. The latter corresponds to the known interpretation of the Somigliana dislocation as infinitesimal translational dislocations continuously distributed on the surface S. Movement of translational dislocations produces plastic deformation, whereas their accumulation on the boundaries contributes to misorientation. The question then arises as to how to incorporate this new effect into the continuum theory of finite inelastic deformations of a single crystal. The extension of the notion of Burgers circuit for disclinations, discussed by Lardner [30] appears to be helpful in solving this problem. Let us consider the reference configuration K and the current configuration + produced during the process of inelastic deformation under applied loads. Let us then choose a certain neighborhood N(X) of a point X with position vector dX and the triad ui. This neighborhood transforms into the neighborhood of the point x with the position vector dx = F dX, where F is the deformation gradient with distorted triad ei. Let us conduct through point x a surface ,S(t) with an outer normal N and the boundary L(t). Then choose a certain piecewise smooth Burgers circuit % piercing the surface s(t) at x. Let us suppose that a local unloading configuration 4 exist for each neighborhood of the point from the circuit % and that such a neighborhood can be chosen in a continuous way. If two neighborhoods overlap, then we require that two local unloading configurations be fitted together without any free rotation. Starting with the orientation for the local unloading configuration I&, yhich is chosen in the same manner as the reference configuration, we can now construct the sequence of unloading configurations at points along the circuit %. The question now arises whether the unloading configuration that we end up with, 4, is the same as the one we started out with, I$. If the configuration is not the same, the change in making a circuit can only consist of a closure failure and a relative rigid body rotation, for the local unloading configurations are, by definition, in the natural state. In such a case the surface s(t) corresponds to Somigliana dislocation. The closure failure pertains to the total translational Burgers vector produced by the Somigliana dislocation and the translational dislocations distributed continuously in the body, which are enclosed by circuit % [3O]: &We, x, t> = 4x, t)n dS,
(1)
R. B. PFCHERSKI
110
where dS is enclosed by surface % with the normal n, and the total dislocation density tensor ar(x, t) is determined as follows: a(x, t) =
2 p”‘(x, r)b”’ CT’(“‘,
(21
Ci)
The symbol p(“(x, t) denotes the scalar dislocation density of the ith type, b”’ is the Burgers vector and tCn the direction of the transIationa1 dislocation line of the ith type. The scalar dislocation density p’” is understood as the total length of the dislocation lines of the ith type averaged over a certain volume of crystalline material. The relative rigid body rotation, on the other hand, pertains to the lattice misorientation at the point x determined by the orthogonal tensor R,{(e): ei = R,(S)&,
i = 1, 2, 3,
(3)
where @iremains parallel to ui. In accordance with the procedure discussed for circuit “e, the deformation gradient F can be decomposed in the following way [21]: F = AR,(%)P,
(4)
where the tensors A and P correspond to elastic and viscoplastic distortions. If making the circuit % produces closure failure only, without rigid body rotation, the Somigliana dislocation degenerates into a dislocation of the translational type and decomposition (4) takes the form of that proposed originally by Lee and Liu 1311and Mandel [22]: F = AP.
0)
Let us consider the deformation of the neighborhood N(X) with the position vector dX. When, in the course of the glide process the translation dislocations with the Burgers vector db, pass through the vector dX, the latter is mapped into its intermediate configuration dP: dff = dX + dh,
(6)
or, with an account of the transformation, df = PdX, dl;, = (P - 1) dX.
(7) (8)
Then the local relative rigid body rotation transforms the vector dX into the con~guration d%as follows: dir = R,,(%)P dX (9) and dt;, = d% - df = (R,(q) - l)P dX,
(10)
where db, is the increment of the Burgers vector produced by the local relative lattice rotation at the point x determined by the misorientation tensor R,(q). Thus, the total increment of the Burgers vector dB, is given by dB, = dl;, + dt;,.
(11)
dB, = (R,(%)P - 1) dX
(12)
Owing to (9) and (6) and (7), we have
771
Sufficient condition for plastic flow localization
and the rate of the Burgers vector increment
db, = or, owing to transformation figuration I&is obtained:
(R,(%)P
dB,: +
R,(%)i') dX;
(9), the following relation corresponding
dR, = (R,(%)R,
’ (Se) + R,(%)P P-
(13) to the intermediate
'R,'(Se)) d6.
con-
(14)
This relates the rate of the total Burgers vector increment dR, to continuum measures of inelastic distortion and lattice misorientation rates. On the other hand, vector dR, is determined by the microscopic mechanisms of inelastic glide in active slip systems and lattice misorientation induced by the fragmentation process. Thus, according to Kroner and Teodosiu [32], db, = c j(a)g(u) @ n’a’ dff, (u)
(15)
where +@) is the inelastic shear strain rate in the cxth active slip system and gCcr),iiCa)denote, respectively, the glide direction and normal to the slip plane of the &h slip system in the intermediate configuration $. The rate of the Burgers vector increment dR, can be expressed by means of a moving Somigliana dislocation, Applying the derivation of Volkov et al. [27], we have dB, =
ah
{[(V x T)@BB,]G(L(~)) + (N *V)V,B,+ NC3 z
- V. VsBt)]WW)}
di (16)
where V is the velocity of Somigliana dislocation, T is the unit tangent vector to the boundary L(t), the operator V, denotes the surface gradient and @L(t)), s(s(t>) are generalized 6 functions concentrated on the line L(t) and surface S(t). Furthermore, dB/dt denotes the rate of change of the Burgers vector B on the surface S(t). Equation (16) holds for any motion of Somigliana dislocation accompanied by any change in B and can be considered the starting point for the theoretical description of many particular situations, e.g. plastic glide, boundary sliding and mobility of boundaries, as well as development of fragmented structure and terminated bands. The latter can be modelled as a movement of partial disclinations dipole. A simple two-dimensional model of straight partial disclination dipoles of the wedge type was considered by Vladimirov and Romanov [33]. The dipole is represented by two parallel semiplanes limited by straight disclination lines. Physically, this planes can be interpreted as tilt boundaries formed by edge dislocation lines of opposite signs and parallel Burgers vectors. The authors have related the velocity of partial disclination dipole Vd with the rearrangement of edge dislocations at the front of the dipole: Vd =
- Sph KAY*) dy* A, In (1 - wlbpo,h,)
’
QJ < bpo,L,
(17)
where g, h determine the region of nonzero edge dislocation velocity u,, A, is the mean free path of the edge dislocation, b is the length of the Burgers vector, pee is the density of the edge dislocations and 6.1denotes the power of the partial wedge disclination corresponding to the misorientation angle of the tilt boundary. Equation (17) can be transformed as follows:
v, =
-
jbl
peb2h, In (1 - olbp,oA,)
- I’
(18)
where (19)
772
R. B, P~CHERSKl
‘i?,
is the mean value of the dislocation velocity and if and i(* are the shear strain rate with respect to a fixed and a moving coordinate system, respectively. Furthermore, (20)
corresponds to the mean fragment diameter. The physical model discussed above can be related to the effect of the moving Somigliana dislocation given by (16). Let us consider, then, the movement of a ham-plane with normal N, nonstation~y Burgers vector B(x, t) and velocity Vd, which is perpendicular to N. In such a case (16) takes the form [34] $
- Vd. V,B,
&S(t))
dx.
(21)
According to (1) and (2), we have B,(%) = c p@)(@@. n)b(“’ dS, (OLJ
(22)
where ptu) is the scalar density of the translational dislocations determined in the domain 93 x [O, d,] containing the surface s(t), where 93 is the region in the Euclidean space occupied by the crystalline body and dp is the duration of the deformation process. According to the interpretation of the two-dimensional model of straight partial disclination dipoles as the tilt boundaries formed by edge dislocations lines of parallel Burgers vectors, &(“)+ n = 1 and B,(X) = 2 p@)bfafdS. (a)
(23)
From (23) and the theory of moving discontinuity surfaces in continuum, the following relations can be derived under the assumption that the functions p’“’are piecewise smooth in the domain 93 x [O, dp] and have limits on the discontinuity surface s(t):
(z -Vd)]dS &S(t)), -
(1 - N @ N)
1
dS ~(~(f)),
(24) c-3)
where B = B, s(s(t>) denotes the total Burgers vector of the translational dislocations distributed on the surface s(t). The specification of the relation (21) is based on the following experimental observations. Rubtsov and Rybin 1131revealed that in the tensile test of highly work hardened crystals of nickel and molybdenum the number of partial disclinations increases monotonically as necking develops. This produces the so-called knife edge high-angle (w 2 IO”)terminated boundaries, which are oriented almost precisely along the axis of tension and not at an angle of 45” to the axis, as one might expect from traces of slip bands. Such boundaries are the traces of disclination dipoles and usually have a large tilt component. The term “knife edge” was used because the boundaries have practically no width on foils perpendicular to the axis of tension. These boundaries are described theoretically by means of the moving half-plane with normal N, nonstationary Burgers vector B(x, t) and velocity Vd. Rybin [14] argued that vector N rotates until the rotations of two neighboring fragments have been completed. In the laboratory reference frame, however, there is a set of stable orientations of N. They correspond to the boundaries in which disclinations do not experience forces parallel to the normal N. Only along these planes is it possible for wedge disclinations to move under the applied stress. This reasoning justifies
773
Suffkient condition for plastic flow localization
the hypothesis
that in the case of complex stress states in the principal directions
where u1 > u2 2 u3, that is, the planes of partial disclination dipoles are parallel to the maximum principal stress, the velocity of partial disclination dipoles is as follows: Vd
(27)
VdA,,
=
whereas the normal N lies in the plane determined by the principal directions A2 and X3. Relation (21) can be transformed, after simple calculations taking into account eqns (23)(27), into the following form: &,
= c
rj’“’
_
$
(4
vd
>
* jj’“‘)
b'"'N ‘8 gcp) - v#)(Al
+
ap) y-g- N 8 Al 1
a+*)
FN@A,
ap
+ =N@A3
2
3
(1 - N@N) >
1
dSS(S)d%,
(28)
where the derivatives @‘“‘laxi are expressed in the Cartesian coordinate system ki, which is chosen in such a way that ki = Xi when the principal directions Xi are determined uniquely or kl = A1 only, when u2 = u3. In the simple model proposed in [33], which relates the movement of partial disclination dipoles to rearrangement of edge dislocations, the gradient of p’“’ in the direction Ai plays the decisive role. Therefore, owing to (18) expressed for the (Yslip systems acting in the critical zone ahead of the moving partial disclination dipoles and the evolution equation [21] i)(cr)
=
Gb)(#@),
p’P’,
Tr))+‘“‘,
(2%
the relation (28) yields dd, = c (./t/lia’N %Igca) + J#“‘N @ Al)+‘“’ dS s(S(t)) dST, (a)
(30)
where
(31)
and T’,) is the athermal strength in the p slip system. Let us assume that there are many dipoles of partial disclination expanding along the surfaces with the normal unit vectors N in a certain volume AV. The averaging procedure over the volume AV leads to the following relation: (32) where xl”’ and xSa) represent the mean values of the material functions A\“’ and J4iaa, over the volume AV, and N is the average direction of the normal vectors N, whereas
PPD
=
S(t)
F
(33)
714
R. B. PQCHERSKI
is the scalar density of partial disclination dipoles representing, boundaries. From (10) and (32) one can obtain R,(%)R,‘(%)
+ R,(%)PP-‘R,‘(%)
physically, density of terminated
- PP-‘R,‘((e>
= x M(“‘p&@) (U)
dS,
(34)
where (35) Dividing both sides of (35) by dS and assuming that the limit iirn, b
(R,(%)R;‘(%)
+ R,(%)PP-‘R,‘(Z)
- PP-‘R;‘(%)> = &R,’
+ R,PP-‘R,’
-
PP-‘R,’
(36)
exists, we have + R,pp-‘R,
fi,R,’
1 -
tip-‘R,’
=
c
R;z’“‘ppDj(u),
(37)
(a) where R,,, is the misorientation tensor independent Ma) corresponds to the configuration 4. 4. CONSTITUTIVE
of the choice of circuit %, and tensor
MODEL OF ADVANCED VISCOPLASTIC LATTICE MISORIENTATION
DEFORMATIONS
WITH
From (1 l), (14), (15) and (37) the relation between continuum measures of inelastic deformations to lattice misorientation and pertinent microscopic mechanisms can be obtained in the following form: R,R,’
+ R,,$P-
In the current configuration AR,R,*A-i
‘R, ’ = 2 +@)(gca) @Ificu) + ppD&a))a (a)
+ (38) transforms
(38)
as follows:
+ AR,PP-‘R;‘A-’
= x jca)(gca) @ n’“’ + ppDMcU)), (Cl)
(39)
where [18] (01)
g
=
A$“’
and
#X’
as well as M(“) = x\a’N @ gca) + xi% @ hi. The decomposition of the gradient of deformation velocity gradient: L = AA-’ Taking the symmetric
+ AR,R,‘A-’
and antisymmetric
D = (AA-‘), for the total rate of deformation
=
,j’“‘A-
1,
(40)
(4) leads to the following equation for the
+ AR,PP-‘R&IA-‘.
(41)
parts of (41), we obtain
+ (AR,R,‘A-‘),
+ (AR,PP-‘R,‘A-‘).s
(42)
775
Suffkient condition for plastic flow localization
W = (AA-‘),
+ (Ali,R,‘A-‘),
for the total spin. The plastic parts of the rates of deformation
+ (AR,i’P-‘R,‘A-‘)a
(43)
and spin are as follows:
M = (Ali,R,‘A-‘),
+ (AR,f’P-‘R,‘A-‘),,
(44)
Wp = (Al$,,R,‘A-‘),
+ (AR,i’P-‘R,‘A-‘)a,
(45)
and from (39) we have I’y
=
x
+‘“‘(I
+ ppDM$a)N-‘(a))N(a’)
(46)
+ ppDMha)fi - ‘ca))(nw,
(47)
(4
wp = 2 +ca)(l (4
where M$Cd =
;(M’d
+
M(d),
N’“’ = ‘( (a) @ @’ tg
@a’
=
+ II’“’ 63 g’“‘),
Thus, the total rate of deformation
$(M(‘d
fp)
_
Mb)T),
= b(g’“’ @ “‘“’
and the total spin are decomposed
_ “b’ @ g’“‘).
(48)
as follows:
D = D’ + Dp,
(49)
w = W’ + wp,
(50)
where D’ = (AA-‘),,
W’ = (AA-‘),.
(51)
By this decomposition W and Wp are interpreted physically as those parts of the rates of deformation and spin, respectively, which are produced by the translational and rotational modes of deformation typical for the large plastic strains in crystalline solids. The elementary carriers of the translational mode are dislocations moving in the active slip systems, determined by the current lattice direction g’“’ in a plane with current normal da), whereas the elementary carriers of the rotational deformation mode are partial disclination dipoles. Movement of the partial disclination dipoles results in reciprocal rotations of individual microvolumes and displacement of certain parts of the body with respect to others. The motion of these partial disclination dipoles is connected with a cooperative glide in the dislocation systems in the active zone ahead of the partial disclination dipole. This effect is represented in eqns (46) and (47) by the tensor MC”) and the scalar density of the partial disclination dipoles PPD. Equation (38) can be interpreted, owing to (40), (49) and (50), as the expression for the rate of residual distortion which remains on the local elastic unloading. The elastic constitutive relation is expressed as follows [35]:
ae+ u
tr (D’) = L . De,
(52)
where uve -- b - W’a
+ UW’
(53)
is the Zaremba-Jauman rate of Cauchy stress as seen by an observer who rotates with the lattice according to W’, L is the fourth-rank tensor of the elastic moduli, (L * De)ij = LijklD& and (uWe)ij = UisWzja It is assumed here that elastic properties are unaffected by slip.
R. B. PFCHERSKI
116
The material rotational
rate of Cauchy stress, on the other hand, is given by ; = o - Wa + aw.
The difference
(54)
between (53) and (54) is, in view of (47), (55)
where p@) = (1 + ppDM;m)fi-
l(a))@a)u
-
~(1
+
ppDM;a)(n-
l’a))&-fol).
(56)
The tensor pea) consists of two parts: (57) where (58) corresponds
to the relative rotation of the material and underlying lattice [35], and
P!? =
-
PPDM,~
(59)
flp~UM,
pertains to the local rotation of the material volume together with the crystal lattice, with respect to its neighborhood. The tensor Pm is responsible for the nonuniformity of these relative rotations produced by propagation of partial disclination dipoles of density PPD. Note that, if ppD vanishes, (56) transforms into the relation (58) discussed earlier by Asaro [35]. On the other hand, when there is no relative rotation of the material and underlying lattice and when ppD # 0, the tensor PO vanishes and in eqn (57) the second term only remains. Combining (46), (52) and (56) with an account of the decomposition (49) yields g + u tr (D) = L * D -
x (It(“) + ppDK(OL))jAU), (a)
(60)
where R(a)
Let us observe Asaro [35]:
=
L
.
N(U)
+
f&a’,
K(a)
=
M’“’
+ s
that for ppD = 0, eqns (61) transform
g + u tr (D) = L . D -
To complete the constitutive The following viscoplastic
M(du u
_
&fk-d
a
into the well-known
x R(a)j(a), (a)
*
(61) form derived by
(62)
law, the specification of the shear strain rates +‘“I is necessary. relation in a single glide system is considered [21, 361:
(63)
where q is the viscosity parameter, $, is the quasi-static strain rate pertinent to the athermal strength 7P, 7 is the resolved stress, A(T,J denotes the material function and Q(e) is known from the theory of viscoplasticity as the excess stress function [37], with the property a(O) = 0. The
111
Sufficient condition for plastic flow localization
symbol (Q(n)) is defined as follows:
The relation of viscoplastic glide in a single slip system, derived from the physical considerations presented in [36], gives the uniform description of plastic deformation in an extensive range of shear strain rates and encompasses rate-sensitive as well as rate-independent flows of material. The limit case when r + rcr is discussed in [21] and [36]. When we apply (63) to (60), the following constitutive equation of finite elastic-viscoplastic deformations with lattice misorientation can be obtained: 8 + u tr (D) = L . D - x (R(“) + Pi&) (U)
1 ‘“)“’o2,I &
[A(+?
(F
- I)]),
(65)
where Z2 is the second invariant of the visoplastic rate of deformation tensor and lo1 is called the quasi-static value of the viscoplastic rate of deformation measure. The constitutive equation (65) should be supplemented by the evolution equation for strain hardening, i.e. for the structural parameter T’,“‘. It is assumed that, similar to the rate-independent approach, the evolution equation has the form [24] ;f’
=
x
h(a@j@),
(66)
(8)
where the hcQ8)are the physical slip system hardening moduli depending on y = x y’“). (a) For pPD = 0, the constitutive equation (65) transforms into 8+
u tr (D) = L - D -
x R@) 1 _“;’ o2,I 2 (a (01)
[a($?)
(y
-
I)]),
(67)
which differs from that discussed by Asaro [35] in the specification of the viscoplastic relation for Jo only. The limit transition when u * N + T’,“’and the transformation of (65) into the constitutive equation describing rate-independent plastic flow with lattice disorientation are considered in a separate paper [38]. 5. SUFFICIENT CONDITION FOR PLASTIC FLOW LOCALIZATION
The constitutive relation (65) contains the term which is responsible for high lattice misorientation due to the development of the terminated boundaries of density PPD. The development of nonuniform lattice rotations causes a geometrical softening of the predominant slip systems. Under the concept of geometrical softening of a slip system, we understand, after Lisiecki et al. [ 193, a rotation of the lattice that produces an increase in the resolved shear stress in a fixed overall stress state. In the case of viscoplastic material, where the resolved shear stress governs the rate of plastic shearing on each slip system, geometrical softening leads to a higher shear strain rate on the softening system. Thus, as noted by Lisiecki et al. [19] and Asaro [24, 351, geometrical softening produces intensification of the shear strains on that system and thus acts as an orientation imperfection which triggers localized shearing. The existence of the orientation imperfection can be considered a sufficient condition for localization, Peirce, Asaro and Needleman considered rate-independent [18], as well as rate-dependent 1391, tinite-element modelling of nonuniform deformation modes in ductile single crystals. The nonuniformity, introduced by the assumption of surface geometrical imperfection, led to necking, which was shown to cause nonuniform lattice rotations and geometrical softening that triggered localized shearing. Thus, in that case the orientation imperfection in the crystal had
718
R. B. PECHERSKI
its origin in the geometrical imperfection of its surface. Equation (67) pertains to such a situation. In our case, however, the orientation imperfection can be produced by development of a critical material structure in the deformed crystal, irrespective of the effects of surface irregularities. This is represented in (65) by the nonuniform distribution of the density of partial disclination dipoles (terminated boundaries) PPD. Certain critical values of ppD appear to be sufficient to produce, in a microvolume of the cystal, a relative lattice rotation such that this microvolume can play the role of orientation imperfection from which the necking and localized shearing develop. Thus, the constitutive equation (63, with a given distribution function of critical values of the density of the partial disclination dipoles F(p&), can be used for the modelling of nonuniform deformation modes in ductile crystals and the development of deformation instability: g + u tr (D) = L - D - 2 [R’“’ + F(p&)K’“‘]
(a)
5 02
(@
[ACT:),
(9
-
I)]).
(68)
2
Let us note that the tensor K’“’ can be specified, for a given state of stress and at least for the simplified form of the evolution equation (29), from the known physical data, and the only function which should be additionaly determined is the distribution function F( pj&). Acknowledgemenrs-The author is indebted to Professor Erwin Stein for his kind hospitality at Hannover University, where this paper has been written. The support of the Alexander von Humboldt Foundation is gratefully acknowledged.
REFERENCES HI J. R. Rice, The localization of plastic deformation. Proc. 14th IUTAM Congress, Delft (1976); Theoretical and Applied Mechanics (Edited by W. T. Koiter). North-Holland, Amsterdam (1977). PI K. W. Neale, Phenomenological constitutive law in finite plasticity. S. M. Arch. 6, 79-128 (1981). [31 H. Petryk, On the onset of instability in elastic-plastic solids. Proc. Symp. on Current Trends and Results in Plasticity, Udine (1983); Plasticify Today (Edited by A. Sawczuk). Applied Science Publishers, London (in press). [41 L. Anand and W. A. Spitzig. Initiation of localized shear bands in plane strain. J. Mech. Phys. Solids 28, 113128 (1980). [51 K. W. Neale and S. C. Shrivastava, Finite elastoplastic torsion of a circular bar, Engng Fracture Mech. This issue, pp. 00-00. [61 A. S. Keh and S. Weissmann, Deformation substructure in body-centered cubic metals, in Electron Microscopy and the Strength ofCrystals, p. 231. WileyiInterscience, New York (1963). 171 J. Gill Sevillano, P. Van Houtte and E. Aernoudt, Large strain work-hardening and textures, in Progress in Materials Science (Edited by J. W. Christian, P. Haasen and T. B. Massalski), Vol. 25, p. 69. Pergamon Press. Oxford (1982). [81 A. E. Romanov and V. I. Vladimirov, Disclinations in solids. Phys. Stat. Sol. (a) 78, 1l-34 (1983). [91 A. N. Vergazov, V. A. Likhachev and V. V. Rybin, Characteristic elements of the dislocation structure in deformed polycrystalline molybdenum. Phys. Met. Metall. 42, 126-133 (1977). [lOI U. F. Kocks, T. Hasegawa and R. 0. Scattergood, On the origin of cell walls and of lattice misorientation during deformation. Scripta Metnll. 14, 449-454 (1980). [Ill H. Chandra, J. D. Embury and U. F. Kocks, On the formation of high angle grain boundaries during the deformation of aluminum single crystals. Scripta Metall. 16, 493-497 (1982). in cold worked metals. Proc. 6th Ini. Conf. on Strength of [12] F. J. Torrealdea and J. Gil Sevillano, “Microbands” Metals and Alloys, Melbourne (Edited by R. Gifkins), Vol. II. p. 547. Pergamon Press, Oxford (1982). [13] A. S. Rubtsov and V. V. Rybin, Structural features of plastic deformation at the stage of localized flow. Phys. Met. Metall. 44, 139-149 (1978). [14] V. V. Rvbin. Phvsical model of the phenomenon
of loss of mechanical stability and necking. Phys. Met. Metall. 44, 149-157 (1978). u51 L. G. Orlov and G. F. Shitikova, Structure changes during necking in monocrystalline silicon iron under tension. Phys. Met. Metall. 52, 169-173 (1981).
[I61 S. Saimoto, W. F. Hosford, Jr. and W. A. Backofen, Ductile fracture in copper single crystals. Phil. Mug. 12, 319-333
(1965).
[I71 Y. W. Chang and R. J. Asaro, An experimental Acfa Metall. 29, 241-257
study of shear localization in aluminum-copper
single crystals.
(1981).
[I81 D. Peirce, R. J. Asaro and A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, 1087-l 119 (1982). [191 L. L. Lisiecki, D. 0. Nelson and R. J. Asaro, Lattice rotations, necking and localized deformation in fee single crystals. Scripta Metall. 16, 441-448 (1982). WI K. S. Havner, On the mechanics of crystalline solids. J. Mech. Phys. Solids 21, 383-394 (1973). ..^ Pll R. B. Ppcherski, Relation of microscopic observations to constitutive modelhng tor advanced detormauons ana fracture initiation of viscoplastic materials. Arch. Mech. 35, 257-277 (1983). [22] J. Mandel, Relation de compartement des milieux elastiques-plastiques et elastiques-viscoplastiques. Notion de repere directeur, in Foundations ofPlasticity (Edited by A. Sawczuk), p. 387. Nordhoff, Leyden (1972).
Sufficient condition for plastic flow localization
119
[23] R. Hill and K. S. Havner, Perspectives in the mechanics of elasto-plastic crystals. J. Mech. Whys. So/ids 30, 522 (1982). [24] R. J. Asaro, Micromechanics of crystals and polycrystals, in Advances of Applied Mechanics, Vol. 23 (in press). [25] J. D. Eshelby, Elastic inclusions and inhomogeneities, in Progress in Solid Mechanics (Edited by I. N. Sneddon and R. Hill), Vol. II. North Holland, Amsterdam (1961). [26] R. J. Asaro, Somigliana dislocations and internal stresses with application to second phase hardening. Znt. J. Engng Sci. 13, 271-286 (1975). [27] A. Ye. Volkov, V. A. Likhachev 56-64 (1979) ( in Russian).
and C. S. Shikhobalov, Izv. Akad. Nauk SSSR, Mekhanika Twerdogo Tela 10,
[28] B. A. Bilby, Continuous distribution of dislocations, in Progress in Solid Mechanics (Edited by I. N. Sneddon and R. Hill), Vol. I. North Holland, Amsterdam (1964). [29] E. Kriiner, Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin-Giittingen-Heidelberg (1958). [30] R. W. Lardner, Mathematical Theory of Dislocations and Fracture. University of Toronto Press (1974). [31] E. H. Lee and D. T. Liu, Finite-strain elastic plastic theory with application to plane wave analysis. J. Appl. Phys. 38, 19-27 (1967). [32] E. KrGner and C. Teodosiu,
[33] [34] [35]
[36] [37] [38] [39]
Lattice defect approach to plasticity and viscoplasticity, in Problems ofPlasticity (Edited by A. Sawczuk), p. 45. Noordhoff, Leyden (1974). V. I. Vladimirov and A. E. Romanov, Movement of partial disclinations dipole by plastic deformation. So/id State Physics 20, 3114-3116 (1978) (in Russian). A. Ye. Volkov, V. A. Likhachev and Shikhobalov, Theory of grain boundaries as autonomous imperfections of a crystal. Phys. Met. Metall. 47, l-12 (1980). R. J. Asaro, Micro and macromechanics of crystalline plasticity, in Plasticity of Metals at Finite Strain: Theory, Computation and Experiment, Proc. Research Workshop, Stanford, 1981 (Edited by E. H. Lee and R. L. Mallett). Division of Applied Mechanics, Stanford University, Stanford, California, and Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, New York (1982). P. Perzyna and R. B. Pccherski, Modified theory of viscoplasticity. Physical foundations and identification of material functions for advanced strain. Arch. Mech. 35, 423-436 (1983). P. Perzyna, Fundamental problems of viscoplasticity. Advan. Appl. Mech. 9, 243-377 (1966). R. B. Pccherski,Constitutive modelling for advanced strains and deformation instability of crystals (in preparation). D. Pierce, R. J. Asaro and A. Needleman, Material rate dependence and localized deformation in crystalline solids. Report of Division of Engineering, Brown University, Providence, Rhode Island. Research Grant NSF/MEA8101948/2 and Research Grant NSF/DMR79-26190/4, April (1983).
Engineering Fracture Mechanics Vol. 21, No. 4, pp. 761-179. 1985
Printedin the U.S.A.
DISCUSSION OF SUFFICIENT CONDITION PLASTIC FLOW LOCALIZATION
FOR
RYSZARD B. PECHERSKI Institute of Fundamental Technological Research, Warsaw, Poland? Abstract-The strong sensitivity of the predicted deformation instability initiation to the different finite-strain Jz theories commonly applied in studies of deformation instability exhibits the weakness of these theories and reveals the necessity for constitutivemodellingcorresponding to key physical mechanisms of large plastic strains, viz. formation of cell structure, fragmentation and development of terminated boundaries which produce large rotations of adjacent microvolumes. The aim of this paper is to incorporate these mechanisms into the continuum model of finite elastic-viscoplastic deformations of a single crystal and discuss a sufficient criterion for localization. The effect of the relative rotations of the crystal lattice is modelled by means of Somigliana dislocation, whereas development of terminated boundaries is described in terms of the movement of partial disclination dipoles. The existence of an “orientation” imperfection is considered a sufficient condition for localization. The orientation imperfection is represented by a critical value of the scalar density of partial disclination dipoles, which produce lattice misorientation and “geometrical softening” sufficient for the onset of localization.
1. INTRODUCTION THE
studies of plastic strain localization reviewed by Rice [I] are based mainly on Jz-type flow theory with smooth yield surface, deformation theory or theories taking into account such “destabilizing” effects as yield surface vertices, deviation from plastic “normality” and the dilatational plastic flow due to nucleation and growth of voids (cf. [21). This approach provides satisfactory qualitative predictions of plastic flow instabilities based on the necessary condition expressed as the loss of ellipticity of the governing incremental equations. However, quantitatively, the necessary condition leads to the upper bounds of the critical stress and strain or the lower bound of the strain hardening rate.S This could explain, at least in part, the discrepancy with experimental results discussed by Anand and Spitzig [4]. Furthermore, applications of different models of plastic flow for analyzing instability phenomena reveal strong sensitivity of the predicted deformation instability initiation to the constitutive relations adopted. This fact has been pointed out by Neale and Schrivastava [5]; it exhibits the weakness of existing finite-strain J2 theories commonly applied in studies of deformation instability and shows the importance of proper constitutive modelling for advanced strains. Adequate constitutive modelling for advanced strains and localization should correspond to the key physical mechanisms responsible for large inelastic strains and the onset of localization, viz. formation of dislocation cell structure with misorientation of neighboring cells, fragmentation of crystalline lattice and further development of bands propagating from the boundaries and terminating abruptly within the cell or fragment interior. The bands turn the crystal lattice by equal angles of opposite signs. The latter mechanism produces very high lattice misorientation. This can lead to achange of Schmid factors, in certain volumes of crystal, such that local “geometrical softening” occurs, triggering the onset of plastic flow instability. The aim of this paper is to identify and describe theoretically the most important properties of the above-mentioned mechanisms, incorporate them into the continuum model of finite elastic-viscoplastic deformations of a single crystal at advanced strains and formulate a pertinent criterion for plastic flow localization. The criterion is the sufficient one, for it derives from physical mechanisms originating at the onset of localization. The effect of the relative rotations of the crystal lattice in the co&se of the development of the cellular and fragmented ton leave at the Institut fiir Baumechanik und Numerische Mechanik, Universitlt Hannover, as a Felloi of the Alexander von Humboldt Foundation. $4 similar conclusion can be drawn from the results obtained by Pctryk [31, who has proved that the mentioned critenon for localization provides an upper bound to the onset of instability of the deformation process in the energy sense.
768
R. B. Pf$CHERSKI
structure is modelled by means of Somigliana dislocation, whereas the critical deformation instability effect of the formation of terminated bands is described in terms of the movement of partial disclination dipoles. The novel result arrived at is the incorporation of these new effects into the continuum theory of finite inelastic deformations. 2. AN OUTLINE OF PHYSICAL FOUNDATIONS Recently, there have become available a number of thorough review works in which a vast quantity of experimental results and data relating to the evolution of the microstructure in the course of large plastic strains of different metals is systematically analyzed, e.g. [6-81. Therefore, only the main conclusions necessary to the considerations of this paper will be discussed here. Our considerations are further limited to the behavior of metals at moderate temperatures where the mechanism of plastic glide prevails. One of the most important microscopic mechanisms responsible for the accommodation of large strains is the formation of a dislocation cell structure. At an earlier stage of deformation the average cellular misorientation is rather small, and the glide dislocations can cut through nearly all of the cell walls. At higher strains, however, some cell walls become impenetrable and form a new substructure. The network of impenetrable walls separates volumes of high lattice misorientation. The process of new subdivision is called the fragmentation of the crystalline lattice, and the volume elements are the fragments. It has been found that one of the main ways in which fragmentation develops is through the formation of band structures propagating from the cell or fragment boundary into the interior. Of great importance here, as noted by Vergazov et al. [9], is the fact that the neighboring boundaries of the bands terminate abruptly within the cell or fragment interior and turn the crystal lattice by equal angles of opposite signs. Similar phenomena have been analyzed by Kocks et al. [IO]. They have discussed pairs of subboundaries of opposite sign which produce large lattice misorientation and argued that the material volume containing such pairs of polarized subboundaries plays the role of small “nuclei” from which “shear bands” observed in free compression and in rolling might originate. Also Chandra et al. [l l] have found that unstable oriented crystals reveal fragmented bands of high o~entation. Torrealdea and Gil Sevillano 1121have analyzed similar structures of “double walled parallelogram-type cellular dislocation arrays forming banded structures,” called by them “microbands.” They have also noted that the formation of structures is assigned to localized unstable shear flow. The aforementioned mechanisms for the formation of terminated bands and polarized pairs of subboundaries and fragmented bands, as well as the “microbands,” are of similar character and can be interpreted by means of some cooperative forms of highly concentrated dislocation motions and the onset of collective degrees of freedom. This produces the local rotations of material volumes that eventually lead to the development of partial disclination dipoles and loops; cf. Rubtsov and Rybin [13], Rybin [14] and Orlov and Shitikova 1151, as well as Romanov and Vladimirov [8]. These authors consider the formation of partial disclination dipoles and loops as a critical state, very characteristic of the onset of plastic flow localization. On the other hand, the importance of the inhomogeneous lattice rotations for the onset of necking and shear band formation in single crystals have been recognized by Saimoto et al. [16], Chang and Asaro [17] and Peirce et al. [18] as well as by Lisiecki et al. 1191. Summing up the above discussion, we can note that the formation of partial disclination dipoles and loops produces very high lattice misorientation, which in certain volumes of crystal can lead to such a change of Schmid factors that local “geometrical softening” of the predominant slip systems operating in these volumes occurs (cf. 1191). 3. CONTINUUM MODEL OF FINITE INELASTIC DEFORMATIONS LATTICE MISORIENTATION
WITH HIGH
Let us consider a crystalline material as a continuous body. By the material point X of this body we understand a certain minimum lattice volume V which is sufficient for a valid continuum mechanics description of gross elastic-viscoplastic behavior (cf. 120, 211). Furthermore, as considered earlier by Mandel [22], it is assumed that the dominant lattice orientation in the
Sufficient condition for plastic flow localization
169
volume V is represented by the triad of director vectors ui, i = 1, 2, 3, attached to the material point x. Let us assume that the triad ui is orthonormal and remains the same throughout the body in the reference configuration K. Then it transforms into the piecewise continuous vector field of elastically deformed and misoriented director vectors e;, i = 1, 2, 3, determined on the current configuration + of the body. A comprehensive review of the main ideas concerning the continuum theory of finite inelastic deformations of crystalline materials with pertinent references was recently provided by Hill and Havner [23] and Asaro [24] (cf. also [21]). It is commonly assumed that dislocations traversing a volume element produce a change in its shape but not in its lattice orientation. This means that the material moves with respect to its underlying crystal lattice, whereas the lattice itself only undergoes elastic deformation. In our case the situation is different because advanced deformations produce, locally, high lattice misorientation because of the fragmentation process. The concept of Somigliana dislocation will be used to describe theoretically the fragmented structure of a strongly deformed single crystal near the onset of localization. The importance of Somigliana dislocation for the theoretical modelling of different kinds of interfaces within a crystal have been recognized by Eshelby [25] and Asaro [261, as well as by Volkov et al. [271. A Somigliana dislocation can be constructed as follows (cf. [25]): make a cut over a surface S and give the two faces of the cut an arbitrarily small relative displacement, removing material where there would be interpenetration, till in the gaps and weld the material together again. Let B(x) be the relative displacement at the point x of the cut. Then we have a Somigliana dislocation over S specified by the displacement jump B(x). Let us assume that, in the current configuration 4 of the continuous body, the vector fields ei, i = 1, 2, 3, suffer discontinuity on certain surfaces. These surfaces can be interpreted by means of Somigliana dislocations and they describe theoretically the formation of the fragmented structure. Furthermore, three groups of translational dislocations within the body are assumed (cf. [28, 291): those which leave it or are annihilated in it, those which remain continuously distributed in it and those which remain on the surface of the discontinuity. The latter corresponds to the known interpretation of the Somigliana dislocation as infinitesimal translational dislocations continuously distributed on the surface S. Movement of translational dislocations produces plastic deformation, whereas their accumulation on the boundaries contributes to misorientation. The question then arises as to how to incorporate this new effect into the continuum theory of finite inelastic deformations of a single crystal. The extension of the notion of Burgers circuit for disclinations, discussed by Lardner [30] appears to be helpful in solving this problem. Let us consider the reference configuration K and the current configuration + produced during the process of inelastic deformation under applied loads. Let us then choose a certain neighborhood N(X) of a point X with position vector dX and the triad ui. This neighborhood transforms into the neighborhood of the point x with the position vector dx = F dX, where F is the deformation gradient with distorted triad ei. Let us conduct through point x a surface ,S(t) with an outer normal N and the boundary L(t). Then choose a certain piecewise smooth Burgers circuit % piercing the surface s(t) at x. Let us suppose that a local unloading configuration 4 exist for each neighborhood of the point from the circuit % and that such a neighborhood can be chosen in a continuous way. If two neighborhoods overlap, then we require that two local unloading configurations be fitted together without any free rotation. Starting with the orientation for the local unloading configuration I&, yhich is chosen in the same manner as the reference configuration, we can now construct the sequence of unloading configurations at points along the circuit %. The question now arises whether the unloading configuration that we end up with, 4, is the same as the one we started out with, I$. If the configuration is not the same, the change in making a circuit can only consist of a closure failure and a relative rigid body rotation, for the local unloading configurations are, by definition, in the natural state. In such a case the surface s(t) corresponds to Somigliana dislocation. The closure failure pertains to the total translational Burgers vector produced by the Somigliana dislocation and the translational dislocations distributed continuously in the body, which are enclosed by circuit % [3O]: &We, x, t> = 4x, t)n dS,
(1)
R. B. PFCHERSKI
110
where dS is enclosed by surface % with the normal n, and the total dislocation density tensor ar(x, t) is determined as follows: a(x, t) =
2 p”‘(x, r)b”’ CT’(“‘,
(21
Ci)
The symbol p(“(x, t) denotes the scalar dislocation density of the ith type, b”’ is the Burgers vector and tCn the direction of the transIationa1 dislocation line of the ith type. The scalar dislocation density p’” is understood as the total length of the dislocation lines of the ith type averaged over a certain volume of crystalline material. The relative rigid body rotation, on the other hand, pertains to the lattice misorientation at the point x determined by the orthogonal tensor R,{(e): ei = R,(S)&,
i = 1, 2, 3,
(3)
where @iremains parallel to ui. In accordance with the procedure discussed for circuit “e, the deformation gradient F can be decomposed in the following way [21]: F = AR,(%)P,
(4)
where the tensors A and P correspond to elastic and viscoplastic distortions. If making the circuit % produces closure failure only, without rigid body rotation, the Somigliana dislocation degenerates into a dislocation of the translational type and decomposition (4) takes the form of that proposed originally by Lee and Liu 1311and Mandel [22]: F = AP.
0)
Let us consider the deformation of the neighborhood N(X) with the position vector dX. When, in the course of the glide process the translation dislocations with the Burgers vector db, pass through the vector dX, the latter is mapped into its intermediate configuration dP: dff = dX + dh,
(6)
or, with an account of the transformation, df = PdX, dl;, = (P - 1) dX.
(7) (8)
Then the local relative rigid body rotation transforms the vector dX into the con~guration d%as follows: dir = R,,(%)P dX (9) and dt;, = d% - df = (R,(q) - l)P dX,
(10)
where db, is the increment of the Burgers vector produced by the local relative lattice rotation at the point x determined by the misorientation tensor R,(q). Thus, the total increment of the Burgers vector dB, is given by dB, = dl;, + dt;,.
(11)
dB, = (R,(%)P - 1) dX
(12)
Owing to (9) and (6) and (7), we have
771
Sufficient condition for plastic flow localization
and the rate of the Burgers vector increment
db, = or, owing to transformation figuration I&is obtained:
(R,(%)P
dB,: +
R,(%)i') dX;
(9), the following relation corresponding
dR, = (R,(%)R,
’ (Se) + R,(%)P P-
(13) to the intermediate
'R,'(Se)) d6.
con-
(14)
This relates the rate of the total Burgers vector increment dR, to continuum measures of inelastic distortion and lattice misorientation rates. On the other hand, vector dR, is determined by the microscopic mechanisms of inelastic glide in active slip systems and lattice misorientation induced by the fragmentation process. Thus, according to Kroner and Teodosiu [32], db, = c j(a)g(u) @ n’a’ dff, (u)
(15)
where +@) is the inelastic shear strain rate in the cxth active slip system and gCcr),iiCa)denote, respectively, the glide direction and normal to the slip plane of the &h slip system in the intermediate configuration $. The rate of the Burgers vector increment dR, can be expressed by means of a moving Somigliana dislocation, Applying the derivation of Volkov et al. [27], we have dB, =
ah
{[(V x T)@BB,]G(L(~)) + (N *V)V,B,+ NC3 z
- V. VsBt)]WW)}
di (16)
where V is the velocity of Somigliana dislocation, T is the unit tangent vector to the boundary L(t), the operator V, denotes the surface gradient and @L(t)), s(s(t>) are generalized 6 functions concentrated on the line L(t) and surface S(t). Furthermore, dB/dt denotes the rate of change of the Burgers vector B on the surface S(t). Equation (16) holds for any motion of Somigliana dislocation accompanied by any change in B and can be considered the starting point for the theoretical description of many particular situations, e.g. plastic glide, boundary sliding and mobility of boundaries, as well as development of fragmented structure and terminated bands. The latter can be modelled as a movement of partial disclinations dipole. A simple two-dimensional model of straight partial disclination dipoles of the wedge type was considered by Vladimirov and Romanov [33]. The dipole is represented by two parallel semiplanes limited by straight disclination lines. Physically, this planes can be interpreted as tilt boundaries formed by edge dislocation lines of opposite signs and parallel Burgers vectors. The authors have related the velocity of partial disclination dipole Vd with the rearrangement of edge dislocations at the front of the dipole: Vd =
- Sph KAY*) dy* A, In (1 - wlbpo,h,)
’
QJ < bpo,L,
(17)
where g, h determine the region of nonzero edge dislocation velocity u,, A, is the mean free path of the edge dislocation, b is the length of the Burgers vector, pee is the density of the edge dislocations and 6.1denotes the power of the partial wedge disclination corresponding to the misorientation angle of the tilt boundary. Equation (17) can be transformed as follows:
v, =
-
jbl
peb2h, In (1 - olbp,oA,)
- I’
(18)
where (19)
772
R. B, P~CHERSKl
‘i?,
is the mean value of the dislocation velocity and if and i(* are the shear strain rate with respect to a fixed and a moving coordinate system, respectively. Furthermore, (20)
corresponds to the mean fragment diameter. The physical model discussed above can be related to the effect of the moving Somigliana dislocation given by (16). Let us consider, then, the movement of a ham-plane with normal N, nonstation~y Burgers vector B(x, t) and velocity Vd, which is perpendicular to N. In such a case (16) takes the form [34] $
- Vd. V,B,
&S(t))
dx.
(21)
According to (1) and (2), we have B,(%) = c p@)(@@. n)b(“’ dS, (OLJ
(22)
where ptu) is the scalar density of the translational dislocations determined in the domain 93 x [O, d,] containing the surface s(t), where 93 is the region in the Euclidean space occupied by the crystalline body and dp is the duration of the deformation process. According to the interpretation of the two-dimensional model of straight partial disclination dipoles as the tilt boundaries formed by edge dislocations lines of parallel Burgers vectors, &(“)+ n = 1 and B,(X) = 2 p@)bfafdS. (a)
(23)
From (23) and the theory of moving discontinuity surfaces in continuum, the following relations can be derived under the assumption that the functions p’“’are piecewise smooth in the domain 93 x [O, dp] and have limits on the discontinuity surface s(t):
(z -Vd)]dS &S(t)), -
(1 - N @ N)
1
dS ~(~(f)),
(24) c-3)
where B = B, s(s(t>) denotes the total Burgers vector of the translational dislocations distributed on the surface s(t). The specification of the relation (21) is based on the following experimental observations. Rubtsov and Rybin 1131revealed that in the tensile test of highly work hardened crystals of nickel and molybdenum the number of partial disclinations increases monotonically as necking develops. This produces the so-called knife edge high-angle (w 2 IO”)terminated boundaries, which are oriented almost precisely along the axis of tension and not at an angle of 45” to the axis, as one might expect from traces of slip bands. Such boundaries are the traces of disclination dipoles and usually have a large tilt component. The term “knife edge” was used because the boundaries have practically no width on foils perpendicular to the axis of tension. These boundaries are described theoretically by means of the moving half-plane with normal N, nonstationary Burgers vector B(x, t) and velocity Vd. Rybin [14] argued that vector N rotates until the rotations of two neighboring fragments have been completed. In the laboratory reference frame, however, there is a set of stable orientations of N. They correspond to the boundaries in which disclinations do not experience forces parallel to the normal N. Only along these planes is it possible for wedge disclinations to move under the applied stress. This reasoning justifies
773
Suffkient condition for plastic flow localization
the hypothesis
that in the case of complex stress states in the principal directions
where u1 > u2 2 u3, that is, the planes of partial disclination dipoles are parallel to the maximum principal stress, the velocity of partial disclination dipoles is as follows: Vd
(27)
VdA,,
=
whereas the normal N lies in the plane determined by the principal directions A2 and X3. Relation (21) can be transformed, after simple calculations taking into account eqns (23)(27), into the following form: &,
= c
rj’“’
_
$
(4
vd
>
* jj’“‘)
b'"'N ‘8 gcp) - v#)(Al
+
ap) y-g- N 8 Al 1
a+*)
FN@A,
ap
+ =N@A3
2
3
(1 - N@N) >
1
dSS(S)d%,
(28)
where the derivatives @‘“‘laxi are expressed in the Cartesian coordinate system ki, which is chosen in such a way that ki = Xi when the principal directions Xi are determined uniquely or kl = A1 only, when u2 = u3. In the simple model proposed in [33], which relates the movement of partial disclination dipoles to rearrangement of edge dislocations, the gradient of p’“’ in the direction Ai plays the decisive role. Therefore, owing to (18) expressed for the (Yslip systems acting in the critical zone ahead of the moving partial disclination dipoles and the evolution equation [21] i)(cr)
=
Gb)(#@),
p’P’,
Tr))+‘“‘,
(2%
the relation (28) yields dd, = c (./t/lia’N %Igca) + J#“‘N @ Al)+‘“’ dS s(S(t)) dST, (a)
(30)
where
(31)
and T’,) is the athermal strength in the p slip system. Let us assume that there are many dipoles of partial disclination expanding along the surfaces with the normal unit vectors N in a certain volume AV. The averaging procedure over the volume AV leads to the following relation: (32) where xl”’ and xSa) represent the mean values of the material functions A\“’ and J4iaa, over the volume AV, and N is the average direction of the normal vectors N, whereas
PPD
=
S(t)
F
(33)
714
R. B. PQCHERSKI
is the scalar density of partial disclination dipoles representing, boundaries. From (10) and (32) one can obtain R,(%)R,‘(%)
+ R,(%)PP-‘R,‘(%)
physically, density of terminated
- PP-‘R,‘((e>
= x M(“‘p&@) (U)
dS,
(34)
where (35) Dividing both sides of (35) by dS and assuming that the limit iirn, b
(R,(%)R;‘(%)
+ R,(%)PP-‘R,‘(Z)
- PP-‘R;‘(%)> = &R,’
+ R,PP-‘R,’
-
PP-‘R,’
(36)
exists, we have + R,pp-‘R,
fi,R,’
1 -
tip-‘R,’
=
c
R;z’“‘ppDj(u),
(37)
(a) where R,,, is the misorientation tensor independent Ma) corresponds to the configuration 4. 4. CONSTITUTIVE
of the choice of circuit %, and tensor
MODEL OF ADVANCED VISCOPLASTIC LATTICE MISORIENTATION
DEFORMATIONS
WITH
From (1 l), (14), (15) and (37) the relation between continuum measures of inelastic deformations to lattice misorientation and pertinent microscopic mechanisms can be obtained in the following form: R,R,’
+ R,,$P-
In the current configuration AR,R,*A-i
‘R, ’ = 2 +@)(gca) @Ificu) + ppD&a))a (a)
+ (38) transforms
(38)
as follows:
+ AR,PP-‘R;‘A-’
= x jca)(gca) @ n’“’ + ppDMcU)), (Cl)
(39)
where [18] (01)
g
=
A$“’
and
#X’
as well as M(“) = x\a’N @ gca) + xi% @ hi. The decomposition of the gradient of deformation velocity gradient: L = AA-’ Taking the symmetric
+ AR,R,‘A-’
and antisymmetric
D = (AA-‘), for the total rate of deformation
=
,j’“‘A-
1,
(40)
(4) leads to the following equation for the
+ AR,PP-‘R&IA-‘.
(41)
parts of (41), we obtain
+ (AR,R,‘A-‘),
+ (AR,PP-‘R,‘A-‘).s
(42)
775
Suffkient condition for plastic flow localization
W = (AA-‘),
+ (Ali,R,‘A-‘),
for the total spin. The plastic parts of the rates of deformation
+ (AR,i’P-‘R,‘A-‘)a
(43)
and spin are as follows:
M = (Ali,R,‘A-‘),
+ (AR,f’P-‘R,‘A-‘),,
(44)
Wp = (Al$,,R,‘A-‘),
+ (AR,i’P-‘R,‘A-‘)a,
(45)
and from (39) we have I’y
=
x
+‘“‘(I
+ ppDM$a)N-‘(a))N(a’)
(46)
+ ppDMha)fi - ‘ca))(nw,
(47)
(4
wp = 2 +ca)(l (4
where M$Cd =
;(M’d
+
M(d),
N’“’ = ‘( (a) @ @’ tg
@a’
=
+ II’“’ 63 g’“‘),
Thus, the total rate of deformation
$(M(‘d
fp)
_
Mb)T),
= b(g’“’ @ “‘“’
and the total spin are decomposed
_ “b’ @ g’“‘).
(48)
as follows:
D = D’ + Dp,
(49)
w = W’ + wp,
(50)
where D’ = (AA-‘),,
W’ = (AA-‘),.
(51)
By this decomposition W and Wp are interpreted physically as those parts of the rates of deformation and spin, respectively, which are produced by the translational and rotational modes of deformation typical for the large plastic strains in crystalline solids. The elementary carriers of the translational mode are dislocations moving in the active slip systems, determined by the current lattice direction g’“’ in a plane with current normal da), whereas the elementary carriers of the rotational deformation mode are partial disclination dipoles. Movement of the partial disclination dipoles results in reciprocal rotations of individual microvolumes and displacement of certain parts of the body with respect to others. The motion of these partial disclination dipoles is connected with a cooperative glide in the dislocation systems in the active zone ahead of the partial disclination dipole. This effect is represented in eqns (46) and (47) by the tensor MC”) and the scalar density of the partial disclination dipoles PPD. Equation (38) can be interpreted, owing to (40), (49) and (50), as the expression for the rate of residual distortion which remains on the local elastic unloading. The elastic constitutive relation is expressed as follows [35]:
ae+ u
tr (D’) = L . De,
(52)
where uve -- b - W’a
+ UW’
(53)
is the Zaremba-Jauman rate of Cauchy stress as seen by an observer who rotates with the lattice according to W’, L is the fourth-rank tensor of the elastic moduli, (L * De)ij = LijklD& and (uWe)ij = UisWzja It is assumed here that elastic properties are unaffected by slip.
R. B. PFCHERSKI
116
The material rotational
rate of Cauchy stress, on the other hand, is given by ; = o - Wa + aw.
The difference
(54)
between (53) and (54) is, in view of (47), (55)
where p@) = (1 + ppDM;m)fi-
l(a))@a)u
-
~(1
+
ppDM;a)(n-
l’a))&-fol).
(56)
The tensor pea) consists of two parts: (57) where (58) corresponds
to the relative rotation of the material and underlying lattice [35], and
P!? =
-
PPDM,~
(59)
flp~UM,
pertains to the local rotation of the material volume together with the crystal lattice, with respect to its neighborhood. The tensor Pm is responsible for the nonuniformity of these relative rotations produced by propagation of partial disclination dipoles of density PPD. Note that, if ppD vanishes, (56) transforms into the relation (58) discussed earlier by Asaro [35]. On the other hand, when there is no relative rotation of the material and underlying lattice and when ppD # 0, the tensor PO vanishes and in eqn (57) the second term only remains. Combining (46), (52) and (56) with an account of the decomposition (49) yields g + u tr (D) = L * D -
x (It(“) + ppDK(OL))jAU), (a)
(60)
where R(a)
Let us observe Asaro [35]:
=
L
.
N(U)
+
f&a’,
K(a)
=
M’“’
+ s
that for ppD = 0, eqns (61) transform
g + u tr (D) = L . D -
To complete the constitutive The following viscoplastic
M(du u
_
&fk-d
a
into the well-known
x R(a)j(a), (a)
*
(61) form derived by
(62)
law, the specification of the shear strain rates +‘“I is necessary. relation in a single glide system is considered [21, 361:
(63)
where q is the viscosity parameter, $, is the quasi-static strain rate pertinent to the athermal strength 7P, 7 is the resolved stress, A(T,J denotes the material function and Q(e) is known from the theory of viscoplasticity as the excess stress function [37], with the property a(O) = 0. The
111
Sufficient condition for plastic flow localization
symbol (Q(n)) is defined as follows:
The relation of viscoplastic glide in a single slip system, derived from the physical considerations presented in [36], gives the uniform description of plastic deformation in an extensive range of shear strain rates and encompasses rate-sensitive as well as rate-independent flows of material. The limit case when r + rcr is discussed in [21] and [36]. When we apply (63) to (60), the following constitutive equation of finite elastic-viscoplastic deformations with lattice misorientation can be obtained: 8 + u tr (D) = L . D - x (R(“) + Pi&) (U)
1 ‘“)“’o2,I &
[A(+?
(F
- I)]),
(65)
where Z2 is the second invariant of the visoplastic rate of deformation tensor and lo1 is called the quasi-static value of the viscoplastic rate of deformation measure. The constitutive equation (65) should be supplemented by the evolution equation for strain hardening, i.e. for the structural parameter T’,“‘. It is assumed that, similar to the rate-independent approach, the evolution equation has the form [24] ;f’
=
x
h(a@j@),
(66)
(8)
where the hcQ8)are the physical slip system hardening moduli depending on y = x y’“). (a) For pPD = 0, the constitutive equation (65) transforms into 8+
u tr (D) = L - D -
x R@) 1 _“;’ o2,I 2 (a (01)
[a($?)
(y
-
I)]),
(67)
which differs from that discussed by Asaro [35] in the specification of the viscoplastic relation for Jo only. The limit transition when u * N + T’,“’and the transformation of (65) into the constitutive equation describing rate-independent plastic flow with lattice disorientation are considered in a separate paper [38]. 5. SUFFICIENT CONDITION FOR PLASTIC FLOW LOCALIZATION
The constitutive relation (65) contains the term which is responsible for high lattice misorientation due to the development of the terminated boundaries of density PPD. The development of nonuniform lattice rotations causes a geometrical softening of the predominant slip systems. Under the concept of geometrical softening of a slip system, we understand, after Lisiecki et al. [ 193, a rotation of the lattice that produces an increase in the resolved shear stress in a fixed overall stress state. In the case of viscoplastic material, where the resolved shear stress governs the rate of plastic shearing on each slip system, geometrical softening leads to a higher shear strain rate on the softening system. Thus, as noted by Lisiecki et al. [19] and Asaro [24, 351, geometrical softening produces intensification of the shear strains on that system and thus acts as an orientation imperfection which triggers localized shearing. The existence of the orientation imperfection can be considered a sufficient condition for localization, Peirce, Asaro and Needleman considered rate-independent [18], as well as rate-dependent 1391, tinite-element modelling of nonuniform deformation modes in ductile single crystals. The nonuniformity, introduced by the assumption of surface geometrical imperfection, led to necking, which was shown to cause nonuniform lattice rotations and geometrical softening that triggered localized shearing. Thus, in that case the orientation imperfection in the crystal had
718
R. B. PECHERSKI
its origin in the geometrical imperfection of its surface. Equation (67) pertains to such a situation. In our case, however, the orientation imperfection can be produced by development of a critical material structure in the deformed crystal, irrespective of the effects of surface irregularities. This is represented in (65) by the nonuniform distribution of the density of partial disclination dipoles (terminated boundaries) PPD. Certain critical values of ppD appear to be sufficient to produce, in a microvolume of the cystal, a relative lattice rotation such that this microvolume can play the role of orientation imperfection from which the necking and localized shearing develop. Thus, the constitutive equation (63, with a given distribution function of critical values of the density of the partial disclination dipoles F(p&), can be used for the modelling of nonuniform deformation modes in ductile crystals and the development of deformation instability: g + u tr (D) = L - D - 2 [R’“’ + F(p&)K’“‘]
(a)
5 02
(@
[ACT:),
(9
-
I)]).
(68)
2
Let us note that the tensor K’“’ can be specified, for a given state of stress and at least for the simplified form of the evolution equation (29), from the known physical data, and the only function which should be additionaly determined is the distribution function F( pj&). Acknowledgemenrs-The author is indebted to Professor Erwin Stein for his kind hospitality at Hannover University, where this paper has been written. The support of the Alexander von Humboldt Foundation is gratefully acknowledged.
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