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The Microscopic Theory of Cracks
CHAPTER 3B
The Microscopic Theory of Cracks V.L. INDENBOM
A.V. Shubnikov Institute of Crystallography Academy of Sciences of the USSR Leninsky Prospect 59, Moscow, 117333, USSR
Elastic Strain Fields and Dislocation Mobility Edited by V.L. Indenbom and J. Lothe
Elsevier Science Publishers B.V., 1992
253
Contents 1. Introduction 2. Microscopic model of a crack
255 255
3. 4. 5. 6.
257 259 260 263
Governing equations Surface energy of the crack Analogy between the generalized and classic Griffith criteria Examples
7. The model of the crack tip References
264 267
254
1. Introduction The Griffith energy approach based on the surface energy concept plays a funda mental role in modern theories of macroscopic crack propagation in solids. The surface energy of the crack is, by definition, numerically equal to the work for overcoming the interaction force between the two faces. The crack is called macroscopic if the main contribution to the surface energy is due to those portions of the surfaces where the distance across the crack is so large that the interaction may be neglected. In this case, one may ignore analysis of the portions with significant interaction forces and confine oneself to a description of the crack as a finite cut, with faces which are assumed not to interact. In the framework of the macroscopic energy approach, an energy γ, identical for all portions of the surface, is assigned to the unit crack surface. The energy density y is numerically equal to half of the area bounded by the graph of the non-linear interaction law between the two faces. It represents a material constant and can be evaluated experimentally as the work needed for breaking the solid [see, e.g., Obreimoff (1930)]. In the planar case, the total surface energy per unit crack width (Γ) is equal to twice the surface energy density y0 multiplied by the crack length L, the latter being determined from the conditions of energy balance of the solid with a crack. On the other hand, the total surface energy being known, the crack length may be estimated from the equation L = Γ/ly. No direct extension of the outlined macroscopic approach can be found for the case of microscopic cracks since for these the surface energy density at all points of the crack is essentially dependent on the distance between the crack faces. This important circumstance is not taken into account even in the dislocation theories of fracture (Stroh 1957, Indenbom 1961) which claim to analyse the crack initiation processes. These theories would generally use the macroscopic energy approach which facilitates finding the fracture criteria but which leaves open to questioning the applicability of the results derived. In Blekherman and Indenbom (1974a), the expression for the total surface energy of a crack is obtained without assumptions as to whether the crack is micro- or macroscopic. This expression coincides asymptotically with the ex pression for the total surface energy of a crack given by the classic Griffith approach.
2. Microscopic model of a crack The problem of the microscopic description of a crack is in many respects analogous to that of the description of the dislocation core. As is shown in chapter 3A, section 5, for the microscopic analysis of the dislocation, Peierls (1940) proposed and Nabarro (1947, 1952, 1967) and Elliott (1947) developed a model in which a solid with a dislocation is represented as two linear elastic 255
V.L. Indenbom
256
half-spaces bonded along the slip plane of the dislocation by non-linear forces depending periodically on the relative tangential displacements of the half-space faces and degenerating for small displacements into linear elastic forces (Hooke's law). The scheme of the Peierls-Nabarro model for the case of an edge disloca tion in a cubic lattice is demonstrated in fig. la. The modification of the Peierls-Nabarro model for the case of the crack, due to Elliot (1947) and to Bilby and Eshelby (1968), consists in representing a solid with a crack as two linear elastic half-spaces separated in equilibrium by the interatomic distance a and interacting in tension according to a certain non linear law σ(ιι/α). Here, σ is the normal component of the stress tensor and u is the relative normal displacement of the half-space faces. For small displace ments, the law σ(ιι/α) reduces to Hooke's law. From a geometric point of view, the only difference between this and the Peierls-Nabarro model is that the cut is made not parallel but perpendicular to the dislocation slip plane (fig. lb). As a matter of fact, the same modification of the Peierls-Nabarro model was used by van der Merwe (1950) in an analysis of a tilt boundary (i.e., a vertical dislocation wall). It is interesting to note that results given by the van der Merwe model are in good agreement with experimental data even in case of large tilt angles, i.e., when the distance between dislocations amount to only a few lattice parameters (Hirth and Lothe 1968). The model described infig.lb was used also by Blekherman and Indenbom (1974a), Entov and Salgonic (1968), Blekherman and Natzvlishvili (1968), Blekherman and Indenbom (1970), and Andersson and Bergkvist (1970). In the outlined microscopic model of the crack, the density of the surface energy assigned to the surface of the half-spaces is equal to y|>(x)] = -
'u(x)
σ[ - )dv.
-Λ_φ— ( k_>>_ < W>-^V -<ν-Λ—A
(a) - φ — φ
(1)
Φ—6—e>-
φ—φ
φ- (b)
Fig. 1. On the analogy between (a) the dislocation core model and (b) the microscopic model of a crack.
The microscopic theory of cracks
a
►
Fig. 2. Qualitative form of the interplane interaction law.
The faces of the half-spaces are, here, treated as atomic planes bordering the crack, and a(u/a) as a law for the interplane interaction in tension is character ized by a graph of the type shown infig.2. The maximum value of the interaction force (i.e., the theoretical tension strength of the material am) is reached at the lability point u = um. For displacements exceeding the radius of the interatomic interaction, the stresses σ rapidly decrease and the density of the surface energy tends to the macroscopic value y = lim«^^ y(u). The important advantage of this model is that it permits a unified description of the process of crack evolution from submicroscopic to macroscopic size. By analogy to the fact that the conventional Peierls-Nabarro model is better justified the larger is the splitting of the dislocation in the slip plane, it may be expected that the discussed model would describe the atomic structure of the crack better the larger the tip zone is. It should be also noted that, as far as the description of the tip zones of the macroscopic cracks is concerned, this model appears to be suitable only in cases when the dislocation density around the crack tip is not too large (brittle fracture). The first attempts of direct calcu lations of the atomic arrangement at the crack tip [see, e.g., Goodier (1968), Gehlen and Kanninen (1970)] and the electron microscopic observations of cracks available (Blekherman and Natzvlishvili 1968) are qualitatively consis tent with the adopted model. 3. Governing
equations
Let us consider a crack induced by an elastic wedge of width B embedded between half-spaces. This is equivalent to a crack generated by an edge disloca tion with Burgers vector B. Let a crack be placed in the plane z = 0 and the
V.L. Indenbom
258
wedge front in the plane x = 0. A normal traction p(x) is applied to the crack. Provided B = 0, this dislocation crack degenerates into an ordinary force (lens-shaped) crack being opened up only by the traction p{x). The law of half-space interaction above the wedge front represents the above-discussed law a(u/a) and in the region of the extra planes of the dislocation it converts to the law a*[(u — B)/a~\, which describes the stresses resulting from the interaction between the half-spaces and the dislocation wedge as a function of the relative displacement of the half-space faces u. In the limiting case (Hooke's law), we have B du
\a
u=o
\_du
\
a
where n = B/a is the power (number of extra planes) of the dislocation. Withing the framework of the discussed model, the equation governing the equilibrium of the solid with a crack is of the form D π
t- x
= g\ L
a
J
-
v)
P(X),
where g[-\ a )
= a(-\ W
χ<0, for
—^— 1,
ΒΦ^
x>0,
= *(-)
(3) for
B = 0.
In eq. (2), D = E/4(l — v 2 ), E is Young's modulus and v is Poisson's ratio. The Cauchy principal value of the integral is taken. The singular integro-differential equation, eq. (2), may be rewritten in the form
--Γ
dt = u(x).
(4)
nD)The solution of eq. (2) should satisfy the boundary conditions M(_oo)
= 0,
u(+oo) = £.
(5)
Let the external traction \p(x)\ decrease with |x| ^ o o n o t slower than A/\x\, where A > 0 is an arbitrary constant. This restriction is of purely mathematical nature since the behaviour of the external tractions for large |x| does not change anything in terms of physics. According to the Hubert transform theory
The microscopic theory of cracks
259
(Andersson and Bergkvist 1970), we have lim
l
x
M-oo L
'
t — X
J-oo
= -B.
(6)
J
Thus, at large distances from the wedge front, a crack is equivalent to a macro scopic dislocation with Burgers vector B. Since for large |x| we have assumed p(x) = 0 ( l / x ) the relation #[w(x)/a] = 0 ( l / x ) is also valid at |x| -+oo.
4. Surface energy of the crack The surface energy of the crack is, by definition, numerically equal to the work necessary for splitting the atomic planes bordering the crack. To start with, let us consider the case B φ 0. In this case, it is convenient to exclude from consideration the wedge extra planes by attributing their elastic energy to the half-space edges disposed behind the wedge front. The elastic energy of the wedge is, therefore, included into the surface energy of the crack. Then, the total surface energy of the crack may be written in the form
r
dx
d + dx
d
- L Γ '(s) ° i." Γ °i^) °
(7)
The first term in the right-hand part of eq. (7) corresponds to the energy of surfaces above the wedge front and the second term corresponds to the energy of surfaces behind the front (i.e., to the elastic energy of the wedge). Integration of the right-hand component of eq. (7) by parts gives Γ=χ
u(x)
σ - ) dt; \a
o
f00
dv
+ x
l~ w wi H u
x
9\
a
u
(8)
(x)dx.
J-- L Due to the validity of theJrelation g\_u(x)la~\ = 0 ( l / x ) for large |x| and the linearity of a(u/a) and a*[(w — B)la~\ for u -* 0 and u -► B, respectively, the surface energy density is subject to the rule y[w(x)] = 0 ( l / x 2 ) for |x| ^ooand, therefore, the two first terms in the right-hand side of eq. (8) vanish. Now, if we substitute in the third term of eq. (8) the expression for g\_u(x)/a~\ from eq. (2) and use the relation xw'(x)i/(i)did
- = \[m
Γ f- i - + -i-)«'(x)u'(i)dxdt
2J-00 )-*>\t-x
= -i(r
«'Wdxj ,
x-tj
(9
V.L. Indenbom
260
we find through eq. (5) that Γ=
DB2 2π
f00 J-oo
xp(x)u'(x)dx.
(10)
These calculations are analogous to the calculations by Elliott (1947) for the misifit energy. In the case B = 0, the total surface energy of the crack, Γ, may be written in the form
-r-e
dx\
Γ= -
00
σ -
dt;.
(11)
Repeating the above calculations, we again obtain expression (10) where in this case B should be put to zero. It is important to note that no assumptions about the crack size have been used when deducing expression (10). Thus, eq. (10) gives the expression for the total surface energy of both micro- and macroscopic cracks. It is of interest that in the absence of external tractions the total surface energy of a wedge-shaped crack, eq. (10), is independent of the particular form of the non-linear interaction law of splitted atomic planes. Moreover, the total surface energy is independent also of the maximum surface energy density y. This fact, previously referred to without proof (Goodier 1968), may be illustrated by the following example (rebinder effect in the microscopic theory of cracks). Let the specific surface energy of micro- or macroscopic cracks be reduced via introduc tion of a surface-active substance. Then, according to eq. (10), the crack length should increase so that the initial value of total surface energy is reestablished. If the total surface energy of the crack is known, then, by analogy with the macroscopic theory, one can introduce the effective crack length
i = f-
(12)
2y This relationship may be treated as a generalization of the Griffith criterion valid both for micro- and macroscopic cracks under brittle fracture.
5. Analogy between the generalized and classic Griffith criteria Let us calculate the surface energy of a macroscopic crack in terms of the classical linear elastic approach. Let the crack be placed in the plane z = 0 between the lines x = —L and x = 0. Then, the equation governing the equilib rium of the solid with a crack has the form
The microscopic theory of cracks
261
where uo(x) is the relative normal displacement of the faces of the cut modelling the crack. Note that u and u0 represent different physical quantities: u is the displacement of real atomic planes, whereas u0 is the displacement of certain fictitious planes shifted against the atomic planes by the distance ±\a. This circumstance is of great importance when comparing the microscopic theory of cracks with the macroscopic theories (Barenblatt 1959, Leonov and Panasjuk 1959, Indenbom and Orlov 1970, 1977, Schmidt and Woltersdorf 1970). The solution of eq. (13) should satisfy the boundary conditions uo(-L) = 0,
uo(0) = B.
(14)
It may be shown (Schmidt and Woltersdorf 1970) that this solution is of the form ub(x) =
,
° «WLiD^
Dny/-x(L + x)J-i<
l x
~
B n^/-x(L + x)
(15)
For 0 ^ x + L <^ L, we have uo(x) = ( ^ + ^ f ) y/Lo(L + *) + 0[(x + L 0 ) 3 / 2 ].
(16)
Here, σ is the weighted average stress
t'iL&uJJFn*·
(17)
Using the evaluating of Cauchy type integrals near the ends of integration line (Muskhehshvili 1968), it may be shown that the stress σ in the region 0 < —L — x <ζ Lis
"w"U
+
i)VlTi + 0 [ ( - L - ) t ) " I : l
"8)
The elastic energy emanating in the course of crack propagation, per unit length, is equal to (Irwin 1957) 1 C~L+E F = lim — σ(χ — e)u0(x) dx.
(19)
Substitution of expressions (16) and (18) in eq. (19) gives „
πΖ,/
2DB\2
Particular cases of eq. (20) were analysed by Blekherman and Indenbom (1974a) in the course of deducing fracture criteria in various versions of the dislocation theory of strength.
V.L. Indenbom
262
Setting F equal to the doubled surface energy density, F = 2γ,
(21)
gives the classical Griffith criterion for the determination of the length of an equilibrium macroscopic crack under brittle fracture. Since the total surface energy of a macroscopic crack is defined as Γ0 = 2yL one has Γ0 = FL. Now let us show that Γ0 =
Ό
DB 2 2π
xp(x)u'0(x)dx.
(22)
In fact, substituting u'Q(x) from eq. (15) into eq. (22) and symmetrizing the resulting expression with respect to t and x, we find DB2 2π
+
B π
0
Ρ{χ)
'
-x
^τ-Γχάχ
2M-J-lWiTWzTidxd'·
,23)
wherefrom, due to eq. (17) and the relation Γ = FL, expression (20) immediately follows. For macroscopic cracks, the difference between u and w0, as well as the contribution of the tip zone to the total surface energy, may be neglected. So, for sufficiently large L, eqs. (10) and (22) coincide asymptotically and consequently criterion (12) turns into criterion (21). It is important to emphasize the essential distinction between the microscopic and macroscopic approaches to the problem of crack equilibrium. In the macroscopic theory the equilibrium solution, i.e., a function minimizing the total energy of the solid, is obtained in two steps. In the first step only the elastic energy of the solid is minimized (i.e., the solution of the elastic problem of equilibrium of the solid with a cut of a given length is searched for). In the second step, from the condition of a minimum of the total energy of the solid (under brittle fracture this energy involves elastic and surface energies), the equilibrium length of the crack is obtained. This minimization is conducted within the class of functions ensuring minimum elastic energy. Thus, in the macroscopic theory, the solution of the corresponding elastic problem is not yet a solution of the fracture problem. In the microscopic theory, where the crack length is not a proper parameter of the problem, the minimization of the total energy is a one-step procedure and, therefore, the solution of eq. (2) is automatically the solution of the problem of the theory of cracks. In this way, in the macroscopic theory, absence of an equilibrium solution corresponds to absence of real positive roots of the algebraic equation, eq. (21), whereas in the microscopic theory absence of an equilibrium solution is equivalent to absence of a solution
The microscopic theory of cracks
263
of the integral equation, eq. (2). The situation just noted is typical of any microscopic theory: the direct consideration of the interatomic interaction makes it needless to introduce into the theory an additional criterion such as the Griffith criterion and gives directly the crack configuration satisfying the condi tion of a minimum of the total energy of the solid. In this sense, the Griffith criterion does not matter in the microscopic theory of cracks.
6. Examples The statements presented above can be illustrated by the results of the numerical calculations of the principal characteristics of the dislocation crack obtained by the procedure of Blekherman and Indenbom (1970). In fig. 3a, a family of interaction laws with the same initial slope (i.e., the same elastic modulus), the same area (i.e., the same maximum surface energy density y) and different values of the theoretical strength am are shown. The crack tip configurations corre sponding to these laws are displayed in fig. 3b. Figures 3c and d show the stress
r
(b)
~SJ a •4-
^S^ <**
2/
/ y
3
\yf /
ΉΔ
k
5 \A/a-
.
-30
-20
(d)
,
/
<-/ 1
7
-10
1
1
1
10
20
30
X/Q
*
f γ 10a =5 0.7.5
j/ 2/J -0.05 -30
-20
-10
0
10
20 x/a
30'30 >
-20
-10
0
I
10
L.
20 x/a
I,
30 ►
Fig. 3. Microscopic characteristics of the crack tip zone: (a) interplane interaction laws, (b) forms of crack tip, (c) stress distributions, and (d) surface energy density distributions.
V.L. Indenbom
264
distributions and the surface energy density distributions, respectively. The broken lines correspond to the results obtained by the methods of the macro scopic theory. The calculations were performed for a dislocation crack with a Burgers vector equal to 20a. It is readily seen that with localization of the interaction law (i.e., with decrease of interatomic force radius), the stress distribution and the surface energy density distribution tend to the corresponding distributions obtained by use of the macroscopic theory and the crack opening u(x) tends to u0(x) + a. Thus, the macroscopic approach may be treated as a limiting case of the microscopic theory which corresponds to the interaction law a{u) = 2yS(u), where δ(u) is the Dirac delta-function. As is known, Griffith himself advanced an analogous suggestion but he was not able to prove it, which resulted in a long-standing macroscopic theory of cracks. In the microscopic theory, the distinction between these approaches is eliminated in a natural way. From a practical point of view the macroscopic theory of cracks applies in linear fracture mechanics, whereas the microscopic theory applies in non-linear fracture mechanics. The well-known BCS method (Bilby et al. 1963) is a simple approximation of our method (Blekherman and Indenbom 1970,1974a), with the cohesive force-distance curve offig.2 replaced by a constant value, as in the theory of Leonov and Panasjuk (1959).
7. The model of the crack tip The microscopic theory gives also a model for the dislocation structure of the crack tip by quasi-brittle fracture. Let us estimate the characteristic length of the crack tip defined by means of the interaction force dispersion 2
i . ■2
\_ a \
2σmax
dx.
(24)
From eq. (1), we have
2y[u(x)] = J o U < %Qd, = | % [ ^ ] u ' ( t ) d ,
(25)
The theory of the Hubert transformation gives g\ —
J-» L 2
£> Γ
a
\u'(x)dx = Q,
(26)
J
[u'(x)] 2 dx= Γ
g2\—
Idx.
(27)
265
The microscopic theory of cracks
From eqs. (26) and (27) and Gelder's inequality, it follows that
2y[«(x)]=[X 0p^lu'(t)dt
1/2
1 ^2
1/2
2
1
20
-
L
a
(28)
J
Taking into consideration that y(u) < y, we arrive at ά>
2yD
(29)
and for typical values of D and y we have d^(10-102)a, which means that the crack tip really depends on the dislocation behaviour. For an atomic model of a crack tip, we can use the generalized model of the dislocation core, fig. 4, which is a composition of the models in figs, la and b. It has four quadrants, two cuts along the slip plane with Burgers vector bu and a cut along the extra plane with Burgers vector b2 = b — bu where b is the Burgers vector of dislocation. When w(x, y) denotes the projection of the
o o o
i
O
o // 6
1
Φ T
O
O
°
>
A*o-o-o
o m o
o
o—o—oAz
Q
0
Vf o
o
ό
o
o
Fig. 4. Generalized dislocation core model.
V.L. Indenbom
266
displacement on the axis x along b and [w(x, y)] the jump of w, then
f-° d
Γ00 d
J-oodx
J+0dx
f-° d
f00 d
J-oody
J+ody
— [II(X, 0)] dx +
— [ii(x, 0)] dx = -i> x ,
Let us introduce the displacements of the atoms on the cut sides,
x < 0,
= [K(X, 0)] + \b = Vx(x) + i 6 l 9
x > 0,
^ 2 W = [«(0,y)]=K2(y),
(31)
}><0,
= lu(0,y)]-b=V2(y)-b29
y>0,
and the dislocation densities in the cuts p(x)=-V[(x);
ß(y)=V'2(y).
(32)
The boundary conditions are Φι(±οο) = 0,
Kii-oo) = i& l s
K^+oo) = - ifti,
φ 2 (+οο) = 0,
K 2 (-oo) = 0,
F 2 (+oo) = ft2.
(33)
The interaction law for quadrants I-IV and II—III, ^(φι), determines the slip resistance and is a periodic function φ χ with period a, where a is the lattice period. For small φί9 0ι(<Ρι)Ξ -Κί—,
Ψι
χ->+οο,
(34)
where Χ χ = 2G/(3 — 2ν), G the shear modulus, and v the Poisson coefficient. The interaction law for quadrants I-II and III-IV, # 2 (φ 2 ), determines fracture resistance and for small φ2, g2((P2) = K2±(^9 y^±
Γχ = -
[>i(x)
dx
Λοο
dx'flfxix'),
J - o o J O
r
2
= - \
[>2(y)
dy\
dy'g2(y%
J - o o J o
and due to eq. (31), it follows that Γ = Γ1+Γ2
Db 2 = —,
(36)
The microscopic theory of cracks
267
where D = £/4(l - v2) = G/2(l - v). The result, given in eq. (36), means that the core energy does not depend on the laws gi((pi) and 02(^2) nor on the correlation between b± and b2. In the case of the Peierls-Nabarro model (μ = 0), we have D π
V[(t)dt - T J - + ffi(Vi) = 0-
(37)
In the case of the van der Merwe model [p(x) = 0], we have D f00 V'2(t)dt ^'<>J -j o- o*
*t-— y
+
(38)
For 0ι(φι) = ~ ^ - s i n ^ p x / f c ) , we arrive at Peierls solution (chapter 3A, section 3). When the dislocations in a pile-up draw together, the first (head) dislocation splits off into the extra plane and the value of bx grows. In a powerful pile-up a superdislocation with macroscopic bx and b2 forms. Further growth of bu at a constant or slow rate, leads to crack formation. This process is discussed in detail by Blekherman and Indenbom (1974b), where if is clearly shown that plastic and brittle rupture can be distinguished not only from the behaviour on a macroscopic scale but also from the behaviour on a microscopic scale. The atomics of fracture of different materials is reviewed by Thomson (1983).
References Andersson, H., and H. Bergkvist, 1970, J. Mech. Phys. Sol. 18, 1. Barenblatt, G.I., 1959, Prikl. Mat. Mekh. 23, 706. Bilby, B.A., and J.D. Eshelby, 1968, in: Fracture, Vol. 1, ed. Liebowitz (Academic Press, New York) p. 99. Bilby, B.A., A.H. Cottrell and K.H. Swinden, 1963, Proc. R. Soc. London, Ser. A, 272, 304. Blekherman, M.Kh., and V.L. Indenbom, 1970, J. Prikl. Mekh. & Tekh. Fiz. 1, 96. Blekherman, M.Kh., and V.L. Indenbom, 1974, Phys. Status Solidi A 23, 729. Blekherman, M.Kh., and V.L. Indenbom, 1974b, Sov. Phys.-Solid State Phys. 16, 1733. Blekherman, M.Kh., and G.I. Natzvlishvili, 1968, Kristallografiya 14, 351. Elliott, H.A., 1947, Proc. Phys. Soc. A 59, p. II, 208. Entov, V.M., and R.V Salgonic, 1968, Inz. J. Mech. Tverd. Tela, N6, 87. Gehlen, P.C., and M.F. Kanninen, 1970, in: Inelastic Behavior of Solids, ed. M.F. Kanninen, et. al. (McGraw-Hill, New York) p. 587. Goodier, J.N., 1968, in: Fracture, Vol. 2, ed. Liebowitz (Academic Press, New York) p. 1. Hirth, J.P, and J. Lothe, 1968, Theory of Dislocations (McGraw-Hill, New York). Indenbom, V.L., 1961, Fiz. Tverd. Tela 3, 2071. Indenbom, V.L., and A.N. Orlov, 1970, Problemi Prochnosti No. 12, p. 3. Indenbom, V.L., and A.N. Orlov, 1977, Proc. VII Magdeburg Symp. Deformation and Fracture.
268
V.L. Indenbom
Irwin, G.R., 1957, J. Appl. Mech. 24, 361. Leonov, M.Ya., and V.V Panasjuk, 1959, Prikl. Mekh. 5, 391. Muskhelishvili, N.I., 1968, Singular Integral Equation (Izd. Nauka, Moscow). Nabarro, F.R.N., 1947, Proc. Phys. Soc. A 59, 256. Nabarro, F.R.N., 1952, Adv. Phys. 1, 269. Nabarro, F.R.N., 1967, Theory of Crystal Dislocations (Clarendon Press, Oxford). Obreimoff, J.W., 1930, Proc. R. Soc. A 127, 290. Peierls, R.E., 1940, Proc. Phys. Soc. A 52, 34. Schmidt, V, and J. Woltersdorf, 1970, Phys. Status Solidi 41, 565. Stroh, A.N., 1957, Adv. Phys. 6, 418. Thomson, R., 1983, in: Atomistics of Fracture, eds R.M. Latanision and I.R. Pickens (Plenum Press, New York) p. 167. van der Merwe, J.H., 1950, Proc. Phys. Soc. A 63, 616.
The Microscopic Theory of Cracks V.L. INDENBOM
A.V. Shubnikov Institute of Crystallography Academy of Sciences of the USSR Leninsky Prospect 59, Moscow, 117333, USSR
Elastic Strain Fields and Dislocation Mobility Edited by V.L. Indenbom and J. Lothe
Elsevier Science Publishers B.V., 1992
253
Contents 1. Introduction 2. Microscopic model of a crack
255 255
3. 4. 5. 6.
257 259 260 263
Governing equations Surface energy of the crack Analogy between the generalized and classic Griffith criteria Examples
7. The model of the crack tip References
264 267
254
1. Introduction The Griffith energy approach based on the surface energy concept plays a funda mental role in modern theories of macroscopic crack propagation in solids. The surface energy of the crack is, by definition, numerically equal to the work for overcoming the interaction force between the two faces. The crack is called macroscopic if the main contribution to the surface energy is due to those portions of the surfaces where the distance across the crack is so large that the interaction may be neglected. In this case, one may ignore analysis of the portions with significant interaction forces and confine oneself to a description of the crack as a finite cut, with faces which are assumed not to interact. In the framework of the macroscopic energy approach, an energy γ, identical for all portions of the surface, is assigned to the unit crack surface. The energy density y is numerically equal to half of the area bounded by the graph of the non-linear interaction law between the two faces. It represents a material constant and can be evaluated experimentally as the work needed for breaking the solid [see, e.g., Obreimoff (1930)]. In the planar case, the total surface energy per unit crack width (Γ) is equal to twice the surface energy density y0 multiplied by the crack length L, the latter being determined from the conditions of energy balance of the solid with a crack. On the other hand, the total surface energy being known, the crack length may be estimated from the equation L = Γ/ly. No direct extension of the outlined macroscopic approach can be found for the case of microscopic cracks since for these the surface energy density at all points of the crack is essentially dependent on the distance between the crack faces. This important circumstance is not taken into account even in the dislocation theories of fracture (Stroh 1957, Indenbom 1961) which claim to analyse the crack initiation processes. These theories would generally use the macroscopic energy approach which facilitates finding the fracture criteria but which leaves open to questioning the applicability of the results derived. In Blekherman and Indenbom (1974a), the expression for the total surface energy of a crack is obtained without assumptions as to whether the crack is micro- or macroscopic. This expression coincides asymptotically with the ex pression for the total surface energy of a crack given by the classic Griffith approach.
2. Microscopic model of a crack The problem of the microscopic description of a crack is in many respects analogous to that of the description of the dislocation core. As is shown in chapter 3A, section 5, for the microscopic analysis of the dislocation, Peierls (1940) proposed and Nabarro (1947, 1952, 1967) and Elliott (1947) developed a model in which a solid with a dislocation is represented as two linear elastic 255
V.L. Indenbom
256
half-spaces bonded along the slip plane of the dislocation by non-linear forces depending periodically on the relative tangential displacements of the half-space faces and degenerating for small displacements into linear elastic forces (Hooke's law). The scheme of the Peierls-Nabarro model for the case of an edge disloca tion in a cubic lattice is demonstrated in fig. la. The modification of the Peierls-Nabarro model for the case of the crack, due to Elliot (1947) and to Bilby and Eshelby (1968), consists in representing a solid with a crack as two linear elastic half-spaces separated in equilibrium by the interatomic distance a and interacting in tension according to a certain non linear law σ(ιι/α). Here, σ is the normal component of the stress tensor and u is the relative normal displacement of the half-space faces. For small displace ments, the law σ(ιι/α) reduces to Hooke's law. From a geometric point of view, the only difference between this and the Peierls-Nabarro model is that the cut is made not parallel but perpendicular to the dislocation slip plane (fig. lb). As a matter of fact, the same modification of the Peierls-Nabarro model was used by van der Merwe (1950) in an analysis of a tilt boundary (i.e., a vertical dislocation wall). It is interesting to note that results given by the van der Merwe model are in good agreement with experimental data even in case of large tilt angles, i.e., when the distance between dislocations amount to only a few lattice parameters (Hirth and Lothe 1968). The model described infig.lb was used also by Blekherman and Indenbom (1974a), Entov and Salgonic (1968), Blekherman and Natzvlishvili (1968), Blekherman and Indenbom (1970), and Andersson and Bergkvist (1970). In the outlined microscopic model of the crack, the density of the surface energy assigned to the surface of the half-spaces is equal to y|>(x)] = -
'u(x)
σ[ - )dv.
-Λ_φ— ( k_>>_ < W>-^V -<ν-Λ—A
(a) - φ — φ
(1)
Φ—6—e>-
φ—φ
φ- (b)
Fig. 1. On the analogy between (a) the dislocation core model and (b) the microscopic model of a crack.
The microscopic theory of cracks
a
►
Fig. 2. Qualitative form of the interplane interaction law.
The faces of the half-spaces are, here, treated as atomic planes bordering the crack, and a(u/a) as a law for the interplane interaction in tension is character ized by a graph of the type shown infig.2. The maximum value of the interaction force (i.e., the theoretical tension strength of the material am) is reached at the lability point u = um. For displacements exceeding the radius of the interatomic interaction, the stresses σ rapidly decrease and the density of the surface energy tends to the macroscopic value y = lim«^^ y(u). The important advantage of this model is that it permits a unified description of the process of crack evolution from submicroscopic to macroscopic size. By analogy to the fact that the conventional Peierls-Nabarro model is better justified the larger is the splitting of the dislocation in the slip plane, it may be expected that the discussed model would describe the atomic structure of the crack better the larger the tip zone is. It should be also noted that, as far as the description of the tip zones of the macroscopic cracks is concerned, this model appears to be suitable only in cases when the dislocation density around the crack tip is not too large (brittle fracture). The first attempts of direct calcu lations of the atomic arrangement at the crack tip [see, e.g., Goodier (1968), Gehlen and Kanninen (1970)] and the electron microscopic observations of cracks available (Blekherman and Natzvlishvili 1968) are qualitatively consis tent with the adopted model. 3. Governing
equations
Let us consider a crack induced by an elastic wedge of width B embedded between half-spaces. This is equivalent to a crack generated by an edge disloca tion with Burgers vector B. Let a crack be placed in the plane z = 0 and the
V.L. Indenbom
258
wedge front in the plane x = 0. A normal traction p(x) is applied to the crack. Provided B = 0, this dislocation crack degenerates into an ordinary force (lens-shaped) crack being opened up only by the traction p{x). The law of half-space interaction above the wedge front represents the above-discussed law a(u/a) and in the region of the extra planes of the dislocation it converts to the law a*[(u — B)/a~\, which describes the stresses resulting from the interaction between the half-spaces and the dislocation wedge as a function of the relative displacement of the half-space faces u. In the limiting case (Hooke's law), we have B du
\a
u=o
\_du
\
a
where n = B/a is the power (number of extra planes) of the dislocation. Withing the framework of the discussed model, the equation governing the equilibrium of the solid with a crack is of the form D π
t- x
= g\ L
a
J
-
v)
P(X),
where g[-\ a )
= a(-\ W
χ<0, for
—^— 1,
ΒΦ^
x>0,
= *(-)
(3) for
B = 0.
In eq. (2), D = E/4(l — v 2 ), E is Young's modulus and v is Poisson's ratio. The Cauchy principal value of the integral is taken. The singular integro-differential equation, eq. (2), may be rewritten in the form
--Γ
dt = u(x).
(4)
nD)The solution of eq. (2) should satisfy the boundary conditions M(_oo)
= 0,
u(+oo) = £.
(5)
Let the external traction \p(x)\ decrease with |x| ^ o o n o t slower than A/\x\, where A > 0 is an arbitrary constant. This restriction is of purely mathematical nature since the behaviour of the external tractions for large |x| does not change anything in terms of physics. According to the Hubert transform theory
The microscopic theory of cracks
259
(Andersson and Bergkvist 1970), we have lim
l
x
M-oo L
'
t — X
J-oo
= -B.
(6)
J
Thus, at large distances from the wedge front, a crack is equivalent to a macro scopic dislocation with Burgers vector B. Since for large |x| we have assumed p(x) = 0 ( l / x ) the relation #[w(x)/a] = 0 ( l / x ) is also valid at |x| -+oo.
4. Surface energy of the crack The surface energy of the crack is, by definition, numerically equal to the work necessary for splitting the atomic planes bordering the crack. To start with, let us consider the case B φ 0. In this case, it is convenient to exclude from consideration the wedge extra planes by attributing their elastic energy to the half-space edges disposed behind the wedge front. The elastic energy of the wedge is, therefore, included into the surface energy of the crack. Then, the total surface energy of the crack may be written in the form
r
dx
d + dx
d
- L Γ '(s) ° i." Γ °i^) °
(7)
The first term in the right-hand part of eq. (7) corresponds to the energy of surfaces above the wedge front and the second term corresponds to the energy of surfaces behind the front (i.e., to the elastic energy of the wedge). Integration of the right-hand component of eq. (7) by parts gives Γ=χ
u(x)
σ - ) dt; \a
o
f00
dv
+ x
l~ w wi H u
x
9\
a
u
(8)
(x)dx.
J-- L Due to the validity of theJrelation g\_u(x)la~\ = 0 ( l / x ) for large |x| and the linearity of a(u/a) and a*[(w — B)la~\ for u -* 0 and u -► B, respectively, the surface energy density is subject to the rule y[w(x)] = 0 ( l / x 2 ) for |x| ^ooand, therefore, the two first terms in the right-hand side of eq. (8) vanish. Now, if we substitute in the third term of eq. (8) the expression for g\_u(x)/a~\ from eq. (2) and use the relation xw'(x)i/(i)did
- = \[m
Γ f- i - + -i-)«'(x)u'(i)dxdt
2J-00 )-*>\t-x
= -i(r
«'Wdxj ,
x-tj
(9
V.L. Indenbom
260
we find through eq. (5) that Γ=
DB2 2π
f00 J-oo
xp(x)u'(x)dx.
(10)
These calculations are analogous to the calculations by Elliott (1947) for the misifit energy. In the case B = 0, the total surface energy of the crack, Γ, may be written in the form
-r-e
dx\
Γ= -
00
σ -
dt;.
(11)
Repeating the above calculations, we again obtain expression (10) where in this case B should be put to zero. It is important to note that no assumptions about the crack size have been used when deducing expression (10). Thus, eq. (10) gives the expression for the total surface energy of both micro- and macroscopic cracks. It is of interest that in the absence of external tractions the total surface energy of a wedge-shaped crack, eq. (10), is independent of the particular form of the non-linear interaction law of splitted atomic planes. Moreover, the total surface energy is independent also of the maximum surface energy density y. This fact, previously referred to without proof (Goodier 1968), may be illustrated by the following example (rebinder effect in the microscopic theory of cracks). Let the specific surface energy of micro- or macroscopic cracks be reduced via introduc tion of a surface-active substance. Then, according to eq. (10), the crack length should increase so that the initial value of total surface energy is reestablished. If the total surface energy of the crack is known, then, by analogy with the macroscopic theory, one can introduce the effective crack length
i = f-
(12)
2y This relationship may be treated as a generalization of the Griffith criterion valid both for micro- and macroscopic cracks under brittle fracture.
5. Analogy between the generalized and classic Griffith criteria Let us calculate the surface energy of a macroscopic crack in terms of the classical linear elastic approach. Let the crack be placed in the plane z = 0 between the lines x = —L and x = 0. Then, the equation governing the equilib rium of the solid with a crack has the form
The microscopic theory of cracks
261
where uo(x) is the relative normal displacement of the faces of the cut modelling the crack. Note that u and u0 represent different physical quantities: u is the displacement of real atomic planes, whereas u0 is the displacement of certain fictitious planes shifted against the atomic planes by the distance ±\a. This circumstance is of great importance when comparing the microscopic theory of cracks with the macroscopic theories (Barenblatt 1959, Leonov and Panasjuk 1959, Indenbom and Orlov 1970, 1977, Schmidt and Woltersdorf 1970). The solution of eq. (13) should satisfy the boundary conditions uo(-L) = 0,
uo(0) = B.
(14)
It may be shown (Schmidt and Woltersdorf 1970) that this solution is of the form ub(x) =
,
° «WLiD^
Dny/-x(L + x)J-i<
l x
~
B n^/-x(L + x)
(15)
For 0 ^ x + L <^ L, we have uo(x) = ( ^ + ^ f ) y/Lo(L + *) + 0[(x + L 0 ) 3 / 2 ].
(16)
Here, σ is the weighted average stress
t'iL&uJJFn*·
(17)
Using the evaluating of Cauchy type integrals near the ends of integration line (Muskhehshvili 1968), it may be shown that the stress σ in the region 0 < —L — x <ζ Lis
"w"U
+
i)VlTi + 0 [ ( - L - ) t ) " I : l
"8)
The elastic energy emanating in the course of crack propagation, per unit length, is equal to (Irwin 1957) 1 C~L+E F = lim — σ(χ — e)u0(x) dx.
(19)
Substitution of expressions (16) and (18) in eq. (19) gives „
πΖ,/
2DB\2
Particular cases of eq. (20) were analysed by Blekherman and Indenbom (1974a) in the course of deducing fracture criteria in various versions of the dislocation theory of strength.
V.L. Indenbom
262
Setting F equal to the doubled surface energy density, F = 2γ,
(21)
gives the classical Griffith criterion for the determination of the length of an equilibrium macroscopic crack under brittle fracture. Since the total surface energy of a macroscopic crack is defined as Γ0 = 2yL one has Γ0 = FL. Now let us show that Γ0 =
Ό
DB 2 2π
xp(x)u'0(x)dx.
(22)
In fact, substituting u'Q(x) from eq. (15) into eq. (22) and symmetrizing the resulting expression with respect to t and x, we find DB2 2π
+
B π
0
Ρ{χ)
'
-x
^τ-Γχάχ
2M-J-lWiTWzTidxd'·
,23)
wherefrom, due to eq. (17) and the relation Γ = FL, expression (20) immediately follows. For macroscopic cracks, the difference between u and w0, as well as the contribution of the tip zone to the total surface energy, may be neglected. So, for sufficiently large L, eqs. (10) and (22) coincide asymptotically and consequently criterion (12) turns into criterion (21). It is important to emphasize the essential distinction between the microscopic and macroscopic approaches to the problem of crack equilibrium. In the macroscopic theory the equilibrium solution, i.e., a function minimizing the total energy of the solid, is obtained in two steps. In the first step only the elastic energy of the solid is minimized (i.e., the solution of the elastic problem of equilibrium of the solid with a cut of a given length is searched for). In the second step, from the condition of a minimum of the total energy of the solid (under brittle fracture this energy involves elastic and surface energies), the equilibrium length of the crack is obtained. This minimization is conducted within the class of functions ensuring minimum elastic energy. Thus, in the macroscopic theory, the solution of the corresponding elastic problem is not yet a solution of the fracture problem. In the microscopic theory, where the crack length is not a proper parameter of the problem, the minimization of the total energy is a one-step procedure and, therefore, the solution of eq. (2) is automatically the solution of the problem of the theory of cracks. In this way, in the macroscopic theory, absence of an equilibrium solution corresponds to absence of real positive roots of the algebraic equation, eq. (21), whereas in the microscopic theory absence of an equilibrium solution is equivalent to absence of a solution
The microscopic theory of cracks
263
of the integral equation, eq. (2). The situation just noted is typical of any microscopic theory: the direct consideration of the interatomic interaction makes it needless to introduce into the theory an additional criterion such as the Griffith criterion and gives directly the crack configuration satisfying the condi tion of a minimum of the total energy of the solid. In this sense, the Griffith criterion does not matter in the microscopic theory of cracks.
6. Examples The statements presented above can be illustrated by the results of the numerical calculations of the principal characteristics of the dislocation crack obtained by the procedure of Blekherman and Indenbom (1970). In fig. 3a, a family of interaction laws with the same initial slope (i.e., the same elastic modulus), the same area (i.e., the same maximum surface energy density y) and different values of the theoretical strength am are shown. The crack tip configurations corre sponding to these laws are displayed in fig. 3b. Figures 3c and d show the stress
r
(b)
~SJ a •4-
^S^ <**
2/
/ y
3
\yf /
ΉΔ
k
5 \A/a-
.
-30
-20
(d)
,
/
<-/ 1
7
-10
1
1
1
10
20
30
X/Q
*
f γ 10a =5 0.7.5
j/ 2/J -0.05 -30
-20
-10
0
10
20 x/a
30'30 >
-20
-10
0
I
10
L.
20 x/a
I,
30 ►
Fig. 3. Microscopic characteristics of the crack tip zone: (a) interplane interaction laws, (b) forms of crack tip, (c) stress distributions, and (d) surface energy density distributions.
V.L. Indenbom
264
distributions and the surface energy density distributions, respectively. The broken lines correspond to the results obtained by the methods of the macro scopic theory. The calculations were performed for a dislocation crack with a Burgers vector equal to 20a. It is readily seen that with localization of the interaction law (i.e., with decrease of interatomic force radius), the stress distribution and the surface energy density distribution tend to the corresponding distributions obtained by use of the macroscopic theory and the crack opening u(x) tends to u0(x) + a. Thus, the macroscopic approach may be treated as a limiting case of the microscopic theory which corresponds to the interaction law a{u) = 2yS(u), where δ(u) is the Dirac delta-function. As is known, Griffith himself advanced an analogous suggestion but he was not able to prove it, which resulted in a long-standing macroscopic theory of cracks. In the microscopic theory, the distinction between these approaches is eliminated in a natural way. From a practical point of view the macroscopic theory of cracks applies in linear fracture mechanics, whereas the microscopic theory applies in non-linear fracture mechanics. The well-known BCS method (Bilby et al. 1963) is a simple approximation of our method (Blekherman and Indenbom 1970,1974a), with the cohesive force-distance curve offig.2 replaced by a constant value, as in the theory of Leonov and Panasjuk (1959).
7. The model of the crack tip The microscopic theory gives also a model for the dislocation structure of the crack tip by quasi-brittle fracture. Let us estimate the characteristic length of the crack tip defined by means of the interaction force dispersion 2
i . ■2
\_ a \
2σmax
dx.
(24)
From eq. (1), we have
2y[u(x)] = J o U < %Qd, = | % [ ^ ] u ' ( t ) d ,
(25)
The theory of the Hubert transformation gives g\ —
J-» L 2
£> Γ
a
\u'(x)dx = Q,
(26)
J
[u'(x)] 2 dx= Γ
g2\—
Idx.
(27)
265
The microscopic theory of cracks
From eqs. (26) and (27) and Gelder's inequality, it follows that
2y[«(x)]=[X 0p^lu'(t)dt
1/2
1 ^2
1/2
2
1
20
-
L
a
(28)
J
Taking into consideration that y(u) < y, we arrive at ά>
2yD
(29)
and for typical values of D and y we have d^(10-102)a, which means that the crack tip really depends on the dislocation behaviour. For an atomic model of a crack tip, we can use the generalized model of the dislocation core, fig. 4, which is a composition of the models in figs, la and b. It has four quadrants, two cuts along the slip plane with Burgers vector bu and a cut along the extra plane with Burgers vector b2 = b — bu where b is the Burgers vector of dislocation. When w(x, y) denotes the projection of the
o o o
i
O
o // 6
1
Φ T
O
O
°
>
A*o-o-o
o m o
o
o—o—oAz
Q
0
Vf o
o
ό
o
o
Fig. 4. Generalized dislocation core model.
V.L. Indenbom
266
displacement on the axis x along b and [w(x, y)] the jump of w, then
f-° d
Γ00 d
J-oodx
J+0dx
f-° d
f00 d
J-oody
J+ody
— [II(X, 0)] dx +
— [ii(x, 0)] dx = -i> x ,
Let us introduce the displacements of the atoms on the cut sides,
x < 0,
= [K(X, 0)] + \b = Vx(x) + i 6 l 9
x > 0,
^ 2 W = [«(0,y)]=K2(y),
(31)
}><0,
= lu(0,y)]-b=V2(y)-b29
y>0,
and the dislocation densities in the cuts p(x)=-V[(x);
ß(y)=V'2(y).
(32)
The boundary conditions are Φι(±οο) = 0,
Kii-oo) = i& l s
K^+oo) = - ifti,
φ 2 (+οο) = 0,
K 2 (-oo) = 0,
F 2 (+oo) = ft2.
(33)
The interaction law for quadrants I-IV and II—III, ^(φι), determines the slip resistance and is a periodic function φ χ with period a, where a is the lattice period. For small φί9 0ι(<Ρι)Ξ -Κί—,
Ψι
χ->+οο,
(34)
where Χ χ = 2G/(3 — 2ν), G the shear modulus, and v the Poisson coefficient. The interaction law for quadrants I-II and III-IV, # 2 (φ 2 ), determines fracture resistance and for small φ2, g2((P2) = K2±(^9 y^±
Γχ = -
[>i(x)
dx
Λοο
dx'flfxix'),
J - o o J O
r
2
= - \
[>2(y)
dy\
dy'g2(y%
J - o o J o
and due to eq. (31), it follows that Γ = Γ1+Γ2
Db 2 = —,
(36)
The microscopic theory of cracks
267
where D = £/4(l - v2) = G/2(l - v). The result, given in eq. (36), means that the core energy does not depend on the laws gi((pi) and 02(^2) nor on the correlation between b± and b2. In the case of the Peierls-Nabarro model (μ = 0), we have D π
V[(t)dt - T J - + ffi(Vi) = 0-
(37)
In the case of the van der Merwe model [p(x) = 0], we have D f00 V'2(t)dt ^'<>J -j o- o*
*t-— y
+
(38)
For 0ι(φι) = ~ ^ - s i n ^ p x / f c ) , we arrive at Peierls solution (chapter 3A, section 3). When the dislocations in a pile-up draw together, the first (head) dislocation splits off into the extra plane and the value of bx grows. In a powerful pile-up a superdislocation with macroscopic bx and b2 forms. Further growth of bu at a constant or slow rate, leads to crack formation. This process is discussed in detail by Blekherman and Indenbom (1974b), where if is clearly shown that plastic and brittle rupture can be distinguished not only from the behaviour on a macroscopic scale but also from the behaviour on a microscopic scale. The atomics of fracture of different materials is reviewed by Thomson (1983).
References Andersson, H., and H. Bergkvist, 1970, J. Mech. Phys. Sol. 18, 1. Barenblatt, G.I., 1959, Prikl. Mat. Mekh. 23, 706. Bilby, B.A., and J.D. Eshelby, 1968, in: Fracture, Vol. 1, ed. Liebowitz (Academic Press, New York) p. 99. Bilby, B.A., A.H. Cottrell and K.H. Swinden, 1963, Proc. R. Soc. London, Ser. A, 272, 304. Blekherman, M.Kh., and V.L. Indenbom, 1970, J. Prikl. Mekh. & Tekh. Fiz. 1, 96. Blekherman, M.Kh., and V.L. Indenbom, 1974, Phys. Status Solidi A 23, 729. Blekherman, M.Kh., and V.L. Indenbom, 1974b, Sov. Phys.-Solid State Phys. 16, 1733. Blekherman, M.Kh., and G.I. Natzvlishvili, 1968, Kristallografiya 14, 351. Elliott, H.A., 1947, Proc. Phys. Soc. A 59, p. II, 208. Entov, V.M., and R.V Salgonic, 1968, Inz. J. Mech. Tverd. Tela, N6, 87. Gehlen, P.C., and M.F. Kanninen, 1970, in: Inelastic Behavior of Solids, ed. M.F. Kanninen, et. al. (McGraw-Hill, New York) p. 587. Goodier, J.N., 1968, in: Fracture, Vol. 2, ed. Liebowitz (Academic Press, New York) p. 1. Hirth, J.P, and J. Lothe, 1968, Theory of Dislocations (McGraw-Hill, New York). Indenbom, V.L., 1961, Fiz. Tverd. Tela 3, 2071. Indenbom, V.L., and A.N. Orlov, 1970, Problemi Prochnosti No. 12, p. 3. Indenbom, V.L., and A.N. Orlov, 1977, Proc. VII Magdeburg Symp. Deformation and Fracture.
268
V.L. Indenbom
Irwin, G.R., 1957, J. Appl. Mech. 24, 361. Leonov, M.Ya., and V.V Panasjuk, 1959, Prikl. Mekh. 5, 391. Muskhelishvili, N.I., 1968, Singular Integral Equation (Izd. Nauka, Moscow). Nabarro, F.R.N., 1947, Proc. Phys. Soc. A 59, 256. Nabarro, F.R.N., 1952, Adv. Phys. 1, 269. Nabarro, F.R.N., 1967, Theory of Crystal Dislocations (Clarendon Press, Oxford). Obreimoff, J.W., 1930, Proc. R. Soc. A 127, 290. Peierls, R.E., 1940, Proc. Phys. Soc. A 52, 34. Schmidt, V, and J. Woltersdorf, 1970, Phys. Status Solidi 41, 565. Stroh, A.N., 1957, Adv. Phys. 6, 418. Thomson, R., 1983, in: Atomistics of Fracture, eds R.M. Latanision and I.R. Pickens (Plenum Press, New York) p. 167. van der Merwe, J.H., 1950, Proc. Phys. Soc. A 63, 616.