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Atomic theory of fracture of crystalline materials: cleavage and dislocation emission criteria
Materials Science and Engineering, A176 (1994) 255-261
255
Atomic theory of fracture of crystalline materials: cleavage and dislocation emission criteria K. M a s u d a - J i n d o a, S. J. Z h o u b, R. T h o m s o n c a n d A. E. C a r l s s o n b
~'~Depanment of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227 (Japan) ~'Department of Physics, Washington University, St. Louis, MO 63130 (USA) ~Laboratory Jbr Materials Science and Engineering, National Institute of Standards and Technology; Gaithersburg, MD 20899 (USA)
Abstract Using the lattice Green function approach, we study the crack-dislocation effects in the fracture of crystalline materials. Firstly, we calculate the Green function for the defective lattice, with dislocation and crack, by solving the Dyson equation. After the lattice Green functions have been determined, the relaxation problem for the reconstituted bonds in the cohesive zone is solved. The external force F with tensile and shear components is applied, as a pair of forces, to the atoms at the center of the crack. In this calculation the dislocation emission is chosen to be on a cleavage plane as well as on a glide plane of the two-dimensional lattice (hexagonal and graphite lattices). We compare the cleavage and dislocation emission criteria of the atomistic simulation with those of linear elasticity theory.
1. Introduction Recently there has been a considerable interest in the study of dislocation-crack interaction in crystalline materials [1-4]. This is due to the fact that a major factor in determining the ductile vs. brittle behavior of materials is the ease of emission of dislocations from a crack tip. If dislocation emission is sufficiently easy, then a crack will response to an applied stress by such emission and concomitant plastic deformation rather than by crack extension. Most of the theoretical works presented so far for the dislocation-crack problem, however, are based on continuum elasticity theory and the underlying assumptions are inaccurate and often contain empirical parameters which cannot be obtained from the theory. It is the purpose of the present paper to propose a more precise criterion for dislocation emission based on an atomistic simulation. We use the lattice Green function method [1, 5-7] to study the fracture behavior of two-dimensional (2D) materials, namely a hexagonal lattice with isotropic atomic bonding and a honeycomb lattice with directional bonding. The 2D graphite lattice (with sp2-hybrid orbitals) is introduced to examine the effects of the covalent nature of the atomic bonding on the fracture behavior. Although the analysis for 2D materials is an idealized one, it allows us to see in the simplest possible fashion some of the effects which 0921-5093/94/$7.00 ,S?;I)I 0921-5093(93)02558-K
must also be present in more complex systems. In particular, mode I and mode II loadings act independently within elasticity theory, but in our atomistic,model it is possible to identify directly the interactions between the different loadings. We obtain numerical estimates of the critical stress K,c required for dislocation emission for several pair-potential-type force laws and find that the dislocation maximum strain q (~1 = b/~w, w being the dislocation halfwidth) describes our results very accurately. We then compare our results with those of Schoeck [3] and Rice [4] and finally develop a crack stability diagram which illustrates the interactions between mode I and mode II loadings.
2. Principle of calculations In this section we develop a general formalism for treating a crack as a defect in a crystalline lattice in terms of the defect Green function. Firstly the force constant operator q~ij(l, l') is defined as an appropriate second derivative of the total strain energy of the lattice. The harmonic spring force on a reference atom located at l caused by the displacements of its neighbors must be balanced by an external force F applied to the atom, i.e. ¢i,(l, r ) . j ( l ' ) = - F , ( l )
(1)
© 1994 - Elsevier Sequoia. All rights reserved
256
K. Masuda-Jindo et al. / Atomic theory of crystalfracture
Here fki/l, l') is the force in the i direction exerted on the atom at lattice point l by the atom at l' when it has unit displacement in the j direction; uj(l') is the assumed displacement from equilibrium of the atom at l' in the j direction and summation over repeated indices is implied; El(1) is the externally applied force on the atom at l. The force constant matrix ~ * of the cracked lattice is obtained from that of the perfect crystal by introducing the force terms on the cleavage surface that annihilate the bonds there. These corrections to the force constant matrix ¢b of the perfect lattice are simply the negative values of the perfect crystal forces from the second term in a Taylor expansion of the potential energy of the lattice. Thus ~ * can be written formally as
¢~*= ¢ ~ - 6 ~
iF/
FII f ~ -
lc :[ lmax"
Fig. 1. Crack geometry and application of forces in 2D hexagonal lattice. 0.10 0.08 0.06 0.04
(2)
0.02 0.00
The formal solution of the problem is given by the Dyson equation [5]
-0.02
G * = G + G f q ~ G*
-0.
(3)
-0.04 -0.06 O~
-0.1(
together with the "master equation" u = G*F
-0.12
(4)
-0.14
\.,,/
-0.16
FC
F ~= -k(r F =
= -k(r
--
a)e-('-*)=l#' a))e-('-*P/#.
-0.18
for the Green function, where u and F represent the displacement and external force vectors respectively. We use the constraint that the sum of all the forces acting on each atom must vanish, i.e. from eqn. ( 1 ) F+ f + ~ * u = 0
(5)
Here f denote forces which are exerted by atoms whose bonds have been stretched into their non-linear regimes. The atomic displacements u are given by solutions of the coupled equation u = G ' F + G*f{u}
(6)
After the appropriate Green functions have been determined, the relaxation problem for the reconstituted bonds in the cohesive zone, eqn. (6) above, is solved with force laws appropriate to the problem [8].
3. Results and discussion 3.1. Crack properties in 2D hexagonal lattice When the crack is created in the 2D hexagonal lattice, each atom (in the middle region of the crack) on one side of the cleavage plane is disconnected from two atoms on the opposite side. In Fig. 1 we show a schematic drawing of the crack in the 2D hexagonal lattice under mode I and mode II loadings. The results of our atomistic simulation are given in terms of stress intensity factors, which form elasticity theory for a finite
-0.20
0.8
1 .0
1 .2
1 .4 Distance
1 .6
1 .8
2.0
Fig. 2. Force laws used in the atomistic simulations.
crack are given by
F,,. (C+t] U2
KI, II-
1,1/2 Nrc )
\~--t/
(7)
where c and t are the half-length of the broken bond region and the lattice position where the external force F~ or F~ is applied [1] relative to the crack tip respectively. The supercell size is chosen to be 4 × 1 0 6 and the crack length is taken to be 2lmax+1 = 101 lattice spacing. The length of the cohesive zone at the righthand side of the crack is about 20, but at the left-hand side all the bonds up to the end of the crack are assumed to be broken. The force laws used in the present calculations are shown in Fig. 2 [8-10]. These force laws are changed in the context of the universal binding energy relation (UBER) with fixed spring constant k, but dimensionless length scale parameters l are varied from 0.05 to 0.24. In our lattice the shear modulus and Poisson ratio are p = k(3) U2 and v = k respectively. 3.2. Dislocation emission criterion In Fig. 3 we show the atomic configuration in the cohesive zone at the right end of the crack under mode
K. Masuda-Jindo et al. (a)
/
257
Atomic theory of crystal fracture
i Crack Tip
0..30 I 028
(a)
/./"
0.26
r0
0,24
Fb Fe present 2722 S~ho~k
022 020 0
,i 8
r,.; 0
16
/"
v
F uBER
v
/,
/"
I
y.~//
0 . 14
(b)
0.12
CrackTip
0
LEmorglng Dislocation
10
,
(
0.08 006 o.
04
0.02
./." i~
0 O0 /' 0 O0
.
. . 010
.
. . 0.20
.
. . 0..30
0.40
0.50
0.60
0.70
0,30
IC r a c k
e)
(b) Tip
Dislocation
0,28 0.26
0
0.24 0.22 s
0.20 0.18 * =
Fig. 3. Crack geometries of 2D hcxagonal lattice:(a) mode I crack; (b) equilibrium crack just before emission; (c) equilibrium crack afteremission.
~.
016 0,14
~
0,12
K,Ic_ 0.69ff
(r/b)l/2
(8)
1-v
Schoeck's model provides an evaluation of the proportionality constant relating Knc to 7/~/=. We see in Fig. 4(a) that his constant provides agreement with the atomistic results at roughly the 10% level. On the other hand, comparison of the present atomistic results with Rice's continuum results is given in Fig. 4(b). Rice introduced a solid state parameter, the unstable stacking energy y~, and derived the emission criterion as [4] (2yuJ~) '/2 Kn~ = \ 1 - v /
(9)
I~ ~
0.10 /J 0,08 006
,----A----~,
/"
0,04~j
I and mixed mode loadings. For the UBER force law in Fig. 2 we have obtained the following: (i) a small lattice trapping of about 9%; (ii) the ratio of the critical stress intensity for mode II emission relative to the Griffith stress intensity is 0.38. Similar (but numerically different) results are also obtained for the other force laws shown in Fig. 2. In Fig. 4(a) we compare the present atomistic results with Schoeck's elasticity criterion [3]. By evaluating the energy of the emerging dislocation distribution as a function of position, Schoeck was able to calculate the value of K. at which there was no energy barrier to emission. The numerically obtained result is
II ~
j T J
-J ~"
002 0.00
..... 0.0
0.2
04
0.8
M o d e I I , F UsER equil ibrlum, F UBER equi I ibrlum, Fa equi I lbrlurn, Fb Mode II, F= equilibrium, F© Rice 0,8
1 .0
Fig. 4. Atomistic results of K.e v s . square root of (a) dislocation maximum strain ~/ and (b) stacking energy gu~. Units of K are kf/1'2
In Fig. 4(b) we give the Kncvalues as a function of the square root of the unstable stacking energy Vu~. For pure mode II loading we find that K.c exceeds Rice's values by 13%-28% depending on the force law parameters. In the equilibrium crack the mode I loading reduces K.e. Here one can see that Kneis closer to Rice's values than for the pure mode II case. Thus one can see that Rice's model actually gives more accurate results for the equilibrium crack. The trends with changing cut-off parameter l are obtained more accurately in the equilibrium crack, showing weak increases with increasing I in both the atomistic results and Rice's results. 3.3. Mode I-Mode H coupling We now discuss the interaction between mode I and mode II loadings. For this calculation, cleavage and emission are restricted to the cleavage plane of the 2D hexagonal lattice. In linear elasticity theory these act independently [11]. Dislocation emission from a crack
258
K. Masuda-Jindo et al.
/
tip along the cleavage plane is controlled exclusively by mode II loading, while cleavage is controlled exclusively by mode I loading. In addition, the shielding of the crack by an edge dislocation on the cleavage plane is pure mode II shielding, with no shielding of the mode I stress singularity. Here it is noted that in linear elasticity theory a pure mode II crack will never cleave, since there is no tensile force to open up the crack; therefore dislocation emission is always favored. It is convenient to discuss mixed mode loading with a "crack stability" diagram [11]. In linear elasticity the cleavage curve in such a diagram (see Fig. 5) is straight and normal to the K~ axis and the emission curve is straight and parallel to the K, axis. In the present atomistic calculation we have found a strong mode I - m o d e II interaction. Figure 5 shows the results for the U B E R force law, appropriate for ductile materials. In the case of fixed mode I loading and increasing mode II loading we see that KtIo is reduced by the mode I load. This effect may be understood in a simple fashion in terms of the 7u, parameter. As the mode I load is increased, the two blocks whose relative displacement defines 7u~ are separated from each other, which as mentioned above reduces the corrugations in the potential as a function of parallel displacement. Thus 7us and in turn KtIo are reduced by mode I loading. On the other hand, we also see in Fig. 5 that Ktc is reduced by the presence of mode II loading. This result may be due to the effect of mode II loading on the surface energy 7,. For comparison we also present in Fig. 5 the prediction (chain curve) by standard elastic analysis that a critical value of the elastic energy release
Atomic theory o f crystal fracture
should represent the cleavage line [12]. This analysis assumes that the crack is always able to reach its absolute, lowest energy state and will not be trapped in a metastable, higher energy state. 3.4. M o d e l c r a c k
In the previous subsection we discussed the dislocation emission from a mode II crack (the Burgers vector of the emitted dislocation is entirely in the crack plane). We will now discuss the dislocation emission problem of mode I loading where the emitted dislocation has a Burgers vector component orthogonal to the crack plane. For this mode I case we will show that the Peierls analysis is incomplete. We use two types of force laws, Ff and Fdf, to study dislocation nucleation from a mode I crack: Ff= -
ku[exp( - u / l ) - b]
(11)
1-b ku{exp[- ( u /l) 3/2] - bd}
Fdf = 1 --
(12)
bd
where k and I are the spring constant and range parameter in the force laws respectively and
exp( ) u0 =(31/2- 1)a
Gc = (K, 2 + K,,2)(1 - v ) = 27~ 2~
(10)
0.20 0
18
0
16
0
14
0
12
.......... :...-..-.£..- . . . . . . . ........
o.lo
~, . . . . . .
A- . . . . . . . . . . .
.a.'.•
....
0.08
(15)
These force laws are basically linear forces cut off by exponentials, except that they are also modified so they go to zero at the cut-off range u 0. It is obvious that both critical configurations are not at a / 2 but nearly at a / 6 . In Table 1 we present the calculated stress intensity factors Kic together with the unstable stacking energy 7us and the surface energy 7s. The atomic configurations for a mode I crack are shown in Fig. 6. One can see in Table 1 that Kic changes in the same way as the surface energy 7s and that the unstable stacking energy 7us varies relatively little. For the force law Ff the ratio
0.06 0.04
TABLE 1. Calculated stress intensity factors Kte
h
0.02 0.00 0.00
's. 0.04
0.08
O. 12
O. 16
0.20
K,
Fig. 5. Crack stability diagram of 2D hexagonal lattice. F UBER force law with l = 0.551(Cu) is used. Chain curve indicates results of Sinclair-Finnis energy release approach.
Unit Ff Fdf
Y~
)'o~
K.¢a
k/(Cu)2 1.805 2.109
kl(Cu) 2
ka 1/2
ka 1/2
a
0.1632 0.1654
0.4074 0.4536
0.2215 0.2230
0.1697 0.1557
~Present calculation; bRice's calculation.4
Kle b
Critical state
K. Masuda-Jindo et al.
Atomic theo O, of co'stal fracture
Ca)
259
materials. We focus our discussion on brittle materials with directional chemical bonding, such as covalent semiconductors or ceramic materials. As a typical example we consider the 2D graphite lattice and try to estimate the stress intensity factors for dislocation emission at the crack tip region. Firstly we estimate the force law for the 2D graphite lattice using the LCAO electronic theory [13]. For the perfect graphite lattice the two-center hopping integrals sso, spo, ppo and pp~ can be determined so as to fit the eigenvalues of the band structure calculations [14]. We assume that the two-center integrals vary exponentially with the interatomic distance Ri/as b = b 0 e x p ( - qR,i)
(16)
To stabilize the crystal, one generally adds a repulsive term E~ at short distances: E~= ('o exp(-pRii )
(17)
The parameters C0, p and q can be fitted to the experimental values of the cohesive energy E c and the stretching force constant so as to satisfy the equilibrium condition. We use the following parameters for the graphite lattice [12, 13]: E, = - I 1.655. pR o =4,
Fig. 6. Drawings of equilibrium crack (a) just before emission and (b) after emission.
of Rice's prediction to our measured K~e is 54%. Here we still use Rice's definition to calculate Yus-At a/6 the "y~]' value is about five times (i.e. 0.163/0.033) smaller than that at b/2, which will change the ratio KI~ above into (54/51/=)% or (49/51/2)%. This means that ys is the true dominant factor rather than Yu,,.This conclusion is not surprising, since unlike the mode II case, dislocation emission from a mode I crack also involves breaking the bonds to form a ledge (as shown in Fig. 6) besides shearing the bonds. Therefore the critical configuration for emission is not necessarily at a/2 and additional terms involving the ledge formation energy must be included in the new criterion. 3.5. DMocation emission in brittle materials" So far we have discussed dislocation emission and cleavage fracture of ductile materials with isotropic atomic bondings. In this subsection we briefly discuss the dislocation emission and cleavage fracture in brittle
Ep = - 7.711
(18)
qRl~ = 2
Then, by differentiating the total energy with respect to the atomic displacement, one can derive the force law for the graphite lattice. We calculate the "stacking fault" energies Yu~ as a function of the shear atomic displacement R~h (rigid body translation along the glide plane as shown in Fig. 7) for both type A and type B atomic displacements. The unstable stacking energies, i.e. the maximum values of Yu~ (shown in Fig. 8), for type A and type B shear displacements are estimated to be 0.7 and 1.08 eV/do respectively. This indicates that the frictional force for dislocation motion on the (01) plane is approximately 40% larger than that on the (11) plane. We have also calculated the surface tension y~ of the 2 D graphite lattice and found that the ratios yus/2 ys are 0.29 and 0.69 on the ( 11 ) and (10) planes respectively. For the 2D graphite lattice we have obtained the following characteristic features. (1) There is strong coupling between mode I and mode II loadings and Kne values are significantly reduced (by about 20%) by mode ! loading. Klc is also reduced by the presence of mode II loading. (2) The dislocation emission is much easier on the (11) plane compared with the (10) plane and is controlled by the parameter value of Yu~. From these calculations we come to the conclusion that the ductility of the 2D graphite lattice depends strongly on the orientation of the external loads.
K. Masuda-Jindo et al.
260
(a)
type
/ Atomic theory of crystalfracture
B
(a)
oo
/ \ /
t o
\
04
/ \
/
\
> e ...R- ~-
.-
04
".,/"
-o
/
'. .,.
o'5
;
o's
R s h / d o --~
(b)
I
(b)
12
t
08
\
/
o
\
>
r 04
\
r Q,
-" " . " • '
04
I
-"; ........
0
". . . . . . . . .
0'4
R s h / d 0 "->
t..
(c)
Fig. 7. (a) Schematic drawing of atomic displacements for type A and type B rigid body translations on (11) and (01) planes in 2D graphite lattice. Also shown in (b) and (c) are the cleavage and glide planes.
4. C o n c l u s i o n s
We have used the lattice Green function approach and studied the cleavage and dislocation emission criteria for cracks in 2D materials. It has been shown that some of the assumptions underlying the continuum theories are incorrect in the problem of
Fig. 8. Unstable stacking energy 7us as a function of the shift Rs, of the atomic displacement for type A and type B shearing deformation. Dashed curves show the variation in the interlayer spacing just across the glide plane (indicated by arrows in Fig. 7).
crack-dislocation interaction. T h e important theoretical findings are as follows. (1) M o d e I - m o d e II interactions play a significant role for both ductile and brittle materials in determining the crack stability diagram. (2) T h e stress intensity factors KIIe for dislocation emission are correlated well with the unstable stacking energy 7us as well as the dislocation width w (proportional to the inverse of the dislocation maximum strain). (3) T h e ductility of the 2D graphite lattice depends strongly on the orientation of the external loads.
References
1 2 3 4 5 6
R. Thomson, SolidState Phys., 39(1986) 1. J.R. Rice and R. Thomson, Philos. Mag., 29 (1974) 73. G. Schoeck, Philos. Mag., 63(1991) 111. J.R. Rice, J. Mech. Phys. Solids, 40(1992) 239. V.K. Tewary, Adv. Phys., 22 (1973) 757. C. Hsieh and R. Thomson, J. Appl. Phys., 44 (1973) 2051.
K. Masuda-,lindo et al.
/
7 K. Masuda-Jindo, V. K. Tewary and R. Thomson, J. Mater. Res., 6(1991) 1553. 8 J. H. Rose, J. R. Smith and J. Ferrante, l'hvs. Rev. B, 28 (1983) 1835. 9 R. Thomson, S. J. Zhou, A. E. Carlsson and V. K. Tewary, Phys. Rev. B, 46{1992) 10613. 10 S.J. Zhou, A. E. Carlsson and R. Thomson, Phys. Rev. B, 47 (1993) 7710.
Atomic theo O' gf cO'stal.~acture
261
11 I.-H. Lin and R. Thomson, Acta Metall., 34 (1986) 187. 12 J. E. Sinclair and M. W. Finnis, in R. Latanision and J.
Pickens (eds.), Atomistics q¢" Fracture, Plenum, New York, 1983, p. 1047. 13 C. Priester, G. Allan and J. Conard, Ph3w. Rev. B, 2~ (1982) 4680. 14 G.S. Paintcr and D. E. Ellis, Phys. Rev. B. l (1970) 4747.
255
Atomic theory of fracture of crystalline materials: cleavage and dislocation emission criteria K. M a s u d a - J i n d o a, S. J. Z h o u b, R. T h o m s o n c a n d A. E. C a r l s s o n b
~'~Depanment of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227 (Japan) ~'Department of Physics, Washington University, St. Louis, MO 63130 (USA) ~Laboratory Jbr Materials Science and Engineering, National Institute of Standards and Technology; Gaithersburg, MD 20899 (USA)
Abstract Using the lattice Green function approach, we study the crack-dislocation effects in the fracture of crystalline materials. Firstly, we calculate the Green function for the defective lattice, with dislocation and crack, by solving the Dyson equation. After the lattice Green functions have been determined, the relaxation problem for the reconstituted bonds in the cohesive zone is solved. The external force F with tensile and shear components is applied, as a pair of forces, to the atoms at the center of the crack. In this calculation the dislocation emission is chosen to be on a cleavage plane as well as on a glide plane of the two-dimensional lattice (hexagonal and graphite lattices). We compare the cleavage and dislocation emission criteria of the atomistic simulation with those of linear elasticity theory.
1. Introduction Recently there has been a considerable interest in the study of dislocation-crack interaction in crystalline materials [1-4]. This is due to the fact that a major factor in determining the ductile vs. brittle behavior of materials is the ease of emission of dislocations from a crack tip. If dislocation emission is sufficiently easy, then a crack will response to an applied stress by such emission and concomitant plastic deformation rather than by crack extension. Most of the theoretical works presented so far for the dislocation-crack problem, however, are based on continuum elasticity theory and the underlying assumptions are inaccurate and often contain empirical parameters which cannot be obtained from the theory. It is the purpose of the present paper to propose a more precise criterion for dislocation emission based on an atomistic simulation. We use the lattice Green function method [1, 5-7] to study the fracture behavior of two-dimensional (2D) materials, namely a hexagonal lattice with isotropic atomic bonding and a honeycomb lattice with directional bonding. The 2D graphite lattice (with sp2-hybrid orbitals) is introduced to examine the effects of the covalent nature of the atomic bonding on the fracture behavior. Although the analysis for 2D materials is an idealized one, it allows us to see in the simplest possible fashion some of the effects which 0921-5093/94/$7.00 ,S?;I)I 0921-5093(93)02558-K
must also be present in more complex systems. In particular, mode I and mode II loadings act independently within elasticity theory, but in our atomistic,model it is possible to identify directly the interactions between the different loadings. We obtain numerical estimates of the critical stress K,c required for dislocation emission for several pair-potential-type force laws and find that the dislocation maximum strain q (~1 = b/~w, w being the dislocation halfwidth) describes our results very accurately. We then compare our results with those of Schoeck [3] and Rice [4] and finally develop a crack stability diagram which illustrates the interactions between mode I and mode II loadings.
2. Principle of calculations In this section we develop a general formalism for treating a crack as a defect in a crystalline lattice in terms of the defect Green function. Firstly the force constant operator q~ij(l, l') is defined as an appropriate second derivative of the total strain energy of the lattice. The harmonic spring force on a reference atom located at l caused by the displacements of its neighbors must be balanced by an external force F applied to the atom, i.e. ¢i,(l, r ) . j ( l ' ) = - F , ( l )
(1)
© 1994 - Elsevier Sequoia. All rights reserved
256
K. Masuda-Jindo et al. / Atomic theory of crystalfracture
Here fki/l, l') is the force in the i direction exerted on the atom at lattice point l by the atom at l' when it has unit displacement in the j direction; uj(l') is the assumed displacement from equilibrium of the atom at l' in the j direction and summation over repeated indices is implied; El(1) is the externally applied force on the atom at l. The force constant matrix ~ * of the cracked lattice is obtained from that of the perfect crystal by introducing the force terms on the cleavage surface that annihilate the bonds there. These corrections to the force constant matrix ¢b of the perfect lattice are simply the negative values of the perfect crystal forces from the second term in a Taylor expansion of the potential energy of the lattice. Thus ~ * can be written formally as
¢~*= ¢ ~ - 6 ~
iF/
FII f ~ -
lc :[ lmax"
Fig. 1. Crack geometry and application of forces in 2D hexagonal lattice. 0.10 0.08 0.06 0.04
(2)
0.02 0.00
The formal solution of the problem is given by the Dyson equation [5]
-0.02
G * = G + G f q ~ G*
-0.
(3)
-0.04 -0.06 O~
-0.1(
together with the "master equation" u = G*F
-0.12
(4)
-0.14
\.,,/
-0.16
FC
F ~= -k(r F =
= -k(r
--
a)e-('-*)=l#' a))e-('-*P/#.
-0.18
for the Green function, where u and F represent the displacement and external force vectors respectively. We use the constraint that the sum of all the forces acting on each atom must vanish, i.e. from eqn. ( 1 ) F+ f + ~ * u = 0
(5)
Here f denote forces which are exerted by atoms whose bonds have been stretched into their non-linear regimes. The atomic displacements u are given by solutions of the coupled equation u = G ' F + G*f{u}
(6)
After the appropriate Green functions have been determined, the relaxation problem for the reconstituted bonds in the cohesive zone, eqn. (6) above, is solved with force laws appropriate to the problem [8].
3. Results and discussion 3.1. Crack properties in 2D hexagonal lattice When the crack is created in the 2D hexagonal lattice, each atom (in the middle region of the crack) on one side of the cleavage plane is disconnected from two atoms on the opposite side. In Fig. 1 we show a schematic drawing of the crack in the 2D hexagonal lattice under mode I and mode II loadings. The results of our atomistic simulation are given in terms of stress intensity factors, which form elasticity theory for a finite
-0.20
0.8
1 .0
1 .2
1 .4 Distance
1 .6
1 .8
2.0
Fig. 2. Force laws used in the atomistic simulations.
crack are given by
F,,. (C+t] U2
KI, II-
1,1/2 Nrc )
\~--t/
(7)
where c and t are the half-length of the broken bond region and the lattice position where the external force F~ or F~ is applied [1] relative to the crack tip respectively. The supercell size is chosen to be 4 × 1 0 6 and the crack length is taken to be 2lmax+1 = 101 lattice spacing. The length of the cohesive zone at the righthand side of the crack is about 20, but at the left-hand side all the bonds up to the end of the crack are assumed to be broken. The force laws used in the present calculations are shown in Fig. 2 [8-10]. These force laws are changed in the context of the universal binding energy relation (UBER) with fixed spring constant k, but dimensionless length scale parameters l are varied from 0.05 to 0.24. In our lattice the shear modulus and Poisson ratio are p = k(3) U2 and v = k respectively. 3.2. Dislocation emission criterion In Fig. 3 we show the atomic configuration in the cohesive zone at the right end of the crack under mode
K. Masuda-Jindo et al. (a)
/
257
Atomic theory of crystal fracture
i Crack Tip
0..30 I 028
(a)
/./"
0.26
r0
0,24
Fb Fe present 2722 S~ho~k
022 020 0
,i 8
r,.; 0
16
/"
v
F uBER
v
/,
/"
I
y.~//
0 . 14
(b)
0.12
CrackTip
0
LEmorglng Dislocation
10
,
(
0.08 006 o.
04
0.02
./." i~
0 O0 /' 0 O0
.
. . 010
.
. . 0.20
.
. . 0..30
0.40
0.50
0.60
0.70
0,30
IC r a c k
e)
(b) Tip
Dislocation
0,28 0.26
0
0.24 0.22 s
0.20 0.18 * =
Fig. 3. Crack geometries of 2D hcxagonal lattice:(a) mode I crack; (b) equilibrium crack just before emission; (c) equilibrium crack afteremission.
~.
016 0,14
~
0,12
K,Ic_ 0.69ff
(r/b)l/2
(8)
1-v
Schoeck's model provides an evaluation of the proportionality constant relating Knc to 7/~/=. We see in Fig. 4(a) that his constant provides agreement with the atomistic results at roughly the 10% level. On the other hand, comparison of the present atomistic results with Rice's continuum results is given in Fig. 4(b). Rice introduced a solid state parameter, the unstable stacking energy y~, and derived the emission criterion as [4] (2yuJ~) '/2 Kn~ = \ 1 - v /
(9)
I~ ~
0.10 /J 0,08 006
,----A----~,
/"
0,04~j
I and mixed mode loadings. For the UBER force law in Fig. 2 we have obtained the following: (i) a small lattice trapping of about 9%; (ii) the ratio of the critical stress intensity for mode II emission relative to the Griffith stress intensity is 0.38. Similar (but numerically different) results are also obtained for the other force laws shown in Fig. 2. In Fig. 4(a) we compare the present atomistic results with Schoeck's elasticity criterion [3]. By evaluating the energy of the emerging dislocation distribution as a function of position, Schoeck was able to calculate the value of K. at which there was no energy barrier to emission. The numerically obtained result is
II ~
j T J
-J ~"
002 0.00
..... 0.0
0.2
04
0.8
M o d e I I , F UsER equil ibrlum, F UBER equi I ibrlum, Fa equi I lbrlurn, Fb Mode II, F= equilibrium, F© Rice 0,8
1 .0
Fig. 4. Atomistic results of K.e v s . square root of (a) dislocation maximum strain ~/ and (b) stacking energy gu~. Units of K are kf/1'2
In Fig. 4(b) we give the Kncvalues as a function of the square root of the unstable stacking energy Vu~. For pure mode II loading we find that K.c exceeds Rice's values by 13%-28% depending on the force law parameters. In the equilibrium crack the mode I loading reduces K.e. Here one can see that Kneis closer to Rice's values than for the pure mode II case. Thus one can see that Rice's model actually gives more accurate results for the equilibrium crack. The trends with changing cut-off parameter l are obtained more accurately in the equilibrium crack, showing weak increases with increasing I in both the atomistic results and Rice's results. 3.3. Mode I-Mode H coupling We now discuss the interaction between mode I and mode II loadings. For this calculation, cleavage and emission are restricted to the cleavage plane of the 2D hexagonal lattice. In linear elasticity theory these act independently [11]. Dislocation emission from a crack
258
K. Masuda-Jindo et al.
/
tip along the cleavage plane is controlled exclusively by mode II loading, while cleavage is controlled exclusively by mode I loading. In addition, the shielding of the crack by an edge dislocation on the cleavage plane is pure mode II shielding, with no shielding of the mode I stress singularity. Here it is noted that in linear elasticity theory a pure mode II crack will never cleave, since there is no tensile force to open up the crack; therefore dislocation emission is always favored. It is convenient to discuss mixed mode loading with a "crack stability" diagram [11]. In linear elasticity the cleavage curve in such a diagram (see Fig. 5) is straight and normal to the K~ axis and the emission curve is straight and parallel to the K, axis. In the present atomistic calculation we have found a strong mode I - m o d e II interaction. Figure 5 shows the results for the U B E R force law, appropriate for ductile materials. In the case of fixed mode I loading and increasing mode II loading we see that KtIo is reduced by the mode I load. This effect may be understood in a simple fashion in terms of the 7u, parameter. As the mode I load is increased, the two blocks whose relative displacement defines 7u~ are separated from each other, which as mentioned above reduces the corrugations in the potential as a function of parallel displacement. Thus 7us and in turn KtIo are reduced by mode I loading. On the other hand, we also see in Fig. 5 that Ktc is reduced by the presence of mode II loading. This result may be due to the effect of mode II loading on the surface energy 7,. For comparison we also present in Fig. 5 the prediction (chain curve) by standard elastic analysis that a critical value of the elastic energy release
Atomic theory o f crystal fracture
should represent the cleavage line [12]. This analysis assumes that the crack is always able to reach its absolute, lowest energy state and will not be trapped in a metastable, higher energy state. 3.4. M o d e l c r a c k
In the previous subsection we discussed the dislocation emission from a mode II crack (the Burgers vector of the emitted dislocation is entirely in the crack plane). We will now discuss the dislocation emission problem of mode I loading where the emitted dislocation has a Burgers vector component orthogonal to the crack plane. For this mode I case we will show that the Peierls analysis is incomplete. We use two types of force laws, Ff and Fdf, to study dislocation nucleation from a mode I crack: Ff= -
ku[exp( - u / l ) - b]
(11)
1-b ku{exp[- ( u /l) 3/2] - bd}
Fdf = 1 --
(12)
bd
where k and I are the spring constant and range parameter in the force laws respectively and
exp( ) u0 =(31/2- 1)a
Gc = (K, 2 + K,,2)(1 - v ) = 27~ 2~
(10)
0.20 0
18
0
16
0
14
0
12
.......... :...-..-.£..- . . . . . . . ........
o.lo
~, . . . . . .
A- . . . . . . . . . . .
.a.'.•
....
0.08
(15)
These force laws are basically linear forces cut off by exponentials, except that they are also modified so they go to zero at the cut-off range u 0. It is obvious that both critical configurations are not at a / 2 but nearly at a / 6 . In Table 1 we present the calculated stress intensity factors Kic together with the unstable stacking energy 7us and the surface energy 7s. The atomic configurations for a mode I crack are shown in Fig. 6. One can see in Table 1 that Kic changes in the same way as the surface energy 7s and that the unstable stacking energy 7us varies relatively little. For the force law Ff the ratio
0.06 0.04
TABLE 1. Calculated stress intensity factors Kte
h
0.02 0.00 0.00
's. 0.04
0.08
O. 12
O. 16
0.20
K,
Fig. 5. Crack stability diagram of 2D hexagonal lattice. F UBER force law with l = 0.551(Cu) is used. Chain curve indicates results of Sinclair-Finnis energy release approach.
Unit Ff Fdf
Y~
)'o~
K.¢a
k/(Cu)2 1.805 2.109
kl(Cu) 2
ka 1/2
ka 1/2
a
0.1632 0.1654
0.4074 0.4536
0.2215 0.2230
0.1697 0.1557
~Present calculation; bRice's calculation.4
Kle b
Critical state
K. Masuda-Jindo et al.
Atomic theo O, of co'stal fracture
Ca)
259
materials. We focus our discussion on brittle materials with directional chemical bonding, such as covalent semiconductors or ceramic materials. As a typical example we consider the 2D graphite lattice and try to estimate the stress intensity factors for dislocation emission at the crack tip region. Firstly we estimate the force law for the 2D graphite lattice using the LCAO electronic theory [13]. For the perfect graphite lattice the two-center hopping integrals sso, spo, ppo and pp~ can be determined so as to fit the eigenvalues of the band structure calculations [14]. We assume that the two-center integrals vary exponentially with the interatomic distance Ri/as b = b 0 e x p ( - qR,i)
(16)
To stabilize the crystal, one generally adds a repulsive term E~ at short distances: E~= ('o exp(-pRii )
(17)
The parameters C0, p and q can be fitted to the experimental values of the cohesive energy E c and the stretching force constant so as to satisfy the equilibrium condition. We use the following parameters for the graphite lattice [12, 13]: E, = - I 1.655. pR o =4,
Fig. 6. Drawings of equilibrium crack (a) just before emission and (b) after emission.
of Rice's prediction to our measured K~e is 54%. Here we still use Rice's definition to calculate Yus-At a/6 the "y~]' value is about five times (i.e. 0.163/0.033) smaller than that at b/2, which will change the ratio KI~ above into (54/51/=)% or (49/51/2)%. This means that ys is the true dominant factor rather than Yu,,.This conclusion is not surprising, since unlike the mode II case, dislocation emission from a mode I crack also involves breaking the bonds to form a ledge (as shown in Fig. 6) besides shearing the bonds. Therefore the critical configuration for emission is not necessarily at a/2 and additional terms involving the ledge formation energy must be included in the new criterion. 3.5. DMocation emission in brittle materials" So far we have discussed dislocation emission and cleavage fracture of ductile materials with isotropic atomic bondings. In this subsection we briefly discuss the dislocation emission and cleavage fracture in brittle
Ep = - 7.711
(18)
qRl~ = 2
Then, by differentiating the total energy with respect to the atomic displacement, one can derive the force law for the graphite lattice. We calculate the "stacking fault" energies Yu~ as a function of the shear atomic displacement R~h (rigid body translation along the glide plane as shown in Fig. 7) for both type A and type B atomic displacements. The unstable stacking energies, i.e. the maximum values of Yu~ (shown in Fig. 8), for type A and type B shear displacements are estimated to be 0.7 and 1.08 eV/do respectively. This indicates that the frictional force for dislocation motion on the (01) plane is approximately 40% larger than that on the (11) plane. We have also calculated the surface tension y~ of the 2 D graphite lattice and found that the ratios yus/2 ys are 0.29 and 0.69 on the ( 11 ) and (10) planes respectively. For the 2D graphite lattice we have obtained the following characteristic features. (1) There is strong coupling between mode I and mode II loadings and Kne values are significantly reduced (by about 20%) by mode ! loading. Klc is also reduced by the presence of mode II loading. (2) The dislocation emission is much easier on the (11) plane compared with the (10) plane and is controlled by the parameter value of Yu~. From these calculations we come to the conclusion that the ductility of the 2D graphite lattice depends strongly on the orientation of the external loads.
K. Masuda-Jindo et al.
260
(a)
type
/ Atomic theory of crystalfracture
B
(a)
oo
/ \ /
t o
\
04
/ \
/
\
> e ...R- ~-
.-
04
".,/"
-o
/
'. .,.
o'5
;
o's
R s h / d o --~
(b)
I
(b)
12
t
08
\
/
o
\
>
r 04
\
r Q,
-" " . " • '
04
I
-"; ........
0
". . . . . . . . .
0'4
R s h / d 0 "->
t..
(c)
Fig. 7. (a) Schematic drawing of atomic displacements for type A and type B rigid body translations on (11) and (01) planes in 2D graphite lattice. Also shown in (b) and (c) are the cleavage and glide planes.
4. C o n c l u s i o n s
We have used the lattice Green function approach and studied the cleavage and dislocation emission criteria for cracks in 2D materials. It has been shown that some of the assumptions underlying the continuum theories are incorrect in the problem of
Fig. 8. Unstable stacking energy 7us as a function of the shift Rs, of the atomic displacement for type A and type B shearing deformation. Dashed curves show the variation in the interlayer spacing just across the glide plane (indicated by arrows in Fig. 7).
crack-dislocation interaction. T h e important theoretical findings are as follows. (1) M o d e I - m o d e II interactions play a significant role for both ductile and brittle materials in determining the crack stability diagram. (2) T h e stress intensity factors KIIe for dislocation emission are correlated well with the unstable stacking energy 7us as well as the dislocation width w (proportional to the inverse of the dislocation maximum strain). (3) T h e ductility of the 2D graphite lattice depends strongly on the orientation of the external loads.
References
1 2 3 4 5 6
R. Thomson, SolidState Phys., 39(1986) 1. J.R. Rice and R. Thomson, Philos. Mag., 29 (1974) 73. G. Schoeck, Philos. Mag., 63(1991) 111. J.R. Rice, J. Mech. Phys. Solids, 40(1992) 239. V.K. Tewary, Adv. Phys., 22 (1973) 757. C. Hsieh and R. Thomson, J. Appl. Phys., 44 (1973) 2051.
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7 K. Masuda-Jindo, V. K. Tewary and R. Thomson, J. Mater. Res., 6(1991) 1553. 8 J. H. Rose, J. R. Smith and J. Ferrante, l'hvs. Rev. B, 28 (1983) 1835. 9 R. Thomson, S. J. Zhou, A. E. Carlsson and V. K. Tewary, Phys. Rev. B, 46{1992) 10613. 10 S.J. Zhou, A. E. Carlsson and R. Thomson, Phys. Rev. B, 47 (1993) 7710.
Atomic theo O' gf cO'stal.~acture
261
11 I.-H. Lin and R. Thomson, Acta Metall., 34 (1986) 187. 12 J. E. Sinclair and M. W. Finnis, in R. Latanision and J.
Pickens (eds.), Atomistics q¢" Fracture, Plenum, New York, 1983, p. 1047. 13 C. Priester, G. Allan and J. Conard, Ph3w. Rev. B, 2~ (1982) 4680. 14 G.S. Paintcr and D. E. Ellis, Phys. Rev. B. l (1970) 4747.