- Email: [email protected]
alumina interfaces: an analysis based on a micromechanical model
Vol. 43, No. 12, pp. 4301-4307, 1995 Elsevier Science Ltd Copyright 9 1995 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 0956-7151/95 $9.50 + 0.00
Acta metall, mater.
eergamo. 0956-7151(95)00125-5
FRACTURE TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES: AN ANALYSIS BASED ON A MICROMECHANICAL MODEL J. Y. SHUt, J. A. BLUME and C. F. SHIH Division of Engineering, Brown University, Providence, RI 02912, U.S.A. (Received 12 October 1994; in revised.form 8 March 1995)
Abstract--The fracture toughness is predicted for an interface between niobium and alumina. It is assumed that voids at the triple points of grain junctions on the interface are the sites of cleavage decohesion. These voids are treated as microcracks and the plastic zone is modelled by a cohesive zone of constant shear stress. The interaction between the microcrack and the main crack is studied using an integral equation approach. The fracture toughness is calculated via various fracture criteria and compared against experimental data. Scatter in the experimental data is related to the variability in microdefect size.
1. INTRODUCTION Metal-ceramic interfaces can be found in many modern material systems including layered structural composites, coating and thin film-substrate systems commonly used in the microelectronics industry. Mechanical failure is often due to interfacial fracture. Unfortunately, the understanding of the fracture toughnesses of such metal-ceramic interfaces is limited. In a homogeneous brittle material, cracks advance in mode I. In contrast, the fracture toughness of a metal--ceramic interface has a strong dependence on the mode mixity of a crack on the interface, In general, the toughness increases very rapidly for large phase angles [1]. The niobium-alumina material system provides a model metal-ceramic interface that has been investigated extensively (see, for example, [2]). Recently, a joint theoretical and experimental study on the interface was carried out by O'Dowd et al. in [1]. Different bending specimens were used to measure the fracture toughness of the niobium-alumina interface. They considered the role of plasticity around the crack tip and obtained a toughness curve from finite element calculations based on the criterion that crack growth initiates when the hoop stress at a point on the interface reaches a critical value. The curve so-obtained fits experimental data much better than that by an elastic analysis [3]. Tvergard and Hutchinson [4] came to the same conclusion by performing an
tCurrent address for correspondence: Chemistry and Material Science Department, L-370, Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A. AM43/12--~
elastic-plastic finite element analysis using a cohesive zone at the crack tip. Although the toughness curve calculated in [1] represents a significant improvement in fitting experimental data, it is still not satisfactory. For example, an arbitrary distance r c from the crack tip has to be chosen in [1] to generate the toughness curve. Different choices of r e lead to modestly different fracture toughness curves. A micromechanical model is needed with a length scale of physical meaning much smaller than the plastic zone size. It was observed in [1] that voids at the triple points of alumina grain junctions exist at the interface between the alumina and niobium. A model for interfacial cleavage decohesion was proposed by assuming that these voids on the interface are sites for cleavage decohesion. As long as these microdefects grow and join the main crack before significant plastic deformation occurs around the microdefects, high stress can be sustained and cleavage decohesion may prevail. An estimate of the fracture stress was made by idealizing these defects as penny-shaped cracks and was found to be close to the stress calculated by the finite element method. However, the calculation did not consider the interaction between the main crack and the penny-shaped cracks. This study intends to provide an elastic-plastic analysis based on a simplified version of the model in Ref. [1] by taking account of the interaction between the ci'acks. This allows us to calculate the dependence of fracture toughness of the interface on the phase angle and microdefect size and spacing. Good agreement is observed between predictions and the experimental data from [1]. Furthermore, our calculation provides an account for the scatter of the experimental toughness data by the distribution in the microdefect size.
4301
4302
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
2. ELASTIC CRACK TIP FIELD AND PHASE ANGLE For a simply connected system under plane strain condition of two isotropic, homogeneous, linearly elastic materials with traction prescribed on its boundary, Dundurs [5] proved that the stresses depend on the material properties of the bimaterial pair through the Dundurs parameters ~ and fl
~ le iv
_•K
Rl-172 EIWJ~2 ' /~'1 1 " 2v2 fl = E'~ + R 2 2 - 2v2
/~2 1 -- 2v t El + E 2 2 - 2v,"
(1)
Here E = E/(1 - v2), E is the Young's modulus and v the Poisson's ratio. Subscripts 1 and 2 refer to the upper and lower materials respectively. Consider a semi-infinite crack lying on the interface between the two materials, as shown in Fig. 1, subjected to a singular stress field of K| ~ aoo + itLo = (2~zr)l/2
(2)
on the interface where K | = K ~ + i K ~ (i = x ~ ) is the complex stress intensity factor and E is the oscillation index defined as E = ~ In 1 +----fl"
(3)
The phase angle of the K%field is defined as tan~b = (~,0"~
Im(K ~r') \aoo,/o = o = Re(K~ "
(4)
It can be seen that when e # 0 , ~, depends on the distance r ahead of the crack tip. In this study, however, we make the simplification that fl = E = 0. This is partially justified by the fact that is usually very small (e = 0.02 for a niobium-alumina interface) so that phase angles based on two moderately different length scales would not differ signifi-
~ - i K ~' leiV Fig. 2. Schematic illustration of the micromechanical model. The cohesive zone is subjected to a constant bridging shear stress. cantly. We write the complex K | form
in the polar
K ~ = Ig~le i~
(5)
where the magnitude IK~176 is a real number. 3. A MICROMECHANICAL MODEL It has been observed that grain boundary triple points are often poorly bonded on the interface [1]. In a niobium-alumina system, the average grain size of alumina is about 3 #m. Hence these poorly bonded regions are at a spacing of about 3-5 # m and suffer high tensile stresses when located near the crack tip. In the model proposed in [1], the interface is pictured as populated by penny-shaped microcracks at a spacing of the grain size and these cracks are the sites of cleavage decohesion. Here, instead of penny-shaped cracks, we use 2D through-thickness cracks under plane strain condition, and consider only one microcrack located near the crack tip.
c~ [e/u
_•K
9
~
Xl
~ l / C | lei* Fig. I. Conventions of an interface crack and K~176
3. I. M o d e l i n g o f the plastic zone
To model plasticity near the main crack tip in the niobium, the plastically deformed region is idealized as a line of cohesive zone inclined at an angle 0c from the main crack. Along the cohesive zone, there is a continuous distribution of edge dislocations. The Burgers vector of the dislocations has a non-zero component only in the direction parallel to the cohesive zone, as shown in Fig. 2. The cohesive zone is assumed to be subject to a constant shear stress. The material elsewhere is assumed to be linearly elastic. Such a model of plasticity around a crack tip was first proposed by Bilby and Swindon [6]. One motivation for this approach is the fact that plasticity is caused by the slip of dislocations driven by a shear traction acting parallel to their slip direction.
S H U et al.:
TOUGHNESS
OF NIOBIUM/ALUMINA
The inclination of the cohesive zone from the main crack was chosen arbitrarily in [6]. By suitable choice of dislocation density along the cohesive zone, the shear traction is constant, and the singular term of the shear stress along the cohesive zone vanishes. However, as shown below, this scheme does not cancel the - 1 / 2 singularity of stresses in other directions. This is not a desirable feature since it is commonly accepted that the stresses within the plastic zone are either bounded or has a singularity weaker than - 1/2, such as H R R singularity. In the following, a more rational choice of the inclination angle of the cohesive zone is proposed. For the time being, we will neglect the presence of the microcrack. In this paper, for a continuous distribution of dislocations, we define the Burgers vector density B as the Burgers vector accumulated over a unit length of distribution multiplied by a factor of El/(8n). For continuously distributed dislocations on the cohesive zone at an angle 0c from the main crack, the complex Burgers vector density is written as B 1 + iB 2 =/~e ~~
(6)
where the magnitude B is a real number. (Note that the component of the Burgers vector in the direction normal to the cohesive zone vanishes.) Substitution of equation (6) into (A9) and (A10) of the Appendix gives the stress intensity factor at the main crack tip induced by the dislocation distribution as
Kl + iK2 = - ( 1 - ~ ) x / ~ c o s ~ {3sinO~+i(3cosO~-l)}
f0
B(r)r-l/2dr.
(7)
The stress intensity factor given by equation (7) will satisfy K l + iK2 = - K ~ if the inclination 0~ of the cohesive zone is chosen to satisfy cos 0 " (3 cos 0~ - 1) = sin 0 " 3 sin 0~
(8)
INTERFACES
remote K~~ by
4303
Then, 0c is related to the phase angle
cos ~k 9cos ~ . (3 cos 0c - 1) 0c - s i n ~ ' sin-~. (5 + 9 cos 0~)= 0.
(11)
From Fig. 3, it can be seen that the inclination 0~ found from equation (10) coincides with that found from equation (11) for the limiting cases of pure mode I or pure mode II. We shall adopt relations (9) and (10) since stresses will remain finite at the tip of the main crack. When the microcrack is present, an additional stress intensity factor will be introduced by the microcrack. But the size of the microcrack is usually much smaller (0.25-1 ~m) than the cohesive zone size which, according to [1], ranges from more than 10 #m for a small phase angle to more than 100 ~m for a large phase angle. Also the microcrack is at a distance of 3-5 pm from the main crack tip, so the additional stress intensity factor at the main crack tip caused by the microcrack is likely to be much smaller than the contribution from the cohesive zone and will be ignored. 3.2. Formulation o f the model
The stress field can be regarded as the superposition of a remote K~-field with a stress intensity factor as described in equation (4) and that due to dislocations continuously distributed on the cohesive zone and the microcrack. It is assumed that the cohesive zone is subjected to a constant bridging shear stress equal to av/2 where a v is the tensile yield strength; the microcrack is taken to be traction-free. Denote the shear and hoop stresses due to the distributions of dislocations by ar0 and ~r00respectively; further, denote the shear component of the crack tip K~ field by ~r0.1 and the shear component of the crack tip K~ field by ~r0,2" Then,
and the Burgers vector density/~ satisfies
(1
oo
o'PB(r)r-l/2dr = l K %
.
.
.
(9)
The physically meaningful solution to equation (8) is 0c = cos-l(1/3 cos 4) - ~k.
.
9 K - field relaxing direction
~ ) 2 x / ~ / 4 _ 3 cos2 ~0~ cos ~0~
(10)
An arbitrary choice of 0r other than that from equation (10) will not cancel the singular K~-field completely. Thus, it is reasonable to relate the orientation 0c of the cohesive zone to the phase angle ~ of the applied K~-field through equation (10). An alternative viewpoint is to assume that the cohesive zone is inclined along the direction of maximum shear stress tr,0 associated with the
" - . , , ._
~17625 !
2 ~
o
I 0
I
I,
i,, 25
,
,
/'["r-.,~ 50
75
(degrees)
Fig. 3. Orientation angle of the cohesive zone vs the phase angle of the K|
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
4304
along the surface of the cohesive zone (0 = 0r total shear traction is given by Ig~l
arO -~-~
(COS
the
Kn = (2/t)3/2/~u+ l-
O'y
~larO,l -IV sin t#g~0,:) = -~-,
0 < r < rp.
(12)
On the surface of the microcrack (0 = 0), the total hoop stress vanishes
aoo+ [K~,~-I cos~p=O, x/ 2xr
a
(13)
IK~I a,0 + ~ r sin 0 = 0,
a
(14)
i=l .... N+I.
ri + 1 -- Fi
+Bi+ I r - r i ~ for ri<<.r<~r,+1. (15b) Fi+ 1 -- riJ Here, B~ is a set of unknown real numbers. Similarly, the microcrack (a < r < a + L) is discretized also into N = 30 elements between the nodes N
and for the right-hand tip of the microcrack by k I = (1 -- ~)(2x)3!2L-mB~+ 1,2, k2 = (1 - cr
1.1.
(19)
By dimensional analysis, the stress intensity factors can be written as
gll~fl(~/,~,L,o~)[gm}~-f2(~l,r~,L,o~)ffya|/2
(20) for the outer tip of the cohesive zone and
(15a)
/~ = ( r p - r) I/2(Biri~-lZr-
l+cos
(18)
-'-'~a a tK I+A 0 , ~ - , ~ , ~ ~ a '2,
( i - l)x 2N '
The Burgers vector density is interpolated over each element as
r~=a+
Since only edge dislocations with Burgers vector co-linear with the cohesive zone are used to represent the cohesive zone, the mode I stress intensity factor associated with this dislocation distribution vanishes. The stress intensity factors for the left-hand tip of the microcrack are given by
k 2 = - - ( 1 - ~)(27r)3i2L -~/2BI, 1
Here, a and L are the defect spacing and microcrack length respectively (see Fig. 2). We have assumed that the interface crack oscillation index ~ is zero. The Burgers vector density B~ + iB 2 for both the cohesive zone and the microcrack are determined by an integral equation approach. The stresses due to a single dislocation in a bi-material system with a semi-infinite crack on the interface is derived in the Appendix and integral equations are set up to enforce traction conditions (12)-(14). The cohesive zone is discretized into N = 30 elements between the nodes
\
(17)
k~ = - ( 1 - ~ ) (2~)3/2L - 1/2B1,2,
and the total shear stress vanishes
ri=rpsin
stress intensity factor K u at the outer tip of the cohesive zone is calculated via
~ ,
i=I,...N+I. (16a)
k2 = f s \,t{p[, "ar p, aL
~),K~I+ff(qj,r2,La
a' ~ ava'/2 (21)
for the left-hand tip of the microcrack. The stress intensity factors for the right-hand tip of the microcrack are found to be smaller than those of the left-hand tip and will not be discussed further. Here f~(i = 1. . . . . 6) are dimensionless real functions which can be calculated by the procedure described above. It is stipulated that the net mode II stress intensity factor at the cohesive zone tip should be zero in order to have non-singular stresses. For a given cohesive zone size rp, from equation (20)
The two components of the Burgers vector density on the microcrack are interpolated as ri
Bj = (L2/4 -- (r - a - L/2) 2) ,/2(~ i d - - -+l- -
r
r, + 1 -- ri
r - r i "X
+ Bi+tj~)
for
ri<~r <~ri+1 (16b)
where B#(j = 1, 2) are unknowns to be found. The traction conditions (12)-(14) are enforced at the mid-point of each element of the cohesive zone and the microcrack. The Burgers vector density of the microcrack is subjected to the constraints that the integral of Bj over the length of the microcrack vanishes;/~ is subjected to the constraint equation (9). Thus we have 3N equations for 3N unknowns. The
The above equation gives the relation between r o and the magnitude ]K~I of the applied loading. The nominal fracture toughness Kc of the interface is the value of IK~[ evaluated for a critical value of % which in ,turn is attained when a fracture criterion is satisfied. Fracture criteria will be discussed below. Substitution of equation (22) into (21) gives
kl=(f4-f3fyl)aYau2=-ffyaU2gl(~l,
ra,Z,ot ),
k2=(f6-fsffy~)ffyal/2=-cryab'2f2(~b,~,Z,c~). (23)
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES Here F, F~, F2 are dimensionless real functions of
(~k, rp/a, L /a, or). 3.3. Fracture criteria Let K~ designate the critical value of IK~[ when the microcrack begins to advance. The stress intensity factors at the left-hand tip are generally greater than those at the right-hand tip, hence cleavage decohesion will occur at the left-hand tip first. Therefore the stress intensity factors for the left-hand tip of the microcrack as given by equation (23) will be used in the relevant fracture criteria. Three different criteria for the initiation of cleavage decohesion are discussed below. 3.3.1. Critical energy release rate. The elasticbrittle Griffith model for the crack growth stipulates that when cleavage decohesion is initiated G = 2~t
(24)
in which 2~nt is the work of reversibly separating a unit area of interface. The energy release rate of the interface microcrack (e = 0) is [9]
k~ +/,22
4305
Hence, the above two equations and equation (23) give F~ = (1 - ~) 2 ~ 2 . ~rva
(30)
Similarly, the fracture toughness Kr can then be determined. In the above two fracture criteria, both 2[nt and k~c are assumed to be intrinsic material constants for a microdefect-free bi-material system. In contrast, the nominal fracture toughness K~ depends on the microstructure of the material, e.g. the size and spacing of microdefects. Estimates of 2,rnt for several material combinations have been made [7, 8]. 3.3.3. Critical hoop stress. A fracture criterion used in previous studies is the attainment of a critical hoop stress tr~ on the interface at a distance r~ from the main crack tip. The interface ahead of the main crack tip is assumed to be perfectly bonded. By dimensional analysis /
rp rc IK~I
"~
a~176176176162 a' ava l/z' ct)av=tr c (31)
1 4a._z
G = (1 + ~)Tgz= 1 + ~ ~
('/r + F2)'
(25)
Upon applying the Griffith criterion, one reaches a critical condition r~ + r~ = (1 + ~) z ~
o-ya
(26)
from which a critical value of rv can be found. Substitution of this critical value into (22) obtains the fracture toughness K~ via
( L Kc=oyal/2Fc_~//,
( 1 + 0 0 -2~n' - ~'2 ~ ). a' o'}a'
\
(27)
Here F~ is the F in equation (22) evaluated at the critical value of %/a. In the limit ~O= 0 we designate K~ by the standard notation K~. 3.3.2. Critical k~. As suggested in an elastic analysis [3], a criterion based on a critical hoop stress on the ligrnent of the main crack suggests a K~/K~ vs ff curve which is closer to the experimental measurement than that based on the energy release rate. When the effects of plasticity are included in the analysis [1, 4], a similar criterion was employed. All these analyses suggest that the opening mode of a mixed mode K~-field may play a greater role than the shear mode in determining the toughness of the interface. We propose that cleavage decohesion initiates at the left-hand tip of the microcrack when the local stress intensity factor kl attains a local fracture toughness k~, giving kl = kit.
(28)
Under pure mode-I loading, k~c is related to the cleavage energy 2~t via 1
kl2~
a . . . . 1 +or .E2
2~,t.
(29)
from which a critical value of rp is determined to be
rp = s2(~b, tr~ 'K~' rc ) a av' ayal/2 ' a ' at .
(32)
Substitution of the critical value of rp into equation (31) gives a critical value K~ of IK~I via
gc= eya'/2F3(~b, trc rc
O'v' a ' ~ '
(33)
Here, Sl, $2 and F3 are dimensionless functions. It can be seen that Kc depends on the choice of r~. An elastic analysis by Suo and Hutchinson [3] gives a K~/K~r which is independent of the choice of re. 4. FRACTURE TOUGHNESS
4.1. Comparison of fracture criteria In previous studies, the fracture criteria used to predict the toughness vs phase angle curves are either the critical energy rate G-criterion or the critical hoop stress criterion. In this paper, an additional critical k~-criterion is proposed in Section 3.3.2. The toughness curves generated by these three criteria are compared below. For the niobium/alumina interface, we take niobium and alumina to be material 1 and 2 respectively. (Note that in [1] the alumina is taken to be material 1.) The following material properties are assumed from [1] 2~nt ~ 1 Jm -2, ay = 250 MPa,
E 2 = 328 GPa, ~ = -0.527.
v2 = 0.22, (34)
A representative micro geometrical dimension of a = 3.0/~m, L = 0.5/~m is taken for our calculations. The toughness curves from the G-criterion and kl-criteflon are calculated with the current model, and the
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
4306
3.0
3.0
/I
2.5
2.5
2.0
9
experimental critical . . . . . critical G ~ . critical hoop stress
2.0 1.5 1.0
~
0
1.5 1.0 0.5
I
9
experimental
a=0.5gm a=l.5gm . . . . . . . a=6.0gm ._
|
0
0.5
25 I
I
I
I
I
I
I
I
I
[
25
I
I
I I 75
1
50
I
I
I
~g (degrees) Fig. 4. Comparison of fracture toughness curves obtained by various fracture criteria.
toughness curve with critical hoop stress criterion is taken from [1]. Figure 4 shows that the G-criterion is quite poor in that it can not predict the rapid increase of the fracture toughness Kc with an increase of the phase angle ~. This is not surprising considering that, if both plasticity and microdefects are neglected, the G-criterion would predict no increase in the toughness and the toughness curve would be a horizontal line at Kc/K~ = 1. The curves generated using the other two criteria do capture the general trend of the experimental data, and the k~-criterion seems to be the best fit to the experiment results. More importantly, the numerical results confirm the feasibility of the current model which provides a more plausible physical picture of the process of cleavage decohesion.
4.2. Influence of geometrical parameters Having shown that the critical k~-criterion best fits the experimental data, we shall adopt this criterion for the following study about the effect of geometrical parameters on the fracture toughness. As stated in Section 3.3, the toughness of a niobium-alumina interface is a function of the phase angle ~b, microdefect size L, defect spacing a, and material constants. For a niobium-alumina interface, a is 3-5/~m and L is about 0.5/~m. L is largely affected by the process of bonding, while a depends on the heat treatment of
ff
j
3.5
w
35
ili!l
3.5 ~3.0 ~2.5 ~ 2.0
50
75
(degrees) Fig. 6. Fracture toughness for different defect spacing.
the materials. The effect of a and L on the interface toughness needs to be investigated. In Fig. 5, a is fixed at 3 # m while L is given different values. It shows that the toughness is affected significantly by the magnitude of L, at large phase angles. The general trend of the toughness curves is very close to the experimental data. It can be seen that the smaller L is, the better the numerical results fit the experimental data. A surprising result is that for a fixed defect size L, the effect of defect spacing a on the toughness curve is not significant. Only when alL is as small as unity, can the difference between the toughness curves be noticed, see Fig. 6. It implies that the large scatter of the experimental data for toughness is not due primarily to the variance of grain size of alumina or defect spacing which is about the same as the grain size. From Fig. 7, it can be seen that for a between 3 and 5/~m, which is typical in this niobium-alumina system, the range in value of K]cfiTya 1/2 is very small and K~c itself varies from about 2.1 to 2.7 MPa m s/2. The effect of flaw size L upon macroscopic mode I toughness K~c is shown in Fig. 8 for a fixed crack spacing of 3 #m. This suggests that the scatter in L may be the source of the scatter in measured Kit. It is plausible that the microdefect size L displays scatter from specimen to specimen in [l]. 5. CONCLUSION A micromechanical model for the fracture toughness of an interface between a ductile and a brittle material is given. A simple model of the plastic zone is also proposed. Experimental data of fracture
o"
5.25
9 9 experimental .... L=0.25gm . . . . . . . L=0.40gm L=0.50pm ~.~ L=0.60gm
1.5 1.0 9
0.5
I
0
I
I
I
I
[
25
I
I
I
I
[
50
I
I
I
I I
I
eq
5.00 L = 0.5gm
~- 4.75
I
75
(degrees) Fig. 5. Fracture toughness for different microdefect size.
4.50
n t 0
i I
I 2.5
t
I
tl
i LILI
5.0 a (/.tin)
t,i,
7.5
Fig. 7. Dependence of Knc on defect spacing.
I
10.0
SHU et al.: 10.0
_
~3
TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
-
4307
two half planes of different materials. The second Dundurs parameter fl is assumed to be zero, implying that e vanishes as well. tween
7.5 --
~
Stresses due to the Interaction
~
Following Suo [9], stress and displacement fields are expressed by two analytic functions ~(z) and f~(z) as 5.0
trll + tr22= 2[~(Z) + ~(z)] 022 + icq2 = q'(z) + ~(z) + (~ - z ) ~ ' ( z )
2.5 0.00
I I I I
I
I
0.25
I
I I I
I
~ I I
0.50
t
0.75
I
I
[ I
I
I
I
I t
2p~-(ul--iu2)=K~(z)-f~(z)-(~-z)~'(z oxl
I
1.00
L (~m) Fig. 8. Dependence of K~c on microdefect size. t o u g h n e s s versus phase angle of remote K :~ is best fitted by a critical local k : c r i t e r i o n . The model
the experimental data even better. F u r t h e r m o r e , 3D effects o f the p e n n y - s h a p e d microcracks a h e a d o f the m a i n crack m a y be important, Acknowledgements--Authors acknowledge the support of the Brown University Materials Research Group on Plasticity and Fracture, funded by the National Science Found a t i o n . Suggestions from Dr N. A. Fleck were very helpful and greatly appreciated.
~o(z ) = ~ ~o(Z ) - ~ ~ ( I - ~)r
(z),
x2 > 0 x2 < 0 '
f~(z) = ~(1 - ~)~0(z), 5D.o(Z) - 7~0(z),
x2 < >0
(A2)
where ~o(Z) -
Z
~0-z0 flo(z)=Bo~+ b~ + ib2 B0 = B2 - iB~ = nic~
B~ - -
2"0
/~0 z -Zo
(A3)
(A4)
Here, c I = 8 / E l . By cancelling the traction on the interfacial crack surface caused by potentials in equation (A2), it can be proven that the additional potentials due to the interaction of the dislocation with the interface crack are 9 (z) = - ~ ( z ) ,
f~(z) = - ~ ( z )
(AS)
where I -~
~b(z) = ~
_
[BoF(z, zo) + BoF(z , z0)
+B0(z~0- zo)G(z, zo)]/X(z) REFERENCES 1. N. P. O'Dowd, M. G. Stout and C. F. Shih, Phil. Mag. A 6 6 , 1037 (1992). 2. W. Mader and M. Rfihle, Acta metall. 37, 853 (1989). 3. Z. Suo and J. W. Hutchinson, Mater. Sci. Engng A107, 135 (1989). 4. V. Tvergaard and J. W. Hutchinson, J. Mech. Phys. Solids 41, 1119 (1993). 5. J. Dundurs, in Mathematical Theory o f Dislocations, pp. 70-115. Am. Soe. Mech. Engng, New York (1968). 6. B. A. Bilby and K. H. Swindon, Proc. R. Sci. A285, 22 (1965). 7. J. R. Rice and J.-S. Wang, Mater. Sci. Engng A107, 23 (1989). 8. J. R. Rice, Z. Suo and J.-S. Wang, in Metal-Ceramic Interfaces, Acta-Scripta Metall. Proc., Vol. 4, p. 269. Pergamon Press, Oxford (1990). 9. Z. Suo, Int. J. Solids Struct. 25, 1133 (1989).
APPENDIX In this appendix, the interaction is studied of an edge dislocation with a semi-infinite crack on the interface be-
(AI)
where z = x t + ix2. ~ is the shear modulus and K = 3 - 4v under plane strain condition. When the two half planes are perfectly bonded, the potentials due to a dislocation with Burgers vector of b = (b~, b2) at an arbitrary point Zo in the upper half plane or on the interface are found to be
suggests t h a t the toughness curve depends strongly o n microdefect size a n d weakly on the defect spacing. The variability in microdefect size is t h o u g h t to play a n i m p o r t a n t role in the large scatter o f the experimental toughness data. A l t h o u g h the current model predicts trends t h a t are in good agreement with experimental data, it is still a p p r o x i m a t e in the way it treats the plastic zone at the m a i n crack tip. A more rigorous finite element analysis with a n accurate representation o f plastic d e f o r m a t i o n m a y yield a toughness curve which fits
)
(A6)
in which F(z, zo)
X ( z ) - X(Zo)
z - z0 X(z) - X(zo) - (z - zo)X'(zo) (A7) (z - z0)2 with X ( z ) = z I/2. The total potentials are the sums of potentials given by equations (A2) and (A5). G(z, zo)
Stress Intensity Factor
is
As z--,0, the singular part of the complex potential q~(z) ~b(z) = ~bz-i/2
(A8)
where -~B0 z?~ 4~ =
B0 1 +
X/zo /
kZo
(A9) .
Upon substituting equations (A5), (A8) and (A9) into (A1), the complex stress intensity factor for the interface crack is obtained as Kl + iK2 = - 2 v / ~ q ~ .
(Al0)
Acta metall, mater.
eergamo. 0956-7151(95)00125-5
FRACTURE TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES: AN ANALYSIS BASED ON A MICROMECHANICAL MODEL J. Y. SHUt, J. A. BLUME and C. F. SHIH Division of Engineering, Brown University, Providence, RI 02912, U.S.A. (Received 12 October 1994; in revised.form 8 March 1995)
Abstract--The fracture toughness is predicted for an interface between niobium and alumina. It is assumed that voids at the triple points of grain junctions on the interface are the sites of cleavage decohesion. These voids are treated as microcracks and the plastic zone is modelled by a cohesive zone of constant shear stress. The interaction between the microcrack and the main crack is studied using an integral equation approach. The fracture toughness is calculated via various fracture criteria and compared against experimental data. Scatter in the experimental data is related to the variability in microdefect size.
1. INTRODUCTION Metal-ceramic interfaces can be found in many modern material systems including layered structural composites, coating and thin film-substrate systems commonly used in the microelectronics industry. Mechanical failure is often due to interfacial fracture. Unfortunately, the understanding of the fracture toughnesses of such metal-ceramic interfaces is limited. In a homogeneous brittle material, cracks advance in mode I. In contrast, the fracture toughness of a metal--ceramic interface has a strong dependence on the mode mixity of a crack on the interface, In general, the toughness increases very rapidly for large phase angles [1]. The niobium-alumina material system provides a model metal-ceramic interface that has been investigated extensively (see, for example, [2]). Recently, a joint theoretical and experimental study on the interface was carried out by O'Dowd et al. in [1]. Different bending specimens were used to measure the fracture toughness of the niobium-alumina interface. They considered the role of plasticity around the crack tip and obtained a toughness curve from finite element calculations based on the criterion that crack growth initiates when the hoop stress at a point on the interface reaches a critical value. The curve so-obtained fits experimental data much better than that by an elastic analysis [3]. Tvergard and Hutchinson [4] came to the same conclusion by performing an
tCurrent address for correspondence: Chemistry and Material Science Department, L-370, Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A. AM43/12--~
elastic-plastic finite element analysis using a cohesive zone at the crack tip. Although the toughness curve calculated in [1] represents a significant improvement in fitting experimental data, it is still not satisfactory. For example, an arbitrary distance r c from the crack tip has to be chosen in [1] to generate the toughness curve. Different choices of r e lead to modestly different fracture toughness curves. A micromechanical model is needed with a length scale of physical meaning much smaller than the plastic zone size. It was observed in [1] that voids at the triple points of alumina grain junctions exist at the interface between the alumina and niobium. A model for interfacial cleavage decohesion was proposed by assuming that these voids on the interface are sites for cleavage decohesion. As long as these microdefects grow and join the main crack before significant plastic deformation occurs around the microdefects, high stress can be sustained and cleavage decohesion may prevail. An estimate of the fracture stress was made by idealizing these defects as penny-shaped cracks and was found to be close to the stress calculated by the finite element method. However, the calculation did not consider the interaction between the main crack and the penny-shaped cracks. This study intends to provide an elastic-plastic analysis based on a simplified version of the model in Ref. [1] by taking account of the interaction between the ci'acks. This allows us to calculate the dependence of fracture toughness of the interface on the phase angle and microdefect size and spacing. Good agreement is observed between predictions and the experimental data from [1]. Furthermore, our calculation provides an account for the scatter of the experimental toughness data by the distribution in the microdefect size.
4301
4302
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
2. ELASTIC CRACK TIP FIELD AND PHASE ANGLE For a simply connected system under plane strain condition of two isotropic, homogeneous, linearly elastic materials with traction prescribed on its boundary, Dundurs [5] proved that the stresses depend on the material properties of the bimaterial pair through the Dundurs parameters ~ and fl
~ le iv
_•K
Rl-172 EIWJ~2 ' /~'1 1 " 2v2 fl = E'~ + R 2 2 - 2v2
/~2 1 -- 2v t El + E 2 2 - 2v,"
(1)
Here E = E/(1 - v2), E is the Young's modulus and v the Poisson's ratio. Subscripts 1 and 2 refer to the upper and lower materials respectively. Consider a semi-infinite crack lying on the interface between the two materials, as shown in Fig. 1, subjected to a singular stress field of K| ~ aoo + itLo = (2~zr)l/2
(2)
on the interface where K | = K ~ + i K ~ (i = x ~ ) is the complex stress intensity factor and E is the oscillation index defined as E = ~ In 1 +----fl"
(3)
The phase angle of the K%field is defined as tan~b = (~,0"~
Im(K ~r') \aoo,/o = o = Re(K~ "
(4)
It can be seen that when e # 0 , ~, depends on the distance r ahead of the crack tip. In this study, however, we make the simplification that fl = E = 0. This is partially justified by the fact that is usually very small (e = 0.02 for a niobium-alumina interface) so that phase angles based on two moderately different length scales would not differ signifi-
~ - i K ~' leiV Fig. 2. Schematic illustration of the micromechanical model. The cohesive zone is subjected to a constant bridging shear stress. cantly. We write the complex K | form
in the polar
K ~ = Ig~le i~
(5)
where the magnitude IK~176 is a real number. 3. A MICROMECHANICAL MODEL It has been observed that grain boundary triple points are often poorly bonded on the interface [1]. In a niobium-alumina system, the average grain size of alumina is about 3 #m. Hence these poorly bonded regions are at a spacing of about 3-5 # m and suffer high tensile stresses when located near the crack tip. In the model proposed in [1], the interface is pictured as populated by penny-shaped microcracks at a spacing of the grain size and these cracks are the sites of cleavage decohesion. Here, instead of penny-shaped cracks, we use 2D through-thickness cracks under plane strain condition, and consider only one microcrack located near the crack tip.
c~ [e/u
_•K
9
~
Xl
~ l / C | lei* Fig. I. Conventions of an interface crack and K~176
3. I. M o d e l i n g o f the plastic zone
To model plasticity near the main crack tip in the niobium, the plastically deformed region is idealized as a line of cohesive zone inclined at an angle 0c from the main crack. Along the cohesive zone, there is a continuous distribution of edge dislocations. The Burgers vector of the dislocations has a non-zero component only in the direction parallel to the cohesive zone, as shown in Fig. 2. The cohesive zone is assumed to be subject to a constant shear stress. The material elsewhere is assumed to be linearly elastic. Such a model of plasticity around a crack tip was first proposed by Bilby and Swindon [6]. One motivation for this approach is the fact that plasticity is caused by the slip of dislocations driven by a shear traction acting parallel to their slip direction.
S H U et al.:
TOUGHNESS
OF NIOBIUM/ALUMINA
The inclination of the cohesive zone from the main crack was chosen arbitrarily in [6]. By suitable choice of dislocation density along the cohesive zone, the shear traction is constant, and the singular term of the shear stress along the cohesive zone vanishes. However, as shown below, this scheme does not cancel the - 1 / 2 singularity of stresses in other directions. This is not a desirable feature since it is commonly accepted that the stresses within the plastic zone are either bounded or has a singularity weaker than - 1/2, such as H R R singularity. In the following, a more rational choice of the inclination angle of the cohesive zone is proposed. For the time being, we will neglect the presence of the microcrack. In this paper, for a continuous distribution of dislocations, we define the Burgers vector density B as the Burgers vector accumulated over a unit length of distribution multiplied by a factor of El/(8n). For continuously distributed dislocations on the cohesive zone at an angle 0c from the main crack, the complex Burgers vector density is written as B 1 + iB 2 =/~e ~~
(6)
where the magnitude B is a real number. (Note that the component of the Burgers vector in the direction normal to the cohesive zone vanishes.) Substitution of equation (6) into (A9) and (A10) of the Appendix gives the stress intensity factor at the main crack tip induced by the dislocation distribution as
Kl + iK2 = - ( 1 - ~ ) x / ~ c o s ~ {3sinO~+i(3cosO~-l)}
f0
B(r)r-l/2dr.
(7)
The stress intensity factor given by equation (7) will satisfy K l + iK2 = - K ~ if the inclination 0~ of the cohesive zone is chosen to satisfy cos 0 " (3 cos 0~ - 1) = sin 0 " 3 sin 0~
(8)
INTERFACES
remote K~~ by
4303
Then, 0c is related to the phase angle
cos ~k 9cos ~ . (3 cos 0c - 1) 0c - s i n ~ ' sin-~. (5 + 9 cos 0~)= 0.
(11)
From Fig. 3, it can be seen that the inclination 0~ found from equation (10) coincides with that found from equation (11) for the limiting cases of pure mode I or pure mode II. We shall adopt relations (9) and (10) since stresses will remain finite at the tip of the main crack. When the microcrack is present, an additional stress intensity factor will be introduced by the microcrack. But the size of the microcrack is usually much smaller (0.25-1 ~m) than the cohesive zone size which, according to [1], ranges from more than 10 #m for a small phase angle to more than 100 ~m for a large phase angle. Also the microcrack is at a distance of 3-5 pm from the main crack tip, so the additional stress intensity factor at the main crack tip caused by the microcrack is likely to be much smaller than the contribution from the cohesive zone and will be ignored. 3.2. Formulation o f the model
The stress field can be regarded as the superposition of a remote K~-field with a stress intensity factor as described in equation (4) and that due to dislocations continuously distributed on the cohesive zone and the microcrack. It is assumed that the cohesive zone is subjected to a constant bridging shear stress equal to av/2 where a v is the tensile yield strength; the microcrack is taken to be traction-free. Denote the shear and hoop stresses due to the distributions of dislocations by ar0 and ~r00respectively; further, denote the shear component of the crack tip K~ field by ~r0.1 and the shear component of the crack tip K~ field by ~r0,2" Then,
and the Burgers vector density/~ satisfies
(1
oo
o'PB(r)r-l/2dr = l K %
.
.
.
(9)
The physically meaningful solution to equation (8) is 0c = cos-l(1/3 cos 4) - ~k.
.
9 K - field relaxing direction
~ ) 2 x / ~ / 4 _ 3 cos2 ~0~ cos ~0~
(10)
An arbitrary choice of 0r other than that from equation (10) will not cancel the singular K~-field completely. Thus, it is reasonable to relate the orientation 0c of the cohesive zone to the phase angle ~ of the applied K~-field through equation (10). An alternative viewpoint is to assume that the cohesive zone is inclined along the direction of maximum shear stress tr,0 associated with the
" - . , , ._
~17625 !
2 ~
o
I 0
I
I,
i,, 25
,
,
/'["r-.,~ 50
75
(degrees)
Fig. 3. Orientation angle of the cohesive zone vs the phase angle of the K|
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
4304
along the surface of the cohesive zone (0 = 0r total shear traction is given by Ig~l
arO -~-~
(COS
the
Kn = (2/t)3/2/~u+ l-
O'y
~larO,l -IV sin t#g~0,:) = -~-,
0 < r < rp.
(12)
On the surface of the microcrack (0 = 0), the total hoop stress vanishes
aoo+ [K~,~-I cos~p=O, x/ 2xr
a
(13)
IK~I a,0 + ~ r sin 0 = 0,
a
(14)
i=l .... N+I.
ri + 1 -- Fi
+Bi+ I r - r i ~ for ri<<.r<~r,+1. (15b) Fi+ 1 -- riJ Here, B~ is a set of unknown real numbers. Similarly, the microcrack (a < r < a + L) is discretized also into N = 30 elements between the nodes N
and for the right-hand tip of the microcrack by k I = (1 -- ~)(2x)3!2L-mB~+ 1,2, k2 = (1 - cr
1.1.
(19)
By dimensional analysis, the stress intensity factors can be written as
gll~fl(~/,~,L,o~)[gm}~-f2(~l,r~,L,o~)ffya|/2
(20) for the outer tip of the cohesive zone and
(15a)
/~ = ( r p - r) I/2(Biri~-lZr-
l+cos
(18)
-'-'~a a tK I+A 0 , ~ - , ~ , ~ ~ a '2,
( i - l)x 2N '
The Burgers vector density is interpolated over each element as
r~=a+
Since only edge dislocations with Burgers vector co-linear with the cohesive zone are used to represent the cohesive zone, the mode I stress intensity factor associated with this dislocation distribution vanishes. The stress intensity factors for the left-hand tip of the microcrack are given by
k 2 = - - ( 1 - ~)(27r)3i2L -~/2BI, 1
Here, a and L are the defect spacing and microcrack length respectively (see Fig. 2). We have assumed that the interface crack oscillation index ~ is zero. The Burgers vector density B~ + iB 2 for both the cohesive zone and the microcrack are determined by an integral equation approach. The stresses due to a single dislocation in a bi-material system with a semi-infinite crack on the interface is derived in the Appendix and integral equations are set up to enforce traction conditions (12)-(14). The cohesive zone is discretized into N = 30 elements between the nodes
\
(17)
k~ = - ( 1 - ~ ) (2~)3/2L - 1/2B1,2,
and the total shear stress vanishes
ri=rpsin
stress intensity factor K u at the outer tip of the cohesive zone is calculated via
~ ,
i=I,...N+I. (16a)
k2 = f s \,t{p[, "ar p, aL
~),K~I+ff(qj,r2,La
a' ~ ava'/2 (21)
for the left-hand tip of the microcrack. The stress intensity factors for the right-hand tip of the microcrack are found to be smaller than those of the left-hand tip and will not be discussed further. Here f~(i = 1. . . . . 6) are dimensionless real functions which can be calculated by the procedure described above. It is stipulated that the net mode II stress intensity factor at the cohesive zone tip should be zero in order to have non-singular stresses. For a given cohesive zone size rp, from equation (20)
The two components of the Burgers vector density on the microcrack are interpolated as ri
Bj = (L2/4 -- (r - a - L/2) 2) ,/2(~ i d - - -+l- -
r
r, + 1 -- ri
r - r i "X
+ Bi+tj~)
for
ri<~r <~ri+1 (16b)
where B#(j = 1, 2) are unknowns to be found. The traction conditions (12)-(14) are enforced at the mid-point of each element of the cohesive zone and the microcrack. The Burgers vector density of the microcrack is subjected to the constraints that the integral of Bj over the length of the microcrack vanishes;/~ is subjected to the constraint equation (9). Thus we have 3N equations for 3N unknowns. The
The above equation gives the relation between r o and the magnitude ]K~I of the applied loading. The nominal fracture toughness Kc of the interface is the value of IK~[ evaluated for a critical value of % which in ,turn is attained when a fracture criterion is satisfied. Fracture criteria will be discussed below. Substitution of equation (22) into (21) gives
kl=(f4-f3fyl)aYau2=-ffyaU2gl(~l,
ra,Z,ot ),
k2=(f6-fsffy~)ffyal/2=-cryab'2f2(~b,~,Z,c~). (23)
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES Here F, F~, F2 are dimensionless real functions of
(~k, rp/a, L /a, or). 3.3. Fracture criteria Let K~ designate the critical value of IK~[ when the microcrack begins to advance. The stress intensity factors at the left-hand tip are generally greater than those at the right-hand tip, hence cleavage decohesion will occur at the left-hand tip first. Therefore the stress intensity factors for the left-hand tip of the microcrack as given by equation (23) will be used in the relevant fracture criteria. Three different criteria for the initiation of cleavage decohesion are discussed below. 3.3.1. Critical energy release rate. The elasticbrittle Griffith model for the crack growth stipulates that when cleavage decohesion is initiated G = 2~t
(24)
in which 2~nt is the work of reversibly separating a unit area of interface. The energy release rate of the interface microcrack (e = 0) is [9]
k~ +/,22
4305
Hence, the above two equations and equation (23) give F~ = (1 - ~) 2 ~ 2 . ~rva
(30)
Similarly, the fracture toughness Kr can then be determined. In the above two fracture criteria, both 2[nt and k~c are assumed to be intrinsic material constants for a microdefect-free bi-material system. In contrast, the nominal fracture toughness K~ depends on the microstructure of the material, e.g. the size and spacing of microdefects. Estimates of 2,rnt for several material combinations have been made [7, 8]. 3.3.3. Critical hoop stress. A fracture criterion used in previous studies is the attainment of a critical hoop stress tr~ on the interface at a distance r~ from the main crack tip. The interface ahead of the main crack tip is assumed to be perfectly bonded. By dimensional analysis /
rp rc IK~I
"~
a~176176176162 a' ava l/z' ct)av=tr c (31)
1 4a._z
G = (1 + ~)Tgz= 1 + ~ ~
('/r + F2)'
(25)
Upon applying the Griffith criterion, one reaches a critical condition r~ + r~ = (1 + ~) z ~
o-ya
(26)
from which a critical value of rv can be found. Substitution of this critical value into (22) obtains the fracture toughness K~ via
( L Kc=oyal/2Fc_~//,
( 1 + 0 0 -2~n' - ~'2 ~ ). a' o'}a'
\
(27)
Here F~ is the F in equation (22) evaluated at the critical value of %/a. In the limit ~O= 0 we designate K~ by the standard notation K~. 3.3.2. Critical k~. As suggested in an elastic analysis [3], a criterion based on a critical hoop stress on the ligrnent of the main crack suggests a K~/K~ vs ff curve which is closer to the experimental measurement than that based on the energy release rate. When the effects of plasticity are included in the analysis [1, 4], a similar criterion was employed. All these analyses suggest that the opening mode of a mixed mode K~-field may play a greater role than the shear mode in determining the toughness of the interface. We propose that cleavage decohesion initiates at the left-hand tip of the microcrack when the local stress intensity factor kl attains a local fracture toughness k~, giving kl = kit.
(28)
Under pure mode-I loading, k~c is related to the cleavage energy 2~t via 1
kl2~
a . . . . 1 +or .E2
2~,t.
(29)
from which a critical value of rp is determined to be
rp = s2(~b, tr~ 'K~' rc ) a av' ayal/2 ' a ' at .
(32)
Substitution of the critical value of rp into equation (31) gives a critical value K~ of IK~I via
gc= eya'/2F3(~b, trc rc
O'v' a ' ~ '
(33)
Here, Sl, $2 and F3 are dimensionless functions. It can be seen that Kc depends on the choice of r~. An elastic analysis by Suo and Hutchinson [3] gives a K~/K~r which is independent of the choice of re. 4. FRACTURE TOUGHNESS
4.1. Comparison of fracture criteria In previous studies, the fracture criteria used to predict the toughness vs phase angle curves are either the critical energy rate G-criterion or the critical hoop stress criterion. In this paper, an additional critical k~-criterion is proposed in Section 3.3.2. The toughness curves generated by these three criteria are compared below. For the niobium/alumina interface, we take niobium and alumina to be material 1 and 2 respectively. (Note that in [1] the alumina is taken to be material 1.) The following material properties are assumed from [1] 2~nt ~ 1 Jm -2, ay = 250 MPa,
E 2 = 328 GPa, ~ = -0.527.
v2 = 0.22, (34)
A representative micro geometrical dimension of a = 3.0/~m, L = 0.5/~m is taken for our calculations. The toughness curves from the G-criterion and kl-criteflon are calculated with the current model, and the
SHU et al.: TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
4306
3.0
3.0
/I
2.5
2.5
2.0
9
experimental critical . . . . . critical G ~ . critical hoop stress
2.0 1.5 1.0
~
0
1.5 1.0 0.5
I
9
experimental
a=0.5gm a=l.5gm . . . . . . . a=6.0gm ._
|
0
0.5
25 I
I
I
I
I
I
I
I
I
[
25
I
I
I I 75
1
50
I
I
I
~g (degrees) Fig. 4. Comparison of fracture toughness curves obtained by various fracture criteria.
toughness curve with critical hoop stress criterion is taken from [1]. Figure 4 shows that the G-criterion is quite poor in that it can not predict the rapid increase of the fracture toughness Kc with an increase of the phase angle ~. This is not surprising considering that, if both plasticity and microdefects are neglected, the G-criterion would predict no increase in the toughness and the toughness curve would be a horizontal line at Kc/K~ = 1. The curves generated using the other two criteria do capture the general trend of the experimental data, and the k~-criterion seems to be the best fit to the experiment results. More importantly, the numerical results confirm the feasibility of the current model which provides a more plausible physical picture of the process of cleavage decohesion.
4.2. Influence of geometrical parameters Having shown that the critical k~-criterion best fits the experimental data, we shall adopt this criterion for the following study about the effect of geometrical parameters on the fracture toughness. As stated in Section 3.3, the toughness of a niobium-alumina interface is a function of the phase angle ~b, microdefect size L, defect spacing a, and material constants. For a niobium-alumina interface, a is 3-5/~m and L is about 0.5/~m. L is largely affected by the process of bonding, while a depends on the heat treatment of
ff
j
3.5
w
35
ili!l
3.5 ~3.0 ~2.5 ~ 2.0
50
75
(degrees) Fig. 6. Fracture toughness for different defect spacing.
the materials. The effect of a and L on the interface toughness needs to be investigated. In Fig. 5, a is fixed at 3 # m while L is given different values. It shows that the toughness is affected significantly by the magnitude of L, at large phase angles. The general trend of the toughness curves is very close to the experimental data. It can be seen that the smaller L is, the better the numerical results fit the experimental data. A surprising result is that for a fixed defect size L, the effect of defect spacing a on the toughness curve is not significant. Only when alL is as small as unity, can the difference between the toughness curves be noticed, see Fig. 6. It implies that the large scatter of the experimental data for toughness is not due primarily to the variance of grain size of alumina or defect spacing which is about the same as the grain size. From Fig. 7, it can be seen that for a between 3 and 5/~m, which is typical in this niobium-alumina system, the range in value of K]cfiTya 1/2 is very small and K~c itself varies from about 2.1 to 2.7 MPa m s/2. The effect of flaw size L upon macroscopic mode I toughness K~c is shown in Fig. 8 for a fixed crack spacing of 3 #m. This suggests that the scatter in L may be the source of the scatter in measured Kit. It is plausible that the microdefect size L displays scatter from specimen to specimen in [l]. 5. CONCLUSION A micromechanical model for the fracture toughness of an interface between a ductile and a brittle material is given. A simple model of the plastic zone is also proposed. Experimental data of fracture
o"
5.25
9 9 experimental .... L=0.25gm . . . . . . . L=0.40gm L=0.50pm ~.~ L=0.60gm
1.5 1.0 9
0.5
I
0
I
I
I
I
[
25
I
I
I
I
[
50
I
I
I
I I
I
eq
5.00 L = 0.5gm
~- 4.75
I
75
(degrees) Fig. 5. Fracture toughness for different microdefect size.
4.50
n t 0
i I
I 2.5
t
I
tl
i LILI
5.0 a (/.tin)
t,i,
7.5
Fig. 7. Dependence of Knc on defect spacing.
I
10.0
SHU et al.: 10.0
_
~3
TOUGHNESS OF NIOBIUM/ALUMINA INTERFACES
-
4307
two half planes of different materials. The second Dundurs parameter fl is assumed to be zero, implying that e vanishes as well. tween
7.5 --
~
Stresses due to the Interaction
~
Following Suo [9], stress and displacement fields are expressed by two analytic functions ~(z) and f~(z) as 5.0
trll + tr22= 2[~(Z) + ~(z)] 022 + icq2 = q'(z) + ~(z) + (~ - z ) ~ ' ( z )
2.5 0.00
I I I I
I
I
0.25
I
I I I
I
~ I I
0.50
t
0.75
I
I
[ I
I
I
I
I t
2p~-(ul--iu2)=K~(z)-f~(z)-(~-z)~'(z oxl
I
1.00
L (~m) Fig. 8. Dependence of K~c on microdefect size. t o u g h n e s s versus phase angle of remote K :~ is best fitted by a critical local k : c r i t e r i o n . The model
the experimental data even better. F u r t h e r m o r e , 3D effects o f the p e n n y - s h a p e d microcracks a h e a d o f the m a i n crack m a y be important, Acknowledgements--Authors acknowledge the support of the Brown University Materials Research Group on Plasticity and Fracture, funded by the National Science Found a t i o n . Suggestions from Dr N. A. Fleck were very helpful and greatly appreciated.
~o(z ) = ~ ~o(Z ) - ~ ~ ( I - ~)r
(z),
x2 > 0 x2 < 0 '
f~(z) = ~(1 - ~)~0(z), 5D.o(Z) - 7~0(z),
x2 < >0
(A2)
where ~o(Z) -
Z
~0-z0 flo(z)=Bo~+ b~ + ib2 B0 = B2 - iB~ = nic~
B~ - -
2"0
/~0 z -Zo
(A3)
(A4)
Here, c I = 8 / E l . By cancelling the traction on the interfacial crack surface caused by potentials in equation (A2), it can be proven that the additional potentials due to the interaction of the dislocation with the interface crack are 9 (z) = - ~ ( z ) ,
f~(z) = - ~ ( z )
(AS)
where I -~
~b(z) = ~
_
[BoF(z, zo) + BoF(z , z0)
+B0(z~0- zo)G(z, zo)]/X(z) REFERENCES 1. N. P. O'Dowd, M. G. Stout and C. F. Shih, Phil. Mag. A 6 6 , 1037 (1992). 2. W. Mader and M. Rfihle, Acta metall. 37, 853 (1989). 3. Z. Suo and J. W. Hutchinson, Mater. Sci. Engng A107, 135 (1989). 4. V. Tvergaard and J. W. Hutchinson, J. Mech. Phys. Solids 41, 1119 (1993). 5. J. Dundurs, in Mathematical Theory o f Dislocations, pp. 70-115. Am. Soe. Mech. Engng, New York (1968). 6. B. A. Bilby and K. H. Swindon, Proc. R. Sci. A285, 22 (1965). 7. J. R. Rice and J.-S. Wang, Mater. Sci. Engng A107, 23 (1989). 8. J. R. Rice, Z. Suo and J.-S. Wang, in Metal-Ceramic Interfaces, Acta-Scripta Metall. Proc., Vol. 4, p. 269. Pergamon Press, Oxford (1990). 9. Z. Suo, Int. J. Solids Struct. 25, 1133 (1989).
APPENDIX In this appendix, the interaction is studied of an edge dislocation with a semi-infinite crack on the interface be-
(AI)
where z = x t + ix2. ~ is the shear modulus and K = 3 - 4v under plane strain condition. When the two half planes are perfectly bonded, the potentials due to a dislocation with Burgers vector of b = (b~, b2) at an arbitrary point Zo in the upper half plane or on the interface are found to be
suggests t h a t the toughness curve depends strongly o n microdefect size a n d weakly on the defect spacing. The variability in microdefect size is t h o u g h t to play a n i m p o r t a n t role in the large scatter o f the experimental toughness data. A l t h o u g h the current model predicts trends t h a t are in good agreement with experimental data, it is still a p p r o x i m a t e in the way it treats the plastic zone at the m a i n crack tip. A more rigorous finite element analysis with a n accurate representation o f plastic d e f o r m a t i o n m a y yield a toughness curve which fits
)
(A6)
in which F(z, zo)
X ( z ) - X(Zo)
z - z0 X(z) - X(zo) - (z - zo)X'(zo) (A7) (z - z0)2 with X ( z ) = z I/2. The total potentials are the sums of potentials given by equations (A2) and (A5). G(z, zo)
Stress Intensity Factor
is
As z--,0, the singular part of the complex potential q~(z) ~b(z) = ~bz-i/2
(A8)
where -~B0 z?~ 4~ =
B0 1 +
X/zo /
kZo
(A9) .
Upon substituting equations (A5), (A8) and (A9) into (A1), the complex stress intensity factor for the interface crack is obtained as Kl + iK2 = - 2 v / ~ q ~ .
(Al0)