INFLUENCE
OF ELASTIC ANISOTROPY
ON THE EDGE PROBLEM
S. Schmauder
Max-Planck-Institut
für Metallforschung, Institut für Werkstoffwissenschaft,
Seestraße 92, D - 7000 Stuttgart 1, FRG
Introduction
Fracture in metaljceramic joints frequently initiates at the interface edge (1, 2) or at a defect situated at the
edge region of the interface. This phenomenon is weIl known as the edge problem. It is due to singular
stresses at the edge if internal or external load is applied (3). In the literature, crack nuc1eation at interface
edges in metaljceramic
bimaterials with a dome-like crack path in the ceramic has been reported.
Bimaterials may fai! at the interface edge even in the absence of thermal stresses (4). The problem can be
reduced
by introducing
interlayers is with
a proper
thermal
coefficientIn (5),
by rounding
the of
edges
(6).
However,
such a procedure
difficult
and in
many expansion
cases impossible.
this orpaper,
the effect
the
elastic anisotropy on the edge problem will be iIluminated for the case of Al203/Nb. This system is used as a
model because there is no thermal mismatch. The main idea of the applied procedure is to estimate the
singularity power of interface edge stresses for isotropie metals as well as for single and polycrystalline Nb
bonded to isotropie alumina by examining the energy release rates of very small edge cracks at the interface.
General Aspects of Interfaces
[1]
[2]
I
CI -
r-1/2
CI
-
r-A
A = A (lX,ß)
2w
crack
~ 0.41
(8aJY, 1975)
FIG. 1.
The plane geometry is considered
here. At the interface edge and at
the tip of an interface
crack
different stress singularities prevail.
are often used to describe bimaterials by only two elastic constants. The range of a and ß is given by a
parallelogram in the right half of which all bimaterials can be represented, eventually by changing the
numbering of the two materials so that a ~ O. Many of the technically relevant bimaterial combinations
satisfy the relation (8) .
[3]
/3 ~ a/4.
In this scheme different material combinations with the same elastic behaviour are described by the same
Dundurs' parameters. The singularity power --\of the stresses at the interface edges is the same for applied
and residual stresses (2), and is a function of a and ß which is not larger than 0.41 oompared to --\ = 0.5 for
the interface crack (Fig. 1).
An implicit analytical solution for the determination of --\from a and ß for abimaterial
oonstituents was given by Bogy (9) as
D (a,
ß; --\)
=
(cos2
(--\11"f2)
-
(1---\)2)2
ß2
[4]
+
(oos2
+ 2(1(1---\)
- --\)2 «(1--\)L(--\11"/2)
1) a2 - (1---\)2) aß
+ cos2 (--\11"/2) sin2 (--\11"/2) = O.
Lines of equal singularity power --\in the a-ß-diagram
with isotropie elastie
are shown in Fig. 2. The value for the bimaterial
Al203/Nb is indicated as a cross belonging to a singularity order of --\~ 0.14.
ß
~:
.......
cr-r-A1a,ß)
~.•.....
~
0.25
0.20
0.10
0.00
-0.10
-0,20
-0.25
FIG. 2.
Lines of equal singularity power of
interface edge stresses.
Method and Model
Numerical
edgein stresses
are performed
by (10).
meansInofthe
themodel
finite the
element
The methodinvestigations
is described of
in the
details
the literature,
e.g. in ref.
crack method
is opened(FEM).
along
the interface and the elastic energy U is calculated for every crack length a. The strain energy release rate for
a plate of unit thickness is calculated acoording to
1 dU
G = 2 loa-l-
[5]
The energy release rate is related to the stress intensity factor for the interface crack
K = (Je •
r;;. YK(a/h)
[6]
by
G =1#K2
[7]
where (Je is the external applied stress and Y K is a oorrection for the finite geometry. For an exterior crack in
a homogeneous material YK has the form (12)
horn
YK
(ajh) = 1.12 - 0.23 (ajh) + 10.55 (ajh)2 - 21.71 (ajh)3 + 30.8 (ajh)4
[8]
~nd E* is a mean value of the Young's moduli El and E2 of the involved materials (8)
P=2
El+E2'1 J
1
1 [1
[9]
The procedure to obtain the correction function is as follows: The function U(ajh) is fitted by a 5th order
polynomial and the strain energy release rate G(a/h) is obtained by analytically derivating U(a/h) according
to eqn. [5]. Then Y K (a/h) is calculated through eqns. [6] and [7].
Metal and ceramic are both treated as linear elastic with the elastic data of the materials given in Table 1.
Material Data
v TABLE
0.361
0.293
0.27
0.44
0.24
0.345
0.397
0.34
325
375
71
70[GPa]1
3.9
120
115
105
E
Material
A tensile specimen of width h = w = 100 mm and geometry as shown in Fig. 1 is modelIed by finite elements
(compare Fig. 3). The upper part is assumed to consist of the stifter phase.
The finite element model with interface cracks of length a/w = 0.04 and a/w = 0.21 which is externally
loaded perpendicular to the interface with 1 N/mm2 is shown in Fig. 3 for Al203/Nb. In order to calculate the
energies of very small interface cracks the model is refined at the interface edge as shown in Fig. 4.
FIG.3.
Deformation (magnified 25 OOOx)of an
Al203/Nb bimaterial with interface cracks
of length (a) a/w = 0.04 and (b) a/w = 0.21.
FIG. 4.
Close-up view of the mesh at
interface edge to calculate
elastic energies for short interface cracks.
Results and Discussion
The strain energy U is an increasing function of the crack length for both, cracks in homogeneous materials as
weil as in the bimaterial, with slightly higher values far the latter one. The geometry function of the crack in
the homogeneous material (eqn. (8]) and in Al203jNb is shown in Fig. 5.
o
..,
o
"
GO
oz
~
<>:0
~~j\-.
~
~~
o
o
GO
FIG. 5.
o'"
°0.00
0.04
0.08
0.\2
REL. CRACK
0.16
LENGTH
0.20
0.21
0.24
A/H
Correction function Y K for a crack in a
homogeneous material (triangles ) and in
A1203jNb (circles) where H = w.
The influence of the stress singularity at the edge is seen from the increase of the curve for the composite in
the range ajw ~ 0.1, while the influence of the dissimilarity of the materials is present as long as the condition
ajh ~ 0.25 is fulfil1ed. For larger crack lengths the correction function is no longer dominated by the
dissimilarity of the materials.
The ratio R of the correction functions of the bimaterial and the homogeneous material (Fig. 6) at a short
crack length is used as a measure, e.g. R(ajh = 0.01). The ratio for A1203jNb is given by (compare Fig. 6)
RNbj A1203 = 1.4.
[10]
.
~ .•
•.
o
0.01
0.12
0.11
0.20
REL. CRACK LENGTH A/H
0.24
0.21
FIG.6.
Ratio
of correction
functions
YK (A1203/Nb)/YK (A1203).
The
composite
A1203/Nbby
obeys
of linear
A = 0.138
accordingthrough
to eqn. these
[4] while
homogeneous
material
is characterized
(R, aA) singularity
= (1, 0). Apower
simple
interpolation
two apairs
of values
leads to the equation
A
= 0.345 (R-I).
[11]
A comparison of analytieaily determined singularity orders from eqn. [4] with values from eqn. [11] and R
:'rom the FEM -ealculations is given in Table 2 for a wide range of different material properties.
The discernibility between the bimaterials A1203/Nb and Al203/Si demonstrates the sensibility of the
method.
0.01
0.001
0.11
0.138
0.0564
0.25
0.0085
0.0819
0.108
0.138
0.16
0.139
0.2072
0.1454
0.26
0.165
0.1382
0.8895
0.0647
0.4906
0.5292
-0.4789
0.6693
1\
[11]
ß
A 0.0028
[2]
[4]
Cl: [1]
TABLE 2
n of the Stress Singularity Powers
The ratio of the correction funetions for two Al203/Nb bimaterials where a (100) and a (110) type Nb plane
are assumed to be bonded to elastieally isotropie Al203 as weil as the correction funetion for homogeneous
materials are shown in Fig. 7 together with the corresponding ratio of the correction functions for the
bimaterial with isotropie constituents. In both of these anisotropie calculations the crack propagates in a [001]
direction. The interesting result is the fact that A is a function of the Nb orientation and with respect to the
interface is distinctively higher or lower compared to the case of isotropie constituents. The systematology of
this dramatic change in A remains a topie for further studies. However, this result of the anisotropie
calculations implies a higher probability for crack nucleation at the edge for different Nb orientations. In that
sense the Al203/Nb(110) eomposite should be most sensible for interface edge crack nucleation.
yinh
k /yhom
k
1.6
1.4
1.2
1.0
yhom
k
0..0.2
0..10
a/w
0.20.
Al2
°3/
1110lNb
A
-0..17
Nb
0..14
110.0lNb
-0.,04
FIG.7.
Ratio of correction functions for
elastically anisotropie behaviour of
Nb. Estimated singularity powers of
stresses at the interface edge are
tabulated (for details see text).
Conclusions
Aside from the quantitative information given in Table 2 and in the Figures, the results of the paper may be
summarized as follows:
A new method is proposed to determine the singularity power of interface edge stresses.
The method is successful in the case of isotropie constituents.
The method is applied to Al203(isotropic)/Nb(anisotropic)
composites with a (100) and a (110) type plane of
Nb bonded to Al203 and crack propagation in a [100] direction.
The sin~larity power of the stresses at the interface edge shows a strong orientation dependence. A lower
singulanty power of A = 0.04 for the (100) Nb orientation and a higher singularity power of A = 0.17 for the
(110) Nb orientation compared to the corresponding bimaterial with isotropie constituents is found.
The stress singularity power for bimaterials of isotropie constituents is known from eqn. [4]. Thus, solving
eqn. [11]for R immediately leads to the crack driving force for these bimaterials
R = 1 + 2.9 A.
[12]
References
1.
2.
3.
4.
5.
RD. Adams and J. Coppendale, J. Adh. 10, 49 (1979)
T. Suga and G. Elssner, Proceedings of the Japanese MRS (1988), to be published
RS. Alwar and Y.R Nagaraja, J. Adh. 7, 279 (1976)
A. Piva and E. Viola, Engng. Fract. Mech. 13, 143 (1980)
K. Suganuma, T. Okamoto, M. Koizumi and M. Shimada, J. Mat. Sei. Let. 4, 648 (1985)
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7.
8.
9.
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11.
12.
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J.
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