- Email: [email protected]
Mode I crack problem in an inhomogeneous orthotropic medium
0
PII:
ht. 1.Engng Sci.Vol. 35.No. 9,pp.869-883,1997 1997 Elsevier Science Limited. All rights reserved
SOO20-7225(97)00014-l
Printed in Great Britain.
0020-7225/97$17.00+0.00
MODE I CRACK PROBLEM IN AN INHOMOGENEOUS ORTHOTROPIC MEDIUM MURAT
OZTURK
and FAZIL
ERDOGAN*
Lehigh University, Bethlehem, PA 18015, U.S.A. Abstract-In
the symmetric crack problem considered the material is both oriented and graded. The properties of the medium is assumed to vary monotonously in the x,-direction, x, and x2 are the principal axes of orthotropy, and the crack is located along the x,-axis The loading is such that x2 = 0 is a plane of symmetry. The mode I crack problem for the inhomogeneous orthotropic plane is formulated and the solution is obtained for various loading conditions and material parameters In the formulation four inde endent engineering constants, E,,, Ezz, G,* and Y,r, are replaced by a stiffness and a shear parameter E = e El&, a stiffness ratio c= (ElllE22)“4, a Poisson’s ratio Y = G parameter K = (E/2G,J - Y. The results show that the stress intensity factors are independent of E and c and generally the effect of K and Y on the stress intensity factors is not very significant. ‘Ihe exception is the values of K approaching - 1. where the physical range of K is - 1K
1. INTRODUCTION
In high temperature applications such as advanced turbine systems, aircraft engines and a great variety of combustion chambers, the demand for improved thermal efficiency requires the development of increasingly more heat-resistant materials and coatings. Because of the level of operating temperatures involved (e.g. approximately 1260°C rotor inlet temperature in gas-fired stationary turbines), in these applications the use of structural ceramics for the purpose of shielding metallic components against excessive heat is becoming almost a necessity [l-5]. Some other important industrial applications of ceramic coatings may be found in the area of prevention of corrosion and wear. From a view point of failure mechanics the homogeneous ceramic coatings, however, have certain shortcomings among which one may mention poor interfacial bonding strength, high thermal and residual stresses, low toughness and consequent tendency toward cracking and spallation. An alternative concept which may be applied to overcome some of these shortcomings appears to be using coatings or interfacial zones with graded compositions and hence, with continuously varying thermomechanical properties. Such composites with smoothly varying volume fractions are known as functionally graded materials [6-8]. It has been shown that grading material properties reduces the magnitude of residual stresses [9] and significantly increases the bonding strength [lo]. Even though grading the microstructure and composition of the medium can be a very useful tool in material design with almost unlimited potential applications, in the near future the main applications of the graded materials will most likely be limited to thermal barrier coatings, interfacial zones to’ enhance bonding and wear and corrosion-resistant coatings. Because of the nature of the techniques used in processing, the graded materials are seldom isotropic. For example, the materials processed by using a plasma spray technique have generally a lamellar structure [ll], whereas processing by electron beam physical vapor deposition would lead to a highly columnar structure [12]. Thus, in studying the mechanics of many of the graded materials, an appropriate model would be an inhomogeneous orthotropic elastic continuum. In this study such a model is used to solve the basic mode I crack problem in * Author to whom all correspondence
should be addressed. 869
870
M. OZTURK
and F. ERDOGAN
z 0
a
x1(5)
Fig. 1. Geometry and notation of the crack problem.
a graded material (Fig. 1). The medium is assumed to be under plane strain or generalized plane stress conditions. The corresponding crack problems for the isotropic inhomogeneous materials were considered in Refs [13, 141. 2. FORMULATION
OF THE
ELASTICITY
PROBLEM
Consider the plane elasticity problem for an inhomogeneous orthotropic medium containing a crack along x2 = 0, - a
$ = s,,(=
E=~,v=~,c4=
22
v21
E 2G,2
-Il.
Equation (1) are valid for plane stress conditions. For plane strain equation replaced by
E,,& v,3v3,)(1
E,,
(1
-
(v,z -
(1
v23v32)
~23~32)
E22 (1 - v,~v~,) ’ K = K
E -”
+
lJ,3+2)(v2,
-
vnv’3,)(1
+ -
v23v3,)
v23v32)
is (1) must be
1’2 ’
(2)
In a general inhomogeneous orthotropic material the parameters E, v, c and K would be functions of x, and x2. However, to simplify the formulation of the problem and to emphasize the spatial variations in the stiffness parameters, the problem will be solved under certain restrictive assumptions regarding the distribution of the elastic constants. Previously it was shown that the effect of Poison’s ratio on the stress intensity factors is not very significant [17, 181. Thus, in the present study v is assumed to be a constant rather than a function of xl and x2. It is also assumed that the material stiffnesses E,,, E22 and G,Z will vary proportionately. This means that c and K may also be considered as constants and the inhomogeneity of the medium may be represented by variation in E only. Note that the special case of c= 1 and K = 1 corresponds to an isotropic material. Also, in a homogeneous orthotropic medium - 1 < K
871
Mode I crack problem
By using now c as a scaling constant and defining X =x,&, a,(x,y)
y = fix*,
u(x,y) = Ur(XrJ*)fi,
= a,,(-w*)k
the stress-displacement
g&Y)
u(x,y) = U,(X,J*)&
= c~&lJ*)r
%&TY)
= ~dh-4
(3)
relations may be expressed as
(44
t4b)
cyky) =
1
&Y
U(X?Y)
-& 2(K+
+
-&
,
u(x,y)
(4c)
I’)
where a&Y)
=
J%Jx,).
(5)
From the equilibrium equations it then follows that
a2u a2u a2u PI aB au + +--=-+p2---y+P,-E ax ( ax axay ax2 ay a2u -++pI7 ax2
a2u +--~4 aE ?!!.+,,~)++$(~+~)=0,(6b) +/32axay E ay ( ay ay a2u
where 2(K PI
=
+ U)
1 -v2
,P2=1+yP,.
(7)
In this problem an additional assumption concerns symmetry and the choice of the function E(x,y). To restrict the consideration to a mode I crack problem it is assumed that the material parameters are independent of x2. It is further assumed that E(x,y) is a monotonously increasing function of x, and in the crack region it may be approximated by an exponential function as follows: E(x,,x2) = E(x,)
= E,,ealXt
= l?(x)
= EoeYX,
y = a,&.
(8)
From equations (6a), (6b) and (8) it may then be seen that
a2u a2u a2f4 +p2+ PlY g+.---y+p17 axay ax ay (
au ay )
=o,
a2u a2u -+p,--_T + P2$-+y(~+g)=R ax2 ay Note that for y=O, equations homogeneous medium [ 181.
(9a) and (9b) would reduce to the differential
(94
Pb) equations for a
872
M. OZTURK and F. ERDOGAN
If we now assume the solution of equations (9a) and (9b) in the form r I -x
F(y,k)emikxdk,
x
I
G(y,k)eeik”
dk,
-Cc
then from equations (9a), (9b), (lOa) and (lob) it may be shown that Fb,k)
= i
G(k,Y)
bj(k)Bj(k)e*T,
= i
Wa)
Bj(k)e*j’,
@lb)
1
where A, j= l,..., 4 are the roots of A4- (vy2+2(k2+
iyk)K)A2+(k2+
iyk)2=0,
02)
given by
A, = -A3=
+‘+2Kv)+
A2 = - A4 =
;(a2+2K$
-
$t/(62+2K9)2-
47J2
lR,
034
+q((62+2K9)2-
4q2
In,
(13b)
S2 = v y2, VI= k2 + iyk
(14)
and
(W-d%v)
b.=
I
A;
_
,jq
A,
j=l
I’
4
‘**-’
*
(15)
The roots Aj are ordered in such a way that %(Aj) > 0, j = 1,2. It is assumed that the inhomogeneous medium contains a through crack along x2 =O, - a 0) of the medium needs to be considered, giving
U(X,JJ)= &Is
ibj(k)Bj(k)e’fl-‘“dk, -= 3
(1%
(16b)
Mode I crack problem
E(x)
flxx(X,Y)
=-
+&,y)
= ~ l-v2
%ykY)
=
-
1-v*
2a
E(x)
a4
1
1 2rr
-!-
2(K + VI 27TI
873
C(~hj-ibjk)B,e~~-‘~~dk,
(174
C (Aj-ivbjk)B,e”‘Yikxdk,
t17b)
r i(Ajbj-ik)B,e”“-““dk,O
The two unknown functions B,(k) and B,(k) are determined y=o. 3. THE
(17c)
3
INTEGRAL
from the boundary conditions at
EQUATION
The elasticity problem formulated in the previous section must be solved under the following boundary conditions: a,*(x,,O) = 0, - 03
+
u&+0)=
0)
=
p(x,),
-
Q
(18)
< 4
(194
~v(x,+o)=o,a
Wb)
where p(xr) is a known function. From equations (3) (17~) and (18) it follows that i
( Ajbj - i k)Bj = 0.
(20)
A second equation to determine B, and B4 may be obtained condition equations (19a) and (19b) by introducing l)(x) =
from the mixed boundary
-&u(x, + 0) = y-$u*(x,, +0) = g(x,).
(21)
1
From equations (16b), (19b) and (21) it may be seen that ( - ik)(B3 + B4) =
O’ +(S)eiksds, a’ = al&. I -a’
(22)
The new unknown function I,+(S)is determined from (19a). Thus, solving (20) and (22) for B3 and B4, from (17b) and (19a) it can be shown that lim
y_+O
J%,y> 1
l-vz
2rr
L(x,y,.r)~(s)ds=cp(~x),
- a’ <~
(23)
where
H(y,k) eik(s-4
H(y,k) =
1 i k( A4b4 - A3b3)
dk,
[(A3b3 - ik)h4b4eAfl - (h4b4 - ik)h3b3eAfl].
(24)
(25)
a74
M. OZTURK
and F. ERDOGAN
Any singular behavior of the kernel L must be due to the behavior of H&k) at lk~l-_, 03which may be obtained as’ 1 --Y2
Ikl
(26)
K,=~,S*=~,S2=~.
(27)
By observing that
H&G)
1 -v2
e i&-x) dk -
2& (S2y)2
2K1
and by adding and subtracting H, in equation expressed as
-
1 (I’ @r(s) ds+ IT I _-(I,s -x
-&
- x) +
(S
-
22(S X)2
-
(Sly)’
+
-
X)
(S
-
(24) the integral equation
L,(xs)+(r)ds
=
(s, +
X)’
’
(28)
(23) may now be
S2)C p(Ax),
IXl
(29)
E’(X)
where the Fredholm kernel L, is given by L,(x,s) =
x2[%(h(k))cos(k(s
- x)) - s(h(k))sin(k(s
- x))] dk,
I0
(sl + s2)(k+ iy) _
-i
Al
+
A2
h(k) = 2(k + iy)
-i vy2
+
4(k2 + iyk)
1
13 1 -1
,
(30)
K#l
(31) K=l.
K < CC and s, + s2 is always real. Also note that for y = 0, A, = ks,, Note that in real materials - 1 CC A2= ks2, h(k) =O, L,(x,s) =0 and equation (29) would reduce to the integral equation for a homogeneous orthotropic plane. From equations (19b) and (21) it may be seen that the integral equation (29) must be solved under the following single-valuedness condition:
a’
I
#(s) ds = 0. --(I’
In the special case of isotropic c=(~?,,/E22)“~,
K=l,U’=U,X=Xl,S,=l,
(32)
inhomogeneous materials with constant Poisson’s ratio, s2 = 1, and the Fredholm kernel in equation (29) can be
evaluated in closed form as follows:
L,(xs) = ye2 v. {
+,(v,lCI)+Ko(voi51)
1
--9
2 s-x
’ From equation (27) it may be seen that for K 5 - 1 s, and sz are purely imaginary. Consequently, the integral in equation (24) does not exist and the problem has no physically meaningful solution. Indeed, in corresponding homogeneous orthotropic solids it is known that for K 5 - 1 the elasticity matrix is not positive definite [19] and. therefore, in real materials the range of the shear parameter is - 1 < K
875
Mode I crack problem
6 = g (s - x), ug =
ViTT,
(34)
where K,(z) and K,(z) are the modified Bessel functions. For the plane strain case in equation (34) v should be replaced by u/(1 - v). Referring to equations (3) and (21) in terms of the real physical variables the integral equation (29) may be expressed as
~
+ M,(x,,t,)
g(r,)dt,
=
I
(s, + %)C e-alxp(xl), E 0
lx,1
(35)
where Ml(X,Jl)
=
-
L,(x,/xfc,t,/~).
l
(36)
2V% which, for the isotropic case, becomes M,(x,,t,) = + i
v,~Kl(~“l~l)+K”(~,151) - -&-A= $(4-x*).
1
1
(37)
1
In physical variables observing that x1 = fix t, = 6, (or = r/X&, k = l&e, ( - 33< e coo), it can be shown that the kernel Ml (x,,t,) depends on the inhomogeneity parameter CX,and the elastic constants u and K, but not on c and EO, where c = (E,l/E22)1’4, m = EOealxl and u, K, c and E. are assumed to be independent of xl and x2. From equation (27) it may also be seen that s1 and s2 depend on the shear parameter K only. After solving equation (35) and obtaining g(t,), the crack surface displacement and the stress intensity factors may be obtained as follows: *I u2(x,r
+
0)
=
(38)
&)df,,
I --a
k,(a) =
k,( - a) =
lim ~cr,,(x,,+O) x,-a+0
lim x,--a-O
= -
V2( - xl - a)a,,(x,,
lim E(xl) _g(x,), xI_ a _ o 61 + s2)c
E(xl)
+ 0) = .,f’:“.+,
4.
(Sl
+
V5(ZYrjg(x1).
(39)
s2)c
THE SOLUTION
To solve the integral equation we first introduce the following normalized quantities: t = t,la, r = x,k
44) = g(4), M2(r,f) = aM,(x,,4),
f(r) = P(X~)@~, - 1< (r,O < 1, - a < (x1,4)
(40) where p0 is a constant corresponding to the amplitude of the external loading. The integral equation (35) and the single-valuedness condition equation (32) may then be expressed as
&
+
M2(r,f)
1
NW=
@I +4CPo e-m,a’f(r) <1, 7(r( E
0
(41)
M. OZTURK
and F. ERDOGAN
qS(t)dt = 0.
(42)
The solution of equation (41) is of the form
(43)
where the bounded function w(t) may be expanded into a series of orthogonal associated with the weight function l/m as follows:
w(t) =
(%+;;)cpo ~A.T,(t), - l<
polynomials
(44)
t<1
0
where Ao, A,,... are unknown dimensionless constants. From equations (43) and (44) and the orthogonality conditions of the Chebyshev polynomials
-
1
’ ~~(Wt&)
dt_
7T I -,c7-o’
:;, ,
{
; 1;
;;
(45) ’
m # n,
it may be seen that A0 = 0. By using the following properties,
TM (t
-
dt=
0,
n = 0, Irl < 1
U,_,(r),
n 2 l,Irl
G,(r),
n =
r)vT7
T,(t) = cos (n e), U,(t) =
G,,(r) = -
It-l I
sin@ + 1)6J sine
(46)
OJ)...) Irl > 1,
, cose = t,
(47)
(I - (Irllr)v7T)n v7T
’
(48)
the integral equation (41) may be reduced to
(49)
where
817
Mode I crack problem
The functional equation (49) is solved by method of reduction [20]. From equations (38), (39), (43) and (44) it can be shown that
k,(a) - -
enla
ko
k,( - a) =
k.
UZ(X,r + 0) = -Vl
W)
1
( - l)“A,, k,
ecu@ i
=po&,
Wb)
1
- (x,,&
VO
$L
+U,_,(x&),V,=
(s,
+
%)CP&
E
.
(52)
0
1
It should again be emphasized that the coefficients A, are dependent on the inhomogeneity parameter (Y, and the elastic constants u and K, but not on E. and c. From equations (51a) and (51b) it is then clear that, within the confines of the assumptions made in this study regarding the distribution of the elastic constants, the stress intensity factors are independent of c and Eo. If we observe that the left-hand side of equation (35) represent the stress a&,0) for lx,1>a as well as for Ix,i
5. RESULTS
AND
DISCUSSION
The mode I crack problem described in Pig. 1 is solved under two types of external forces, namely the crack surface tractions and fixed-grip loading. In the results given the crack surface traction is assumed to be
P(h) = -
PO- Pl
( 3_P2(3’-P3(33,,x,,
(53)
where po, p,, p2 and p3 are known constants having the dimension of stress. In the fixed-grip loading the untracked medium is assumed to be under a uniform strain given by E22(X,J2)
Through superposition,
=
Eo
+
&I~).
(54)
for this loading condition it may be shown that
p(x,) = -
!!$Iem+1 (,,+E,( ~)),lx,ku.
(55)
In the solution E. and c appear as multiplicative constants and do not contribute to the distribution of the calculated quantities. Similarly the half crack-length a is the only length parameter which is used as a scaling constant in presenting the results. Thus, by expressing E=m = E. exp(cu,x,)= E, exp((cu,a)(x,lu)), it may be seen that the inhomogeneity parameter CY,enters the analysis only through the dimensionless constant (~,a. The arbitrary variables in the problem are, therefore, the Poisson’s ratio v = G, the shear parameter
M. OZTURK
878
and F. ERDOGAN
K = (ELZG,,) - v and CY,Q.In the case of a homogeneous medium subjected to arbitrary crack surface loading ~&,,0) =p(x,) = -f,(zc), x=x,/u, the stress intensity factors are given by
dx,k,(
Thus, forf,(x)=p,,P,
f,(x)
- a) = -
V-i-G’
dx
(56)
n=O,l,,.., we find k,(a) = ( - l)“k,( - a) = ( - l)“C&$G,
(57)
where c, = 1, c, = c2 = 112,c, = c, = 3/S, ...
(58)
Some sample results for the mode I stress intensity factor k, and the crack surface displacement u2 in an inhomogeneous medium subjected to uniform crack surface pressure p. are shown in Figs 2-5*. The influence of v and K on k, and u2 appears to be less significant than that of a,u. Figures 2 and 3 show that the stress intensity factor on the stiffer side of the medium is always greater than that on the less stiff side. The explanation of this seemingly counterintuitive result may be found in Fig. 6, where the crack surface displacement is given in an inhomogeneous medium with (~~a= 1 or E(x,) = Eoexp(@) and in homogeneous planes having the stiffness From equation (39) one may observe that E( - a) = &/exp( l), E(0) = E,, and E(u)=E,exp(l).
Fig. 2. The normalized stress intensity factors in an inhomogeneous orthotropic medium subjected to uniform crack surface pressure, (T,,,,(x, + 0) = - p,,, v = 0.3.
1.6
-!""""'......................,.....~
1.2
--I____________
kl
_,,_,._.._,._.J.._..-..-..-.._..-...-..-..-..-..-..-
la7,"
:: j
-1 Fig. 3. The
0
1
2 lc
3
4
5
normalized stress intensity factors in an inhomogeneous orthotropic medium subjected to uniform crack surface pressure, (+.“.” (x, + 0) = -p,,, Y = 0.9.
2 Note that in orthotropic materials the Poisson’s ratio v can be greater than 0.5.
Mode I crack problem
819
Fig. 4. Crack surface displacement u2 in an inhomogeneous orthotropic medium under uniform pressure p,, applied to the crack surfaces. (I+= (sI +s,)ap&E,, Y = 0.3, K = 0.5).
the stress intensity factor is proportional to the stiffness I? and the displacement derivative g evaluated at the crack tip. Furthermore, in homogeneous orthotropic planes, the stress intensity factor k, =pO& is independent of 8 (or g is inversely proportional to E). Thus, if we compare the inhomogeneous plane solution with that of a homogeneous plane having the same stiffness at the crack tip X, =a, one would expect that, since the crack surface displacement derivative near x1 =a in the inhomogeneous medium is greater than that in the corresponding homogeneous medium, k,(a) would be greater than pO&. Similarly from the relative magnitudes of the crack opening near x1 = - a it may be concluded that k,( - a)
1 .o
Fig. 5. Crack surface displacement u2 in an inhomogeneous orthotropic medium under uniform pressure p. applied to the crack surfaces. (v,, = (s, +s2)ap,~lE,,, Y = 0.3, a,a = 0.5).
Fig. 6. Crack surface displacement IQ in an inhomogeneous orthotropic medium under uniform pressure p,, applied to the crack surfaces. (v,, = (s, +s2)ap,,c/E,,, Y = 0.3, K = 0.5, ala = 1.0).
880
M. OZTURK
0.0
0.5
and F. ERDOGAN
1 .o
ma
1.5
2.0
Fig. 7. The normalized stress intensity factors in an inhomo$eneous orthotropic medium subjected to fixed-grip loading E,,;k,, = (e&c )&, Y =0.3.
limit for (~,a+ 0 the stress intensity factors become kl (a) = k,( - a) + (EWE&‘) G and a + 1/2(~,E&~) fi. For an unusually large value of u the variation of kr with k,(a)= - k,( - ) the shear parameter K is shown in Figs 9 and 10. Figures 7-10 show that, except for K-_, - 1, under fixed-grip loading the influence of K on the stress intensity factors is rather insignificant. Similarly, from Fig. 11 it may be seen that the effect of v on /cl(a) and k,( - a) is also very small. To give some idea about the influence of the nature of crack surface tractions on the stress intensity factors, Table 1 shows some results obtained by using the loading given by equation (53). Such results may be useful if the stresses obtained from the untracked medium can be approximated by a polynomial of degree three. Similarly, Table 2 shows the effect of the Poisson ratio v = G on the stress intensity factors. Based on these results the assumption of constant v throughout the medium appears to be rather realistic. 3
0 0.0
0.5
ofi
1.5
2.0
Fig. 8. The normalized stress intensity factors in an inhomo eneous orthotropic to fixed-grip loading l,; k,,=(c,E,Jc 5 )V%, v =0.3.
Fig. 9. ‘Ihe normalized
stress intensity
subjected
factors in an inhomogeneous orthotropic medium subjected to fixed-grip loading c,,: k,, = (a,&JcZ)~, Y = 0.9.
881
Mode I crack problem 1.5.
8
,
I
,
I
,
I
,
. .
1.0
5
0.5
kl(a)/ko
--__________~2~0. ola=0.25
-..-..-..-..-..-..-..-..-..kl(-a)/kl,
a,a=0.25 -___----_____a,o=O~5 ---. ..........................................~~.~.~.?.~.?...
0.01 ’ -1
Fig. 10. The normalized
I
)
-
-
ko
1
~'!?:.!:I)...
. . . . . . . . . . . . . . . . .._..............._......_.
’
’
’
0
1
’
( K
2
’
’
I
3
stress intensity factors in an inhomogeneous to fixed-grip loading e,, k,,= (r,E,@)&,
’
’
4
J
5
orthotropic Y = 0.9.
medium
subjected
In conclusion it may be pointed out that in the mode I crack problem for an orthotropic inhomogeneous medium formulated in terms of the material orthotropy parameters E, v, K and c: (i) the Poisson ratio v has only a negligible influence on the stress intensity factors; (ii) the influence of the shear parameter K may be significant only for K < - 0.5; (iii) the parameters E and c have no influence on stresses and affect the displacements in the usual manner; (iv) the 4
3
I
Fig. 11. The normalized
Table 1. The normalized
stress
stress intensity factors in an inhomogeneous to fixed-grip loading E,,: k,, = (e,,&/cZ)~, intensity factors for an inhomogeneous loading p(x,)= -p,(x,/a)“, n =0,1,2,3,
orthotropic K = 0.5. orthotropic Y =0.3
medium
medium
subjected
under
crack
surface
K = - 0.25 0.00 0.01 0.10 0.25 0.50 0.75 1.00 1.50 2.00
1.0 1.0025 1.0246 1.0604 1.1177 1.1720 1.2235 1.3184 1.4043
1.0 0.9975 0.9747 0.9364 0.8740 0.8154 0.7616 0.6701 0.5979
0.5 0.5000 0.4998 0.4990 0.4963 0.4923 0.4873 0.4753 0.4621
-
- 0.5 0.5000 0.4998 0.4989 0.4954 0.4898 0.4824 0.4642 0.4439
0.5 0.5006 0.5061 0.5151 0.5295 0.5433 0.5565 0.5815 0.6047
0.5 0.4994 0.4937 0.4841 0.4684 0.4535 0.4395 0.4148 0.3938
0.375 0.3750 0.3749 0.3745 0.3732 0.3711 0.3686 0.3626 0.3560
-
0.375 0.3750 0.3749 0.3744 0.3727 0.3699 0.3662 0.3570 0.3465
K=5.0 0.00 0.01 0.10 0.25 0.50 0.75 1.00 1.50 2.00
1.0 1.0025 1.0231 1.0531 1.0946 1.1281 1.1556 1.1979 1.2290
1.0 0.9975 0.9733 0.9306 0.8594 0.7932 0.7339 0.6367 0.5636
0.5 0.5000 0.4998 0.4989 0.4961 0.4925 0.4884 0.4799 0.4718
-
- 0.5 0.5000 0.4998 0.4984 0.4935 0.4858 0.4762 0.4538 0.4302
0.5 0.5006 0.5058 0.5133 0.5238 0.5324 0.5396 0.5513 0.5603
0.5 0.4994 0.4933 0.4826 0.4646 0.4476 0.4319 0.4047 0.3823
0.375 0.3750 0.3749 0.3742 0.3731 0.3712 0.3692 0.3648 0.3609
-
0.375 0.3750 0.3749 0.3742 0.3717 0.3679 0.3630 0.3515 0.3392
882
M. OZTURK
and F. ERDOGAN
Table2. The effect of the Poisson ratio v = G on the stress intensity factors in an orthotropic inhomogeneous medium under fixed-grip loading c,) and consta_nt crack surface pressure p0 [see equation (53)equation (54)]; E0 = EJc’,
K = 0.5
Fixed-load K=o.s Y
0.1 0.2 0.3 0.4 0.5 0.7 0.9
a,a=0.5 - k,(a)
k,( - a) ___
cr,a=l.O - k,(a)
PO&
PO6
Plfi
1.0958 1.1007 1.1053 1.1096 1.1137 1.1215 1.1287
0.8602 0.8633 0.8661 0.8689 0.8715 0.8764 0.8809
0.4961 0.4962 0.4962 0.4962 0.4963 0.4964 0.4965
k,( - a) ___ PlG
-
0.4936 0.4940 0.4943 0.4947 0.4950 0.4957 0.4964
- k,(a)
kr( - a) ___
- k,(a)
PO&
PO&
Plfi
1.1594 1.1739 1.1874 1.2001 1.2121 1.2343 1.2546
0.7354 0.7413 0.7468 0.7520 0.7569 0.7661 0.7746
0.4883 0.4879 0.4877 0.4875 0.4874 0.4873 0.4873
k,( - a) Plfi
-
0.4765 0.4778 0.4790 0.4802 0.4813 0.4835 0.4855
-
0.1796 0.1795 0.1794 0.1793 0.1793 0.1793 0.1793
Fixed-grip ff,a=0.5
0.1 0.2 0.3 0.4 0.5 0.7 0.9
1.4183 1.4233 1.4280 1.4325 1.4368 14449 1.4524
0.6647 0.6676 0.6704 0.6730 0.6755 0.6802 0.6846
cr,a=l.O
0.8138 0.8144 0.8150 0.8156 0.8162 0.8173 0.8184
-
influence of the material inhomogeneity
0.3009 0.3009 0.3009 0.3010 0.3010 0.3011 0.3012
parameter
1.9991 2.0151 2.0301 2.0442 2.0576 2.0826 2.1056
0.4265 0.4319 0.4368 0.4415 0.4459 0.4541 0.4616
for the orthotropic
medium
having the same inhomogeneity
1.3084 1.3143 1.3198
on the stresses and displacements
significant; and (v) within the practical range of material parameters problem
1.2954 1.2988 1.3022 1.3053
may be approximated
is quite
the solution of the crack
by that of an isotropic
medium
parameters.
Acknowledgements-This study was supported by AR0 under the grant no. DAAH04-95-1-0232 and by AFOSR under the grant no. F49620-93-l-0252.
REFERENCES Houck, D. L. (ed.), Proceedings
of the National
Thermal Spray Conference. ASM International,
1987.
:: Batakis, A. P and Vogan, J. W., Rocket thrust chamber thermal barrier coatings. NASA Contractor Report 1750222, 1985. 3. DeMasi, J. T., Sheffler, K. D. and Ortiz. M., Thermal barrier coating life prediction model development. NASA Contractor Report 182230, 1989. 4. Meier, S. M., Nissley, D. M. and Sheffler, K. D., Thermal barrier coating life prediction model development. NASA Contractor Report 189111, 1991. 5. Brindley, W. J. (Compiler), Proceedings of the Thermal Barrier Coating Workshop. NASA-Lewis, Cleveland, OH, 1995. 6. Yamanouchi, M., Koizumi, M., Hirai, T. and Shiota, I. (ed.), FGM-90, Proceedings of the First International Symposium on Functionally Graded Materials, FGM Forum. Tokyo, Japan, 1990. 7. Holt, J. B., Koizumi, M., Hirai, T. and Munir, Z. A., (ed.), Proceedings of the Second International Symposium on Functionally Graded Materials, Ceramic Transactions, Vol. 34. American Ceramic Society, Westerville, OH. 1993. 8. Ilschner, B. and Cherradi, N. (ed.), Proceedings of the Third International Symposium on Structural and Functional Gradient Materials. Presses Polytechniques et Universitaires Romands, Lausanne, Switzerland. 1995. 9. Lee, Y. D. and Erdogan, E, International Journal of Fracture 1995,69, 145. 10. Kurihara, K., Sasaki, K. and Kawarada, M., Adhesion Improvement of Diamond Films, in FGM-90, Proceedings of the First International Symposium on Functionally Graded Materials, FGM Forum, Yamanouchi, M., Hirai, T.. Koizumi, M. and Shiota, I. (ed.), Tokoyo; Japan, 1990. 11. Sampath, S., Herman, H., Shimoda, N. and Saito, T., M.R.S. Bulletin 1995, 20, 1 27. 12. Kaysser, W. A. and Ilschner, B., M.R.S. Bulletin 1995, 20, 1 22. 13. Delale. E and Erdogan, F., ASME Journal of Applied Mechanics 1977, 44, 237. 14. Konda, N. and Erdogan, F., Journal of Engineering Fracture Mechanics 1994, 47,533. 15. Krenk, S., Journal of Composite Materials 1979, 13, 108. 16. Cinar, A. and Erdogan, F., International Journal of Fracture 1983, 19, 83.
Mode I crack problem 17. Delale. E and Erdogan, F., ASME Journal of Applied Mechanics 1983, 50,609. 18. Erdogan, F. and Wu, B. H., Journal of Mechanics, Physics and Solids 1993,41,889. 19. Chou, Y. T., Journal of Applied Physics 1%2, 33, 2747. 20. Kantorovich, L. V. and Krylov, V. I., Approximate Methods of Higher analysis. Interscience, New York, 1985. 21. Krenk. S., International Journal of Solids and Structures 1975, 11,449. (Received 5 September 19%; accepted 2 October 19%)
883
PII:
ht. 1.Engng Sci.Vol. 35.No. 9,pp.869-883,1997 1997 Elsevier Science Limited. All rights reserved
SOO20-7225(97)00014-l
Printed in Great Britain.
0020-7225/97$17.00+0.00
MODE I CRACK PROBLEM IN AN INHOMOGENEOUS ORTHOTROPIC MEDIUM MURAT
OZTURK
and FAZIL
ERDOGAN*
Lehigh University, Bethlehem, PA 18015, U.S.A. Abstract-In
the symmetric crack problem considered the material is both oriented and graded. The properties of the medium is assumed to vary monotonously in the x,-direction, x, and x2 are the principal axes of orthotropy, and the crack is located along the x,-axis The loading is such that x2 = 0 is a plane of symmetry. The mode I crack problem for the inhomogeneous orthotropic plane is formulated and the solution is obtained for various loading conditions and material parameters In the formulation four inde endent engineering constants, E,,, Ezz, G,* and Y,r, are replaced by a stiffness and a shear parameter E = e El&, a stiffness ratio c= (ElllE22)“4, a Poisson’s ratio Y = G parameter K = (E/2G,J - Y. The results show that the stress intensity factors are independent of E and c and generally the effect of K and Y on the stress intensity factors is not very significant. ‘Ihe exception is the values of K approaching - 1. where the physical range of K is - 1K
1. INTRODUCTION
In high temperature applications such as advanced turbine systems, aircraft engines and a great variety of combustion chambers, the demand for improved thermal efficiency requires the development of increasingly more heat-resistant materials and coatings. Because of the level of operating temperatures involved (e.g. approximately 1260°C rotor inlet temperature in gas-fired stationary turbines), in these applications the use of structural ceramics for the purpose of shielding metallic components against excessive heat is becoming almost a necessity [l-5]. Some other important industrial applications of ceramic coatings may be found in the area of prevention of corrosion and wear. From a view point of failure mechanics the homogeneous ceramic coatings, however, have certain shortcomings among which one may mention poor interfacial bonding strength, high thermal and residual stresses, low toughness and consequent tendency toward cracking and spallation. An alternative concept which may be applied to overcome some of these shortcomings appears to be using coatings or interfacial zones with graded compositions and hence, with continuously varying thermomechanical properties. Such composites with smoothly varying volume fractions are known as functionally graded materials [6-8]. It has been shown that grading material properties reduces the magnitude of residual stresses [9] and significantly increases the bonding strength [lo]. Even though grading the microstructure and composition of the medium can be a very useful tool in material design with almost unlimited potential applications, in the near future the main applications of the graded materials will most likely be limited to thermal barrier coatings, interfacial zones to’ enhance bonding and wear and corrosion-resistant coatings. Because of the nature of the techniques used in processing, the graded materials are seldom isotropic. For example, the materials processed by using a plasma spray technique have generally a lamellar structure [ll], whereas processing by electron beam physical vapor deposition would lead to a highly columnar structure [12]. Thus, in studying the mechanics of many of the graded materials, an appropriate model would be an inhomogeneous orthotropic elastic continuum. In this study such a model is used to solve the basic mode I crack problem in * Author to whom all correspondence
should be addressed. 869
870
M. OZTURK
and F. ERDOGAN
z 0
a
x1(5)
Fig. 1. Geometry and notation of the crack problem.
a graded material (Fig. 1). The medium is assumed to be under plane strain or generalized plane stress conditions. The corresponding crack problems for the isotropic inhomogeneous materials were considered in Refs [13, 141. 2. FORMULATION
OF THE
ELASTICITY
PROBLEM
Consider the plane elasticity problem for an inhomogeneous orthotropic medium containing a crack along x2 = 0, - a
$ = s,,(=
E=~,v=~,c4=
22
v21
E 2G,2
-Il.
Equation (1) are valid for plane stress conditions. For plane strain equation replaced by
E,,& v,3v3,)(1
E,,
(1
-
(v,z -
(1
v23v32)
~23~32)
E22 (1 - v,~v~,) ’ K = K
E -”
+
lJ,3+2)(v2,
-
vnv’3,)(1
+ -
v23v3,)
v23v32)
is (1) must be
1’2 ’
(2)
In a general inhomogeneous orthotropic material the parameters E, v, c and K would be functions of x, and x2. However, to simplify the formulation of the problem and to emphasize the spatial variations in the stiffness parameters, the problem will be solved under certain restrictive assumptions regarding the distribution of the elastic constants. Previously it was shown that the effect of Poison’s ratio on the stress intensity factors is not very significant [17, 181. Thus, in the present study v is assumed to be a constant rather than a function of xl and x2. It is also assumed that the material stiffnesses E,,, E22 and G,Z will vary proportionately. This means that c and K may also be considered as constants and the inhomogeneity of the medium may be represented by variation in E only. Note that the special case of c= 1 and K = 1 corresponds to an isotropic material. Also, in a homogeneous orthotropic medium - 1 < K
871
Mode I crack problem
By using now c as a scaling constant and defining X =x,&, a,(x,y)
y = fix*,
u(x,y) = Ur(XrJ*)fi,
= a,,(-w*)k
the stress-displacement
g&Y)
u(x,y) = U,(X,J*)&
= c~&lJ*)r
%&TY)
= ~dh-4
(3)
relations may be expressed as
(44
t4b)
cyky) =
1
&Y
U(X?Y)
-& 2(K+
+
-&
,
u(x,y)
(4c)
I’)
where a&Y)
=
J%Jx,).
(5)
From the equilibrium equations it then follows that
a2u a2u a2u PI aB au + +--=-+p2---y+P,-E ax ( ax axay ax2 ay a2u -++pI7 ax2
a2u +--~4 aE ?!!.+,,~)++$(~+~)=0,(6b) +/32axay E ay ( ay ay a2u
where 2(K PI
=
+ U)
1 -v2
,P2=1+yP,.
(7)
In this problem an additional assumption concerns symmetry and the choice of the function E(x,y). To restrict the consideration to a mode I crack problem it is assumed that the material parameters are independent of x2. It is further assumed that E(x,y) is a monotonously increasing function of x, and in the crack region it may be approximated by an exponential function as follows: E(x,,x2) = E(x,)
= E,,ealXt
= l?(x)
= EoeYX,
y = a,&.
(8)
From equations (6a), (6b) and (8) it may then be seen that
a2u a2u a2f4 +p2+ PlY g+.---y+p17 axay ax ay (
au ay )
=o,
a2u a2u -+p,--_T + P2$-+y(~+g)=R ax2 ay Note that for y=O, equations homogeneous medium [ 181.
(9a) and (9b) would reduce to the differential
(94
Pb) equations for a
872
M. OZTURK and F. ERDOGAN
If we now assume the solution of equations (9a) and (9b) in the form r I -x
F(y,k)emikxdk,
x
I
G(y,k)eeik”
dk,
-Cc
then from equations (9a), (9b), (lOa) and (lob) it may be shown that Fb,k)
= i
G(k,Y)
bj(k)Bj(k)e*T,
= i
Wa)
Bj(k)e*j’,
@lb)
1
where A, j= l,..., 4 are the roots of A4- (vy2+2(k2+
iyk)K)A2+(k2+
iyk)2=0,
02)
given by
A, = -A3=
+‘+2Kv)+
A2 = - A4 =
;(a2+2K$
-
$t/(62+2K9)2-
47J2
lR,
034
+q((62+2K9)2-
4q2
In,
(13b)
S2 = v y2, VI= k2 + iyk
(14)
and
(W-d%v)
b.=
I
A;
_
,jq
A,
j=l
I’
4
‘**-’
*
(15)
The roots Aj are ordered in such a way that %(Aj) > 0, j = 1,2. It is assumed that the inhomogeneous medium contains a through crack along x2 =O, - a
U(X,JJ)= &Is
ibj(k)Bj(k)e’fl-‘“dk, -= 3
(1%
(16b)
Mode I crack problem
E(x)
flxx(X,Y)
=-
+&,y)
= ~ l-v2
%ykY)
=
-
1-v*
2a
E(x)
a4
1
1 2rr
-!-
2(K + VI 27TI
873
C(~hj-ibjk)B,e~~-‘~~dk,
(174
C (Aj-ivbjk)B,e”‘Yikxdk,
t17b)
r i(Ajbj-ik)B,e”“-““dk,O
The two unknown functions B,(k) and B,(k) are determined y=o. 3. THE
(17c)
3
INTEGRAL
from the boundary conditions at
EQUATION
The elasticity problem formulated in the previous section must be solved under the following boundary conditions: a,*(x,,O) = 0, - 03
+
u&+0)=
0)
=
p(x,),
-
Q
(18)
< 4
(194
~v(x,+o)=o,a
Wb)
where p(xr) is a known function. From equations (3) (17~) and (18) it follows that i
( Ajbj - i k)Bj = 0.
(20)
A second equation to determine B, and B4 may be obtained condition equations (19a) and (19b) by introducing l)(x) =
from the mixed boundary
-&u(x, + 0) = y-$u*(x,, +0) = g(x,).
(21)
1
From equations (16b), (19b) and (21) it may be seen that ( - ik)(B3 + B4) =
O’ +(S)eiksds, a’ = al&. I -a’
(22)
The new unknown function I,+(S)is determined from (19a). Thus, solving (20) and (22) for B3 and B4, from (17b) and (19a) it can be shown that lim
y_+O
J%,y> 1
l-vz
2rr
L(x,y,.r)~(s)ds=cp(~x),
- a’ <~
(23)
where
H(y,k) eik(s-4
H(y,k) =
1 i k( A4b4 - A3b3)
dk,
[(A3b3 - ik)h4b4eAfl - (h4b4 - ik)h3b3eAfl].
(24)
(25)
a74
M. OZTURK
and F. ERDOGAN
Any singular behavior of the kernel L must be due to the behavior of H&k) at lk~l-_, 03which may be obtained as’ 1 --Y2
Ikl
(26)
K,=~,S*=~,S2=~.
(27)
By observing that
H&G)
1 -v2
e i&-x) dk -
2& (S2y)2
2K1
and by adding and subtracting H, in equation expressed as
-
1 (I’ @r(s) ds+ IT I _-(I,s -x
-&
- x) +
(S
-
22(S X)2
-
(Sly)’
+
-
X)
(S
-
(24) the integral equation
L,(xs)+(r)ds
=
(s, +
X)’
’
(28)
(23) may now be
S2)C p(Ax),
IXl
(29)
E’(X)
where the Fredholm kernel L, is given by L,(x,s) =
x2[%(h(k))cos(k(s
- x)) - s(h(k))sin(k(s
- x))] dk,
I0
(sl + s2)(k+ iy) _
-i
Al
+
A2
h(k) = 2(k + iy)
-i vy2
+
4(k2 + iyk)
1
13 1 -1
,
(30)
K#l
(31) K=l.
K < CC and s, + s2 is always real. Also note that for y = 0, A, = ks,, Note that in real materials - 1 CC A2= ks2, h(k) =O, L,(x,s) =0 and equation (29) would reduce to the integral equation for a homogeneous orthotropic plane. From equations (19b) and (21) it may be seen that the integral equation (29) must be solved under the following single-valuedness condition:
a’
I
#(s) ds = 0. --(I’
In the special case of isotropic c=(~?,,/E22)“~,
K=l,U’=U,X=Xl,S,=l,
(32)
inhomogeneous materials with constant Poisson’s ratio, s2 = 1, and the Fredholm kernel in equation (29) can be
evaluated in closed form as follows:
L,(xs) = ye2 v. {
+,(v,lCI)+Ko(voi51)
1
--9
2 s-x
’ From equation (27) it may be seen that for K 5 - 1 s, and sz are purely imaginary. Consequently, the integral in equation (24) does not exist and the problem has no physically meaningful solution. Indeed, in corresponding homogeneous orthotropic solids it is known that for K 5 - 1 the elasticity matrix is not positive definite [19] and. therefore, in real materials the range of the shear parameter is - 1 < K
875
Mode I crack problem
6 = g (s - x), ug =
ViTT,
(34)
where K,(z) and K,(z) are the modified Bessel functions. For the plane strain case in equation (34) v should be replaced by u/(1 - v). Referring to equations (3) and (21) in terms of the real physical variables the integral equation (29) may be expressed as
~
+ M,(x,,t,)
g(r,)dt,
=
I
(s, + %)C e-alxp(xl), E 0
lx,1
(35)
where Ml(X,Jl)
=
-
L,(x,/xfc,t,/~).
l
(36)
2V% which, for the isotropic case, becomes M,(x,,t,) = + i
v,~Kl(~“l~l)+K”(~,151) - -&-A= $(4-x*).
1
1
(37)
1
In physical variables observing that x1 = fix t, = 6, (or = r/X&, k = l&e, ( - 33< e coo), it can be shown that the kernel Ml (x,,t,) depends on the inhomogeneity parameter CX,and the elastic constants u and K, but not on c and EO, where c = (E,l/E22)1’4, m = EOealxl and u, K, c and E. are assumed to be independent of xl and x2. From equation (27) it may also be seen that s1 and s2 depend on the shear parameter K only. After solving equation (35) and obtaining g(t,), the crack surface displacement and the stress intensity factors may be obtained as follows: *I u2(x,r
+
0)
=
(38)
&)df,,
I --a
k,(a) =
k,( - a) =
lim ~cr,,(x,,+O) x,-a+0
lim x,--a-O
= -
V2( - xl - a)a,,(x,,
lim E(xl) _g(x,), xI_ a _ o 61 + s2)c
E(xl)
+ 0) = .,f’:“.+,
4.
(Sl
+
V5(ZYrjg(x1).
(39)
s2)c
THE SOLUTION
To solve the integral equation we first introduce the following normalized quantities: t = t,la, r = x,k
44) = g(4), M2(r,f) = aM,(x,,4),
f(r) = P(X~)@~, - 1< (r,O < 1, - a < (x1,4)
(40) where p0 is a constant corresponding to the amplitude of the external loading. The integral equation (35) and the single-valuedness condition equation (32) may then be expressed as
&
+
M2(r,f)
1
NW=
@I +4CPo e-m,a’f(r) <1, 7(r( E
0
(41)
M. OZTURK
and F. ERDOGAN
qS(t)dt = 0.
(42)
The solution of equation (41) is of the form
(43)
where the bounded function w(t) may be expanded into a series of orthogonal associated with the weight function l/m as follows:
w(t) =
(%+;;)cpo ~A.T,(t), - l<
polynomials
(44)
t<1
0
where Ao, A,,... are unknown dimensionless constants. From equations (43) and (44) and the orthogonality conditions of the Chebyshev polynomials
-
1
’ ~~(Wt&)
dt_
7T I -,c7-o’
:;, ,
{
; 1;
;;
(45) ’
m # n,
it may be seen that A0 = 0. By using the following properties,
TM (t
-
dt=
0,
n = 0, Irl < 1
U,_,(r),
n 2 l,Irl
G,(r),
n =
r)vT7
T,(t) = cos (n e), U,(t) =
G,,(r) = -
It-l I
sin@ + 1)6J sine
(46)
OJ)...) Irl > 1,
, cose = t,
(47)
(I - (Irllr)v7T)n v7T
’
(48)
the integral equation (41) may be reduced to
(49)
where
817
Mode I crack problem
The functional equation (49) is solved by method of reduction [20]. From equations (38), (39), (43) and (44) it can be shown that
k,(a) - -
enla
ko
k,( - a) =
k.
UZ(X,r + 0) = -Vl
W)
1
( - l)“A,, k,
ecu@ i
=po&,
Wb)
1
- (x,,&
VO
$L
+U,_,(x&),V,=
(s,
+
%)CP&
E
.
(52)
0
1
It should again be emphasized that the coefficients A, are dependent on the inhomogeneity parameter (Y, and the elastic constants u and K, but not on E. and c. From equations (51a) and (51b) it is then clear that, within the confines of the assumptions made in this study regarding the distribution of the elastic constants, the stress intensity factors are independent of c and Eo. If we observe that the left-hand side of equation (35) represent the stress a&,0) for lx,1>a as well as for Ix,i
5. RESULTS
AND
DISCUSSION
The mode I crack problem described in Pig. 1 is solved under two types of external forces, namely the crack surface tractions and fixed-grip loading. In the results given the crack surface traction is assumed to be
P(h) = -
PO- Pl
( 3_P2(3’-P3(33,,x,,
(53)
where po, p,, p2 and p3 are known constants having the dimension of stress. In the fixed-grip loading the untracked medium is assumed to be under a uniform strain given by E22(X,J2)
Through superposition,
=
Eo
+
&I~).
(54)
for this loading condition it may be shown that
p(x,) = -
!!$Iem+1 (,,+E,( ~)),lx,ku.
(55)
In the solution E. and c appear as multiplicative constants and do not contribute to the distribution of the calculated quantities. Similarly the half crack-length a is the only length parameter which is used as a scaling constant in presenting the results. Thus, by expressing E=m = E. exp(cu,x,)= E, exp((cu,a)(x,lu)), it may be seen that the inhomogeneity parameter CY,enters the analysis only through the dimensionless constant (~,a. The arbitrary variables in the problem are, therefore, the Poisson’s ratio v = G, the shear parameter
M. OZTURK
878
and F. ERDOGAN
K = (ELZG,,) - v and CY,Q.In the case of a homogeneous medium subjected to arbitrary crack surface loading ~&,,0) =p(x,) = -f,(zc), x=x,/u, the stress intensity factors are given by
dx,k,(
Thus, forf,(x)=p,,P,
f,(x)
- a) = -
V-i-G’
dx
(56)
n=O,l,,.., we find k,(a) = ( - l)“k,( - a) = ( - l)“C&$G,
(57)
where c, = 1, c, = c2 = 112,c, = c, = 3/S, ...
(58)
Some sample results for the mode I stress intensity factor k, and the crack surface displacement u2 in an inhomogeneous medium subjected to uniform crack surface pressure p. are shown in Figs 2-5*. The influence of v and K on k, and u2 appears to be less significant than that of a,u. Figures 2 and 3 show that the stress intensity factor on the stiffer side of the medium is always greater than that on the less stiff side. The explanation of this seemingly counterintuitive result may be found in Fig. 6, where the crack surface displacement is given in an inhomogeneous medium with (~~a= 1 or E(x,) = Eoexp(@) and in homogeneous planes having the stiffness From equation (39) one may observe that E( - a) = &/exp( l), E(0) = E,, and E(u)=E,exp(l).
Fig. 2. The normalized stress intensity factors in an inhomogeneous orthotropic medium subjected to uniform crack surface pressure, (T,,,,(x, + 0) = - p,,, v = 0.3.
1.6
-!""""'......................,.....~
1.2
--I____________
kl
_,,_,._.._,._.J.._..-..-..-.._..-...-..-..-..-..-..-
la7,"
:: j
-1 Fig. 3. The
0
1
2 lc
3
4
5
normalized stress intensity factors in an inhomogeneous orthotropic medium subjected to uniform crack surface pressure, (+.“.” (x, + 0) = -p,,, Y = 0.9.
2 Note that in orthotropic materials the Poisson’s ratio v can be greater than 0.5.
Mode I crack problem
819
Fig. 4. Crack surface displacement u2 in an inhomogeneous orthotropic medium under uniform pressure p,, applied to the crack surfaces. (I+= (sI +s,)ap&E,, Y = 0.3, K = 0.5).
the stress intensity factor is proportional to the stiffness I? and the displacement derivative g evaluated at the crack tip. Furthermore, in homogeneous orthotropic planes, the stress intensity factor k, =pO& is independent of 8 (or g is inversely proportional to E). Thus, if we compare the inhomogeneous plane solution with that of a homogeneous plane having the same stiffness at the crack tip X, =a, one would expect that, since the crack surface displacement derivative near x1 =a in the inhomogeneous medium is greater than that in the corresponding homogeneous medium, k,(a) would be greater than pO&. Similarly from the relative magnitudes of the crack opening near x1 = - a it may be concluded that k,( - a)
1 .o
Fig. 5. Crack surface displacement u2 in an inhomogeneous orthotropic medium under uniform pressure p. applied to the crack surfaces. (v,, = (s, +s2)ap,~lE,,, Y = 0.3, a,a = 0.5).
Fig. 6. Crack surface displacement IQ in an inhomogeneous orthotropic medium under uniform pressure p,, applied to the crack surfaces. (v,, = (s, +s2)ap,,c/E,,, Y = 0.3, K = 0.5, ala = 1.0).
880
M. OZTURK
0.0
0.5
and F. ERDOGAN
1 .o
ma
1.5
2.0
Fig. 7. The normalized stress intensity factors in an inhomo$eneous orthotropic medium subjected to fixed-grip loading E,,;k,, = (e&c )&, Y =0.3.
limit for (~,a+ 0 the stress intensity factors become kl (a) = k,( - a) + (EWE&‘) G and a + 1/2(~,E&~) fi. For an unusually large value of u the variation of kr with k,(a)= - k,( - ) the shear parameter K is shown in Figs 9 and 10. Figures 7-10 show that, except for K-_, - 1, under fixed-grip loading the influence of K on the stress intensity factors is rather insignificant. Similarly, from Fig. 11 it may be seen that the effect of v on /cl(a) and k,( - a) is also very small. To give some idea about the influence of the nature of crack surface tractions on the stress intensity factors, Table 1 shows some results obtained by using the loading given by equation (53). Such results may be useful if the stresses obtained from the untracked medium can be approximated by a polynomial of degree three. Similarly, Table 2 shows the effect of the Poisson ratio v = G on the stress intensity factors. Based on these results the assumption of constant v throughout the medium appears to be rather realistic. 3
0 0.0
0.5
ofi
1.5
2.0
Fig. 8. The normalized stress intensity factors in an inhomo eneous orthotropic to fixed-grip loading l,; k,,=(c,E,Jc 5 )V%, v =0.3.
Fig. 9. ‘Ihe normalized
stress intensity
subjected
factors in an inhomogeneous orthotropic medium subjected to fixed-grip loading c,,: k,, = (a,&JcZ)~, Y = 0.9.
881
Mode I crack problem 1.5.
8
,
I
,
I
,
I
,
. .
1.0
5
0.5
kl(a)/ko
--__________~2~0. ola=0.25
-..-..-..-..-..-..-..-..-..kl(-a)/kl,
a,a=0.25 -___----_____a,o=O~5 ---. ..........................................~~.~.~.?.~.?...
0.01 ’ -1
Fig. 10. The normalized
I
)
-
-
ko
1
~'!?:.!:I)...
. . . . . . . . . . . . . . . . .._..............._......_.
’
’
’
0
1
’
( K
2
’
’
I
3
stress intensity factors in an inhomogeneous to fixed-grip loading e,, k,,= (r,E,@)&,
’
’
4
J
5
orthotropic Y = 0.9.
medium
subjected
In conclusion it may be pointed out that in the mode I crack problem for an orthotropic inhomogeneous medium formulated in terms of the material orthotropy parameters E, v, K and c: (i) the Poisson ratio v has only a negligible influence on the stress intensity factors; (ii) the influence of the shear parameter K may be significant only for K < - 0.5; (iii) the parameters E and c have no influence on stresses and affect the displacements in the usual manner; (iv) the 4
3
I
Fig. 11. The normalized
Table 1. The normalized
stress
stress intensity factors in an inhomogeneous to fixed-grip loading E,,: k,, = (e,,&/cZ)~, intensity factors for an inhomogeneous loading p(x,)= -p,(x,/a)“, n =0,1,2,3,
orthotropic K = 0.5. orthotropic Y =0.3
medium
medium
subjected
under
crack
surface
K = - 0.25 0.00 0.01 0.10 0.25 0.50 0.75 1.00 1.50 2.00
1.0 1.0025 1.0246 1.0604 1.1177 1.1720 1.2235 1.3184 1.4043
1.0 0.9975 0.9747 0.9364 0.8740 0.8154 0.7616 0.6701 0.5979
0.5 0.5000 0.4998 0.4990 0.4963 0.4923 0.4873 0.4753 0.4621
-
- 0.5 0.5000 0.4998 0.4989 0.4954 0.4898 0.4824 0.4642 0.4439
0.5 0.5006 0.5061 0.5151 0.5295 0.5433 0.5565 0.5815 0.6047
0.5 0.4994 0.4937 0.4841 0.4684 0.4535 0.4395 0.4148 0.3938
0.375 0.3750 0.3749 0.3745 0.3732 0.3711 0.3686 0.3626 0.3560
-
0.375 0.3750 0.3749 0.3744 0.3727 0.3699 0.3662 0.3570 0.3465
K=5.0 0.00 0.01 0.10 0.25 0.50 0.75 1.00 1.50 2.00
1.0 1.0025 1.0231 1.0531 1.0946 1.1281 1.1556 1.1979 1.2290
1.0 0.9975 0.9733 0.9306 0.8594 0.7932 0.7339 0.6367 0.5636
0.5 0.5000 0.4998 0.4989 0.4961 0.4925 0.4884 0.4799 0.4718
-
- 0.5 0.5000 0.4998 0.4984 0.4935 0.4858 0.4762 0.4538 0.4302
0.5 0.5006 0.5058 0.5133 0.5238 0.5324 0.5396 0.5513 0.5603
0.5 0.4994 0.4933 0.4826 0.4646 0.4476 0.4319 0.4047 0.3823
0.375 0.3750 0.3749 0.3742 0.3731 0.3712 0.3692 0.3648 0.3609
-
0.375 0.3750 0.3749 0.3742 0.3717 0.3679 0.3630 0.3515 0.3392
882
M. OZTURK
and F. ERDOGAN
Table2. The effect of the Poisson ratio v = G on the stress intensity factors in an orthotropic inhomogeneous medium under fixed-grip loading c,) and consta_nt crack surface pressure p0 [see equation (53)equation (54)]; E0 = EJc’,
K = 0.5
Fixed-load K=o.s Y
0.1 0.2 0.3 0.4 0.5 0.7 0.9
a,a=0.5 - k,(a)
k,( - a) ___
cr,a=l.O - k,(a)
PO&
PO6
Plfi
1.0958 1.1007 1.1053 1.1096 1.1137 1.1215 1.1287
0.8602 0.8633 0.8661 0.8689 0.8715 0.8764 0.8809
0.4961 0.4962 0.4962 0.4962 0.4963 0.4964 0.4965
k,( - a) ___ PlG
-
0.4936 0.4940 0.4943 0.4947 0.4950 0.4957 0.4964
- k,(a)
kr( - a) ___
- k,(a)
PO&
PO&
Plfi
1.1594 1.1739 1.1874 1.2001 1.2121 1.2343 1.2546
0.7354 0.7413 0.7468 0.7520 0.7569 0.7661 0.7746
0.4883 0.4879 0.4877 0.4875 0.4874 0.4873 0.4873
k,( - a) Plfi
-
0.4765 0.4778 0.4790 0.4802 0.4813 0.4835 0.4855
-
0.1796 0.1795 0.1794 0.1793 0.1793 0.1793 0.1793
Fixed-grip ff,a=0.5
0.1 0.2 0.3 0.4 0.5 0.7 0.9
1.4183 1.4233 1.4280 1.4325 1.4368 14449 1.4524
0.6647 0.6676 0.6704 0.6730 0.6755 0.6802 0.6846
cr,a=l.O
0.8138 0.8144 0.8150 0.8156 0.8162 0.8173 0.8184
-
influence of the material inhomogeneity
0.3009 0.3009 0.3009 0.3010 0.3010 0.3011 0.3012
parameter
1.9991 2.0151 2.0301 2.0442 2.0576 2.0826 2.1056
0.4265 0.4319 0.4368 0.4415 0.4459 0.4541 0.4616
for the orthotropic
medium
having the same inhomogeneity
1.3084 1.3143 1.3198
on the stresses and displacements
significant; and (v) within the practical range of material parameters problem
1.2954 1.2988 1.3022 1.3053
may be approximated
is quite
the solution of the crack
by that of an isotropic
medium
parameters.
Acknowledgements-This study was supported by AR0 under the grant no. DAAH04-95-1-0232 and by AFOSR under the grant no. F49620-93-l-0252.
REFERENCES Houck, D. L. (ed.), Proceedings
of the National
Thermal Spray Conference. ASM International,
1987.
:: Batakis, A. P and Vogan, J. W., Rocket thrust chamber thermal barrier coatings. NASA Contractor Report 1750222, 1985. 3. DeMasi, J. T., Sheffler, K. D. and Ortiz. M., Thermal barrier coating life prediction model development. NASA Contractor Report 182230, 1989. 4. Meier, S. M., Nissley, D. M. and Sheffler, K. D., Thermal barrier coating life prediction model development. NASA Contractor Report 189111, 1991. 5. Brindley, W. J. (Compiler), Proceedings of the Thermal Barrier Coating Workshop. NASA-Lewis, Cleveland, OH, 1995. 6. Yamanouchi, M., Koizumi, M., Hirai, T. and Shiota, I. (ed.), FGM-90, Proceedings of the First International Symposium on Functionally Graded Materials, FGM Forum. Tokyo, Japan, 1990. 7. Holt, J. B., Koizumi, M., Hirai, T. and Munir, Z. A., (ed.), Proceedings of the Second International Symposium on Functionally Graded Materials, Ceramic Transactions, Vol. 34. American Ceramic Society, Westerville, OH. 1993. 8. Ilschner, B. and Cherradi, N. (ed.), Proceedings of the Third International Symposium on Structural and Functional Gradient Materials. Presses Polytechniques et Universitaires Romands, Lausanne, Switzerland. 1995. 9. Lee, Y. D. and Erdogan, E, International Journal of Fracture 1995,69, 145. 10. Kurihara, K., Sasaki, K. and Kawarada, M., Adhesion Improvement of Diamond Films, in FGM-90, Proceedings of the First International Symposium on Functionally Graded Materials, FGM Forum, Yamanouchi, M., Hirai, T.. Koizumi, M. and Shiota, I. (ed.), Tokoyo; Japan, 1990. 11. Sampath, S., Herman, H., Shimoda, N. and Saito, T., M.R.S. Bulletin 1995, 20, 1 27. 12. Kaysser, W. A. and Ilschner, B., M.R.S. Bulletin 1995, 20, 1 22. 13. Delale. E and Erdogan, F., ASME Journal of Applied Mechanics 1977, 44, 237. 14. Konda, N. and Erdogan, F., Journal of Engineering Fracture Mechanics 1994, 47,533. 15. Krenk, S., Journal of Composite Materials 1979, 13, 108. 16. Cinar, A. and Erdogan, F., International Journal of Fracture 1983, 19, 83.
Mode I crack problem 17. Delale. E and Erdogan, F., ASME Journal of Applied Mechanics 1983, 50,609. 18. Erdogan, F. and Wu, B. H., Journal of Mechanics, Physics and Solids 1993,41,889. 19. Chou, Y. T., Journal of Applied Physics 1%2, 33, 2747. 20. Kantorovich, L. V. and Krylov, V. I., Approximate Methods of Higher analysis. Interscience, New York, 1985. 21. Krenk. S., International Journal of Solids and Structures 1975, 11,449. (Received 5 September 19%; accepted 2 October 19%)
883