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Hygrothermal stresses for a plane crack in a generally anisotropic plate
Engineering Fracrure Mechanics
Vol. 4.5, No.
0013-7944/93
6. pp. X31-841, 1993
$6.00 t- 0.00
IQ 1993Pergamon Press Ltd.
Printed in Great Britain.
HYGROTHERMAL STRESSES FOR A PLANE CRACK A GENERALLY ANISOTROPIC PLATE
IN
C. K. CHAO and R. C. CHANG Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan 106, R.O.C. Abstract-The solutions are presented for the hygrothermal stress field of a generally anisotropic plate under uniform heat flux and moisture concentration transfer obstructed by a hygrothermally insulated crack. For uncoupled diffusion of temperature and moisture, the solutions of both temperature and moisture are obtained directly from the Hilbert problem approach, and are treated as the particular solutions to a pair of nonhomogeneous partiat differential equations for an uncoupled hygrothermoeiastic system. The associated homogeneous solutions are expressed in terms of three stress functions based on the complex variable approach of Lekhnitskii. With some identities concerning the eigenvalues and eigenvectors. the general expressions of the stress and displacement fields can then be found in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are expressed in terms of the heat flow, moisture concentration, material geometry, elastic and hygrothermal anisotropy. The simultaneous existence of mode I, II and III fracture is found to be due to material inherent anisotropy. Special cases for isotropic and orthotropic materials are also discussed.
1. INTRODUCTION
and/or temperature environments is of degree of material degradation can be significant, particularly due to the stress concentration around a material discontinuity in anisotropic material. The stresses induced by heat flow and moisture concentration transfer which are known as hygrothermal stresses will become singular at the crack tips and may finally result in material failure through crack propagation. The general influence of moisture and/or temperature on the stresses and displacements in laminated composite materials has been determined by Shen and Springer [1] and Pipes et al. [2]. It is known that mechanical stiffness and strength are degraded when a laminate absorbs moisture, and recovery is incomplete after desorption. These environmental influences can interact so that the stress state of the material is dependent on both temperature and moisture in its surrounding. A theory of diffusion that incorporates the interaction between temperature and moisture was developed by Henry [3]. Phenomenological arguments leading to coupled equations governing the simultaneous diffusion of moisture and heat were studied by Hartranft and Sih [4,5]. The conditions of suddenly applied temperature and/or of moisture on the plate surface were both considered and the induced stresses tend to zero when the temperature and moisture concentration become uniform for a steady-state condition. The situation when the moisture diffusion coefficient is temperature-dependent was treated by Sih et al. [6] for symmetry boundary conditions that produce no bending. Coupling of moisture and heat was found to be inherent in the case of a transient condition and found to decrease as time increases to achieve a steady-state condition. The present investigation is concerned with the moisture, temperature and stress fields of a generally anisotropic plate containing a hygrothermally insulated crack. Under the assumption of the steady state, the temperature and moisture fields can be solved independently by the Hilbert problem fo~ulation. Starting with a pair of coupled nonhomogeneous partial differential equations for an uncoupled hygrothe~oelastic system, the general solutions are constructed as a linear combination of elasticity-eigenvectors, heat-eigenvector and moisture-eigenvector. Based on the complex variable approach outlined by Savin [7] and Lekhnitskii [S], the stress and displacement fields can be found in terms of three pairs of eigenvalues of the elasticity constants, a pair of eigenvalues of the heat conduction constants as well as a pair of eigenvalues of the moisture diffusion constants. Following Muskhelishvili [9], a plane crack problem can then be expressed in the form of the Hilbert problem which allows us to obtain the solution in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are also obtained in THE DEGRADA~ON of material exposed to moisture considerable importance in composite structures. The
831
832
C. I(. CHAO
and K. C. CHANG
a closed form. The influence of heat flow, moisture concentration transfer. material geometry. elastic and hygrothermal anisotropy on those parameters can then be studied through this explicit form. Special attention is given to orthotropic and isotropic cases. The validity of the fully open crack assumption is also discussed. 2. BASIC
FORMULATION
For uncoupled hygrothermoelasticity. the generalized be expressed in contracted notation as [ 10. 1l] (,=S,,o,+r,I.+fi,C‘
Hooke’s
(i,,j=l.Z
law of anisotropic
material
. . . . . 6).
can
(1)
where repeated indices imply summation, and S,,. SI,, fi,, T and C denote the compliance tensor, thermal expansion coefficients, moisture expansion coefficients. temperature and moisture concentration, respectively. The moisture expansion coefficients 13, can be treated as analogous to the coefficients of thermal expansion, f~,. For two-dimensional problems, the components of displacements U, 23, IV, temperature and moisture concentration are independent of -_. The engineering strains, c,, in eq. (1) are defined in a Cartesian coordinate system by f, CJ =
where a comma For generalized
(2 =
“f,‘ll,,
A,,_ =
!, =
a',,
2,:
1, = =
7'*,
11',,
(q-f_=0
(6 =
I?(,, =
24,! +
Z'*,.
i2)
stands for differentiation. and the stresses, IT,, are defined in an analogous plane deformation. eqs (1) and (7) may reduce to u,,=.$,cr,+s7,T+~,C’
r.,,.=.%,0,+6J+fl~C
It’,) =‘Qa,+~J+fjjC Ii.* +
manner,
(3)
W,=:*T+r,$-9JfijSC 7'., ==3~,q+&T-+&C
and
-(&,o,+%T+L4C)ISv
0, =
9,,= s,,-
For a steady-state
s,3s,,/s,,
s,,IS,,
2, =
I,
pi =
P,- 13A:‘S,,
condition,
-
%I
the equations
Q,*, + T,,, = ()
of equilibrium
without
T,,_, +c+=O
In order to satisfy eqs (5) two stress functions the stress components as follows:
5,. =
y.,
T’,:
operators
=
---
as
to
Y ,I.
L,Y=L,2T+L,2C.
are defined
and related
t,, = -- F,y\
L,FfLZY=L,,Ti-L,IC
where the differential
as (5)
and V/(.U,JJ) are introduced
problem hygrothermoelastic 24. I‘. w: this gives
The governing equations for the uncoupled substituting eqs (6) into (3) and eliminating
L,F+
body forces can be written
5,,,\+?,,,,.=0.
Ffs,~)
c, = F_,,
(T, = F .I/
(4)
(i,.i # 3).
can then be obtained
by
Hygrothe~al
L4 =
833
stress field of an anisotropic plate
$ - 232,-ax3a4ay +(2sI,,-&)--
322
a4
a4 +3”
23
16ax+3
ax2 ay”
"a)J4
(8) These are a pair of nonhomogeneous partial differential equations for the uncoupled hygrothermoeiastic system. Equation (7) can be further decoupled into two individual equations as follows: (&A., - L:F’ = (.Vi, (W, The general homogeneous expressed as
- L:Y
=
@4&t
- &L,I )T + C&L,, - LAW -
L,L)T
+
CL4L
-
LdwP=
(9) (10)
solution to eq. (9) has been given by Lekhnitskii [8], which can be
Fh= k=li @%Zd + f’,GJ), where the overbar denotes the conjugates of complex variables, and zk
=
x+
(12)
pky,
where pk (AZ= I, 2,3) are the roots of the following characte~sti~ equation:
with
l,(P) =
-$?4
/4(p)
$2
+
(3%+ 346k- c3bi+ 356W-t 3td + $1~~.
(14)
The temperature and moisture concentration appearing in nonhomogeneous (IO) can be solved independently from the following equations:
terms of eqs (9) and
=
-2&p
+ (2& + S&i* - 2&G
k,~ T,, + (k,, + MT,xy + k,, T,),, = 0 d, I C,.xx+ (4, + 4dC,.xy + 4, C,,, = 0,
u 5)
where k, and d,, (i,j = 1,2) denote the thermal conduction coefficients and moisture diffusion coefficients, respectively. These are the equations governing the uncoupled diffusion of heat and moisture through a solid. The general solutions to eqs (15) can be expressed as T=O,(Z4)+04(24)
Z,=x-t-u4y
(16)
c=Q5(Z5)+@5(&)
Z,=xfCL5y,
(17)
where S4 and O5 are arbitrary hoiomorphic functions and p4 and ,us are the roots of the following characteristic equations: &A
+ 6%~+ ~Z,)LL~ + k,, = 0
d2211:+(d~2+dz,)~5+d,l=O.
(18) (19)
C. K. CHAO
834
and R. C. CHANG
A particular solution to ey. (9) can then be expressed as 120) with FJ(Z,) = ‘%@,**G)
F,(Z,) = A, oT*(z,),
(21)
where a superscript * denotes integration with respect to its argument and
(27)
with
lr2(jLL=
-/72+ &ji - p,jG.
(23)
Note that the governing equation (9) resembles the force vibration of a discrete system in that the heat-eigenvalue p4 and moisture-eigenvalue y, play the role of forcing frequency while the elasticity-eigenvalues pk (k = 1,2,3) may be identified with the natural frequencies. In the present approach, the heat-eigenvalue or moisture-eigenvalue cannot be identical to the elasticity-eig~nvatues as considered for the resonance case. The complete solution to eq. (9) can then be obtained as F=F,+F,=
i {Fh(Zk)+Fk;.(Zk)). x-i
Similarly, the general solution to eq. (10) can be put in the form 5
Y =
-..~--
1 jqJ;.(Zk) +
k .: I
where the prime stands for differentiation
vkF;(Zk)},
(25)
with respect to its argument and
Using the notations concerning the eigenvalues and corresponding eigenvectors, similar to the expression of Lekhnitskii [S], the stress and displacement fields can be expressed in the form
IPk4k#%(Zk) + PkYlkSt)kvk)r %=ki, T,, =
e--g,
{Pk$k(Zk) +
(30)
-.---
PkQ)kGZk)l.
(31)
835
Hygrothermal stress field of an anisotropic plate
(32)
0’ = k=l
kk4k(Zk)+ qk4kvk)l
(33)
(34) where c$~=F;
(35)
and
where 6, stands for the delta function, which will be 1 as i = j or 0 as i #j, with the usual definition. Note that the genera1 forms appearing in eqs (27)--(34) cannot be specialized directly into isotropic material due to the repeated eigenvalues of the elastic constants.
3. HY GROTHERMAL
STRESS
Consider an infinite anisotropic plate. The uniform heat and moisture flow in the direction of the positive y-axis is disturbed by the presence of a hygrothermally insulated crack lying along y = 0, 1x1
+ ~~k**)~~(z~) + (kz, + P&P~(Z&
(37)
On the crack surface, setting y = 0 and /x/c a, this expression takes on the form K&z,+ cLJW@;(x) + (k2, +
ji&d~Wl+
I- k, + wW@;(x) + &, + ibW@;(X)l- = -(h: + h,,) (38)
(4 Fig. I. The hygrothermoeiastic
plane crack.
09
Fig. 2. Supposition of heat and moisture flux in an unflawed plate and a corrective problem.
836
C. K. CHAO and R. C. CHANG
[(k,, +p~k,~)O~(.u)-
(k,,
c ,&k,,)@j(x)]’
-[(k,, +pLqk?&L3;(.Y)-_(k~,+p,k,2)0;(.\-)]
= -(Ir,’
-II,
).
fZ9)
where the superscripts +, - denote the upper and lower sides of the crack surtace, respectively. The problem of finding a sectionally llolomorphic function in the whole plant subject to the boundary conditions is known as the Hilbert problem [9. 121. For the present problem the solution can be obtained as [(k,, + p,k,z)O;(Z)
+ (k,, -i- ji4k22)@;(Z)]
where 6, and 0, are any holomorphic functions. For the sake of convenience, the soluuon is represented as the sum of a uniform heat and moisture flux in an unflawed plate (Fig. ?a) and a corrective solution (Fig. 2b). The constant and negative heat flux h^is prescribed on the crack surface (Fig. 2b), i.e. 11; = It ,, = -- 1; (42) and, since the heat flux vanishes at infinity. it gives I),(Z) = f&(Z) = 0.
(4i)
Substituting eqs (42) and (43) into eqs (40) and ( 41) yields ^ (44) The solution associated with the uniform heat flux disturbed by the presence of an insulated crack can be obtained by using the technique of superposition, i.e.
Integrating eq. (45) with respect to Z, the temperature
function is found as
The integrating constant is neglected here. which can be treated as a reference temperature. and the temperature will become zero along the x-axis excluding the crack surface. Substituting eq. (46) into (16), the full field solution of temperature can be established. Analogous to the preceding processes. the moisture concentration function can also be expressed as 147)
(d,, -I-$u5d~:)05(z) = _t [\, (Z? -- LP)],
wherepdenotes the constant moisture flux in the direction of the positive j,-axis. Substituting eqs (46) and (47) into (21) and using eq. (35). we have 4,(Z) = b,Q(Z
- ff?
C&(Z) = cJV:(Z’
- u?),
(48)
where -14, -A, h,=-....--------26, + iu,h-,21 (‘<= ~g-T-,&)
.
(49)
In order to find the hygrothermal stresses induced by the presence of temperature and moisture concentration, the traction boundary conditions on the crack surface are considered. Let the tractions on the crack surface be C>.= ;,
?,,. = t,
7,: = i,?
r50j
837
Hygrothermal stress field of an anisotropic plate
and knowing the following properties: lim (bk(Zk) = #t(x)
Jl% $k(Zk) = &(x)
J-t,+
the
(51)
IN< a,
traction boundary conditions on the crack surface can be written as EY&~/c(~)+ Yi!~(Pk(~)l+ +
[Yik4kCX)
+
(521
Yik&kCx)l= ;i’ + iT
lVik4kCx) - Yik#kCX)l+ - i3ik+kCX) - Yik#k(x)l-= ;P - ‘is
(53)
l
wherei=1,2,3,k=1,2
,...,
5, /xl
Y2k =
-pk
Y3k =
(54)
-qk*
The solutions associated with eqs (52) and (53) can be obtained from the Hilbert problem in the form [9, 121 1 Yi*QjklZ)
+
Yj~~~(z~
=
2niJ(Z2
n (t?’ + t,-),,/(f2 - a*) dt + - a’) s -a
Y&Jk(Z) - Yik#AZ) =&
s
u (i: - t^,-)dt _ t_ z +
a
P,(Z) J(Z2 _ a*)
t-z
(55)
Q@L
where P,(Z) and Qi(Z) are holomorphic functions, which can be expressed as polynomials degree rz. Since the crack surfaces are traction free, then i#? E ;; = 0.
of (57)
For the condition of zero stress at infinity, only the constant terms of P,(Z) exist, i.e. P,(Z) = 2e,
Q(Z) = 0.
(58)
With the aid of eqs (57) and (58), eqs (55) and (54) further reduce to yi&k(Z) = ,/(z:’
a2)
(i = 1,233, k = 132,. . . ,5),
where ei are all real constants due to the conjugate condition. There are a total of five stress functions (Pk (k = 1,2, . . . ,5) appearing in eq. (59), but only three functions & (k = 1,2,3) are unknown since 4, and (bs have been given in eq. (48). Hence, the stress functions @j0’ = I, 2,3) can be expressed as cb,(Z) =
J;y:a2) + (h,R + CjP)J(Z”
- a2)
(i,j =L:1,2,3),
(60)
with l$ = -y$
y+Jx$
tW
c,= -yji’yiScSt where the constants b4 and es are shown in eq. (49). It is (i = 1,2,3) in eq. (13) cannot be identical, the determinant be singular. The problem now reduces to the determination required to satisfy the single-valuedness of displacements, ul, = 0 = =
(62) noted that the elasticity-eigenvalues pi of yij exists and, therefore, y,; i will not of the unknown constants ei which are i.e.
@‘6&k) dzk
sa
[u’+(x)
-a
- u’-(x)] dx
= =
efh(fftt-4 -PA~xY
J -0
-h&(xX) -~d&)l-)
dx,
(63)
838
C. K. CHAO and R. C. CHANG
where c is any simple closed contour surroundng x(Z) = ,/(Z’ - a’) has the following property:
the
crack.
The
Piemelj
function
Substituting eq. (60) into (63) and using (64), we have
where Im{ ] denotes imaginary part. Note that both the constants h, and cl in eq. (6.5) are equal to zero as a result of mutual independence between temperature and moisture concentration. Simifarly, the other two constraint conditions associated with the displacements 13and w can be obtained as (66) (67) Solving for eqs (65)-(67), we have (68) where
r,=p,,%f,,
A,=p,,‘H,
(69)
with p,, = Im{p,y,;’ ), p2, = Im{y,)l,; ’ ), p3, = Im{t,;l,, ‘) H, = Im{p,b, i M,=Im{p,c,j
170)
H, = Imjq,b,)
H, = Im(t,b,)
(71)
h4?=Im{qkci:j
Mi=Im{r,c,],
(72)
where i, j = 1,2,3 and k = 1.2, . _. , 5. It is shown that the coefficients A,, F,. pli, Hj and Mj are all real constants and dependent on material properties. The general solutions for a hygrothermal stress field can then be obtained by substituting eqs (48) and (60) into eqs (27)~34). 4. RESULTS
AND DISCUSSION
The stress functions appearing in eq. (60) contain a singular term (Z’ - a’) -’ ’ which results in singular behavior of the stresses near the crack tip. With the usual definition, the stress intensity factors K,, K, and K3 can be obtained as K, = +jYA*‘t,il,h^
+ rJ,
(i = I, 2. 3).
(73)
where + and - stand for the right and left sides of the crack tip, respectively. It is seen that the stress intensity factors are proportional to the powers of i of the crack length and linearly proportional to the heat and moisture flux. It is not surprising that the influence of heat flux on the stress intensity factors through the material constants Ai is exactly the same as that of moisture flux through the moisture constants r, since the process of heat transfer is fundamentally equivalent to that of moisture transfer. Similarly, by integrating eqs (32)-(34) and using the constraint conditions expressed in eqs (65)-(67), the crack opening displacements are obtained by setting y = 0, Ix/ < u. i.e. ADi=
-2s,(cza
- .x*)(f&h” + M,,jl),
(741
839
Hygrothermal stress field of an anisotropic plate
where AD, = u+(x) - u-(x), and AD,, AD, stand for the crack opening displacements a, w, respectively, on the crack surface. It can be seen that ADZ will always be negative in either -a < x -C0 or 0
‘?AD,(<
-A&&)+AD&
-A+,(t)+AD&
-Au)r_&)ld5
= -~[(n,H,+n,H,+n,n,)h^2+(T,M,+T,M,+~~~,))2],
(75)
where the integration variable t represents the distance ahead of the crack tip. It is understood that the energy release rate is proportional to the square of heat flux as well as moisture flux. For orthotropic materials, only the material constants S,, cli, pi (i,j = 1,2,3) and S,, &, &,, exist in eq. (1) and all the coefficients klz, kzl , d,,,, d2, in eq. (15) vanish. The governing equation (7) becomes L,F = L,,T 4 L,zC,
(76)
and the other equation Lz Y = 0 is automatically satisfied in the sense that all the shear stress components in the z-direction vanish. Therefore, the system degenerates into an in-plane problem, and the characteristic equation (13) becomes Id(P) = 0,
(77)
which gives only two pairs of eigenvalues. The eigenvalues puq,ps in eqs (18) (19) and eigenvalues pr, p2 in eq. (77) becomes pure imagina~es. Let PI = i5, LIZ= X2 P4= ii4 ius= iis, where ii (i = 1,2,4,5)
(78)
are all real constants, and
i, + i* = J([2% ia
=
&h
+ $56+ 2(5”,,s?2Y’211%)it iz
1 lk22)
is
=
&d,,
=
J(~22/&
1
(79)
b&2>.
Equations (70), (71) and (72) now reduce to
b-‘I =
Substituting eqs (80)+82) into eqs (69) and (73) the stress intensity factors become K, =o
Kz= ?+$:_ ,(H,l;+Mf) I % (83) II 1 2 where - and + stand for the right and left sides of the crack tip, respectively. This implies that the mode I stress intensity factor K[ would not exist for orthotropic material. The crack opening displacements are also obtained as AD, = -2x&a2
- x’)(HI ii + A&f)
AD, = 0.
(84)
840 It is indicated displacement in the form
C. K. CHAO and R. C. CHANG that there is no tendency for the crack to open or close. However, a relative tangential AD, may exist between the crack faces. Similarly. the energy release rate can be found
For isotropic becomes
materials,
the material
i,+&=’
properties
;,I?=1
are independent
of the direction
and cq. (79)
ii,= CT= I.
(86)
However, as mentioned above, the eigenvalues cannot be identical in the type of genera1 sofution shown in eq. (24). In order to proceed with our discussion, cl and is will be made small perturbations of unity, and then eqs (81) and (X2) become H, = 3/2/i
M, = fli2d.
(87)
It is understood that C?= 01”, . /? = b,, k = k,, and d = L& for isotropic material and Oz= ( 1 + 1’1%. [ = (1 + v)j3, g,,, = (1 - v’)/E for plane strain conditions where k, d E, I’, a and /3 are the heat conductivity, moisture diffusivity, Young’s modulus, Poisson’s ratio, thermal expansion coefficient and moisture expansion coefficient, respectively. The stress intensity factors can then be obtained as
where - and f refer to the right and left sides of the crack tip. respectively, opening displacement can be obtained as
and the energy
Similarly,
the crack
release rate is given by
For the special case_? = 0, the results given in eqs (&S--(90) are shown to be identical given in the literature [ 151.
to the soiutions
5. CONCLUSIONS The general solutions for plane anisotropic hygrothermoelasticity are derived and expressed in an explicit form based on the Hilbert problem formulation. For uncoupled diffusion of heat and moisture through a solid, both temperature and moisture concentration are solved independently from the complex variable approach, and are treated as the particular solutions to a pair of‘ nonhomogeneous partial differential equations for an uncoupled hygrothermoeiastic system. The complete solutions are constructed as a linear co~nbi~lation of the elasticity-ei~envectors, the heat-eigenvector and the moisture-eigenvector. Some new identities concerning the eigenvaiues and eigenvectors are developed in this paper, which are useful in replacing the complex eigenvectors by the basic material constants. The stress intensity factors. crack opening displacement and the strain energy release rate are obtained in a closed form. The simultaneous existence of mode I, II and III fracture is found to be present in this paper due to material inherent anisotropy. It is noted that the influence of moisture flux on these parameters is exactly the same as that of heat flux due to the fact that the process of moisture transfer is fundamentally equivalent to that of heat transfer as stated earlier. Special attention is given to the cases for orthotropic and isotropic materials. It is seen that both the mode I stress intensity factor K, and crack opening displacement AD? are found to vanish and consequently the situation with a partial contact crack would not occur for degenerate
Hygrothermal
stress field of an anisotropic
plate
841
materials. Comparison with the solutions for a special case shows that the solution presented here is exact, simple and general. REFERENCES HI C. H. Shen and G. S. Springer, Moisture absorption and desorption of composite material. I. compos. Maw. lo,2220 (1976). PI R. B. Pipes, J. R. Vinson and T. W. Chou. On the hygrothermal response of laminate systems. I. compos. Mater. 10, 1299148 (1976). [31 P. S. H. Henry, Diffusion in absorbing media. Proc. R. Sot. Land. A171, 215-241 (1939). [41 R. J. Hartranft and G. C. Sih, The influence of the Soret and Dufour effects on the diffusion of heat and moisture in solids. Inr. .I. Engng Sci. 18, 1375-1388 (1980). PI R. J. Hartranft and G. C. Sih, The influence of coupled diffusion of heat and moisture on the state of stress in a plate. J. Mech. Compos. Muler. USSR 1, 5361 (1980). stresses in composites: coupling of moisture and heat 161 G. C. Sih, M. T. Shih and S. C. Chou, Transient hygrothermal with temperature varying diffusivity. Inf. J. Engng Sci. 18, 1942 (1980). [71 G. N. Savin, Stress Concentration Around Holes. Pergamon, New York (1961). S. G. Lekhnitskii, Theory of Elusficity of an Anisotropic Elasric Body. Holden-Day, San Francisco, CA (1963). t;i N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Groningen (1953). [lOI R. J. Hartranft and G. C. Sih. Stresses induced in an infinite medium by the coupled diffusion of heat and moisture from a spherical hole. Engng Fracfure Mech. 14, 261-287 (1981) Reading, MA (1962). PII W. Nowacki. Thermoelasticity. Addison-Wesley, Singular Integral Equations. Noordhoff, Groningen (1953). [121 N. I. Muskhelishvili, u31 F. A. Sturla and J. R. Barber, Thermal stresses due to a plane crack in general anisotropic material. J. uppl. Mech. 55, 372-376 (1988). 1141 G. R. Irwin, Analysis of stresses and strains near the end of a crack transversing a plate. J. appl. Mech. 24, 361-364 (1957). [I51 G. C. Sih. On the singular character of thermal stresses near a crack tip. J. appl. Mech. 29, 238-254 (1962). (Received 27 April 1992)
Vol. 4.5, No.
0013-7944/93
6. pp. X31-841, 1993
$6.00 t- 0.00
IQ 1993Pergamon Press Ltd.
Printed in Great Britain.
HYGROTHERMAL STRESSES FOR A PLANE CRACK A GENERALLY ANISOTROPIC PLATE
IN
C. K. CHAO and R. C. CHANG Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan 106, R.O.C. Abstract-The solutions are presented for the hygrothermal stress field of a generally anisotropic plate under uniform heat flux and moisture concentration transfer obstructed by a hygrothermally insulated crack. For uncoupled diffusion of temperature and moisture, the solutions of both temperature and moisture are obtained directly from the Hilbert problem approach, and are treated as the particular solutions to a pair of nonhomogeneous partiat differential equations for an uncoupled hygrothermoeiastic system. The associated homogeneous solutions are expressed in terms of three stress functions based on the complex variable approach of Lekhnitskii. With some identities concerning the eigenvalues and eigenvectors. the general expressions of the stress and displacement fields can then be found in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are expressed in terms of the heat flow, moisture concentration, material geometry, elastic and hygrothermal anisotropy. The simultaneous existence of mode I, II and III fracture is found to be due to material inherent anisotropy. Special cases for isotropic and orthotropic materials are also discussed.
1. INTRODUCTION
and/or temperature environments is of degree of material degradation can be significant, particularly due to the stress concentration around a material discontinuity in anisotropic material. The stresses induced by heat flow and moisture concentration transfer which are known as hygrothermal stresses will become singular at the crack tips and may finally result in material failure through crack propagation. The general influence of moisture and/or temperature on the stresses and displacements in laminated composite materials has been determined by Shen and Springer [1] and Pipes et al. [2]. It is known that mechanical stiffness and strength are degraded when a laminate absorbs moisture, and recovery is incomplete after desorption. These environmental influences can interact so that the stress state of the material is dependent on both temperature and moisture in its surrounding. A theory of diffusion that incorporates the interaction between temperature and moisture was developed by Henry [3]. Phenomenological arguments leading to coupled equations governing the simultaneous diffusion of moisture and heat were studied by Hartranft and Sih [4,5]. The conditions of suddenly applied temperature and/or of moisture on the plate surface were both considered and the induced stresses tend to zero when the temperature and moisture concentration become uniform for a steady-state condition. The situation when the moisture diffusion coefficient is temperature-dependent was treated by Sih et al. [6] for symmetry boundary conditions that produce no bending. Coupling of moisture and heat was found to be inherent in the case of a transient condition and found to decrease as time increases to achieve a steady-state condition. The present investigation is concerned with the moisture, temperature and stress fields of a generally anisotropic plate containing a hygrothermally insulated crack. Under the assumption of the steady state, the temperature and moisture fields can be solved independently by the Hilbert problem fo~ulation. Starting with a pair of coupled nonhomogeneous partial differential equations for an uncoupled hygrothe~oelastic system, the general solutions are constructed as a linear combination of elasticity-eigenvectors, heat-eigenvector and moisture-eigenvector. Based on the complex variable approach outlined by Savin [7] and Lekhnitskii [S], the stress and displacement fields can be found in terms of three pairs of eigenvalues of the elasticity constants, a pair of eigenvalues of the heat conduction constants as well as a pair of eigenvalues of the moisture diffusion constants. Following Muskhelishvili [9], a plane crack problem can then be expressed in the form of the Hilbert problem which allows us to obtain the solution in an explicit form. The stress intensity factors, crack opening displacements and energy release rate are also obtained in THE DEGRADA~ON of material exposed to moisture considerable importance in composite structures. The
831
832
C. I(. CHAO
and K. C. CHANG
a closed form. The influence of heat flow, moisture concentration transfer. material geometry. elastic and hygrothermal anisotropy on those parameters can then be studied through this explicit form. Special attention is given to orthotropic and isotropic cases. The validity of the fully open crack assumption is also discussed. 2. BASIC
FORMULATION
For uncoupled hygrothermoelasticity. the generalized be expressed in contracted notation as [ 10. 1l] (,=S,,o,+r,I.+fi,C‘
Hooke’s
(i,,j=l.Z
law of anisotropic
material
. . . . . 6).
can
(1)
where repeated indices imply summation, and S,,. SI,, fi,, T and C denote the compliance tensor, thermal expansion coefficients, moisture expansion coefficients. temperature and moisture concentration, respectively. The moisture expansion coefficients 13, can be treated as analogous to the coefficients of thermal expansion, f~,. For two-dimensional problems, the components of displacements U, 23, IV, temperature and moisture concentration are independent of -_. The engineering strains, c,, in eq. (1) are defined in a Cartesian coordinate system by f, CJ =
where a comma For generalized
(2 =
“f,‘ll,,
A,,_ =
!, =
a',,
2,:
1, = =
7'*,
11',,
(q-f_=0
(6 =
I?(,, =
24,! +
Z'*,.
i2)
stands for differentiation. and the stresses, IT,, are defined in an analogous plane deformation. eqs (1) and (7) may reduce to u,,=.$,cr,+s7,T+~,C’
r.,,.=.%,0,+6J+fl~C
It’,) =‘Qa,+~J+fjjC Ii.* +
manner,
(3)
W,=:*T+r,$-9JfijSC 7'., ==3~,q+&T-+&C
and
-(&,o,+%T+L4C)ISv
0, =
9,,= s,,-
For a steady-state
s,3s,,/s,,
s,,IS,,
2, =
I,
pi =
P,- 13A:‘S,,
condition,
-
%I
the equations
Q,*, + T,,, = ()
of equilibrium
without
T,,_, +c+=O
In order to satisfy eqs (5) two stress functions the stress components as follows:
5,. =
y.,
T’,:
operators
=
---
as
to
Y ,I.
L,Y=L,2T+L,2C.
are defined
and related
t,, = -- F,y\
L,FfLZY=L,,Ti-L,IC
where the differential
as (5)
and V/(.U,JJ) are introduced
problem hygrothermoelastic 24. I‘. w: this gives
The governing equations for the uncoupled substituting eqs (6) into (3) and eliminating
L,F+
body forces can be written
5,,,\+?,,,,.=0.
Ffs,~)
c, = F_,,
(T, = F .I/
(4)
(i,.i # 3).
can then be obtained
by
Hygrothe~al
L4 =
833
stress field of an anisotropic plate
$ - 232,-ax3a4ay +(2sI,,-&)--
322
a4
a4 +3”
23
16ax+3
ax2 ay”
"a)J4
(8) These are a pair of nonhomogeneous partial differential equations for the uncoupled hygrothermoeiastic system. Equation (7) can be further decoupled into two individual equations as follows: (&A., - L:F’ = (.Vi, (W, The general homogeneous expressed as
- L:Y
=
@4&t
- &L,I )T + C&L,, - LAW -
L,L)T
+
CL4L
-
LdwP=
(9) (10)
solution to eq. (9) has been given by Lekhnitskii [8], which can be
Fh= k=li @%Zd + f’,GJ), where the overbar denotes the conjugates of complex variables, and zk
=
x+
(12)
pky,
where pk (AZ= I, 2,3) are the roots of the following characte~sti~ equation:
with
l,(P) =
-$?4
/4(p)
$2
+
(3%+ 346k- c3bi+ 356W-t 3td + $1~~.
(14)
The temperature and moisture concentration appearing in nonhomogeneous (IO) can be solved independently from the following equations:
terms of eqs (9) and
=
-2&p
+ (2& + S&i* - 2&G
k,~ T,, + (k,, + MT,xy + k,, T,),, = 0 d, I C,.xx+ (4, + 4dC,.xy + 4, C,,, = 0,
u 5)
where k, and d,, (i,j = 1,2) denote the thermal conduction coefficients and moisture diffusion coefficients, respectively. These are the equations governing the uncoupled diffusion of heat and moisture through a solid. The general solutions to eqs (15) can be expressed as T=O,(Z4)+04(24)
Z,=x-t-u4y
(16)
c=Q5(Z5)+@5(&)
Z,=xfCL5y,
(17)
where S4 and O5 are arbitrary hoiomorphic functions and p4 and ,us are the roots of the following characteristic equations: &A
+ 6%~+ ~Z,)LL~ + k,, = 0
d2211:+(d~2+dz,)~5+d,l=O.
(18) (19)
C. K. CHAO
834
and R. C. CHANG
A particular solution to ey. (9) can then be expressed as 120) with FJ(Z,) = ‘%@,**G)
F,(Z,) = A, oT*(z,),
(21)
where a superscript * denotes integration with respect to its argument and
(27)
with
lr2(jLL=
-/72+ &ji - p,jG.
(23)
Note that the governing equation (9) resembles the force vibration of a discrete system in that the heat-eigenvalue p4 and moisture-eigenvalue y, play the role of forcing frequency while the elasticity-eigenvalues pk (k = 1,2,3) may be identified with the natural frequencies. In the present approach, the heat-eigenvalue or moisture-eigenvalue cannot be identical to the elasticity-eig~nvatues as considered for the resonance case. The complete solution to eq. (9) can then be obtained as F=F,+F,=
i {Fh(Zk)+Fk;.(Zk)). x-i
Similarly, the general solution to eq. (10) can be put in the form 5
Y =
-..~--
1 jqJ;.(Zk) +
k .: I
where the prime stands for differentiation
vkF;(Zk)},
(25)
with respect to its argument and
Using the notations concerning the eigenvalues and corresponding eigenvectors, similar to the expression of Lekhnitskii [S], the stress and displacement fields can be expressed in the form
IPk4k#%(Zk) + PkYlkSt)kvk)r %=ki, T,, =
e--g,
{Pk$k(Zk) +
(30)
-.---
PkQ)kGZk)l.
(31)
835
Hygrothermal stress field of an anisotropic plate
(32)
0’ = k=l
kk4k(Zk)+ qk4kvk)l
(33)
(34) where c$~=F;
(35)
and
where 6, stands for the delta function, which will be 1 as i = j or 0 as i #j, with the usual definition. Note that the genera1 forms appearing in eqs (27)--(34) cannot be specialized directly into isotropic material due to the repeated eigenvalues of the elastic constants.
3. HY GROTHERMAL
STRESS
Consider an infinite anisotropic plate. The uniform heat and moisture flow in the direction of the positive y-axis is disturbed by the presence of a hygrothermally insulated crack lying along y = 0, 1x1
+ ~~k**)~~(z~) + (kz, + P&P~(Z&
(37)
On the crack surface, setting y = 0 and /x/c a, this expression takes on the form K&z,+ cLJW@;(x) + (k2, +
ji&d~Wl+
I- k, + wW@;(x) + &, + ibW@;(X)l- = -(h: + h,,) (38)
(4 Fig. I. The hygrothermoeiastic
plane crack.
09
Fig. 2. Supposition of heat and moisture flux in an unflawed plate and a corrective problem.
836
C. K. CHAO and R. C. CHANG
[(k,, +p~k,~)O~(.u)-
(k,,
c ,&k,,)@j(x)]’
-[(k,, +pLqk?&L3;(.Y)-_(k~,+p,k,2)0;(.\-)]
= -(Ir,’
-II,
).
fZ9)
where the superscripts +, - denote the upper and lower sides of the crack surtace, respectively. The problem of finding a sectionally llolomorphic function in the whole plant subject to the boundary conditions is known as the Hilbert problem [9. 121. For the present problem the solution can be obtained as [(k,, + p,k,z)O;(Z)
+ (k,, -i- ji4k22)@;(Z)]
where 6, and 0, are any holomorphic functions. For the sake of convenience, the soluuon is represented as the sum of a uniform heat and moisture flux in an unflawed plate (Fig. ?a) and a corrective solution (Fig. 2b). The constant and negative heat flux h^is prescribed on the crack surface (Fig. 2b), i.e. 11; = It ,, = -- 1; (42) and, since the heat flux vanishes at infinity. it gives I),(Z) = f&(Z) = 0.
(4i)
Substituting eqs (42) and (43) into eqs (40) and ( 41) yields ^ (44) The solution associated with the uniform heat flux disturbed by the presence of an insulated crack can be obtained by using the technique of superposition, i.e.
Integrating eq. (45) with respect to Z, the temperature
function is found as
The integrating constant is neglected here. which can be treated as a reference temperature. and the temperature will become zero along the x-axis excluding the crack surface. Substituting eq. (46) into (16), the full field solution of temperature can be established. Analogous to the preceding processes. the moisture concentration function can also be expressed as 147)
(d,, -I-$u5d~:)05(z) = _t [\, (Z? -- LP)],
wherepdenotes the constant moisture flux in the direction of the positive j,-axis. Substituting eqs (46) and (47) into (21) and using eq. (35). we have 4,(Z) = b,Q(Z
- ff?
C&(Z) = cJV:(Z’
- u?),
(48)
where -14, -A, h,=-....--------26, + iu,h-,21 (‘<= ~g-T-,&)
.
(49)
In order to find the hygrothermal stresses induced by the presence of temperature and moisture concentration, the traction boundary conditions on the crack surface are considered. Let the tractions on the crack surface be C>.= ;,
?,,. = t,
7,: = i,?
r50j
837
Hygrothermal stress field of an anisotropic plate
and knowing the following properties: lim (bk(Zk) = #t(x)
Jl% $k(Zk) = &(x)
J-t,+
the
(51)
IN< a,
traction boundary conditions on the crack surface can be written as EY&~/c(~)+ Yi!~(Pk(~)l+ +
[Yik4kCX)
+
(521
Yik&kCx)l= ;i’ + iT
lVik4kCx) - Yik#kCX)l+ - i3ik+kCX) - Yik#k(x)l-= ;P - ‘is
(53)
l
wherei=1,2,3,k=1,2
,...,
5, /xl
Y2k =
-pk
Y3k =
(54)
-qk*
The solutions associated with eqs (52) and (53) can be obtained from the Hilbert problem in the form [9, 121 1 Yi*QjklZ)
+
Yj~~~(z~
=
2niJ(Z2
n (t?’ + t,-),,/(f2 - a*) dt + - a’) s -a
Y&Jk(Z) - Yik#AZ) =&
s
u (i: - t^,-)dt _ t_ z +
a
P,(Z) J(Z2 _ a*)
t-z
(55)
Q@L
where P,(Z) and Qi(Z) are holomorphic functions, which can be expressed as polynomials degree rz. Since the crack surfaces are traction free, then i#? E ;; = 0.
of (57)
For the condition of zero stress at infinity, only the constant terms of P,(Z) exist, i.e. P,(Z) = 2e,
Q(Z) = 0.
(58)
With the aid of eqs (57) and (58), eqs (55) and (54) further reduce to yi&k(Z) = ,/(z:’
a2)
(i = 1,233, k = 132,. . . ,5),
where ei are all real constants due to the conjugate condition. There are a total of five stress functions (Pk (k = 1,2, . . . ,5) appearing in eq. (59), but only three functions & (k = 1,2,3) are unknown since 4, and (bs have been given in eq. (48). Hence, the stress functions @j0’ = I, 2,3) can be expressed as cb,(Z) =
J;y:a2) + (h,R + CjP)J(Z”
- a2)
(i,j =L:1,2,3),
(60)
with l$ = -y$
y+Jx$
tW
c,= -yji’yiScSt where the constants b4 and es are shown in eq. (49). It is (i = 1,2,3) in eq. (13) cannot be identical, the determinant be singular. The problem now reduces to the determination required to satisfy the single-valuedness of displacements, ul, = 0 = =
(62) noted that the elasticity-eigenvalues pi of yij exists and, therefore, y,; i will not of the unknown constants ei which are i.e.
@‘6&k) dzk
sa
[u’+(x)
-a
- u’-(x)] dx
= =
efh(fftt-4 -PA~xY
J -0
-h&(xX) -~d&)l-)
dx,
(63)
838
C. K. CHAO and R. C. CHANG
where c is any simple closed contour surroundng x(Z) = ,/(Z’ - a’) has the following property:
the
crack.
The
Piemelj
function
Substituting eq. (60) into (63) and using (64), we have
where Im{ ] denotes imaginary part. Note that both the constants h, and cl in eq. (6.5) are equal to zero as a result of mutual independence between temperature and moisture concentration. Simifarly, the other two constraint conditions associated with the displacements 13and w can be obtained as (66) (67) Solving for eqs (65)-(67), we have (68) where
r,=p,,%f,,
A,=p,,‘H,
(69)
with p,, = Im{p,y,;’ ), p2, = Im{y,)l,; ’ ), p3, = Im{t,;l,, ‘) H, = Im{p,b, i M,=Im{p,c,j
170)
H, = Imjq,b,)
H, = Im(t,b,)
(71)
h4?=Im{qkci:j
Mi=Im{r,c,],
(72)
where i, j = 1,2,3 and k = 1.2, . _. , 5. It is shown that the coefficients A,, F,. pli, Hj and Mj are all real constants and dependent on material properties. The general solutions for a hygrothermal stress field can then be obtained by substituting eqs (48) and (60) into eqs (27)~34). 4. RESULTS
AND DISCUSSION
The stress functions appearing in eq. (60) contain a singular term (Z’ - a’) -’ ’ which results in singular behavior of the stresses near the crack tip. With the usual definition, the stress intensity factors K,, K, and K3 can be obtained as K, = +jYA*‘t,il,h^
+ rJ,
(i = I, 2. 3).
(73)
where + and - stand for the right and left sides of the crack tip, respectively. It is seen that the stress intensity factors are proportional to the powers of i of the crack length and linearly proportional to the heat and moisture flux. It is not surprising that the influence of heat flux on the stress intensity factors through the material constants Ai is exactly the same as that of moisture flux through the moisture constants r, since the process of heat transfer is fundamentally equivalent to that of moisture transfer. Similarly, by integrating eqs (32)-(34) and using the constraint conditions expressed in eqs (65)-(67), the crack opening displacements are obtained by setting y = 0, Ix/ < u. i.e. ADi=
-2s,(cza
- .x*)(f&h” + M,,jl),
(741
839
Hygrothermal stress field of an anisotropic plate
where AD, = u+(x) - u-(x), and AD,, AD, stand for the crack opening displacements a, w, respectively, on the crack surface. It can be seen that ADZ will always be negative in either -a < x -C0 or 0
‘?AD,(<
-A&&)+AD&
-A+,(t)+AD&
-Au)r_&)ld5
= -~[(n,H,+n,H,+n,n,)h^2+(T,M,+T,M,+~~~,))2],
(75)
where the integration variable t represents the distance ahead of the crack tip. It is understood that the energy release rate is proportional to the square of heat flux as well as moisture flux. For orthotropic materials, only the material constants S,, cli, pi (i,j = 1,2,3) and S,, &, &,, exist in eq. (1) and all the coefficients klz, kzl , d,,,, d2, in eq. (15) vanish. The governing equation (7) becomes L,F = L,,T 4 L,zC,
(76)
and the other equation Lz Y = 0 is automatically satisfied in the sense that all the shear stress components in the z-direction vanish. Therefore, the system degenerates into an in-plane problem, and the characteristic equation (13) becomes Id(P) = 0,
(77)
which gives only two pairs of eigenvalues. The eigenvalues puq,ps in eqs (18) (19) and eigenvalues pr, p2 in eq. (77) becomes pure imagina~es. Let PI = i5, LIZ= X2 P4= ii4 ius= iis, where ii (i = 1,2,4,5)
(78)
are all real constants, and
i, + i* = J([2% ia
=
&h
+ $56+ 2(5”,,s?2Y’211%)it iz
1 lk22)
is
=
&d,,
=
J(~22/&
1
(79)
b&2>.
Equations (70), (71) and (72) now reduce to
b-‘I =
Substituting eqs (80)+82) into eqs (69) and (73) the stress intensity factors become K, =o
Kz= ?+$:_ ,(H,l;+Mf) I % (83) II 1 2 where - and + stand for the right and left sides of the crack tip, respectively. This implies that the mode I stress intensity factor K[ would not exist for orthotropic material. The crack opening displacements are also obtained as AD, = -2x&a2
- x’)(HI ii + A&f)
AD, = 0.
(84)
840 It is indicated displacement in the form
C. K. CHAO and R. C. CHANG that there is no tendency for the crack to open or close. However, a relative tangential AD, may exist between the crack faces. Similarly. the energy release rate can be found
For isotropic becomes
materials,
the material
i,+&=’
properties
;,I?=1
are independent
of the direction
and cq. (79)
ii,= CT= I.
(86)
However, as mentioned above, the eigenvalues cannot be identical in the type of genera1 sofution shown in eq. (24). In order to proceed with our discussion, cl and is will be made small perturbations of unity, and then eqs (81) and (X2) become H, = 3/2/i
M, = fli2d.
(87)
It is understood that C?= 01”, . /? = b,, k = k,, and d = L& for isotropic material and Oz= ( 1 + 1’1%. [ = (1 + v)j3, g,,, = (1 - v’)/E for plane strain conditions where k, d E, I’, a and /3 are the heat conductivity, moisture diffusivity, Young’s modulus, Poisson’s ratio, thermal expansion coefficient and moisture expansion coefficient, respectively. The stress intensity factors can then be obtained as
where - and f refer to the right and left sides of the crack tip. respectively, opening displacement can be obtained as
and the energy
Similarly,
the crack
release rate is given by
For the special case_? = 0, the results given in eqs (&S--(90) are shown to be identical given in the literature [ 151.
to the soiutions
5. CONCLUSIONS The general solutions for plane anisotropic hygrothermoelasticity are derived and expressed in an explicit form based on the Hilbert problem formulation. For uncoupled diffusion of heat and moisture through a solid, both temperature and moisture concentration are solved independently from the complex variable approach, and are treated as the particular solutions to a pair of‘ nonhomogeneous partial differential equations for an uncoupled hygrothermoeiastic system. The complete solutions are constructed as a linear co~nbi~lation of the elasticity-ei~envectors, the heat-eigenvector and the moisture-eigenvector. Some new identities concerning the eigenvaiues and eigenvectors are developed in this paper, which are useful in replacing the complex eigenvectors by the basic material constants. The stress intensity factors. crack opening displacement and the strain energy release rate are obtained in a closed form. The simultaneous existence of mode I, II and III fracture is found to be present in this paper due to material inherent anisotropy. It is noted that the influence of moisture flux on these parameters is exactly the same as that of heat flux due to the fact that the process of moisture transfer is fundamentally equivalent to that of heat transfer as stated earlier. Special attention is given to the cases for orthotropic and isotropic materials. It is seen that both the mode I stress intensity factor K, and crack opening displacement AD? are found to vanish and consequently the situation with a partial contact crack would not occur for degenerate
Hygrothermal
stress field of an anisotropic
plate
841
materials. Comparison with the solutions for a special case shows that the solution presented here is exact, simple and general. REFERENCES HI C. H. Shen and G. S. Springer, Moisture absorption and desorption of composite material. I. compos. Maw. lo,2220 (1976). PI R. B. Pipes, J. R. Vinson and T. W. Chou. On the hygrothermal response of laminate systems. I. compos. Mater. 10, 1299148 (1976). [31 P. S. H. Henry, Diffusion in absorbing media. Proc. R. Sot. Land. A171, 215-241 (1939). [41 R. J. Hartranft and G. C. Sih, The influence of the Soret and Dufour effects on the diffusion of heat and moisture in solids. Inr. .I. Engng Sci. 18, 1375-1388 (1980). PI R. J. Hartranft and G. C. Sih, The influence of coupled diffusion of heat and moisture on the state of stress in a plate. J. Mech. Compos. Muler. USSR 1, 5361 (1980). stresses in composites: coupling of moisture and heat 161 G. C. Sih, M. T. Shih and S. C. Chou, Transient hygrothermal with temperature varying diffusivity. Inf. J. Engng Sci. 18, 1942 (1980). [71 G. N. Savin, Stress Concentration Around Holes. Pergamon, New York (1961). S. G. Lekhnitskii, Theory of Elusficity of an Anisotropic Elasric Body. Holden-Day, San Francisco, CA (1963). t;i N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Groningen (1953). [lOI R. J. Hartranft and G. C. Sih. Stresses induced in an infinite medium by the coupled diffusion of heat and moisture from a spherical hole. Engng Fracfure Mech. 14, 261-287 (1981) Reading, MA (1962). PII W. Nowacki. Thermoelasticity. Addison-Wesley, Singular Integral Equations. Noordhoff, Groningen (1953). [121 N. I. Muskhelishvili, u31 F. A. Sturla and J. R. Barber, Thermal stresses due to a plane crack in general anisotropic material. J. uppl. Mech. 55, 372-376 (1988). 1141 G. R. Irwin, Analysis of stresses and strains near the end of a crack transversing a plate. J. appl. Mech. 24, 361-364 (1957). [I51 G. C. Sih. On the singular character of thermal stresses near a crack tip. J. appl. Mech. 29, 238-254 (1962). (Received 27 April 1992)