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An approximate three-dimensional theory of layered plates containing through thickness cracks
AN APPRQ~ATE ~EE-~IME~SrG~AL LAY’EltEI.3 PLATES ~G~~~G THROUGH
THEORY OF THKXNESS CRACKS
R. BADALIANCE and G. C. SIH
2
material constants, etc., however, must be determined from a physic~y plausible ~s~ption that the plane strain condition[4,6] is satisfkd in the interior region of each layer of the laminate. The solution ceases to be valid in regions close to the free surfaces and interfaces. Boundary layers are introduced to exclude the zones in which the plane strain condition prevails. This is analogous to the boundary layer in fluid rnec~~s where potentiat flow is separated from a viscous layer near the wall. FoUowin8 the de&&ion of the stress-intensity factor for three-dimensional crack problems[6,7], numerical results are obtained for various elastic constants and geometrical parameters of a cracked laminate. The results of this study indicate that the proposed method of solution can be extended to laminates with any number of anisotropic layers sim&ting those which are used in existing structures. While the failure problem of delamination and/or through laminar crackin is not answered here, this work of stress analysis is a necessary step toward the development of a fracture theory of huninate cracking. APPROXIMATE THRIWDQUENSIONALTHEORY FOR MULTI-LAYERRDLAMINATE!3 The use of va&ional principles, e.&, the principle of minimum compkmentary potential
energy, to develop approximate sohttions is a standard technique. An approximate form is assumed for the v&ables, in this case the stress components, and it is substituted into the functional. Minimizationof that functional with respect to admissiik varktkns of the assumed functions (stress comments} kads to the choke for those functious which best approximate the exact solution in an avenged sense. Consider a laminated composite plate of uniform thickness made of n layers, where each layer can have dif&ent thickness and material proper&s. It is assumed that the layers are bonded together perfectly, i.e., there exists no interfaciai Baws.A k&ate of thisnatm, whsn subjected to hr-pkne stretching, willdeform out of its pianef8J.This effect is elhrdMed when the laminate is a symmetrk one, Le., the layers are stacked in even ttumbers such that the system possesses mat&al symmetry with respect to the middlepkne of the laminate as shown in Fit. 1.
Loteml
boundary L
h
h,
Cross section
of a symmetric
laminate
Fig. 1. Geometry of IaminaM plate.
3
An approximatethree-dimensionaltheoryof layeredplates
The procedure to be employed for laminated plates is to assume that each of the stress components in each layer can be approximated as a product of a function of the out-of-plane variable z multiplied by a function of the in-plane variables x and y, i.e.,
a:’ = fl’(z)g:P’(x, y) no sum on i, j, p
(1)
The purpose of this assumed form for the stress components is to reduce the governing equations from dependence on three independent variablesto dependence on two variables for which many standard solution techniques exist. Substitution of the assumed form for the components into the equilibrium equations with the additional condition that the equations separate leads to the more specific assumption uza-)= fp(r)Z,‘“‘(x, y)
[ei’, $I=
- fXzNZ%,
(2)
Y1,zy(%, Y)I
= f~t)[S,“‘(X, y), Sr(p’(x, Y1,Ek [u,‘p’,UT’, 71”,‘1
Y)I
The conditions of traction continuity across material interfaces are satisfied by taking rz*‘. &@‘,zpr = [Z,, z,, z,], Mr) = fp+G) and f;(r) = fL+t(z) at the value of z corresponding to the interface between layers p and p + 1. It is reasonable to substitute this assumed form for the stress field into the complementary potential energy functional and minimizethat functional with respect to Z,, Z,, Z, SI’P’,Sr’p’,and 7’$: subject to the conditions of equiliirium in order to develop an approximate solution for lam&r eomposites. It is, however, advantageous to examine the resulting form of the strain components so as to choose the approximate solution which comes closest to satisfying the displacement continuity conditions across the material interfaces. For example, the strain component G for isotropic and homogeneous materials is given by
G=El [ux- vtuy+ a*)1=;
v”(z)(Sx - vS,)- vf(z)Zl
(3)
In order for G and thus u, to be continuous it is necessary, though not sufficient,that SX(“) and ,!$,‘,‘,’ be the same for all layers. Examination of the shearing strain component yXryields a similar condition for CY’.Thus the assumed form for the stress field which will be adopted is: uz@’= fw(zm&
Y)
[T$‘, $)I = -f&‘(zwk
Y)94(&
[U.‘p’,oy@‘,7ryq=f);&)[Sx(x, Yh w,
Y)l
(4)
YIPTxrk Y)l
where &(z) and f:(z) must match their respective values for the adjacent layers. For notational purposes it is convenient to rewrite the assumed form for the stress components in an equivalent manner as:
@‘=d(z)Z*(x, Y)
02
R BAWLIANCB and 0. C. SXH
ms Y)l =rf”(wxlx~ YX 4(x, Y), ws
[Tit’, r(P.'l = -f(~Hz(x, [fr;“‘, a YW’I cl
(5)
Y),
Y)l
where f(z) must be continuous and have a continuous first derivative across the material interfaces. Recall, the complementary potential energy (4) of an isotropic, homogeneous body is de&ted as the strahr energy of that body minus the work done on the portion of the body surface (5) over which dispiacements are speci&d, i.e.,
where the strain energy density is
SubstWion of this assumed form for the stress components, equation (S), into the above complementarypotential energy functional and the equilibriumequations for a laminarplate with edge loading and traction-free surfaces (f = f = 0 on surfaces) yields
(8)
(9) where
and amp’, a*‘, and a*’ are the normal, tangential, and transverse components of the displacements prescribed on the edge (S,“‘) of layer p, respectively (Fig. 1). Lagrange multipliers (&) are employed to insure satisfaction of the equilibrium equations, and the functional
is formed.
An
approximate three-dimensional theof3foflayeredPlates
5
me variation of II with respect to S,, S,, T,,Z,,& and~4th thecodn~ty~~Wthm enforced on f(z) and f’(r) results in
where
(131
The divergencetheoremand the integration-by-parts techniqueareappliedto the va equation 6II = 0 to obtain: HI=
I
[(I, sxf r2sy + r4zi + A Lx )ssx
A
+cr,sy+12sxf
Lzt + A2,YmsY
+(~(~~-I~)T,+AI,,+A,)ST, +(/Z
+A~+As,)sz,
+ (I& + A2 +
(14)
As,, W,
+(I.(& +Sy++Wz +~d~l~
dy
where A”= A,n, +Azny; A. = -AI& + A2ny and (n,, n,) are the directioncosines of the normalto the plate edge (Fig. l),
R. BADAUANCE and 0. C. SM
6 Thei gemming
eqtm?ims
variables can be det~ned
ami
lxmbry
conditions
art
new
qpmn?,
i.e.,
the
in-plane
from 0
r,S*+r2S~+r4z~fhl.x= r‘s,
+ IZS,
+ Lz
+ AZ.,
= 0
2(1,-I*)T,,+A,.,+A2.x
I,(&
z,
=o
bZ
fAlfA3,
=o
I,&
+ A2 + A3.y
= 0
+Sy)+CZ
=
$A3=0
zx..% + &3_,
z,
=
.%a
f
TL,.Y
2,
=
Tx7.x+ Sw
with A. = - II,, A, = - il.,
aud A3=i -P, on C.. In other words, the boundary conditions associated with the in-plane variables S,, S,, . . . can be either of the tractiou type or averaged dis~~ace~nts. On the contour e aioug the piate edge, one can prescribe either S, = .%
or A, = - &
T,=F=
or A,=-&
z=z
or
A39:
-a,
The traction free surface conditions are reflected by the requirement that f(t) and f’(z) must be zero on the plate surfaces. After some nontrivial manipulations the set of governing differential equations (15) can be sepa8W&into the form:
z, - a6v2n
= &
a,V‘A3
(a
,V2A3
+
a2A3)
+ a4Q2A3 + asA3 = 0
where extraneous solutions are eliminated by the condition
aZ,f % = - a7V2A,ax ay
where
aeA3
An approximate threedimensional theory of layered plates
1 [Zs(Z, +
z2z3 -- I, I,, a2 = (zl+zz) (z,*-z22) 2(Z,-b) (z1+z2) 12) + z3z4 - 2z47
I
a, = Z,’- Z,Zs, a5 = - (11’ - I,‘>,
as
=
[Z5(ZI
2Z4(Z, - 12)+ Z~ZS
a4 = -
+ :;:1~I$3
z3 a6 = 2(11-
-
2E)l
The other in-plane variables can be expressed in terms of AS,Z, and & as
A2=
-
z3zY
+Qd
(18) -
z2~+[z,b-z4(z,-z2))~
[Ib
+ Z,(ZI
-
Z2)15}
-[&+I1((k~2)+]
The set of equations (la), (18)and the condition (17)form together with the boundary conditions a complete system of equations for an approximate three-dimensionaltheory of layered plates. A TEROUGE CRACK IN A LAMINATEDPLATE For this analysis, the crack length is taken as small in comparison with the in-plane dimensions of the plate. The stress 6eld in the vicinity of the crack is thus independent of local boundary effects and the plate can be modeled as i&rite in the two, in-plane directions. This assumption simplifiesthe analysis significantlywhile maintain@ the essential characteristics of the cracked laminar plate problem. In particular, the problem of a laminar plate (symmetric about its mid-plane) containing a crack in a uniaxialfar-field,stress state will be considered (Fig. 2). The crack is directed normal to the loading direction, thus modeling the most damaqing type of flaw. This problem can be expressed as the superposition of two auxiliaryproblems. The &st auxiliaryproblem is that of an untracked plate subjected to the same loading conditions as the actual cracked plate. For the second auxiliary probkm, the cracked plate configuration is considered with no boundary loads. Instead, the crack faces are loaded with tractions of equal magnitude but opposite sense to those found at the crack location in the first problem. Thus the superposition of the two problems yields the cracked plate with traction free crack faces and remote loadinq as orighmhy described. The first of the auxiliary problems, an untracked plate in a far, uniform stress field, has nonsingular stress components and as such does not contribute to the crack-tip singuhnity.
R. BADALIANCEand G. C. SIH
8
Fii 2. Three-layer conlpo6itcplatecontainins a throustl crack.
Therefore, the stress Beld in the vicinity of the crack tips for the case of crack face loadinq is identical to that for the remote loading situation of the actual cracked plate. It is mathematicahy advantaqeous to consider the crack face loading problem because it leads to stress fields which die-out far from the crack and can thus be treated by integral transform techniques[9]. The probkm of a composite plate, made up of homogeneous isotropic layers distriited symmetrically about its mid-plane, containing a through-the&ickness crack will be considered for the second a&Gary problem mentioned before. The crack is located in the y = 0 plane and directed along the x-axis as shown in Fig. 2. Consider the region s y, 2 >O. The boundary and symmetry conditions are: on
Y “0,
%Y “?~=0;Q,~p(52),0sX~E4;~=0,X>(I
where p(z, z) is the pressure applied on the crack face which is symmetric in both of its arquments x and z.
and on
z-o,TxI=c711~o;u*=o
The quivalent boundary conditions for the approximate theory are: T,=Z,=O: s,=p(x), &=O,
on
Osxsa
x>a
x ~0, T, =Z=O;A,=O
isrequired to be symmetric in z to model the symmetry conditions on the z = 0 plane. Further f(z) and f’(z) are zero on the plate surface z = f t/2. The governing equations (16)are to besolved subject to the boundary conditions mentioned above. Further, the functions %, S, Tm Z,, Z,, and 2 must die-out at large distances from the crack.
and f(z)
In-plane oariations of the stress field As indicated earlier, the solution procedure will be based on applyingthe Fourier transform to the x variableto determine the nine in-plane variables S;, S,, . . . It is convenient to recognize the
An approximate threedimensional
theory of faycred plates
9
symmetry properties of these variables in advance, thus reducing the Fourier transform to sine and cosine transforms for the odd and even functions, respectively. It is observed that the functions $4,8, Z, Z, AZ,and A3are even in x while 7’,, X, andA8are odd in x. Thereforedefine S,“(s, y) =
~mSx(x, y) cos(sx) dx, etc.
and T:~s,y)=lmbbyfsin(sx)dx,et~. 0 Application of Fourier sine and cosine transforms to the governing set of equations (16)and the condition (17) leads to the following set of equations: tin,’ $A~=+ (u3s4(I,dyl+Gl4 - 2a3s ),Y, 2
ad2 + udh3c = 0
(19)
providedthat
Tlkegeneralsolutionto equations(19)which s&i&esthe asymptoticconditionsat i&ity can bewrittenintheform AS=@, y) = 2RelM~)e-~l
where
and
Z(S,
y) = f B~s)c”~ + RelP&)e”7
r(s,
y) = B(s)eWqyf ReMs)e-‘*I
(21)
10
3. ~AlhLIANCE and G. C. SIH
The additional transformed functions of the in-plane variables can be expressed in terms of AS’, Z.‘, and Z throu& equations (18).In particular, the functions TZ,, S,", and A: are obtained to study the boundary conditions:
The general solution is now substituted into the boundary conditions (transformed appropriately). The condition Z fs, 0) = 0 yields
Whiie the condition T$(s, 0)= 0 relates the real and imaginary parts of A,(s) as
It is convenient to express this result symbolically as Im[A&)l = /3,(~)RefA,(s)Jwhere
(24) _rr
s2_& [
aod
206(
((12-k,a,)(l+askl)-a,a,k3 (I+ uskd’+ (a&d2
An approximate t~~n~~
theory oflayrod plates
11
Tbe set of mixed boundzwyconditions
remain to be satisfied. These conditions can be expressed in terms of the transformed functkms as
Thus the mixed ~~~
conditions can be expressed in the form of a set of dual in
equathns as f
9) d g(s)R(s) cos fsx) ds = -fp(x),x
f0
4.a R(s) cos (sx) ds = 0, x L a
where
This set of dual integral equations can be reduced to a single integral equation by the introduction of an aw&ary function, w(x), de&d by R(s) cos (sx) ds, XC a
R. BADALKANCEand G. C. SIH
12
equation (28)implies that R(s) can be written as R(s) = fI; w(x) cos (SIC) dx far ah x, where w(x) must satisfy
Then
L-k4r
w(x)cos(sx)dx
I
cos(sx)dx =
-+w
(30)
This equation (30)contains the crack tip singularity and corresponding care must be taken in the solution procedure. Exact three-dimensional solutions for symmetric crack probIems[!JJ], indicate that on ah interior points in the immediate vicinity of the leading crack edge the local stress field must be of the form:
02”+frsi*(B)sin r248)I k(Z)0- -&in(,)sin ;g G=Gj=cos2[ 2 ( )I a=$&
[
cos
(31)
where (I; 8) are local p&r coordinates in the plane normal to the crack edge (Fii 2). The auxiliary functi~ w(x) corresponds to h&O) and, as such, is expected to be locahy Pomona to the square root of the distance from the crack tip. This ~~~a~n is used to introducea secondauxiliaryfunction,4(t), ia an attemptto removethe inherentsingubrityfrom qua&n (2%).IMae Jr(t)by the integrai equation (4) (32) Subs~
into equation (30) yiekis, after some s~p~c~ons, fdJtfsin(srfdsldt1
-lp(x)dx,x
(33)
whereJois the zero order Bessel function of the first kind. If the expectations mentioned above are correct, equation (33)will be a nonsinguiar integral equation for g(t) which is amenable to numerical solution procedures.? The function g(s) can be written in the form
go =c +8*(s) s
where g*(s) = ofs-‘) as -+mand C is a constant given in the Appendix. Substi~tion of the above into equation(33)and use of the known safutian to Abei’sequation leads to the Fredholm integral equation (see references [IO, 11, i2l for details) iA
posfuicbccksof tbc IIOIU&~ resuttsdo indaodindicata that e(f) is a weB-bcbavcdfunction.
13
(35) This cqW.ion can be reduced to standard form by the following mmdimensionaiiitions q=&,
t==aT,
x=a&
s=:
(36) where
The accuracy of the numerical solution procedure to be employed is improved by separatkg s*(s) as
where
Rfs)=O(s”) 8!3 s 4~53,
'RR
constants H and IV2are given in the App&ix. Hence, the
kernel K(Q I) can be rewritten as
where Io and Ro are the zero-order mod&d Bessel functions of the 5rst and second kind, respectively. The case of crack face loading which does not vary with x is considered, i.e., p(x) = I? The kernel is now evaluated numerically and the integr8l equation is ~~~a~ by a set of algebraic equations via Simpson’s ale. Numerical solutions for NT) are thereby obtained. The origin8l unknown function, R(s), is obtained by substitution of equation (32)into (29). Carrying out the indicated integrations, R(s) can be written as
The unknown transformed functions S,“, S, etc., can be determined directly from R(s) by back §~bsti~~on.
14
R. &tLMLlANCti
and 0. C. SIH
It is expected that damage resulting from crack propagation will originate in the most highly stressed re8ion, the vicinity of the crack tip. Thus the main objective is to obtain the best approximate sob&ionfor the crack edge stress field. Referring to equation (38)it can be shown that the s~~~~ at the crack tip is associated with +(I) and that int@ expression in this equation has no contribution to it. Because an asymptotic description of the crack tip field is sought, the integmi expression in equation (38) is dropped and the functions S,‘, S,‘, etc., are expressed in terms of the simplifiedequation for R(s). These transformed functions are inverted asymptoticallyfor large s. Adopting polar coordinates r, rI, r2,and 6, 6, & as shown in Fig. 2, it is found that SAX,YP = -~PY(l)C{~sin[~tB~+~$]
-zcos [e*-;(e,+e2$+0(1) S,(x,y)=
+~PY(l)C{~sin[~(BI+8$]
+os[ 80-;(e,+e2~]}+ow ZYCGY)"" -~PY(I)c[~cos[~~e1+Bd]}+0(1~
Zzk Yv = -.+
PY(l)[e’(klb;;
x +3s l
[
k3b1)]
e~-~~e,+~~)]]+ql)
z(x, y)= a~Y(l)~([?j$k~+ X
b&r-?)]
ds
II
X
I0
y.h(us) cos (sx) eety ds
(391
.Jote that Z,(x, y) and Z,.(x,y) are nonsingular. Further, approaching the crack tip (u, 0), i.e., taking the limit r + u, 8 + 0, r2+ 24 & +O, the near tip fields can be written as
th. bz, C, et, ma k,. k, are given in the Appendix.
An approximatethrce-dimcnxior4theoryof layeredplates u,
-
,y$g
[co, (2) +; sin(&I sin(g]
I5
+W W)
T_ 5~~[~*n(B,)cos(~)]+o(1~ a* = KO&M 1/Z (2rY [cos (93
+OW
K =
ae,C*(l)
where (41)
6(Z) = W(z) and
Note that the stress field in the vicinity of the crack tip, as given iu equations @NJ), contains the usual square root siz@arity in the plane normal to the crack front, where the dcpbndcaoe of these stress components OQthe in-plane variables (rr , e,) is in complete agreement with the exact three-dimensional solution for a crack in e elastic medium[5J.Also, it is interesting to no% that thetransvareeshcatstregsts~,andr,~~bythis~~thcoayQ,rmate~aa in-plaae [email protected] From quations (40)it can be seen that the stress intensity fa&rs cga be chrractaizcd as
k*(z)= KPfwa 1/x
(421 H(z) = (b&s + b&d+
KPftz)dR
These stress intensity factors, unlike those used in the curreut theory of fracturepI vary along the crack front in the zdirection through the functions f(z) and p(z). It can be seen from boundary condition that the load distribution over the crack face corresponding to this solution is a&) = W(z). Hence, the first of equations (42) can be writteu as
Thus K can be COnSidered
to be a measure of the idhenCe of ehdc constant8 and geometrical parameters of the cracked laminate on the intensity of the crack edge stress field. The stress stern parameter 15:is directly dependent upon q(l). Where Y(1) is the limit of !P(E)as the dimensionless variable 5 (normalizedagainst the half crack iength) approaches unity, i.e., the crack tip.
tT, and T, will bc singularin the presence of torsionaland transverseshear loads.
16
R. BADAISANCEand G. C. SlH
~OU~E-~~~~ VARIATIONOF THE CR.ACIC~TIF t5lBESS FIELD Exact ~ymptotic analysisI descriid earlier indicates that at all interior points near the leading edge of the crack, the plate is in a state of plane strain, i.e., or = ~(a, + a,). Within each layer, the plane strain condition can be employed to obtain j(z) in the vicinity of the crack edge. Supposition of the requirements that the approximate solution should satisfy the above equation leads to the following differential equation for j,(z):
fxz) + p:fi (2) = 0
(44)
where p: is equal to the asymptotic value of the function -Zzlvc(Sx + S,) at the crack tip, i.e.,
It can be shown in general that the solutions of equation (44)do not possess sufacient number of free parameters to satisfy both of the free surface conditions (f = j’ = 0). Such an analytic solution, however, cannot be expected to hold throughout the thickness of the plate. This is mainly because the experimental observations of 6&e thickness plates show that there exists a layer of ma&al near the plate surface which behaves very differently from that in the bulk. Spectically, the fracture testing of through cracks in finite thickness plates has shown evidence f~twosuriace~yenofthcmaUriaic6~nf~toasths”Jhaartips”thatllnderOoa consi&ra& amount of plastic deformation. However, the interior portion of the fractured plate ~nuiastsssatiagZr~aadinthe~teofpfawstraiaThista~totlst~d~in apipearBcmtheviscwsactioadomiwrttsinalayerof&tidcfosetothe~wsUwbiktthe~ c~~iathcbufkeoitbcedcq~t~ydcsccn’bcdbyusingthspotffntislf~thsory. In order to impose a Emitationon tbe analyticalresult in (41,Btranft and Sib introducad two narrow slices of finite thickness (boundary iayerst) rh12 close to the plate surface, wbac the variation c was chosen as e = l/(2 + G/a). The same approach is carried out here for laminated plates. The nondimensional boundary layer of thickness t (to be chosen later) is in&oducedin tke outer-most layers of the composite plate. In this thin strip the function f(z) is chosen such that it satisfiesthe traction-free condition on the exterior surface and quilibrium in the interior region. The three layer symmetric composite (Fig. 2), is chosen as the simplest nontrivial example. The numeral 1 is used to refer to the interior layer while the outside layers are denoted by 2. The solutions to equa&n (44) which satisfy continuity of j and fi across the material interfaces are: f,(z) = A cos (pa), f*(z)
= A{ cos (pth
0 =iz 4 ht 11 cos [pt(z
- hdl
-~sin(p,h,)sin[p2(2-hJ),h,sh1+k2
See the Appendix for b,, b2, et, m,,, k, and k,. t’!&c exists a small but finite distance to the boundary within which the ~20Iution of experimental mcasmements &nd accuracy
of any num
soWionbreaksdown.
Aa approximate threedimensional theory of layered plates
17
The functions S(z) given by equation (46) which satisfy the plane strain conditions are not general enough to aflow imposition of the free surface conditions at z = ha + hr. A boundary layer is introduced adjacent to the free surface within which an arbitrary function 12(z) is constructed such that the traction-free conditions f&h, + h3 = fl (hr + h3 = 0 are satisfied along with the requirements that f&z) and its first and second derivatives must be continuous at z = irt + ii& -a). This arbitrary function can be represented by the foilowing fourth order poI~Otit%l
The five coefficients f~c,i = 1,2 , . . . ,5) are determined from the conditions mentioned before. The ftmctions X(z) (i = 1( 2) can be expressed soleiy in terms of the parameter pI, which depends on the behavior of the functions S,, S, and Z,(x, y). The dependency of fr(t) on p1 is found by using an iterative procedure, which consists of selecting an initial value of pt that specifies a given set of the coefficients of asymptotic behavior of S,, S, and Z2(x, y). This leads to a new value of pt obtained from the plane strain condition and the procedure is repeated until it convwes to any one of a set of discrete values for pl which satisfy equation WI. The crack edge stress field as given by equations (4@ is now compfetegy determined and there remains the numerical calculattion of the stress intensity factor, k,(z) which can be expressed in the form k,(z) = Kq&)a’~ The newly defined function
represents the equivalent load distribution on the crack faces which is compatibhs with the variational theory. To recapitulate, the in-piane stress components are proportional to f?(z), and the transverse normal stress at is equal to YI(OI+ ap) except in the boundary layer, whik the transverse shear stresses 7, ad 7= are xmn-sm. T&m, the singular stress field in the vicinity of the crack edge can be characterized by the non-dimensional stress intensity parameter K and the parameters pl and p2 which govern the functions L(z). It should be pointed out that the behavior of the laminate may be qualitatively d&&rent in the vicinity of the material interfaces than that within each layer. Such an efEcct may have significant inffuence on the toad transfer from one iayer to the next. This line of reasordng suggests the possibility of introducing additional boundary layers or transition regions at the materird interfaces. However, the introduction of such boundary layers is not required by this formulation, because the function $4~) given in equation (46) do satisfy the traction continuity conditions at the interfaces. In order to carry out the numerical analysis it is necessary to assume a value for t. the non-dimensional boundary layer thickness. For laminates the bo’tndary layer thickness will in general depend on the elastic constants E11E2, wt, vt and geometric parameters halh, and (h, + h&a, but sufficient data for mode&g the functional dependence is not yet avaiiable. Eence, as a first attempt in mode&g kuninates the present formulation includes bomtdary Iayers at the exterior surfaces but no interfacial transition layers. This non-dimensional boundary layer
EFM
vol.
7. No.
1-B
18
R. BADALIANCEand G. C. SW
thickness was assumed to be of the form
Numerical results are obtained for the dependence of the stress intensificationfactor K on the geometric and material properties of the laminate associated with the refined formulation which includes boundary Iayers, Rest&s arc presented graphicallyin the form of plots for .K as a function of hJa at various values of El/&, hdh, with Y]= 19= O-3in Figs. 3-7. Attention is directed to the Muence of the relative stiffness of the inside layer to the outer layers f&I&j on the ~p~it~e of the stress field in the vicinity of the crack edge. Fiies 3-7 indicate that K decreases as EIf& increases if the other parameters are held cm&ant. As E&L increases the load is carried increasinglyby the middie layer. In the Emitas & goes to zero, the
20
R. BADALIANCE and 0. C. SIH
actual plate half thickness is reduced from h I + klZto h I and the stress ~t~sity factor is expected to decrease accordingly,? The infIuenceof h&t, on this effect can be seen by comparing Fiis. 3-7. As hJhl increases, the effect of changes of Et/E2 on K becomes less pronounced. To und~tand these intentions, recall that at finite iSi& increasing kinks camcs the outer layers to carry a larger portion of the applied bad and for very huge kJk, the laminate behavior shouid be almost independent of the material properties of the inside layer. UN ~A~~g OF Tflfs THBURY An ~rox~ate ~~~sion~ theory for lunate plates was developed here. The stress field was assumed in the form of a product of a function of the out-of-plane variable multipliedby a function of the in-planevariables,The functions governingthe in-planevariations of the stress Add were obtained from variationalprinciple of rn~~ ~omplemcn~ potential energy. The function governing the transverse (out-of-plane) variation of the stress field was chosen by enforcing the plane strain pontoon oi = V(D;+ 0~)in the vicinity of the crack edge in the interior region of the pk. Boundary layers were introduced at the free surfaces to satisfy traction free boundary conditions. This approximate theory has been applied to a problem of a laminated plate containing through-the-thickness crack subjected to in-plane stretching. The crack edge stress field was determined to be of the form Q = ~~~~)~~g)~r’~,where ir, 4, z) are the local cylindrical coordinates. The stress ~~s~~~n parameter K, which measures the amplitude of the strzss sing&r@ at the crack edge is found to depend on the material properties and geometric parameters of the cracked composite plate. It is believed that the gross properties of the crack edge stress geld as measured by K are properIy predicted by this approximate theory. On the other hand, the function f(r) describing variation of stress field in the transverse direction turns out to be insensitiveto the materialproperties of the laminate. Physical intuition suggests that the major portion of the load per unit thickness should be baited in the stiffer layer; therefore, the function f(r) should be signScantly in3uenced by the relative sti%ess of the layers of the composite. The reason that the present model does not demonstrate this effect is closely associated with the use of the complementary energy principk which does not require satisfaction of displacement continuity conditions across the material interfaces. A possible approach toward irn~v~ this appear theory whigh is likely to be fruitful is to introduce inter&&al boundary layers into the mathema~ formulati~, In these interfacial boundary layers, it is suggested that the plane strain condition be abandoned, because this condition is not expected to hold in the neighborhood of material interfaces. Instead, a condition relating the relative displacement of the adjacent layers should be enforced in place of the plane strain condition. Introduction of these interfacialboundary layers should permit more freedom in the choice of the function f(r), thus ending a more realistic model of the transverse dis~bution of the crack edge stress field. ~UN~~~~U
~
Ac&nowle&vnent--This research work was sponsored by the Air Force Materials Lab. under contract F336571-C-1429, Wright Paterson Air Force Base, Ohio. The authors also wish to thank Professor P. D. Hilton for many of his helpful suggestions.
tin the present model, it is not possible to account completely for this effect. The toad distribution permitted in this formulation for finite but small& is qualitatively diiercnt tbaa that for E, - 0, is., for E2 finite,
An approximate threedimensional theory of layered plates
21
REFERENCES [I] G. C. Sih, P. D. Hilton, R. Badaliance and G. Villarreal, Fracture mechanics studies of composite systems. Air Force Renort AFML-TR-70.112.Part III (1973). [2] G.-C. Sih and E. P. Chen,.Fracture analysis of unidiictional and angle-ply composites. IFS&f-73-26,NADC-7R-73-I (1973). [3] E. Rcissacr, Gn bending of elastic plates. Q. uppL Math. 5, 55-68 (1947). [4] R. J. Hartranft and G. C. Sii, An approximate three-dimensional theory of plate with application to crack probkms. Jnt. J. Engng Sci 8, 711-729 (1970). [5] R. J. Hartranft and G. C. Slh, The use of eigenfunction expansions in the general solution of threedimensional crack probkms. J. Math. Mech. 19, 123-138(1969). [6] G. C. Sih, A review of the three-dimensional stress problem for a cracked plate. Int. 1. Fmetun Me& 7,3%1(1971). [7] G. C. Sih and H. Liebowitz, Mathematical theories of brittle fracture. In Mdthemoticol Fandommrals ofFmctun, (Ed. H. Liebowitx) pp. 67-190. Academic Press, New York (196(l). [8] E. Reissner and Y. Stavsky, Bending and stretching of certain types of heterogeneous acolotropic elastic plates. I. qpl. Mpch. 28, No. 3. Trans. ASME 83.402408 (l%l). [9] I. N. Sneddon, Fourier Transjorms. McGraw-Hii, New York (1951). [lo] G. C. Sih, P. D. Hilton and R. P. Wei, Exploratory development of fracture mechanics of composite systems. A&Force Report AFML-TR-70-112 (1970). [I I] R. J. Hartranft and G. C. Sih, Effect of plate thickness on the bending stress distribution around through cracks. J. MatA Phys., 47, 276-291 (l96Q. [12] G. C. Sih and J. F. Lo&r, A classof wave diffractionproblemsinvolvinggcomctrically-inducedsingularities.J. Math. Mech. 19, 327-350 (1969). (Received June 1974)
APPENDIX
c, = 52 (k,b,-k,b,)-ok,‘-k,‘)p,
cl = Lb, -2k,&;
Cl =4&a;
8, =2k,c-Lb,;
d,= k,ka,+$(k,b,+k,b,)
1
a,-2(1-+e,);
ml=
-
m,=
-(kl+klet+$$e,)
(k?-k,‘)+k,e,+k,k, C
-et-$(k,‘-Pk,‘)r,]
m~=(J~~J*~[(J~+J~)-e,~,k,+~[~
-1
[v
($k,)
-(I.(J,-J+J,J,)k,]]]
+I.(I,-&)-I&]
R. BADALLWCE
22
k, k, c-_cz-]-Ml 2
2
-
and G. C. SIH
W-l,l,l(k,e,-t)]]
m.=(L'3r,l,[-Lk,6,+~[~[(c1-~c,)k, +&
k,tk,e,+ 6(
e, t(k+k~3,&, 4 )
-;(k.'-3k,'k3r,]-LU, -IA - IMkre3)
2
THEORY OF THKXNESS CRACKS
R. BADALIANCE and G. C. SIH
2
material constants, etc., however, must be determined from a physic~y plausible ~s~ption that the plane strain condition[4,6] is satisfkd in the interior region of each layer of the laminate. The solution ceases to be valid in regions close to the free surfaces and interfaces. Boundary layers are introduced to exclude the zones in which the plane strain condition prevails. This is analogous to the boundary layer in fluid rnec~~s where potentiat flow is separated from a viscous layer near the wall. FoUowin8 the de&&ion of the stress-intensity factor for three-dimensional crack problems[6,7], numerical results are obtained for various elastic constants and geometrical parameters of a cracked laminate. The results of this study indicate that the proposed method of solution can be extended to laminates with any number of anisotropic layers sim&ting those which are used in existing structures. While the failure problem of delamination and/or through laminar crackin is not answered here, this work of stress analysis is a necessary step toward the development of a fracture theory of huninate cracking. APPROXIMATE THRIWDQUENSIONALTHEORY FOR MULTI-LAYERRDLAMINATE!3 The use of va&ional principles, e.&, the principle of minimum compkmentary potential
energy, to develop approximate sohttions is a standard technique. An approximate form is assumed for the v&ables, in this case the stress components, and it is substituted into the functional. Minimizationof that functional with respect to admissiik varktkns of the assumed functions (stress comments} kads to the choke for those functious which best approximate the exact solution in an avenged sense. Consider a laminated composite plate of uniform thickness made of n layers, where each layer can have dif&ent thickness and material proper&s. It is assumed that the layers are bonded together perfectly, i.e., there exists no interfaciai Baws.A k&ate of thisnatm, whsn subjected to hr-pkne stretching, willdeform out of its pianef8J.This effect is elhrdMed when the laminate is a symmetrk one, Le., the layers are stacked in even ttumbers such that the system possesses mat&al symmetry with respect to the middlepkne of the laminate as shown in Fit. 1.
Loteml
boundary L
h
h,
Cross section
of a symmetric
laminate
Fig. 1. Geometry of IaminaM plate.
3
An approximatethree-dimensionaltheoryof layeredplates
The procedure to be employed for laminated plates is to assume that each of the stress components in each layer can be approximated as a product of a function of the out-of-plane variable z multiplied by a function of the in-plane variables x and y, i.e.,
a:’ = fl’(z)g:P’(x, y) no sum on i, j, p
(1)
The purpose of this assumed form for the stress components is to reduce the governing equations from dependence on three independent variablesto dependence on two variables for which many standard solution techniques exist. Substitution of the assumed form for the components into the equilibrium equations with the additional condition that the equations separate leads to the more specific assumption uza-)= fp(r)Z,‘“‘(x, y)
[ei’, $I=
- fXzNZ%,
(2)
Y1,zy(%, Y)I
= f~t)[S,“‘(X, y), Sr(p’(x, Y1,Ek [u,‘p’,UT’, 71”,‘1
Y)I
The conditions of traction continuity across material interfaces are satisfied by taking rz*‘. &@‘,zpr = [Z,, z,, z,], Mr) = fp+G) and f;(r) = fL+t(z) at the value of z corresponding to the interface between layers p and p + 1. It is reasonable to substitute this assumed form for the stress field into the complementary potential energy functional and minimizethat functional with respect to Z,, Z,, Z, SI’P’,Sr’p’,and 7’$: subject to the conditions of equiliirium in order to develop an approximate solution for lam&r eomposites. It is, however, advantageous to examine the resulting form of the strain components so as to choose the approximate solution which comes closest to satisfying the displacement continuity conditions across the material interfaces. For example, the strain component G for isotropic and homogeneous materials is given by
G=El [ux- vtuy+ a*)1=;
v”(z)(Sx - vS,)- vf(z)Zl
(3)
In order for G and thus u, to be continuous it is necessary, though not sufficient,that SX(“) and ,!$,‘,‘,’ be the same for all layers. Examination of the shearing strain component yXryields a similar condition for CY’.Thus the assumed form for the stress field which will be adopted is: uz@’= fw(zm&
Y)
[T$‘, $)I = -f&‘(zwk
Y)94(&
[U.‘p’,oy@‘,7ryq=f);&)[Sx(x, Yh w,
Y)l
(4)
YIPTxrk Y)l
where &(z) and f:(z) must match their respective values for the adjacent layers. For notational purposes it is convenient to rewrite the assumed form for the stress components in an equivalent manner as:
@‘=d(z)Z*(x, Y)
02
R BAWLIANCB and 0. C. SXH
ms Y)l =rf”(wxlx~ YX 4(x, Y), ws
[Tit’, r(P.'l = -f(~Hz(x, [fr;“‘, a YW’I cl
(5)
Y),
Y)l
where f(z) must be continuous and have a continuous first derivative across the material interfaces. Recall, the complementary potential energy (4) of an isotropic, homogeneous body is de&ted as the strahr energy of that body minus the work done on the portion of the body surface (5) over which dispiacements are speci&d, i.e.,
where the strain energy density is
SubstWion of this assumed form for the stress components, equation (S), into the above complementarypotential energy functional and the equilibriumequations for a laminarplate with edge loading and traction-free surfaces (f = f = 0 on surfaces) yields
(8)
(9) where
and amp’, a*‘, and a*’ are the normal, tangential, and transverse components of the displacements prescribed on the edge (S,“‘) of layer p, respectively (Fig. 1). Lagrange multipliers (&) are employed to insure satisfaction of the equilibrium equations, and the functional
is formed.
An
approximate three-dimensional theof3foflayeredPlates
5
me variation of II with respect to S,, S,, T,,Z,,& and~4th thecodn~ty~~Wthm enforced on f(z) and f’(r) results in
where
(131
The divergencetheoremand the integration-by-parts techniqueareappliedto the va equation 6II = 0 to obtain: HI=
I
[(I, sxf r2sy + r4zi + A Lx )ssx
A
+cr,sy+12sxf
Lzt + A2,YmsY
+(~(~~-I~)T,+AI,,+A,)ST, +(/Z
+A~+As,)sz,
+ (I& + A2 +
(14)
As,, W,
+(I.(& +Sy++Wz +~d~l~
dy
where A”= A,n, +Azny; A. = -AI& + A2ny and (n,, n,) are the directioncosines of the normalto the plate edge (Fig. l),
R. BADAUANCE and 0. C. SM
6 Thei gemming
eqtm?ims
variables can be det~ned
ami
lxmbry
conditions
art
new
qpmn?,
i.e.,
the
in-plane
from 0
r,S*+r2S~+r4z~fhl.x= r‘s,
+ IZS,
+ Lz
+ AZ.,
= 0
2(1,-I*)T,,+A,.,+A2.x
I,(&
z,
=o
bZ
fAlfA3,
=o
I,&
+ A2 + A3.y
= 0
+Sy)+CZ
=
$A3=0
zx..% + &3_,
z,
=
.%a
f
TL,.Y
2,
=
Tx7.x+ Sw
with A. = - II,, A, = - il.,
aud A3=i -P, on C.. In other words, the boundary conditions associated with the in-plane variables S,, S,, . . . can be either of the tractiou type or averaged dis~~ace~nts. On the contour e aioug the piate edge, one can prescribe either S, = .%
or A, = - &
T,=F=
or A,=-&
z=z
or
A39:
-a,
The traction free surface conditions are reflected by the requirement that f(t) and f’(z) must be zero on the plate surfaces. After some nontrivial manipulations the set of governing differential equations (15) can be sepa8W&into the form:
z, - a6v2n
= &
a,V‘A3
(a
,V2A3
+
a2A3)
+ a4Q2A3 + asA3 = 0
where extraneous solutions are eliminated by the condition
aZ,f % = - a7V2A,ax ay
where
aeA3
An approximate threedimensional theory of layered plates
1 [Zs(Z, +
z2z3 -- I, I,, a2 = (zl+zz) (z,*-z22) 2(Z,-b) (z1+z2) 12) + z3z4 - 2z47
I
a, = Z,’- Z,Zs, a5 = - (11’ - I,‘>,
as
=
[Z5(ZI
2Z4(Z, - 12)+ Z~ZS
a4 = -
+ :;:1~I$3
z3 a6 = 2(11-
-
2E)l
The other in-plane variables can be expressed in terms of AS,Z, and & as
A2=
-
z3zY
+Qd
(18) -
z2~+[z,b-z4(z,-z2))~
[Ib
+ Z,(ZI
-
Z2)15}
-[&+I1((k~2)+]
The set of equations (la), (18)and the condition (17)form together with the boundary conditions a complete system of equations for an approximate three-dimensionaltheory of layered plates. A TEROUGE CRACK IN A LAMINATEDPLATE For this analysis, the crack length is taken as small in comparison with the in-plane dimensions of the plate. The stress 6eld in the vicinity of the crack is thus independent of local boundary effects and the plate can be modeled as i&rite in the two, in-plane directions. This assumption simplifiesthe analysis significantlywhile maintain@ the essential characteristics of the cracked laminar plate problem. In particular, the problem of a laminar plate (symmetric about its mid-plane) containing a crack in a uniaxialfar-field,stress state will be considered (Fig. 2). The crack is directed normal to the loading direction, thus modeling the most damaqing type of flaw. This problem can be expressed as the superposition of two auxiliaryproblems. The &st auxiliaryproblem is that of an untracked plate subjected to the same loading conditions as the actual cracked plate. For the second auxiliary probkm, the cracked plate configuration is considered with no boundary loads. Instead, the crack faces are loaded with tractions of equal magnitude but opposite sense to those found at the crack location in the first problem. Thus the superposition of the two problems yields the cracked plate with traction free crack faces and remote loadinq as orighmhy described. The first of the auxiliary problems, an untracked plate in a far, uniform stress field, has nonsingular stress components and as such does not contribute to the crack-tip singuhnity.
R. BADALIANCEand G. C. SIH
8
Fii 2. Three-layer conlpo6itcplatecontainins a throustl crack.
Therefore, the stress Beld in the vicinity of the crack tips for the case of crack face loadinq is identical to that for the remote loading situation of the actual cracked plate. It is mathematicahy advantaqeous to consider the crack face loading problem because it leads to stress fields which die-out far from the crack and can thus be treated by integral transform techniques[9]. The probkm of a composite plate, made up of homogeneous isotropic layers distriited symmetrically about its mid-plane, containing a through-the&ickness crack will be considered for the second a&Gary problem mentioned before. The crack is located in the y = 0 plane and directed along the x-axis as shown in Fig. 2. Consider the region s y, 2 >O. The boundary and symmetry conditions are: on
Y “0,
%Y “?~=0;Q,~p(52),0sX~E4;~=0,X>(I
where p(z, z) is the pressure applied on the crack face which is symmetric in both of its arquments x and z.
and on
z-o,TxI=c711~o;u*=o
The quivalent boundary conditions for the approximate theory are: T,=Z,=O: s,=p(x), &=O,
on
Osxsa
x>a
x ~0, T, =Z=O;A,=O
isrequired to be symmetric in z to model the symmetry conditions on the z = 0 plane. Further f(z) and f’(z) are zero on the plate surface z = f t/2. The governing equations (16)are to besolved subject to the boundary conditions mentioned above. Further, the functions %, S, Tm Z,, Z,, and 2 must die-out at large distances from the crack.
and f(z)
In-plane oariations of the stress field As indicated earlier, the solution procedure will be based on applyingthe Fourier transform to the x variableto determine the nine in-plane variables S;, S,, . . . It is convenient to recognize the
An approximate threedimensional
theory of faycred plates
9
symmetry properties of these variables in advance, thus reducing the Fourier transform to sine and cosine transforms for the odd and even functions, respectively. It is observed that the functions $4,8, Z, Z, AZ,and A3are even in x while 7’,, X, andA8are odd in x. Thereforedefine S,“(s, y) =
~mSx(x, y) cos(sx) dx, etc.
and T:~s,y)=lmbbyfsin(sx)dx,et~. 0 Application of Fourier sine and cosine transforms to the governing set of equations (16)and the condition (17) leads to the following set of equations: tin,’ $A~=+ (u3s4(I,dyl+Gl4 - 2a3s ),Y, 2
ad2 + udh3c = 0
(19)
providedthat
Tlkegeneralsolutionto equations(19)which s&i&esthe asymptoticconditionsat i&ity can bewrittenintheform AS=@, y) = 2RelM~)e-~l
where
and
Z(S,
y) = f B~s)c”~ + RelP&)e”7
r(s,
y) = B(s)eWqyf ReMs)e-‘*I
(21)
10
3. ~AlhLIANCE and G. C. SIH
The additional transformed functions of the in-plane variables can be expressed in terms of AS’, Z.‘, and Z throu& equations (18).In particular, the functions TZ,, S,", and A: are obtained to study the boundary conditions:
The general solution is now substituted into the boundary conditions (transformed appropriately). The condition Z fs, 0) = 0 yields
Whiie the condition T$(s, 0)= 0 relates the real and imaginary parts of A,(s) as
It is convenient to express this result symbolically as Im[A&)l = /3,(~)RefA,(s)Jwhere
(24) _rr
s2_& [
aod
206(
((12-k,a,)(l+askl)-a,a,k3 (I+ uskd’+ (a&d2
An approximate t~~n~~
theory oflayrod plates
11
Tbe set of mixed boundzwyconditions
remain to be satisfied. These conditions can be expressed in terms of the transformed functkms as
Thus the mixed ~~~
conditions can be expressed in the form of a set of dual in
equathns as f
9) d g(s)R(s) cos fsx) ds = -fp(x),x
f0
4.a R(s) cos (sx) ds = 0, x L a
where
This set of dual integral equations can be reduced to a single integral equation by the introduction of an aw&ary function, w(x), de&d by R(s) cos (sx) ds, XC a
R. BADALKANCEand G. C. SIH
12
equation (28)implies that R(s) can be written as R(s) = fI; w(x) cos (SIC) dx far ah x, where w(x) must satisfy
Then
L-k4r
w(x)cos(sx)dx
I
cos(sx)dx =
-+w
(30)
This equation (30)contains the crack tip singularity and corresponding care must be taken in the solution procedure. Exact three-dimensional solutions for symmetric crack probIems[!JJ], indicate that on ah interior points in the immediate vicinity of the leading crack edge the local stress field must be of the form:
02”+frsi*(B)sin r248)I k(Z)0- -&in(,)sin ;g G=Gj=cos2[ 2 ( )I a=$&
[
cos
(31)
where (I; 8) are local p&r coordinates in the plane normal to the crack edge (Fii 2). The auxiliary functi~ w(x) corresponds to h&O) and, as such, is expected to be locahy Pomona to the square root of the distance from the crack tip. This ~~~a~n is used to introducea secondauxiliaryfunction,4(t), ia an attemptto removethe inherentsingubrityfrom qua&n (2%).IMae Jr(t)by the integrai equation (4) (32) Subs~
into equation (30) yiekis, after some s~p~c~ons, fdJtfsin(srfdsldt1
-lp(x)dx,x
(33)
whereJois the zero order Bessel function of the first kind. If the expectations mentioned above are correct, equation (33)will be a nonsinguiar integral equation for g(t) which is amenable to numerical solution procedures.? The function g(s) can be written in the form
go =c +8*(s) s
where g*(s) = ofs-‘) as -+mand C is a constant given in the Appendix. Substi~tion of the above into equation(33)and use of the known safutian to Abei’sequation leads to the Fredholm integral equation (see references [IO, 11, i2l for details) iA
posfuicbccksof tbc IIOIU&~ resuttsdo indaodindicata that e(f) is a weB-bcbavcdfunction.
13
(35) This cqW.ion can be reduced to standard form by the following mmdimensionaiiitions q=&,
t==aT,
x=a&
s=:
(36) where
The accuracy of the numerical solution procedure to be employed is improved by separatkg s*(s) as
where
Rfs)=O(s”) 8!3 s 4~53,
'RR
constants H and IV2are given in the App&ix. Hence, the
kernel K(Q I) can be rewritten as
where Io and Ro are the zero-order mod&d Bessel functions of the 5rst and second kind, respectively. The case of crack face loading which does not vary with x is considered, i.e., p(x) = I? The kernel is now evaluated numerically and the integr8l equation is ~~~a~ by a set of algebraic equations via Simpson’s ale. Numerical solutions for NT) are thereby obtained. The origin8l unknown function, R(s), is obtained by substitution of equation (32)into (29). Carrying out the indicated integrations, R(s) can be written as
The unknown transformed functions S,“, S, etc., can be determined directly from R(s) by back §~bsti~~on.
14
R. &tLMLlANCti
and 0. C. SIH
It is expected that damage resulting from crack propagation will originate in the most highly stressed re8ion, the vicinity of the crack tip. Thus the main objective is to obtain the best approximate sob&ionfor the crack edge stress field. Referring to equation (38)it can be shown that the s~~~~ at the crack tip is associated with +(I) and that int@ expression in this equation has no contribution to it. Because an asymptotic description of the crack tip field is sought, the integmi expression in equation (38) is dropped and the functions S,‘, S,‘, etc., are expressed in terms of the simplifiedequation for R(s). These transformed functions are inverted asymptoticallyfor large s. Adopting polar coordinates r, rI, r2,and 6, 6, & as shown in Fig. 2, it is found that SAX,YP = -~PY(l)C{~sin[~tB~+~$]
-zcos [e*-;(e,+e2$+0(1) S,(x,y)=
+~PY(l)C{~sin[~(BI+8$]
+os[ 80-;(e,+e2~]}+ow ZYCGY)"" -~PY(I)c[~cos[~~e1+Bd]}+0(1~
Zzk Yv = -.+
PY(l)[e’(klb;;
x +3s l
[
k3b1)]
e~-~~e,+~~)]]+ql)
z(x, y)= a~Y(l)~([?j$k~+ X
b&r-?)]
ds
II
X
I0
y.h(us) cos (sx) eety ds
(391
.Jote that Z,(x, y) and Z,.(x,y) are nonsingular. Further, approaching the crack tip (u, 0), i.e., taking the limit r + u, 8 + 0, r2+ 24 & +O, the near tip fields can be written as
th. bz, C, et, ma k,. k, are given in the Appendix.
An approximatethrce-dimcnxior4theoryof layeredplates u,
-
,y$g
[co, (2) +; sin(&I sin(g]
I5
+W W)
T_ 5~~[~*n(B,)cos(~)]+o(1~ a* = KO&M 1/Z (2rY [cos (93
+OW
K =
ae,C*(l)
where (41)
6(Z) = W(z) and
Note that the stress field in the vicinity of the crack tip, as given iu equations @NJ), contains the usual square root siz@arity in the plane normal to the crack front, where the dcpbndcaoe of these stress components OQthe in-plane variables (rr , e,) is in complete agreement with the exact three-dimensional solution for a crack in e elastic medium[5J.Also, it is interesting to no% that thetransvareeshcatstregsts~,andr,~~bythis~~thcoayQ,rmate~aa in-plaae [email protected] From quations (40)it can be seen that the stress intensity fa&rs cga be chrractaizcd as
k*(z)= KPfwa 1/x
(421 H(z) = (b&s + b&d+
KPftz)dR
These stress intensity factors, unlike those used in the curreut theory of fracturepI vary along the crack front in the zdirection through the functions f(z) and p(z). It can be seen from boundary condition that the load distribution over the crack face corresponding to this solution is a&) = W(z). Hence, the first of equations (42) can be writteu as
Thus K can be COnSidered
to be a measure of the idhenCe of ehdc constant8 and geometrical parameters of the cracked laminate on the intensity of the crack edge stress field. The stress stern parameter 15:is directly dependent upon q(l). Where Y(1) is the limit of !P(E)as the dimensionless variable 5 (normalizedagainst the half crack iength) approaches unity, i.e., the crack tip.
tT, and T, will bc singularin the presence of torsionaland transverseshear loads.
16
R. BADAISANCEand G. C. SlH
~OU~E-~~~~ VARIATIONOF THE CR.ACIC~TIF t5lBESS FIELD Exact ~ymptotic analysisI descriid earlier indicates that at all interior points near the leading edge of the crack, the plate is in a state of plane strain, i.e., or = ~(a, + a,). Within each layer, the plane strain condition can be employed to obtain j(z) in the vicinity of the crack edge. Supposition of the requirements that the approximate solution should satisfy the above equation leads to the following differential equation for j,(z):
fxz) + p:fi (2) = 0
(44)
where p: is equal to the asymptotic value of the function -Zzlvc(Sx + S,) at the crack tip, i.e.,
It can be shown in general that the solutions of equation (44)do not possess sufacient number of free parameters to satisfy both of the free surface conditions (f = j’ = 0). Such an analytic solution, however, cannot be expected to hold throughout the thickness of the plate. This is mainly because the experimental observations of 6&e thickness plates show that there exists a layer of ma&al near the plate surface which behaves very differently from that in the bulk. Spectically, the fracture testing of through cracks in finite thickness plates has shown evidence f~twosuriace~yenofthcmaUriaic6~nf~toasths”Jhaartips”thatllnderOoa consi&ra& amount of plastic deformation. However, the interior portion of the fractured plate ~nuiastsssatiagZr~aadinthe~teofpfawstraiaThista~totlst~d~in apipearBcmtheviscwsactioadomiwrttsinalayerof&tidcfosetothe~wsUwbiktthe~ c~~iathcbufkeoitbcedcq~t~ydcsccn’bcdbyusingthspotffntislf~thsory. In order to impose a Emitationon tbe analyticalresult in (41,Btranft and Sib introducad two narrow slices of finite thickness (boundary iayerst) rh12 close to the plate surface, wbac the variation c was chosen as e = l/(2 + G/a). The same approach is carried out here for laminated plates. The nondimensional boundary layer of thickness t (to be chosen later) is in&oducedin tke outer-most layers of the composite plate. In this thin strip the function f(z) is chosen such that it satisfiesthe traction-free condition on the exterior surface and quilibrium in the interior region. The three layer symmetric composite (Fig. 2), is chosen as the simplest nontrivial example. The numeral 1 is used to refer to the interior layer while the outside layers are denoted by 2. The solutions to equa&n (44) which satisfy continuity of j and fi across the material interfaces are: f,(z) = A cos (pa), f*(z)
= A{ cos (pth
0 =iz 4 ht 11 cos [pt(z
- hdl
-~sin(p,h,)sin[p2(2-hJ),h,sh1+k2
See the Appendix for b,, b2, et, m,,, k, and k,. t’!&c exists a small but finite distance to the boundary within which the ~20Iution of experimental mcasmements &nd accuracy
of any num
soWionbreaksdown.
Aa approximate threedimensional theory of layered plates
17
The functions S(z) given by equation (46) which satisfy the plane strain conditions are not general enough to aflow imposition of the free surface conditions at z = ha + hr. A boundary layer is introduced adjacent to the free surface within which an arbitrary function 12(z) is constructed such that the traction-free conditions f&h, + h3 = fl (hr + h3 = 0 are satisfied along with the requirements that f&z) and its first and second derivatives must be continuous at z = irt + ii& -a). This arbitrary function can be represented by the foilowing fourth order poI~Otit%l
The five coefficients f~c,i = 1,2 , . . . ,5) are determined from the conditions mentioned before. The ftmctions X(z) (i = 1( 2) can be expressed soleiy in terms of the parameter pI, which depends on the behavior of the functions S,, S, and Z,(x, y). The dependency of fr(t) on p1 is found by using an iterative procedure, which consists of selecting an initial value of pt that specifies a given set of the coefficients of asymptotic behavior of S,, S, and Z2(x, y). This leads to a new value of pt obtained from the plane strain condition and the procedure is repeated until it convwes to any one of a set of discrete values for pl which satisfy equation WI. The crack edge stress field as given by equations (4@ is now compfetegy determined and there remains the numerical calculattion of the stress intensity factor, k,(z) which can be expressed in the form k,(z) = Kq&)a’~ The newly defined function
represents the equivalent load distribution on the crack faces which is compatibhs with the variational theory. To recapitulate, the in-piane stress components are proportional to f?(z), and the transverse normal stress at is equal to YI(OI+ ap) except in the boundary layer, whik the transverse shear stresses 7, ad 7= are xmn-sm. T&m, the singular stress field in the vicinity of the crack edge can be characterized by the non-dimensional stress intensity parameter K and the parameters pl and p2 which govern the functions L(z). It should be pointed out that the behavior of the laminate may be qualitatively d&&rent in the vicinity of the material interfaces than that within each layer. Such an efEcct may have significant inffuence on the toad transfer from one iayer to the next. This line of reasordng suggests the possibility of introducing additional boundary layers or transition regions at the materird interfaces. However, the introduction of such boundary layers is not required by this formulation, because the function $4~) given in equation (46) do satisfy the traction continuity conditions at the interfaces. In order to carry out the numerical analysis it is necessary to assume a value for t. the non-dimensional boundary layer thickness. For laminates the bo’tndary layer thickness will in general depend on the elastic constants E11E2, wt, vt and geometric parameters halh, and (h, + h&a, but sufficient data for mode&g the functional dependence is not yet avaiiable. Eence, as a first attempt in mode&g kuninates the present formulation includes bomtdary Iayers at the exterior surfaces but no interfacial transition layers. This non-dimensional boundary layer
EFM
vol.
7. No.
1-B
18
R. BADALIANCEand G. C. SW
thickness was assumed to be of the form
Numerical results are obtained for the dependence of the stress intensificationfactor K on the geometric and material properties of the laminate associated with the refined formulation which includes boundary Iayers, Rest&s arc presented graphicallyin the form of plots for .K as a function of hJa at various values of El/&, hdh, with Y]= 19= O-3in Figs. 3-7. Attention is directed to the Muence of the relative stiffness of the inside layer to the outer layers f&I&j on the ~p~it~e of the stress field in the vicinity of the crack edge. Fiies 3-7 indicate that K decreases as EIf& increases if the other parameters are held cm&ant. As E&L increases the load is carried increasinglyby the middie layer. In the Emitas & goes to zero, the
20
R. BADALIANCE and 0. C. SIH
actual plate half thickness is reduced from h I + klZto h I and the stress ~t~sity factor is expected to decrease accordingly,? The infIuenceof h&t, on this effect can be seen by comparing Fiis. 3-7. As hJhl increases, the effect of changes of Et/E2 on K becomes less pronounced. To und~tand these intentions, recall that at finite iSi& increasing kinks camcs the outer layers to carry a larger portion of the applied bad and for very huge kJk, the laminate behavior shouid be almost independent of the material properties of the inside layer. UN ~A~~g OF Tflfs THBURY An ~rox~ate ~~~sion~ theory for lunate plates was developed here. The stress field was assumed in the form of a product of a function of the out-of-plane variable multipliedby a function of the in-planevariables,The functions governingthe in-planevariations of the stress Add were obtained from variationalprinciple of rn~~ ~omplemcn~ potential energy. The function governing the transverse (out-of-plane) variation of the stress field was chosen by enforcing the plane strain pontoon oi = V(D;+ 0~)in the vicinity of the crack edge in the interior region of the pk. Boundary layers were introduced at the free surfaces to satisfy traction free boundary conditions. This approximate theory has been applied to a problem of a laminated plate containing through-the-thickness crack subjected to in-plane stretching. The crack edge stress field was determined to be of the form Q = ~~~~)~~g)~r’~,where ir, 4, z) are the local cylindrical coordinates. The stress ~~s~~~n parameter K, which measures the amplitude of the strzss sing&r@ at the crack edge is found to depend on the material properties and geometric parameters of the cracked composite plate. It is believed that the gross properties of the crack edge stress geld as measured by K are properIy predicted by this approximate theory. On the other hand, the function f(r) describing variation of stress field in the transverse direction turns out to be insensitiveto the materialproperties of the laminate. Physical intuition suggests that the major portion of the load per unit thickness should be baited in the stiffer layer; therefore, the function f(r) should be signScantly in3uenced by the relative sti%ess of the layers of the composite. The reason that the present model does not demonstrate this effect is closely associated with the use of the complementary energy principk which does not require satisfaction of displacement continuity conditions across the material interfaces. A possible approach toward irn~v~ this appear theory whigh is likely to be fruitful is to introduce inter&&al boundary layers into the mathema~ formulati~, In these interfacial boundary layers, it is suggested that the plane strain condition be abandoned, because this condition is not expected to hold in the neighborhood of material interfaces. Instead, a condition relating the relative displacement of the adjacent layers should be enforced in place of the plane strain condition. Introduction of these interfacialboundary layers should permit more freedom in the choice of the function f(r), thus ending a more realistic model of the transverse dis~bution of the crack edge stress field. ~UN~~~~U
~
Ac&nowle&vnent--This research work was sponsored by the Air Force Materials Lab. under contract F336571-C-1429, Wright Paterson Air Force Base, Ohio. The authors also wish to thank Professor P. D. Hilton for many of his helpful suggestions.
tin the present model, it is not possible to account completely for this effect. The toad distribution permitted in this formulation for finite but small& is qualitatively diiercnt tbaa that for E, - 0, is., for E2 finite,
An approximate threedimensional theory of layered plates
21
REFERENCES [I] G. C. Sih, P. D. Hilton, R. Badaliance and G. Villarreal, Fracture mechanics studies of composite systems. Air Force Renort AFML-TR-70.112.Part III (1973). [2] G.-C. Sih and E. P. Chen,.Fracture analysis of unidiictional and angle-ply composites. IFS&f-73-26,NADC-7R-73-I (1973). [3] E. Rcissacr, Gn bending of elastic plates. Q. uppL Math. 5, 55-68 (1947). [4] R. J. Hartranft and G. C. Sii, An approximate three-dimensional theory of plate with application to crack probkms. Jnt. J. Engng Sci 8, 711-729 (1970). [5] R. J. Hartranft and G. C. Slh, The use of eigenfunction expansions in the general solution of threedimensional crack probkms. J. Math. Mech. 19, 123-138(1969). [6] G. C. Sih, A review of the three-dimensional stress problem for a cracked plate. Int. 1. Fmetun Me& 7,3%1(1971). [7] G. C. Sih and H. Liebowitz, Mathematical theories of brittle fracture. In Mdthemoticol Fandommrals ofFmctun, (Ed. H. Liebowitx) pp. 67-190. Academic Press, New York (196(l). [8] E. Reissner and Y. Stavsky, Bending and stretching of certain types of heterogeneous acolotropic elastic plates. I. qpl. Mpch. 28, No. 3. Trans. ASME 83.402408 (l%l). [9] I. N. Sneddon, Fourier Transjorms. McGraw-Hii, New York (1951). [lo] G. C. Sih, P. D. Hilton and R. P. Wei, Exploratory development of fracture mechanics of composite systems. A&Force Report AFML-TR-70-112 (1970). [I I] R. J. Hartranft and G. C. Sih, Effect of plate thickness on the bending stress distribution around through cracks. J. MatA Phys., 47, 276-291 (l96Q. [12] G. C. Sih and J. F. Lo&r, A classof wave diffractionproblemsinvolvinggcomctrically-inducedsingularities.J. Math. Mech. 19, 327-350 (1969). (Received June 1974)
APPENDIX
c, = 52 (k,b,-k,b,)-ok,‘-k,‘)p,
cl = Lb, -2k,&;
Cl =4&a;
8, =2k,c-Lb,;
d,= k,ka,+$(k,b,+k,b,)
1
a,-2(1-+e,);
ml=
-
m,=
-(kl+klet+$$e,)
(k?-k,‘)+k,e,+k,k, C
-et-$(k,‘-Pk,‘)r,]
m~=(J~~J*~[(J~+J~)-e,~,k,+~[~
-1
[v
($k,)
-(I.(J,-J+J,J,)k,]]]
+I.(I,-&)-I&]
R. BADALLWCE
22
k, k, c-_cz-]-Ml 2
2
-
and G. C. SIH
W-l,l,l(k,e,-t)]]
m.=(L'3r,l,[-Lk,6,+~[~[(c1-~c,)k, +&
k,tk,e,+ 6(
e, t(k+k~3,&, 4 )
-;(k.'-3k,'k3r,]-LU, -IA - IMkre3)
2