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Closed-form analytical solutions for a Griffith crack in a non-symmetric laminate plate
Composite Structures 21 (1992) 49-55
Closed-form analytical solutions for a Griffith crack in a non-symmetric laminate plate Wilfried Becker Dornier GmbH, DeutscheAerospace, Abteilung TMBF, Pos(fach 1420, D-7990Friedrichshafen-1, FRG Within the framework of linear-elastic laminated plate theory, the problem of a Griffith crack in an infinitely extended unsymmetric laminate plate is treated. Due to the non-symmetric layup, bending extension coupling is admitted and is taken into account by means of a new complex potential method. Closed-form analytical solutions are given for homogeneous in-plane, bending and transversal loading of the laminate plate. The considered bending extension coupling is accompanied by interesting characteristics of the plate behaviour in the crack vicinity, e.g. additional singularities occur at the crack tips.
1 INTRODUCTION In the case of symmetric laminate plates, the static analysis usually uncouples into two different problems, the one of pure in-plane deformations with pure in-plane forces and the other of out-ofplane deformations with bending and transversal forces. In the case of unsymmetric laminate plates, however, in general this is no longer true. Bending extension coupling can occur in various ways which in the constitutive plate relations is reflected by respective coupling stiffnesses. The problem to be considered in the following is that of a simple finite straight crack (a Griffith crack) through such a laminate plate with bending extension coupling. The faces of the crack are assumed to be free of any tractions whereas the plate is assumed to be loaded by prescribed constant remote forces. In the absence of bending extension coupling complex potential methods 1-3 have proved to be very appropriate for the representation of crack solutions; 4'5 the author has also achieved good results with the complex approach. 6-8 In the present case with bending extension coupling, an accordingly generalized complex potential method has been formulated recently. 9 By means of this method, closed-form analytical solutions are given for the present crack problem. The solutions reveal interesting effects of the bending extension coupling in the crack vicinity. In particular, remote in-plane forces may give rise
to crack tip singularities of the plate bending moments. Similarly, remote bending forces may cause crack tip singularities of the membrane stress resultants. 2 COMPLEX POTENTIAL METHOD Within classical laminated plate theory bending extension coupling is taken into account by respective coupling stiffnesses Bij in the laminate stiffness matrix by which the constitutive plate behaviour can be given as follows: l° m
--
Nxy =
Mx
My
_Mxy_ -Aij A~2
Ai2 A2~
A26 B12 B22
AI6
Bl6 026
ex 8y
AI6
A26
A66
Bu
BI2 BI6 DII Dl2
B66
Yxy
D16
Kx
Bl2 816
B22 B26
B26 B66
Bu BI6 D12 DI6
Bl2 B26 D22 D26
026
D66 -
Ky
_Kxy
(1) Here, the strains ex, ey, 7xy and curvatures rx, ry, r~y of the laminate mid-plane result in the following way from the in-plane displacements u, v and
49 Composite Structures 0263-8223/92/S05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
W. Becker
50 the normal displacement w : e x = blx
~¢y= U ,y
Yxy = lgy "l" U ,x
~x ---- -- W,xx
l(y = -- W,yy
~xy----- - 2 W xr
(2)
The quantities N x, Ny, Nxy, Mx, My, Mxy denote the usual membrane stress resultants and plate bending moments, respectively, which on their part have to satisfy the equilibrium conditions:
N u + Nxy,y=O Uxy,x + N,,y = 0
(3)
By successive substitution of relations (1) and (2) into the equilibrium conditions the following three partial differential equations are obtained for the mid-plane displacements u, v, w :
Ai6U,xx
-
a -
-
[
-
Ozi
0 0 -
-
-
0
(6)
ig~iOy-l~iigzi+f~iOzi
If identically satisfied equilibrium is demanded for arbitrarily chosen potentials Fi(zi), this leads to the following relations:
+ Az6/~~)q~-(B~j + 3B.6/xi+ (B,2 + 2B66)/~~
+ BE6,u ~) = 0 +
+ 3B26,u 2 +
+ (m12 +A66)u xy+ A26v yy- BII W, xxx
-- B26 W y)f =
igx
_
+
+
+ 2A2
B22,u 3) = 0
(B,,
2B66)Wxyy
+
+(Bi6 +(B12 + 2B66),u~+ 3B26fl~ +
0
B22,u3)qi
+(AI2 + A66)U xy + A26tl,yy + A66U,xa
- (Ot] +4DI6,ui+ (2Di2 + 4D66),u ~
+ 2A26v xy+ A22va,y- Bl6Wxxx
+ 4Dz6/~ ~+ D22/~.) = 0
B l l U xxx + 3 B i 6 u xxy +
+ B26 U,yyy+
(BI2 +
0
2 B 6 6 ) U xyy
Bl6Vxxx+ (BI2 + 2B66)V~xy
+ 3 B 2 6 u x y y + B22Uyyy -
-4DI6wxxxy-(2Dr2 -- 4D26 W x ~ y -
Dii W xx~x
Q0 + QJkt + Q2fl 2 + Q3,tt3 + Q4,tt4+ Qs/~~
+4D66)W,x~yy
D22 W yyyy = 0
+ Q6# 6 + Q7/~7 + Qs# ~ = 0
(4)
For the intended complex approach, four complex potentials F~(z~), i = 1, ..., 4, are introduced. Each of these potentials is a function of a complex variable of the kind z~= x + #~y where/~ are complex quantities still to be determined. By use of the potentials F i and their first derivatives F'i the following representation is chosen for the displacements u, v and w:
(8)
is obtained whose roots are the needed quantities /~,...,/~4 together with their complex conjugates. With relations (1), (2) and (5) the membrane stress resultants and plate bending moments can be represented by the second derivatives of the potentials F i, as follows: It
Nx = 2Re[aiFi (zi)] Ny = 2Re[biF"(zi)] Nxy = 2Re[ciF"(zi)]
W= 2Re[F~(z]) + F2(z2) + F3(Z 3) + F4(Z4)] u = 2 R e [ p i F ~ i ( zi)]
(7)
For each i the relations (7) represent a non-linear system of three equations from which the constants pi, qg and /~i can be determined. This is described in more detail in Ref. 9. In particular, after elimination of the quantities p~ and q~, a polynomial of the kind
-- (2B66 + Bi2)Wxxy
- 3B26 W,xyy -- B22 W yyy =
,,
+ A22,u2)qi-(Bl6 +(2B66 + Bt2),ui
A66U,yy + A i 6 v xx
-3Bi6W~xy-(BI2 +
a
(A~, + 2ml6kti+ A66,u2i)Pi + (AI6 +(m,2 + m66)/ti
M~.~ + 2M~y,xy + My,yy=0
A~u, xx + 2AI6 U,x.v+
tities Pi and qi are also complex constants still to be determined. To this end the displacement representations (5) are substituted into the equilibrium conditions (4), taking into account
(5)
v = 2Re[qiF'i(z,)] In order to avoid unnecessary lengthiness the two last equations are given in a condensed way by use of the implicit summation convention. The quan-
Mx = 2Re[diF~(zi) ]
(9)
My = 2Re[eiF~(zi)] Mxy= 2Re[ fiF"i ( zi )] Here, the quantities a~, b~, c~, di, e i and f are complex constants which result in the following way
Griffith crack in a non-symmetric laminate
51 oo
from p~, qi and J/i:
Ny
oo Mx7
- ai"
bi ci de ei -A u AI2 Al6 Bu Bt2 BI6
f ~
- Nx
Fig. 1. Laminateplatewithcrackunderin-planeand bendingloading. AI2 A22 A26 Bl2 B22 B26
A16 Bu B12 B16- -Pi A26 A66 BI6 B26 B66
BI2 BI6 Du DI2 Di6
B22 B26 D12 D22 D26
B26 B66 DI6 D26 D66
q ij/ i Pij/i + qi -1
oo
-- 2j/i
Ny --"Ny~
Qy= My,y+ Mxy,x = 2Re[(j/iei + fi)FT(zi)]
2J/J,)Fitit (zi)]
oo
Nxy-"Nxy
co
~
MxMx
co
oo
My-" My~ _.~
oo
(13) oo
Mxy M,y
for
zi ~ oo
In addition, at infinity a rigid body rotation with respect to the z-axis shall be excluded: lgy--*Ux for
zi ~ oo
(14)
For the boundary value problem stated in this way the solution can be given in the form of the following complex potentials:
Qx = Mx,x + Mxy,y= 2Re[( di + j/if)F['(zi)]
Or = Qy + Mxy,x = 2Re[(j/iei +
co
NxNx
Finally, the transversal plate forces Qx, Qj and Kirchhoff's substitute transversal forces Q~, Qy are of interest. They can be represented by the third derivatives of the potentials F~as follows:
...1_
oo
-j/2i
(lO)
__ Qx=Qx+ Mxy.y-2Re[(di
The laminate plate is loaded by prescribed forces N~, Ny , Nxy and moments Mx , My, Mxy at infinity:
(11)
2f)F~"(zi)]
In the complex setting presented so far the underlying bending extension coupling is ultimately taken into account by the actual numerical value of the introduced complex constants j/i, Pi, qi, a~, ...,f/. In this way, the consideration of bending extension coupling is quite convenient. The complex potentials themselves have to be chosen in such a way that given boundary conditions can be fulfilled.
F;(z,)=
z,
+s,
(15)
Here, the quantities ri and si together are eight complex (or, equivalently, 16 real) constants. For their determination, the stated boundary conditions (12)-(14) yield the following 14 real linear equations: 2Re[ib: 5] = 0 2Re [icj 5] = 0 2Re[iej 5] = 0 2Re[i(j/j ej+ 2~)5] = 0 2Re[bjsj] = 0
3 CRACK S O L U T I O N F O R I N - P L A N E A N D BENDING LOADING
2Re[cj sj] = 0 2Re[ejsj] = 0
The problem to be considered now is that of an infinitely extended laminate plate with bending extension coupling containing a straight crack of length 2a (a Griffith crack) along the x-axis in such a way that the origin of the x-y-coordinate system coincides with the middle of the crack (Fig. 1 ). The faces of the crack are assumed to be traction-free which gives:
2Re[aj(~ + sj)] = N~'
(16)
2Re[bj(5+sj)]=Ny 2Re[cj(5+ sj.)]= N~y 2Re[dj(5 + sj)] = M~° 2Re[ej(5+ sj)] = My° 2Re[~( 5 + sj)]= M~
Ny=Nxy=My=9_y=O
for
]x]
y=0
Re[(pjj/j- qj)(5 + sj)] = 0 (12)
(summation over j = 1,..., 4)
52
W.Becker
From this set of 14 linear equations the complex constants can be ascertained in an easy way after an arbitrary one of them has been set to zero (non-uniqueness of the complex potentials). Although formally they look very simple, the complex potentials (15) imply the representation of all considered plate quantities within the cracked laminate plate. It is emphasized that, due to the incorporation of bending extension coupling, specific features are included that do not occur in symmetric laminate plates. As an example, a crack of length 2a = 20 mm in a simple [0°/90°]-cross-ply laminate plate with the following non-zero stiffness coefficients is considered: A tl = A22 = 145 800 N/mm
Fig. 2.
Bending moment M,. in the crack vicinity for loading by N ~.
Fig. 3.
Twist moment M~,. in the crack vicinity for loading by N,~?.
Ai2 =5430 N/mm A¢,r, = 10 000 N/mm Bll = -B22 = - 6 2 800 N
(17)
Dlj =D22=48 600 Nmm DI2= 1810 Nmm
co
D66 = 3330 Nmm As roots of the polynomial (8) the complex quantities/~i take the following values: /~l = i3.78 ~2= - 0 " 5 5 + i0.84 ~ = 0.55 + i0.84.
(18)
/~4 = i0.26 Under pure in-plane remote loading of N~Y= 1 N/mm not only membrane stress resultants but, due to bending extension coupling, also local plate bending moments are induced in the crack vicinity. At the crack tips not only the well known 1/,/r-membranestress singularities occur but also singularities can occur in the plate bending moments. The quantitative discussion of these coupling effects requires nothing but the straightforward exploitation of the complex potentials (15). As an example, Figs 2 and 3 show the resultant bending moment My and the twist moment M~,. in one of the quadrants of the crack vicinity. If, on the other hand, a remote bending moment M~° is applied I/f-r-singularities occur not only i n t h e plate bending moments but also in the membrane stress resultants. It is worthwhile to investigate the occurrence of 1/fr-singularities of the plate quantities in a more systematic manner. A closer inspection of the 14 linear equations (16) for the complex quantities r, and ss reveals that the crack tip singularities are
exclusively induced by the remote forces Ny, Nxy, M,7, and not by N T, MT, or Mxy. Therefore the latter three forces are no longer considered. In the crack vicinity it is convenient to introduce the variable ~i= zi- a = ~ + #it/ which describes the location of a point relative to the right crack tip. In fracture mechanics it is a common practice to approach the crack tip along the positive ~ axis and to characterize the singularity of a plate quantity by an intensity factor which for N x, for example, reads: co
K~x=lim 2 ~
Nx(a +~)
(19)
Here, the subscript N~ for the intensity factor K is used on purpose because in quite the same way intensity factors can be introduced for the other plate sectional forces Ny, Nxy, M~, and so on. The superscript ~ in eqn (19) is used to indicate the approach direction. In many cases the characterization of a 1/~r-singularity only by the intensity factor K.~ may be insufficient. For example, the bending moment Mr in Fig. 2 has a vanishing intensity factor K~, because My is identically zero along the ~-axis, but nevertheless there is a singularity at the crack tip. Such situations can be accounted for by a supplemental approach towards the crack tip along the q-axis and by a respective intensity factor K ~: K~ =lim 2 ~ r / Nx(a+#,rl)
(20)
Griffith crack in a non-symmetric laminate
Formally, the two intensity factors K * and K~ can be summarized by the introduction of a complex intensity factor, as follows:
=lira
,f2-~p[Nx(a+p)+ iNx(a+,ujp)]
p~O
Thus, the given influence quantities characterize the emergence of crack tip singularities in a concise and comprehensive way. The influence quantities k41, ksl, k61, k42, k52, k62, k13, k23 and k33 reflect the underlying bending extension coupling, whereas the other influence quantifies occur also in the case of uncoupled plate problems.
(21)
By use of relations (9) and (15) the complex intensity factors can be represented as
KN~=2,~ {Re[ajrj]+iRe[aj -~ rj]}
(22)
and by analogous relations in the case of the other plate sectional forces. As the intensity factors depend linearly on the oo oo oo applied remote forces Ny, Nxy, My, finally the following representation is introduced:
-kll KN~
53
--_
k13 -
k2~
k22
k23
k31
k32 k42
k33 k43
k4~
ksl k61
mgMxy
k12
Ny
FOR TRANSVERSAL
LOADING Finally, the case is considered that the laminate is loaded by prescribed Kirchhoff's substitute transversal forces Qx, - oo Qy - coat infinity (Fig. 4): -
~
-co
Qx Qx O>,~Qy
for
zi ~oo
(25)
Along the crack faces the presupposition of zero tractions is maintained. For the complex potential solution the following representation is chosen:
Fi(zi)=gi~-7 + hizi t!
k52 k53 k62
4 CRACK SOLUTION
respectively
(26)
k63
(23) Here, the quantities k 0 constitute complex-valued influence coefficients of the remote loadings on the induced singularities. The factor ~ in eqn (23) has been separated in order that the influence quantities k 0 are independent from the crack length. In the case of the considered cross-ply laminate with the stiffnesses (17), the introduced influence quantities take the following numerical values:
m Fi(zi)=gi -
Zi
P-hi
Here again, the quantities g/and h i are eight complex (or 16 real) constants. For their determination, the stated boundary conditions (12) and (25) yield only the following 10 real linear equations:
2Re[ibjgj]=O 2Re[icjgj]=0 2Re[iej gj] = 0
b
kll
k2t k31
kr2 k22 k~2 k42
k13 k23 k33
2Re[bihj] = 0
2Re[i(ktjej+2~)~] = 0
k41 ksj k52
k53
k43
2Re[cj hi ] = 0
k61
k63 _
2Re[ejhj]= 0
k6:
1"28 + 1"00 + - 0"19-
i0"63 - il-43 0"17 + i 0 . 0 9 i 1"55 - i0.32 i0"18 i0"30 1.00 + i0"36 - i0"05 i0"03 - i2"00 - 0 - 6 2 - i0"23 i0"33 i0"69 1"00 + i0-80 i0"09 0.71 + i0"67 - i0-18 (24
(27)
2Re[(/~j ej + 2~)hj)]=0 2Re[( 4 + 2/~j~)(gj+ hi)]= 0 7 2Re[(kt/ei+ 2~)(g/+ hj)] = ()y For a unique determination of the eight complex constants, another six real conditions can be formulated. With some arbitrariness here the fol-
W. Becker
54 Qy
Fig. 4.
Laminate plate with crack under transversal loading. ,k/
Q, in the crack vicinity for loading by Qy.-
Fig. 5.
~o
lowing behaviour is demanded at infinity:
Mx,y~O M V,X ~
r
0
Nxx-'O
N,.,~0
(28)
V, X ' ' ~ 0
Nv,y~ o
for
zi--"oo
-.tj
This gives the following six additional linear equations:
Fig. 6.
Q~in the crack vicinity for loading by Q~. Q
2Re[& 4(g/+ hi)l= 0 Q
2Re[e~(&+ h i)] = 0 2Re[a/(g/+ h/)] = 0 2Re[&a/(gi+ h i )] = 0
(29)
~0
¢
2Re[ bi(& + hi )] = O 2Re[& bj(&+ hy)]= 0 As a consequence, the membrane stress resultants vanish identically at infinity. Due to equilibrium requirements the plate bending moments on the other hand exhibit a linear functional behaviour sufficiently far away from the crack. The already considered cracked [0°/90°]-crossply laminate plate again is to be regarded as an example. A remote transversal loading of 0 ~ = 1 N/mm shall be applied. Then, in the crack vicinity the complex potentials (26) yield the behaviour of Ox and Qy as shown in Figs 5 and 6. A t the crack tips 1/4r-singularities occur in Q~, Qy, Qx, and Qy. The accompanying bending moments, of course, have no singularities. The asymptotically linear behaviour of My can easily be seen in Fig. 7. A very interesting feature of the given complex potential solution is that it also describes the occurrence of membrane stress resultants around the crack. This is remarkable insofar as no inplane forces are applied at infinity. As an example, Fig. 8 shows the behaviour of Nv in the crack
Fig. 7.
Mvin the crack vicinity for loading ' by Q,.. -"
Fig. 8.
N,. in the crack vicinity for loading by Q-v ®°
vicinity. It can be stated that the localization of N~ around the crack is distinctly less pronounced than that of the transversal forces. The same is true for a comparison to the behaviour of the membrane stress resultants and the plate bending moments in the case of applied remote in-plane and/or bending forces.
Griffith crack in a non-symmetric laminate
Finally, the occurring 1/K-singularities of the transversal plate forces are to be considered in some more detail. A closer inspection of the system of 16 real linear equations (27) and (29) for the complex quantities gi and h i reveals that the crack tip singularities are induced exclusively by the applied transversal force Qy-co, and not by Qx.~ For the characterization of the singularities, complex-valued intensity factors analogous to those of definition (21) can be introduced for the transversal plate forces, e.g:
Ke,= K~,+ iK~, =lim 2~p [Qx(a+p)+iQx(a+/ujp)] (30)
55
and by analogous relations for the other transversal force quantifies. Finally, the following representation with respective complex-valued influence coefficients ki is introduced: KQ,/_- k2
KO.,I
k3
Ko,J
k,
-co
(32)
Qy ~
The considered cross-ply laminate with the stiffnesses (17) gives rise to the following actual numerical values of the influence quantities:
p~O
By use of relations (11) and (26) the complex intensity factors can be represented as:
kl I [ - i0"78 ] k2 = 10"89+ i0-40[
k3 1-i0"74 k4
Ke.=2 ~ a {Re[(g +,ui~)g/]
(31)
/
(33)
L1.00 + i0-50j
In the case of vanishing bending extension coupling, in general, none of these quantities would be zero, but of course the actual numerical values of the influence coefficients k i are affected by the underlying bending extension coupling.
REFERENCES 1. Lekhnitskii, S. G., Anisotropic Plates. Gordon and Breach, New York, 1968. 2. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Leyden, 1975. 3. Green, A. E. & Zerna, W., Theoretical Elasticity (2rid edn). Clarendon Press, Oxford, 1968. 4. Sih, G. C., Pads, P. C. & Irwin, G. R., On cracks in rectilinearly anisotropic bodies. Int. Z Fracture Mechanics, 1 (1965) 189-203. 5. Tada, H., Paris, P. C. & Irwin, G. R., The Stress Analysis of Cracks Handbook (2rid edn). Paris Productions, St Louis, 1985.
6. Becker, W. & Gross, D., About the mode II Dugdale crack solution. Int. Z Fracture, 34 (1987) 65-70. 7. Becker, W. & Gross, D., About the Dugdale crack under mixed mode loading. Int. J. Fracture, 37 (1988) 163-70. 8. Becker, W., The problem of a Dugdale crack in a plate under normal bending. Engineering Fracture Mechanics, 34 (1989) 1063-7. 9. Becker, W., A complex potential method for plate problems with bending extension coupling. Archive of Applied Mechanics, 61 ( 1991 ) 318-26. 10. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975.
Closed-form analytical solutions for a Griffith crack in a non-symmetric laminate plate Wilfried Becker Dornier GmbH, DeutscheAerospace, Abteilung TMBF, Pos(fach 1420, D-7990Friedrichshafen-1, FRG Within the framework of linear-elastic laminated plate theory, the problem of a Griffith crack in an infinitely extended unsymmetric laminate plate is treated. Due to the non-symmetric layup, bending extension coupling is admitted and is taken into account by means of a new complex potential method. Closed-form analytical solutions are given for homogeneous in-plane, bending and transversal loading of the laminate plate. The considered bending extension coupling is accompanied by interesting characteristics of the plate behaviour in the crack vicinity, e.g. additional singularities occur at the crack tips.
1 INTRODUCTION In the case of symmetric laminate plates, the static analysis usually uncouples into two different problems, the one of pure in-plane deformations with pure in-plane forces and the other of out-ofplane deformations with bending and transversal forces. In the case of unsymmetric laminate plates, however, in general this is no longer true. Bending extension coupling can occur in various ways which in the constitutive plate relations is reflected by respective coupling stiffnesses. The problem to be considered in the following is that of a simple finite straight crack (a Griffith crack) through such a laminate plate with bending extension coupling. The faces of the crack are assumed to be free of any tractions whereas the plate is assumed to be loaded by prescribed constant remote forces. In the absence of bending extension coupling complex potential methods 1-3 have proved to be very appropriate for the representation of crack solutions; 4'5 the author has also achieved good results with the complex approach. 6-8 In the present case with bending extension coupling, an accordingly generalized complex potential method has been formulated recently. 9 By means of this method, closed-form analytical solutions are given for the present crack problem. The solutions reveal interesting effects of the bending extension coupling in the crack vicinity. In particular, remote in-plane forces may give rise
to crack tip singularities of the plate bending moments. Similarly, remote bending forces may cause crack tip singularities of the membrane stress resultants. 2 COMPLEX POTENTIAL METHOD Within classical laminated plate theory bending extension coupling is taken into account by respective coupling stiffnesses Bij in the laminate stiffness matrix by which the constitutive plate behaviour can be given as follows: l° m
--
Nxy =
Mx
My
_Mxy_ -Aij A~2
Ai2 A2~
A26 B12 B22
AI6
Bl6 026
ex 8y
AI6
A26
A66
Bu
BI2 BI6 DII Dl2
B66
Yxy
D16
Kx
Bl2 816
B22 B26
B26 B66
Bu BI6 D12 DI6
Bl2 B26 D22 D26
026
D66 -
Ky
_Kxy
(1) Here, the strains ex, ey, 7xy and curvatures rx, ry, r~y of the laminate mid-plane result in the following way from the in-plane displacements u, v and
49 Composite Structures 0263-8223/92/S05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
W. Becker
50 the normal displacement w : e x = blx
~¢y= U ,y
Yxy = lgy "l" U ,x
~x ---- -- W,xx
l(y = -- W,yy
~xy----- - 2 W xr
(2)
The quantities N x, Ny, Nxy, Mx, My, Mxy denote the usual membrane stress resultants and plate bending moments, respectively, which on their part have to satisfy the equilibrium conditions:
N u + Nxy,y=O Uxy,x + N,,y = 0
(3)
By successive substitution of relations (1) and (2) into the equilibrium conditions the following three partial differential equations are obtained for the mid-plane displacements u, v, w :
Ai6U,xx
-
a -
-
[
-
Ozi
0 0 -
-
-
0
(6)
ig~iOy-l~iigzi+f~iOzi
If identically satisfied equilibrium is demanded for arbitrarily chosen potentials Fi(zi), this leads to the following relations:
+ Az6/~~)q~-(B~j + 3B.6/xi+ (B,2 + 2B66)/~~
+ BE6,u ~) = 0 +
+ 3B26,u 2 +
+ (m12 +A66)u xy+ A26v yy- BII W, xxx
-- B26 W y)f =
igx
_
+
+
+ 2A2
B22,u 3) = 0
(B,,
2B66)Wxyy
+
+(Bi6 +(B12 + 2B66),u~+ 3B26fl~ +
0
B22,u3)qi
+(AI2 + A66)U xy + A26tl,yy + A66U,xa
- (Ot] +4DI6,ui+ (2Di2 + 4D66),u ~
+ 2A26v xy+ A22va,y- Bl6Wxxx
+ 4Dz6/~ ~+ D22/~.) = 0
B l l U xxx + 3 B i 6 u xxy +
+ B26 U,yyy+
(BI2 +
0
2 B 6 6 ) U xyy
Bl6Vxxx+ (BI2 + 2B66)V~xy
+ 3 B 2 6 u x y y + B22Uyyy -
-4DI6wxxxy-(2Dr2 -- 4D26 W x ~ y -
Dii W xx~x
Q0 + QJkt + Q2fl 2 + Q3,tt3 + Q4,tt4+ Qs/~~
+4D66)W,x~yy
D22 W yyyy = 0
+ Q6# 6 + Q7/~7 + Qs# ~ = 0
(4)
For the intended complex approach, four complex potentials F~(z~), i = 1, ..., 4, are introduced. Each of these potentials is a function of a complex variable of the kind z~= x + #~y where/~ are complex quantities still to be determined. By use of the potentials F i and their first derivatives F'i the following representation is chosen for the displacements u, v and w:
(8)
is obtained whose roots are the needed quantities /~,...,/~4 together with their complex conjugates. With relations (1), (2) and (5) the membrane stress resultants and plate bending moments can be represented by the second derivatives of the potentials F i, as follows: It
Nx = 2Re[aiFi (zi)] Ny = 2Re[biF"(zi)] Nxy = 2Re[ciF"(zi)]
W= 2Re[F~(z]) + F2(z2) + F3(Z 3) + F4(Z4)] u = 2 R e [ p i F ~ i ( zi)]
(7)
For each i the relations (7) represent a non-linear system of three equations from which the constants pi, qg and /~i can be determined. This is described in more detail in Ref. 9. In particular, after elimination of the quantities p~ and q~, a polynomial of the kind
-- (2B66 + Bi2)Wxxy
- 3B26 W,xyy -- B22 W yyy =
,,
+ A22,u2)qi-(Bl6 +(2B66 + Bt2),ui
A66U,yy + A i 6 v xx
-3Bi6W~xy-(BI2 +
a
(A~, + 2ml6kti+ A66,u2i)Pi + (AI6 +(m,2 + m66)/ti
M~.~ + 2M~y,xy + My,yy=0
A~u, xx + 2AI6 U,x.v+
tities Pi and qi are also complex constants still to be determined. To this end the displacement representations (5) are substituted into the equilibrium conditions (4), taking into account
(5)
v = 2Re[qiF'i(z,)] In order to avoid unnecessary lengthiness the two last equations are given in a condensed way by use of the implicit summation convention. The quan-
Mx = 2Re[diF~(zi) ]
(9)
My = 2Re[eiF~(zi)] Mxy= 2Re[ fiF"i ( zi )] Here, the quantities a~, b~, c~, di, e i and f are complex constants which result in the following way
Griffith crack in a non-symmetric laminate
51 oo
from p~, qi and J/i:
Ny
oo Mx7
- ai"
bi ci de ei -A u AI2 Al6 Bu Bt2 BI6
f ~
- Nx
Fig. 1. Laminateplatewithcrackunderin-planeand bendingloading. AI2 A22 A26 Bl2 B22 B26
A16 Bu B12 B16- -Pi A26 A66 BI6 B26 B66
BI2 BI6 Du DI2 Di6
B22 B26 D12 D22 D26
B26 B66 DI6 D26 D66
q ij/ i Pij/i + qi -1
oo
-- 2j/i
Ny --"Ny~
Qy= My,y+ Mxy,x = 2Re[(j/iei + fi)FT(zi)]
2J/J,)Fitit (zi)]
oo
Nxy-"Nxy
co
~
MxMx
co
oo
My-" My~ _.~
oo
(13) oo
Mxy M,y
for
zi ~ oo
In addition, at infinity a rigid body rotation with respect to the z-axis shall be excluded: lgy--*Ux for
zi ~ oo
(14)
For the boundary value problem stated in this way the solution can be given in the form of the following complex potentials:
Qx = Mx,x + Mxy,y= 2Re[( di + j/if)F['(zi)]
Or = Qy + Mxy,x = 2Re[(j/iei +
co
NxNx
Finally, the transversal plate forces Qx, Qj and Kirchhoff's substitute transversal forces Q~, Qy are of interest. They can be represented by the third derivatives of the potentials F~as follows:
...1_
oo
-j/2i
(lO)
__ Qx=Qx+ Mxy.y-2Re[(di
The laminate plate is loaded by prescribed forces N~, Ny , Nxy and moments Mx , My, Mxy at infinity:
(11)
2f)F~"(zi)]
In the complex setting presented so far the underlying bending extension coupling is ultimately taken into account by the actual numerical value of the introduced complex constants j/i, Pi, qi, a~, ...,f/. In this way, the consideration of bending extension coupling is quite convenient. The complex potentials themselves have to be chosen in such a way that given boundary conditions can be fulfilled.
F;(z,)=
z,
+s,
(15)
Here, the quantities ri and si together are eight complex (or, equivalently, 16 real) constants. For their determination, the stated boundary conditions (12)-(14) yield the following 14 real linear equations: 2Re[ib: 5] = 0 2Re [icj 5] = 0 2Re[iej 5] = 0 2Re[i(j/j ej+ 2~)5] = 0 2Re[bjsj] = 0
3 CRACK S O L U T I O N F O R I N - P L A N E A N D BENDING LOADING
2Re[cj sj] = 0 2Re[ejsj] = 0
The problem to be considered now is that of an infinitely extended laminate plate with bending extension coupling containing a straight crack of length 2a (a Griffith crack) along the x-axis in such a way that the origin of the x-y-coordinate system coincides with the middle of the crack (Fig. 1 ). The faces of the crack are assumed to be traction-free which gives:
2Re[aj(~ + sj)] = N~'
(16)
2Re[bj(5+sj)]=Ny 2Re[cj(5+ sj.)]= N~y 2Re[dj(5 + sj)] = M~° 2Re[ej(5+ sj)] = My° 2Re[~( 5 + sj)]= M~
Ny=Nxy=My=9_y=O
for
]x]
y=0
Re[(pjj/j- qj)(5 + sj)] = 0 (12)
(summation over j = 1,..., 4)
52
W.Becker
From this set of 14 linear equations the complex constants can be ascertained in an easy way after an arbitrary one of them has been set to zero (non-uniqueness of the complex potentials). Although formally they look very simple, the complex potentials (15) imply the representation of all considered plate quantities within the cracked laminate plate. It is emphasized that, due to the incorporation of bending extension coupling, specific features are included that do not occur in symmetric laminate plates. As an example, a crack of length 2a = 20 mm in a simple [0°/90°]-cross-ply laminate plate with the following non-zero stiffness coefficients is considered: A tl = A22 = 145 800 N/mm
Fig. 2.
Bending moment M,. in the crack vicinity for loading by N ~.
Fig. 3.
Twist moment M~,. in the crack vicinity for loading by N,~?.
Ai2 =5430 N/mm A¢,r, = 10 000 N/mm Bll = -B22 = - 6 2 800 N
(17)
Dlj =D22=48 600 Nmm DI2= 1810 Nmm
co
D66 = 3330 Nmm As roots of the polynomial (8) the complex quantities/~i take the following values: /~l = i3.78 ~2= - 0 " 5 5 + i0.84 ~ = 0.55 + i0.84.
(18)
/~4 = i0.26 Under pure in-plane remote loading of N~Y= 1 N/mm not only membrane stress resultants but, due to bending extension coupling, also local plate bending moments are induced in the crack vicinity. At the crack tips not only the well known 1/,/r-membranestress singularities occur but also singularities can occur in the plate bending moments. The quantitative discussion of these coupling effects requires nothing but the straightforward exploitation of the complex potentials (15). As an example, Figs 2 and 3 show the resultant bending moment My and the twist moment M~,. in one of the quadrants of the crack vicinity. If, on the other hand, a remote bending moment M~° is applied I/f-r-singularities occur not only i n t h e plate bending moments but also in the membrane stress resultants. It is worthwhile to investigate the occurrence of 1/fr-singularities of the plate quantities in a more systematic manner. A closer inspection of the 14 linear equations (16) for the complex quantities r, and ss reveals that the crack tip singularities are
exclusively induced by the remote forces Ny, Nxy, M,7, and not by N T, MT, or Mxy. Therefore the latter three forces are no longer considered. In the crack vicinity it is convenient to introduce the variable ~i= zi- a = ~ + #it/ which describes the location of a point relative to the right crack tip. In fracture mechanics it is a common practice to approach the crack tip along the positive ~ axis and to characterize the singularity of a plate quantity by an intensity factor which for N x, for example, reads: co
K~x=lim 2 ~
Nx(a +~)
(19)
Here, the subscript N~ for the intensity factor K is used on purpose because in quite the same way intensity factors can be introduced for the other plate sectional forces Ny, Nxy, M~, and so on. The superscript ~ in eqn (19) is used to indicate the approach direction. In many cases the characterization of a 1/~r-singularity only by the intensity factor K.~ may be insufficient. For example, the bending moment Mr in Fig. 2 has a vanishing intensity factor K~, because My is identically zero along the ~-axis, but nevertheless there is a singularity at the crack tip. Such situations can be accounted for by a supplemental approach towards the crack tip along the q-axis and by a respective intensity factor K ~: K~ =lim 2 ~ r / Nx(a+#,rl)
(20)
Griffith crack in a non-symmetric laminate
Formally, the two intensity factors K * and K~ can be summarized by the introduction of a complex intensity factor, as follows:
=lira
,f2-~p[Nx(a+p)+ iNx(a+,ujp)]
p~O
Thus, the given influence quantities characterize the emergence of crack tip singularities in a concise and comprehensive way. The influence quantities k41, ksl, k61, k42, k52, k62, k13, k23 and k33 reflect the underlying bending extension coupling, whereas the other influence quantifies occur also in the case of uncoupled plate problems.
(21)
By use of relations (9) and (15) the complex intensity factors can be represented as
KN~=2,~ {Re[ajrj]+iRe[aj -~ rj]}
(22)
and by analogous relations in the case of the other plate sectional forces. As the intensity factors depend linearly on the oo oo oo applied remote forces Ny, Nxy, My, finally the following representation is introduced:
-kll KN~
53
--_
k13 -
k2~
k22
k23
k31
k32 k42
k33 k43
k4~
ksl k61
mgMxy
k12
Ny
FOR TRANSVERSAL
LOADING Finally, the case is considered that the laminate is loaded by prescribed Kirchhoff's substitute transversal forces Qx, - oo Qy - coat infinity (Fig. 4): -
~
-co
Qx Qx O>,~Qy
for
zi ~oo
(25)
Along the crack faces the presupposition of zero tractions is maintained. For the complex potential solution the following representation is chosen:
Fi(zi)=gi~-7 + hizi t!
k52 k53 k62
4 CRACK SOLUTION
respectively
(26)
k63
(23) Here, the quantities k 0 constitute complex-valued influence coefficients of the remote loadings on the induced singularities. The factor ~ in eqn (23) has been separated in order that the influence quantities k 0 are independent from the crack length. In the case of the considered cross-ply laminate with the stiffnesses (17), the introduced influence quantities take the following numerical values:
m Fi(zi)=gi -
Zi
P-hi
Here again, the quantities g/and h i are eight complex (or 16 real) constants. For their determination, the stated boundary conditions (12) and (25) yield only the following 10 real linear equations:
2Re[ibjgj]=O 2Re[icjgj]=0 2Re[iej gj] = 0
b
kll
k2t k31
kr2 k22 k~2 k42
k13 k23 k33
2Re[bihj] = 0
2Re[i(ktjej+2~)~] = 0
k41 ksj k52
k53
k43
2Re[cj hi ] = 0
k61
k63 _
2Re[ejhj]= 0
k6:
1"28 + 1"00 + - 0"19-
i0"63 - il-43 0"17 + i 0 . 0 9 i 1"55 - i0.32 i0"18 i0"30 1.00 + i0"36 - i0"05 i0"03 - i2"00 - 0 - 6 2 - i0"23 i0"33 i0"69 1"00 + i0-80 i0"09 0.71 + i0"67 - i0-18 (24
(27)
2Re[(/~j ej + 2~)hj)]=0 2Re[( 4 + 2/~j~)(gj+ hi)]= 0 7 2Re[(kt/ei+ 2~)(g/+ hj)] = ()y For a unique determination of the eight complex constants, another six real conditions can be formulated. With some arbitrariness here the fol-
W. Becker
54 Qy
Fig. 4.
Laminate plate with crack under transversal loading. ,k/
Q, in the crack vicinity for loading by Qy.-
Fig. 5.
~o
lowing behaviour is demanded at infinity:
Mx,y~O M V,X ~
r
0
Nxx-'O
N,.,~0
(28)
V, X ' ' ~ 0
Nv,y~ o
for
zi--"oo
-.tj
This gives the following six additional linear equations:
Fig. 6.
Q~in the crack vicinity for loading by Q~. Q
2Re[& 4(g/+ hi)l= 0 Q
2Re[e~(&+ h i)] = 0 2Re[a/(g/+ h/)] = 0 2Re[&a/(gi+ h i )] = 0
(29)
~0
¢
2Re[ bi(& + hi )] = O 2Re[& bj(&+ hy)]= 0 As a consequence, the membrane stress resultants vanish identically at infinity. Due to equilibrium requirements the plate bending moments on the other hand exhibit a linear functional behaviour sufficiently far away from the crack. The already considered cracked [0°/90°]-crossply laminate plate again is to be regarded as an example. A remote transversal loading of 0 ~ = 1 N/mm shall be applied. Then, in the crack vicinity the complex potentials (26) yield the behaviour of Ox and Qy as shown in Figs 5 and 6. A t the crack tips 1/4r-singularities occur in Q~, Qy, Qx, and Qy. The accompanying bending moments, of course, have no singularities. The asymptotically linear behaviour of My can easily be seen in Fig. 7. A very interesting feature of the given complex potential solution is that it also describes the occurrence of membrane stress resultants around the crack. This is remarkable insofar as no inplane forces are applied at infinity. As an example, Fig. 8 shows the behaviour of Nv in the crack
Fig. 7.
Mvin the crack vicinity for loading ' by Q,.. -"
Fig. 8.
N,. in the crack vicinity for loading by Q-v ®°
vicinity. It can be stated that the localization of N~ around the crack is distinctly less pronounced than that of the transversal forces. The same is true for a comparison to the behaviour of the membrane stress resultants and the plate bending moments in the case of applied remote in-plane and/or bending forces.
Griffith crack in a non-symmetric laminate
Finally, the occurring 1/K-singularities of the transversal plate forces are to be considered in some more detail. A closer inspection of the system of 16 real linear equations (27) and (29) for the complex quantities gi and h i reveals that the crack tip singularities are induced exclusively by the applied transversal force Qy-co, and not by Qx.~ For the characterization of the singularities, complex-valued intensity factors analogous to those of definition (21) can be introduced for the transversal plate forces, e.g:
Ke,= K~,+ iK~, =lim 2~p [Qx(a+p)+iQx(a+/ujp)] (30)
55
and by analogous relations for the other transversal force quantifies. Finally, the following representation with respective complex-valued influence coefficients ki is introduced: KQ,/_- k2
KO.,I
k3
Ko,J
k,
-co
(32)
Qy ~
The considered cross-ply laminate with the stiffnesses (17) gives rise to the following actual numerical values of the influence quantities:
p~O
By use of relations (11) and (26) the complex intensity factors can be represented as:
kl I [ - i0"78 ] k2 = 10"89+ i0-40[
k3 1-i0"74 k4
Ke.=2 ~ a {Re[(g +,ui~)g/]
(31)
/
(33)
L1.00 + i0-50j
In the case of vanishing bending extension coupling, in general, none of these quantities would be zero, but of course the actual numerical values of the influence coefficients k i are affected by the underlying bending extension coupling.
REFERENCES 1. Lekhnitskii, S. G., Anisotropic Plates. Gordon and Breach, New York, 1968. 2. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Leyden, 1975. 3. Green, A. E. & Zerna, W., Theoretical Elasticity (2rid edn). Clarendon Press, Oxford, 1968. 4. Sih, G. C., Pads, P. C. & Irwin, G. R., On cracks in rectilinearly anisotropic bodies. Int. Z Fracture Mechanics, 1 (1965) 189-203. 5. Tada, H., Paris, P. C. & Irwin, G. R., The Stress Analysis of Cracks Handbook (2rid edn). Paris Productions, St Louis, 1985.
6. Becker, W. & Gross, D., About the mode II Dugdale crack solution. Int. Z Fracture, 34 (1987) 65-70. 7. Becker, W. & Gross, D., About the Dugdale crack under mixed mode loading. Int. J. Fracture, 37 (1988) 163-70. 8. Becker, W., The problem of a Dugdale crack in a plate under normal bending. Engineering Fracture Mechanics, 34 (1989) 1063-7. 9. Becker, W., A complex potential method for plate problems with bending extension coupling. Archive of Applied Mechanics, 61 ( 1991 ) 318-26. 10. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975.