Pergamon

Mechanics Research Communicatmns, Vol. 28, No. 4. pp. 421428,200l

Copyright 0 2001 Elsewer Science Ltd

Printed I” the USA. All rights reserved

0093-64 13/O I l&see front matter

PII: SOO93-6413(01)00192-6

EXTENSION OF LEKHNITSKII’S COMPLEX POTENTIAL APPROACH TO UNSYMMETRIC COMPOSITE LAMINATES

Chen Puhui, Shen Zhen Aircraft Strength Research Institute, Xian, Shaanxi, 710065, P. R. China e-mail: Phchen@pub.xaonline.com

As well known, the most famous theories for the two-dimensional problems, such as generalized plane strain and generalized plane stress, of anisotropic plates containing elliptic holes and cracks are the Lekhnitskii’s complex potential approach [I, 21 and the Eshelby-Stroh formalism [3, 41. Analytic solutions for an infinite anisotropic plate containing an elliptic hole under uniform loading can be found in the works of Lekhnitskii [I, 21 and Savin [5]. In references [6, 7, 8, 9, IO], the fundamental solutions (Green’s functions) were derived for an infinite anisotropic plate containing an elliptic hole subjected to in-plane concentrated force and moment. These solutions have been widely used in the stress and deformation analysis of symmetric composite laminate containing holes and cracks. However, they can not be applied to unsymmetric composite laminates due to the coupling of bending and extension. Thus, in this paper, the classical Lekhnitskii’s complex potential approach was extended to the unsymmetric laminates.

The governing equations of unsymmetric laminates have been given in [ 1 I]. For a composite laminate, its in-plane displacements u, (i=1,2) depend on all three coordinates x, y and z, that is [ 121

6+v biv u=u,,-z--, v=v,,-z-

c% & where a,,, vO, and w are laminate midplane displacements in x, y and z directions, respectively. Such displacements induce that laminate strains and stresses vary through the laminate thickness

where ~0 and K are laminate midplane strains and curvatures, respectively. Consider the laminate as a homogenous anisotropic plate, the constitutive equation for the laminate can be expressed in terms of the midplane strains and curvatures [ 121

where

423

are extensional, coupling and bending stiffness matrixes, respectively; and

N=[K NV N.k M=[W Mv kr are membrane stress resultants and laminate moments, respectively; and

are laminate midplane strains and curvatures, respectively. Partial inversion of (3) results in

where A’ = A-‘, B’ = -A-‘B , D’ = D - B&B

The equilibrium equations without body forces and the compatibility equation of a laminate are [I21

x +s=O 8N.p aNv d’M, +2a’Kv I d=Mv

ax ay -+-=o, -

‘ax 4, ax2 WJ w

where 9 is the specified load applied normal to the laminate. Introduce the stress function F(x,y) such that in the absence of body forces the stress resultants can be written as

which satisfy the first two equations of (10). Substituting (7) and (12) into (8) and the results into the third equation of (IO) and (I I) yield the following system of simultaneous fourth-order partial differential equations for the problem

L,,,F + L ,,,/, w = 4 1 L,,J - L,,,w = 0 (13)

where Lfifi, L,, and L,,- are differential operators of order four, and given by aA

L,, = Ai2 $ - 2Ak - a'

+(2AG +A6)-- 2A. 8 aI

__ axJay ax7y'

+ A;, - I6 ax*' ag

(144

Thus, the problem is governed by a system of two partial differential equations. For an unsymmetric laminate, the coupling stiffness B,, is non-zero, and thus the two equations of (13) are coupled through the operator L,,-. These equations can be simplified significantly when the laminate is symmetric one with respect to the middle plane. In this case, the coupling stiffness B,, is equal to zero, which induce that no coupling between bending and extension (LlrF = 0), and the two equations of (13) can be solved separately for in-plane loading and bending problems as shown in [l, 21.

The general solutions of (13) are constructed from the homogeneous and particular solutions. The particular solutions, denoted by F,, and w,,, depend on the distribution of the transverse load q. The homogeneous solutions are the major concern in this paper and derived below. By eliminating one of the two functions, the homogeneous equations

L,,,F + L,,,, w = 0 (154

L,,F - L,,, w = 0 (15b)

Lh’SYMiMETRIC COMPOSITE LAMINATE 4?5

are reduced to the following single eighth-order partial differential equation for U (&fJ ,,,,, + &,,; L,,, )U = 0

where U denotes F and W.

(16)

Equation (16) can be solved by the solution method similar to that developed by Lekhnitskii [l, 21. The general solution of (16) can be expressed as

U(Z) = c/(x + m), /J = a + ip, i = G (17)

where z is a generalized complex variable, p is a complex parameter, and a and pare real numbers. By introducing (17) into (16) and using the chain rule of differentiation, the following equation is obtained.

{M/X,,(P) + C,(P)}$ = 0 (18)

where

I,(P) = A,*,P* -2,-f& + (2A;, + A,$,)$ - 2A;,fi + A;, (l9a)

I,<,>$ (P) = D;J + 4%~’ + 2(D;, + 2%)~’ + ~D;,P + D:, (l9b)

UP) = BLP’ + (2& - %)P’ + (B;, + B;, - 28X + WI - B,‘,)P + B;, (l9c)

By setting the quantity enclosed within braces in (18) equal to zero, one can obtain the following characteristic equation

MPX,, (P) + CF (P) = 0 (20)

Assuming the eight roots, ,u~ and pr (k=l, 2, 3, 4), of (20) are distinct, the general form solutions for (16) can be

written as

F(x,y)=2Re~F,(x+~,y)=2Re~~F,(z,) (21a)

w(x,y)=2Re‘~w~(x+~~y)=2Re~w,(z,) (21b)

where zl(k = 1, 2, 3,4) are generalized complex variables, and Fk(zr) and w,(z,) are two holomorphic functions.

According to (I 5), the two functions F, (zl ) and w1 (z, ) have the following relation

w”“(z ) = ‘v, (& ) F’“‘(z ) = /1 (P )F’“‘(z, ) I L l,,,r (Pk) ‘ ’ ‘ ‘ ‘

(22)

where the (“‘I) denotes the fourth derivative of the functions F, (z, ) and w, (z, ) with respect to their arguments. Integration of (22) yields

w, (z, ) = 1, (P, )F, (21) (23) where the arbitrary terms of integration are set equal to zero. If they are retained, they can be embedded in the same power terms of holomorphic function F, (z, ) [13].

The complete solutions of stresses, strains and deformations for an unsymmetric laminate are the sum of the homogenous solutions corresponding to Fk(z,) , and particular solutions corresponding to F,, and wO.

The homogenous solutions are derived below. Introduce new functions p’r (z, ) defined as

$7‘ (z, ) = y (24)

Substituting (2 l) into (7) and (12) and with the aid of (23) and (24), the midplane curvatures and the stress resultants can be obtained as

~~ =-2Re~[~k(~,)~~(zk~l (254 ~~ =-2Re~[~~(~,)~:q,(z,)l Wb) K,” = -4 Re $[A bk )WP~ (z,)l (25~)

-126 C PUHUI and S. ZHEN

N, =2Re$kk(z,)1 NV =2Re&bi(z,)l

Nrv = -2 Re@wi k 11 Substituting (25) and (26) into (8) results in the midplane strains and the laminate moments

&: =2Re$rl,,(~,)(o;(z,)]

4 =2Re&[r~,(~,k(z,)]

4 =2Re~[~,,(~~)a,(z,)] 1-1

MI =2Re&~,,(~,)al(z,)] ,=I

MV =~R~&P,,(P,)P;(~,)] ‘=I

where

[ rl,, % VI‘ PIA hi PUP = _",*, [' 1 ;I k 1 -4 -4 -4d -2Q4y Using (27) and the strain-displacement relationships

au E:‘X”,, & E:‘=_L, Egl = au, + Vn ax ay IY ay ax

the integration of the normal strains, with the aid of (1 1), results in the midplane displacements

u,, =2Re~[~,,(~)~(z~)l+~+~~

v,, =2Re$[Fpk(z,)]-on+v,

where the constants w, u;, vi represent rigid body displacements.

The equilibrium of the laminate yields the transverse shear force components [I]

Q, =!?$+F?!$

and

!?%+aQ,=, ax itv (33)

Substituting (28) into (32) yields

Q, =2Re~[p,,(~~)+~,p,,(p,)~;(r,) (344

Q, =zRe~[p,,(~,)+~,p,,(~,lbl,(z,f (34b)

By using (33), one can obtain another two equivak expressions of the shear force components, namely

Q, =2Re~[p,,(p,)+r,p,,(p,)b;(z,) (354

(264

(26b)

(26~)

(274

W’b)

(274

(28a)

(28b)

(28~)

(29)

(30)

(314

@lb)

(324

Wb)

Q =_~~~~Plr(~r)+~iP,*(~*) ,, A=1

Wb)

L'T\r'SYTvlMETRICCOMPOSITELAMINATE 437

and

WW

(36b)

Once the solutions corresponding to the particular solutions F,, and IV, of the governing equation (13) are determined, the sum of the homogenous and particular solutions gives the complete solutions.

The problem as above formulated is still undetermined. The complex potentials need to be determined subject to certain boundary and jump conditions on the boundary (or surfaces of discontinuity). In the following, assume that an unsymmetric laminate is subjected to in-plane traction components (X,,(s), Y,,(s)), bending moment m,,(s), twisting moment M,,,(s) and transverse shear force Q,,(s) per unit length, acting along the arc of the boundaries. In order to reduce the complexity of the derivation, it is convenient to set the transverse load 9 equal to zero. It should be indicated that no any difftculties arise in the derivation for 4 # 0

The boundary conditions can bc written as X” = N, cos(n, x) + N,, cos(n, y) (37a)

y” = NXV cos(n, x) + NV cos(n, y) (37b)

M* = m.(s) (38)

aM P(S) = Q.(s,+~ as (39)

Integration of (39) yields

Q+Mn, =fG)+c (40) where s is the length of the arc AB in the boundaries, and C is the arbitrary constant, and

Q = lQ& 9 f(s)= @Ws (41)

Substituting (26) and

cos(n,x) = $, cos(n,y) = -$ (42)

into (37), integration of (37) yields

(43b)

where C, and C, are the arbitrary constants, With the aid of the following equations [I]

Mn = M, cos’(n,x)+Mv c0s’(n,y)+2M,~ cos(n,x)cos(n,y) (44)

Mn, = (Mv - M,)cos(n,x)cos(n,y) + M_[cos*(n,x)-cos’(n,y)] (45)

Q, = Q.cos(n,x)+Q"cos(n,y) (46) The boundary conditions (38) and (40) can be expressed as

M, cos*(n,x)+Mvcos’(n,y)+2M~ cos(n,x)cos(n,y) = m.(s) (47)

(M~-M~)cos(n,x)cos(n,y)+M~[cos’(n,x)-cos’(n,y)l+e=f(s)+C (48) Multiplying both sides of (47) and (48) with cos(n,x) and cos(n,y) respectively, and subtracting (48) from (47), one can obtain

MA-(M,, -QW=mndy+U+CW (49) Multiplying both sides of (47) and (48) with cos(n,y) and cos(n,x) respectively, and summing of (47) and (48), one can obtain

(M_ +Q)dy-Mvdr=-m”dx+(f +C)dy (50) Substituting (46) into the first equation of (41), and using (42) lead to

4’ 8 C PUHUI and S. ZHEN

4? = I(tQ,+ - Q&l Substituting (35) into (5 1) yields

(51)

(52)

(53)

Substituting (28) and (52) into (49), integration of (49) yields

2Ret P,,(K) , p&(2,)= J(m&+fdu)+Cx+C; X=l p,

C; is the arbitrary constant.

Substituting (36) into (5 1) yields

Q= -2Re~[p,,(~,)+~,p,,(~,)Ep;(s,) [ 1 1=1 IB

Substituting (28) and (54) into (50), integration of (50) yields

2Retp,,(p,)%(s,)= [(m&-fh)-Cy+C; 1=,

C; is the arbitrary constant.

(54)

(55)

Equations (43) (53) and (55) are the boundary conditions of the problem expressed by the complex potential functions.

The classical Lekhnitskii’s complex potential approach has been extended to the stress and deformation analysis of unsymmetric composite laminates, and the stresses, deformation and boundary conditions have been expressed by four holomorphic functions, The developed method can be applied to the stress and deformation analysis of unsymmetric composite laminates containing holes and cracks subjected to in-plane loading and bending. In future papers, closed-form solutions will be derived for an infinite unsymmetric laminate containing an elliptic hole under in-plane loading and bending respectively. Also, the fundamental solutions (Green’s functions) will be derived for an unsymmetric laminate containing an elliptic hole.

I. 2. 3.

4. 5. 6.

I.

8.

9.

IO.

11.

12. 13.

Lekhniskii, S. G., Anisotropic Plates, Gordon and Brench, New York (1968). Lekhniskii, S. G., Theory ofEhsticity ofh Anisotropic Body, Mir Publishers, Moscow (1981). Eshelby, J. D. et al, Anisotropic Elasticity with Applications to Dislocation Theory, Acta. Metall., Vol.1, p. 25 1 (1953). Stroh, A. N., Dislocation and Cracks in Anisotropic Elasticity, Phil. Mug., Vol. 7, p. 625 (1958). Savin, G. N., Stress Concenrrarion Around Holes, Pergamon Press, London (1961). Kamel, M. and Liaw, B. M., Green’s Functions due to Concentrated Moments Applied in An Anisotropic Plane with An Elliptic Hole or A Crack, Me& Res. Communs l6(5), p. 311 (1989a). Kamel, M. and Liaw, B. M., Analysis of A Loaded Elliptical Hole or Crack in An Anisotropic Plane, Mech. Res. Communs 16(6), p. 379 (1989b). Hwu, C. and Yen, W. J., Green’s Functions of Two-Dimensional Anisotropic Plates Containing An Elliptic Hole, fnt. J. Solids Sfrucfures 27(13), p.1705 (1991). Gao C. F., Problem of An Anisotropic Plate with An Elliptic Hole or A Crack under Concentrated Load, Chinese Journal of Apphed Mechanics, Vol. 10, No. 3, p. 49 (1993). Chen Puhui et al, A solution Method for An Anisotropic Plate with An Elliptic Hole under Concentrated Force and Moment, ACTA MECHANICA SOLIDA SINICA, Vol. 19, S. Issue, p, 34 (1998). Dong, S. B., Pister, K. S. and Tayer, R. L., On the Theory of Laminated Anisotropic Shells and Plates, .I ofthe Aerospace Sciences, Vol. 29, p.969 (1962). Jones, R. M., Mechanics of Composife Materials, Scripta Book Comp., Washington, D. C. (1975) Sosa, H. A., Plane Problems in Piezoelectric Media with Defects, Int. J. Solids Sfrucrures 28(4), p. 491 (1991).