Analysis method for bonded patch repair of a skin with a cutout

Composite Structures 55 (2002) 277±294

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Analysis method for bonded patch repair of a skin with a cutout A. Barut a, J. Hanauska a, E. Madenci a

a,*

, D.R. Ambur

b

Department of Aerospace and Mechanical Engineering, The University of Arizona, P.O. Box 210119, Tucson, AZ 85721, USA b Mechanics and Durability Branch, NASA Langley Research Center, Hampton, VA 23665, USA

Abstract This study presents an analysis method for determining the transverse shear and normal stresses in the adhesive and in-plane stresses in the repair patch and in the repaired skin. The damage to the skin is represented in the form of a cutout. The circular or elliptical cutout can be located arbitrarily under the patch. The patch is free of external tractions while the skin is subjected to general loading along its external edge. The method utilizes the principle of minimum potential energy in conjunction with complex potential functions to analyze a patch-repaired damage con®guration. The present results have been validated against experimental measurements and three-dimensional ®nite element (FE) predictions concerning the patch repair of a circular cutout in a skin under uniform loading. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Adhesive; Patch; Repair; Cutout; Composite

1. Introduction The extensive use of composite materials in primary aircraft components raises questions about their maintenance and repair arising from mechanical and thermal loading. The method of repair is dependent on the extent of the damage and the certi®cation requirements. The objectives of a patch repair are to restore the static strength and durability of the structure and to decrease stress concentrations caused by damage in the form of circular or elliptical cutouts. Patches can be applied to the external or internal surface of the structure. The patch can be either adhesively bonded or bolted to the damaged panel. An important part of the repair process is the prediction of the strength of the patch-repair and its e€ectiveness in reducing the stress concentration around the cutout. Recent applications of bonded patch repairs on composite structures are described and discussed by Baker and his colleagues [1±4]. In the case of a bonded patch repair, its strength depends on the surface preparation, geometry of the patch and damage area, material properties, and the adhesive thickness and its possible variation and exposure to adverse environments. Furthermore, reduction

*

Corresponding author. Tel.: +1-520-621-6113; fax: +1-520-6218191. E-mail address: [email protected] (E. Madenci).

of the transverse and peel stress concentrations along the edges of adhesive is important in order to prevent premature failure of the bonded repair. Peak transverse shear stresses in the adhesive can be reduced by tailoring the sti€ness and by varying the ply lay-up of the patch such that it can strain with the parent laminate (skin). From a structural point of view, a bonded patch repair is the same as a bonded joint. As with a bonded joint, the load is transferred from the skin to the patch by the bond around the damaged portion of the skin. A damage-tolerant design requires accurate assessment of the stress ®eld in the repair area and of the peel and transverse shear stresses in the adhesive. The determination of complete stress and strain ®elds in bonded composite joints is a dicult analytical problem, and an adequate solution must account for the step-wise geometry and material property variations, the laminated construction of the adherents, and the nonlinear behavior of the adhesive. Also, the local stress variations near the ends of the overlap are characterized by very high gradients or even analytically predicted singularities. The sharp gradients of the stress components depend on the elastic properties of the adherents and adhesive, and on the joint geometry. Many analyses have been performed to assess the strength and/or sti€ness of a repaired structure. The majority of the previous analyses concern the determination of the stress state within and along the length of the adhesive bond line because it is the weak link in a

0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 5 8 - 1

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A. Barut et al. / Composite Structures 55 (2002) 277±294

bonded repair or joint. These analyses can be grouped into two categories. The ®rst category utilizes the ®nite element (FE) method with conventional elements. Although the complete stress ®eld in a bonded joint or a patch repair can be achieved by employing three-dimensional conventional FEs, it is dicult to maintain a reasonable aspect ratio of the elements in the adhesive. In order to avoid such modeling diculties, Bagdonavich and Kizhakkethara [5] utilized a coarse global model in conjunction with a submodeling of the adhesive region. An alternative to three-dimensional FE modeling is the use of plate elements for the adherents and spring elements for the adhesive, which was introduced by Loss and Kedward [6]. Sun et al. [7] adopted this approach to investigate a bonded composite patch repair of a cracked metal plate. A variation of this type of modeling, known as the three-layer technique, replaces the spring elements with plate elements [8]. The bonded composite patch repair of a cracked metal plate, including the e€ects of material and geometric nonlinearities, was also considered using this technique [9]. The second category is based on the analytical approach introduced by Goland and Reissner [10] and later extended by Hart-Smith [11]. This approach focuses on the determination of the transverse shear and normal (peel) stresses along the bond line by solving for the beam/ plate bending and in-plane equilibrium equations. These models are limited to cylindrical bending, and the stresses in the adherents are of minor concern. Extensive discussions and extensions of this method were reported by Oplinger [12] and Tsai and Morton [13]. This type of analytical modeling, included in design tools such as the PABST and ESAComp computer programs, is appropriate for iterative design and optimization. An analytical approach for determining the transverse shear and peel stresses in the adhesive and in-plane stresses in the patch and the skin with a cutout under general loading conditions does not exist. Therefore, the present study concerns the development of a bonded patch repair analysis that includes the complexities arising from the presence of a cutout in the skin under general loading conditions. This method, an extension of the approach by Goland and Reissner [10], yields the peel and transverse shear stresses in the adhesive and the in-plane stresses in the patch and skin. It is based on the complex potential theory and the variational formulation in conjunction with the assumption that transverse shear and peel stresses in the adhesive layer are uniform through the thickness. The method is not limited to a particular laminate lay-up, geometric con®guration, or loading condition. A composite skin having a circular cutout that was repaired with a patch is considered under tension. The predictions of the present analysis are veri®ed against those obtained by a three-dimensional ANSYS FE model and the experimental measurements.

2. Problem statement This study concerns the development of an analytical method to determine the stress ®eld in patch-repaired composite materials. The patch repair con®guration shown in Fig. 1 consists of a composite patch bonded to the skin and covering a circular or elliptical hole that represents the damage. The adhesive thickness is uniform. The elliptical cutout, which can be located arbitrarily in the skin under the patch, has a semi-major axis and a semi-minor axis of length a and b, respectively. The special cases of a circular cutout and a line-shaped crack are achieved by a ˆ b and a ˆ 0 or b ˆ 0, respectively. Two coordinate systems whose origins coincide with the center of the cutout are shown in Fig. 1. The global structural coordinates are given by …x; y; z†, and the principal coordinates of the elliptical cutout are given by …xs ; ys ; zs †. The orientation of the cutout with respect to the global structural coordinate frame is de®ned by the angle ws . Although not required, a local coordinate system, …xp ; yp ; zp †, with orientation angle wp is also attached to the patch for consistency with the formulation presented in the following section. The patch and skin are symmetrically laminated. The patch is attached to the skin by the adhesive and, therefore, its external boundaries are traction-free. The exterior edges of the skin are subjected to both in-plane tractions and bending moments. The in-plane tractions include components px ; py , and pz , and the bending tractions include components mx and my . The tractions are de®ned with respect to the …x; y; z† structural coordinates, and their positive-valued directions are shown in Fig. 1. The global displacement components in the x-, y-, and z-directions are denoted by Ux…q† ; Uy…q† , and Uz…q† …q ˆ s; p; a†, respectively. The superscripts p, s, and a denote the patch, skin, and adhesive, respectively. The patch and skin are made of specially orthotropic layers, …q† and each layer has an orientation angle, hk , that is de®ned with respect to the positive x-axis (Fig. 1). Moreover, each layer has thickness tk , elastic moduli EL and ET , shear modulus GLT , and Poisson's ratio mLT , where L and T are the longitudinal (®ber) and transverse principal material directions, respectively, of a given ply. The thicknesses of the patch, adhesive, and skin are denoted by h…q† , with q ˆ p; a; s. The adhesive material is isotropic, homogeneous, and elastic, with Young's modulus, E, and Poisson's ratio, m. …s† As shown in Fig. 1, Cl represents the lth boundary segment of the entire boundary. The unit normal to the …s† lth boundary segment is represented by nl , with com…s† …s† ponents nxl and nyl in the x- and y-directions, respec…s† tively. The unit normal, nl , makes an angle, /l , with respect to the positive x-axis as shown in Fig. 1. Simi…p† …a† larly, the unit normals to the kth boundary, Cl ˆ Cl , …p† …a† of the patch and adhesive are denoted by nk and nk , respectively.

A. Barut et al. / Composite Structures 55 (2002) 277±294

279

Fig. 1. The geometry of a patch-repaired composite skin with an elliptical hole under general loading.

3. Solution method This analysis method for bonded patch repair of composites is based on the principle of minimum potential energy. The displacement components are approximated in terms of the superposition of local and …q† global functions, u…q† a and ua , respectively, as …q†

ua…q† ˆ ua…q† ‡ ua ;

with q ˆ p; s and a ˆ x; y; z:

…1†

u…q† a ,

The local displacement functions, are expressed as Laurent series in terms of complex functions in the form " # Nq2 2   X X …q† …q† …q† …q† ux…q† ˆ 2Re dxk ank Unk zek ; …2a† " uy…q† ˆ 2Re " uz…q†

ˆ 2Re

nˆNq1

kˆ1

2 X kˆ1

Nq2 X

…q†

dyk

nˆ Nq1

Nq2 2 X X kˆ1 nˆNq1

 …q† …q† …q† ank Unk zek

…q† …q† bnk Fnk



…q† zjk

# 

;

# 

;

…2b† …2c†

…q†

ua ˆ

Mq X m X mˆ0 nˆ0

…q† camn Tm …x†Tn …y†;

…q†

…q†

sin w…q† qk ;

…q†

…3a†

…q† dyk

…q† sin w…q† pk

…q† cos w…q† qk ;

…3b†

‡

where the explicit de®nitions of complex functions …q† …q† …q† …q† …q† Unk …zek † and Fnk …zjk † and the complex constants pk …q† and qk are given explicitly in Appendix A. The parameters Nq1 and Nq2 de®ne the extent of the complex series. For the skin, Ns1 ˆ Ns2 , with Ns2 > 0, and the displacement ®elds are, therefore, expressed in

…4†

…q†T

ux…q† ˆ Vx a…q† ; uy…q† ˆ Vy a…q† ;

dxk ˆ cos w…q† pk

with a ˆ x; y; z

…q† in which camn , with a ˆ x; y; z, are the unknown real coecients. The parameter Mq speci®es the order of the series. These local and global displacement functions can be expressed in matrix form as

…q†T

with ˆ

the form of Laurent series. Because there is no hole in the patch, the parameter Np1 is set to 0 and the displacement ®elds are represented in the form of power …q† …q† series. In these series, ank and bnk are the unknown complex coecients. These local functions satisfy the inplane and bending equilibrium equations of an unpatched plate exactly as described by Madenci et al. [14]. …q† The global displacement functions, ua , are expressed as a series in terms of Chebyshev polynomials of the global coordinates …x; y† in the form

…q†T

uz…q† ˆ Vz

…5a†

b…q† ;

and …q†

…q†T

…q†

…q†T

ux ˆ Vx c…q† x ; uy ˆ Vy c…q† y ; …q†

…q†

uz ˆ V z

T

c…q† z

…5b†

280

A. Barut et al. / Composite Structures 55 (2002) 277±294

in which the vectors a…q† and b…q† contain the real and …q† …q† imaginary parts of the unknown coecients ank and bnk , respectively. The vectors ca…q† , with a ˆ x; y; z, contain the …q† …q† real unknown coecients, camn . The known vectors Va …q†

and Va and their corresponding unknown coecient vectors a…q† and b…q† are de®ned explicitly in Appendix A. In matrix form, the approximate displacement representations given in Eq. (1) are rewritten as T

…q† ux…q† ˆ V…q† x qe ; T

q…q† uy…q† ˆ V…q† e ; y

…6†

T

…q† …q† qj u…q† z ˆ Vz

in which the known vectors Va…q† , with a ˆ x; y; z, are de®ned as   …q† T …q† V…q† ˆ V ; V ; 0 ; x x x 

T

V…q† ˆ y T

V…q† ˆ z

…q†

…q†



Vy ; 0; Vy 

…q†

…q†

;

…7†



Vz ; Vz

:

The unknown vectors qe…q† and qj…q† are de®ned as n T o T T …q†T ˆ a…q† ; c…q† q…q† e x ; cy

…8a†

and

n T o T T ˆ b…q† ; c…q† q…q† : j z

…8b†

It is worth noting that the series representation of the displacement components is not required to satisfy any type of kinematic admissibility. 3.1. Kinematic relations The patch and skin interact through the adhesive, supporting transverse and shear deformations but not in-plane deformation. Both the patch and skin are subjected to in-plane and transverse deformations but not shear deformation. However, the transverse normal and shear strain components are disregarded in the patch and skin because they are thin. Therefore, the inplane strain components in the adhesive and the transverse normal and shear strain components in the patch and skin are not included in the derivation of the kinematic relations. In accordance with the Kirchho€ plate theory, the global displacement components, Ux…q† ; Uy…q† , and Uz…q† in the patch and the skin are de®ned as Ua…q† …x; y; z† ˆ ua…q† …x; y†

…q† n…q† h…q† uz;a ;

Uz…q† …x; y; z† ˆ uz…q† …x; y†; …q ˆ p; s† and a ˆ x; y

…9†

Fig. 2. Detailed side view of skin, adhesive, and patch sections. …q† …q† in which u…q† x ; uy , and uz are the displacement components de®ned on mid-surfaces with respect to the global Cartesian coordinates …x; y; z† (Fig. 2). A subscript after a comma indicates di€erentiation with respect to the variable. As shown in Fig. 2, the coordinate n…q† located on each of the mid-planes is de®ned as

n…q† ˆ

z…q†

z

h…q†

;

with q ˆ p; s; a

…10†

and varies in the range 1 6 n…q† 6 1. The half thicknesses of the patch, skin, and adhesive are speci®ed by h…q† , and the location of the mid-surfaces with respect to the global coordinate system, …x; y; z†, are de®ned by z…q† , with q ˆ p; s; a. Substituting for the displacement components from Eqs. (2a)±(2c) in the strain±displacement relations, the strain components become  1  1  …q† …q† …q† …q† …q† …q† Ua;b ‡ Ub;a ˆ ua;b ‡ ub;a n…q† h…q† uz;ab ; eab ˆ 2 2 with q ˆ p; s and a ˆ x; y: …11† These expressions can be rewritten as …q† …q† …q† …q† e…q† aa ˆ eaa ‡ n h jaa ; 1 …q† 1 …q† …q† …q† e…q† xy ˆ cxy ‡ n h jxy ; 2 2 with q ˆ p; s and a ˆ x; y

…12†

in which …q† e…q† xx ˆ ux;x ; …q† jxx ˆ

…q† uz;xx ;

…q† eyy ˆ u…q† y;y ;

j…q† yy ˆ

…q† …q† c…q† xy ˆ ux;y ‡ ux;y ;

u…q† z;yy ;

…q† jxy ˆ

…q† 2uz;xy ;

…13a† …13b†

…q† …q† …q† where exx ; eyy , and cxy represent the in-plane strain re…q† …q† …q† sultants and jxx ; jyy , and jxy represent the bending strain resultants, all along the mid-surfaces. The inplane and bending strain (curvature) resultant vectors, e…q† and j…q† , are de®ned in terms of the in-plane strain resultants and curvatures as

A. Barut et al. / Composite Structures 55 (2002) 277±294

n o T …q† …q† e…q† ˆ e…q† ; xx ; eyy ; exy

with …q ˆ p; s†

…14a†

…a† ˆ caz

and

n o T …q† …q† ; j…q† ; j j…q† ˆ jxx ; yy xy

with …q ˆ p; s†:

…14b†

Substituting for the displacement components and their derivatives from Eq. (6), the in-plane strain and bending strain (curvature) resultant vectors can be expressed as e…q† ˆ Le…q† q…q† e ; …q† j…q† ˆ L…q† j qj ;

with …q ˆ p; s† in which 2

Le…q†

L…q† e

and …q†

Vx;x

6 6 ˆ6 6 4

…15† L…q† j

T

…q†T

Vy;y …q†

T

…q†

Vx;y ‡ Vy;x and

L…q† e ˆ

2

T

are de®ned as 3 …q†T T Vx;x 0 7 …q†T 7 7 T 0 Vy;y 7 5 …q†T …q†T Vx;y Vy;x

3 …q†T Vz;xx 7 …q†T 7 7 Vz;yy 7: T 5 …q† 2Vz;xy

…q†T

6 Vz;xx 6 …q†T 6 6 Vz;yy 4 …q†T 2Vz;xy

…16†

e

…q† …q†

ˆL q ;

with …q ˆ p; s†

…17†

…18†

…a; b ˆ x; y; q ˆ s; p†

…24†

 1  …p† ua …x; y† u…s† a …x; y† …a† 2h i 1 h …s† …s† with a ˆ x; y; ‡ …a† h…p† u…p† z;a ‡ h uz;a ; 2h  1  ˆ …a† u…p† uz…s† …x; y† : z …x; y† 2h

c…a† az ˆ

…a† ezz

…25a† …25b†

…19†

Substituting from Eq. (6) for the displacement components and their derivatives in Eqs. (25a) and (25b) leads to the strain vector containing the transverse shear and normal strain components in the adhesive in matrix notation as

…20†

…p† e…a† ˆ L…p† a q

For the adhesive between the patch and the skin, the displacement components are assumed to vary linearly through the thickness. In order to ensure displacement continuity among the patch, adhesive, and skin, the displacement components for the adhesive are expressed as
1 Ua…a† …x; y; z† ˆ Ua…p† x; y; z…p† h…p† 2
 ‡ Ua…s† x; y; z…s† ‡ h…s† 
1 ‡ n…a† Ua…p† x; y; z…p† h…p† 2
 Ua…s† x; y; z…s† ‡ h…s† ; with a ˆ x; y; z:

under the assumption that because the adhesive is a thin layer. Substituting for the displacement component evaluated at n…q† ˆ 1 for q ˆ p; s leads to

and

n T o T T ; q…q† : q…q† ˆ q…q† e j

with a ˆ x; y; …22a†  1  …a† ˆ …a† Uz…p† …x; y† Uz…s† …x; y† : …22b† ezz 2h The expressions for the transverse shear strain compo…a† nents, caz …a ˆ x; y†, is simpli®ed to
1  …p† c…a† Ua x; y; z…p† h…p† az ˆ …a† 2h
 …23† Ua…s† x; y; z…s† ‡ h…s† …q†

…q†

in which L is de®ned as   L…q† ˆ Le…q† L…q† j



1 h …p† Ua…s† x; y; z…s† ‡ h…s† Ua x; y; z…p† h…p† …a† 2h  
…p† x; y; z…p† h…p† ‡ h…a† 1 ‡ n…a† Uz;a  
i …s† x; y; z…s† ‡ h…s† ; ‡ h…a† 1 n…a† Uz;a

Ua…q†  h…a† Uz;b

Also, these two vectors in Eq. (15) can be combined in the form …q†

281

where

 …p† L…p† a ˆ Lae

L…p† aj

 La…s† ˆ L…s† ae in which …q† Lae ˆ

and …q† Laj ˆ

L…s† aj 2

V…q† x

T

…27a† 

…27b†

3

7 1 6 6 V…q†T 7; 4 5 …a† y 2h T 0 2

…26†



and

…21†

Utilizing these displacement components, the transverse shear strain and the normal strain components in the adhesive can be expressed as

…s† L…s† a q ;

h…q† V…q† z;x

with q ˆ p; s

T

…28†

3

7 1 6 6 h…q† V…q†T 7: 4 5 …a† z;y 2h dqs …q†T … 1† Vz

…29†

282

A. Barut et al. / Composite Structures 55 (2002) 277±294

3.2. Stress±strain relations The external in-plane loads acting along the boundary of the mid-surface of the skin result not only in inplane stresses but also in bending moments in both the patch and skin due the eccentricity between the midsurfaces of the skin and patch and their interaction with the adhesive. The peeling stress in the adhesive is primarily due to the bending deformations arising from this load eccentricity. The in-plane and bending stresses are obtained from N…q† ˆ A…q† e…q† ;

with q ˆ p; s

n o T …a† …a† …a† ; cyz ; ezz ; e…a† ˆ cxz 2 3 G 0 0 60 G 07 7 E…a† ˆ h…a† 6 4 0 0 E5

…36b†

…36c†

…30a†

in which G and E are the shear and Young's moduli, respectively. Representation of e…a† in Eq. (26) permits the stress± strain relations given in Eq. (35) to be expressed as
…p† …s† s…a† ˆ E…a† L…p† L…s† : …37† a q a q

…30b†

3.3. Boundary conditions

and M…q† ˆ D…q† j…q† ;

with q ˆ p; s

in which A…q† and D…q† , de®ned with respect to the global …x; y† coordinates, represent the laminate material properties associated with in-plane deformations and bending deformations, respectively. The in-plane and bending stress vectors are de®ned as n o T N…q† ˆ Nxx…q† ; Nyy…q† ; Nxy…q† …31a† and T

n

o

M…q† ˆ Mxx…q† ; Myy…q† ; Mxy…q† ;

…31b†

where Nxx…q† , Nyy…q† , and Nxy…q† represent the in-plane stress resultants and Mxx…q† , Myy…q† , and Mxy…q† represent the bending stress (moment) resultants. The relations given in Eqs. (31a) and (31b) can be combined in the form s…q† ˆ E…q† e…q† ; …q†

with …q ˆ p; s† …q†

in which s ; E , and e n o T T T s…q† ˆ N…q† ; M…q† ;  …q†  A 0 E…q† ˆ ; 0 D…q† n T o T T e…q† ˆ e…q† ; j…q† :

…q†

…32†

are de®ned as …33a† …33b† …33c†

…34†

Because the adhesive does not support in-plane de…a† formation, the in-plane stress components, r…a† xx , ryy , and …a† …a† …a† rxy , are disregarded. The transverse shear, rxz and ryz , …a† and normal (peel), rzz , stress components are determined by s…a† ˆ E…a† e…a† ;

…35†

where

n o T …a† …a† ; r…a† s…a† ˆ h…a† rxz ; yz ; rzz

u…s† n …s† ut u…s† z u…s† z;n

ˆ …l† u^…s† n ; …l† …s† ˆ u^t ; ˆ …l† u^…s† z ; ˆ …l† u^…s† z;n ;

…s†

on Cl …l ˆ 1; . . . ; L†:

…36a†

…38†

Utilizing the vector representations of the displacement components de®ned in Eq. (6), these prescribed displacements can be expressed in vector form as T

T

…s† …s† …s† cos /l V…s† x qe ‡ sin /l Vy qe T

T

…l† …s† u^n

…s† …s† …s† sin /l V…s† x qe ‡ cos /l Vy qe T

…s† V…s† z qj

…l† …s† u^z

ˆ 0;

…l† …s† u^t

ˆ 0;

ˆ 0;

T

Representation of e…q† in Eq. (18) permits the stress± strain relations given in Eq. (32) to be expressed as s…q† ˆ E…q† L…q† q…q† :

…s†

Along the lth segment of the skin boundary, Cl , as shown in Fig. 1, the prescribed displacement compo…l† …s† nents normal and tangent to the boundary (…l† u^…s† u^t , n , …l† …s† …l† …s† and u^z ) and the slope normal to the boundary ( u^z;n ) can be imposed as

T

…s† …s† …s† cos /l V…s† z;x qj ‡ sin /l Vz;y qj

…l† …s† u^z;n

…39†

ˆ 0:

These equations are rewritten in compact form as …s†T

Vl q…s†

^u…s† n ˆ 0;

…40†

T

…s† where q…s† ˆ fq…s† e ; qj g is given in Eq. (20). The matrix …s† Vl and the vector ^u…s† n are de®ned as …s†T

Vl

2

6 6 ˆ6 6 4

T

‡ sin /l V…s† cos /l V…s† x y T

T

‡ cos /l V…s† sin /l V…s† x y

3

0 T

0 T

V…s† z T T cos /l V…s† ‡ sin /l V…s† z;x z;y

0 0

7 7 7 7 5

…41† and T

^u…s† ˆ l

n

…l† …s† u^n

…l† …s† u^t

…l† …s† u^z

…l† …s† u^z;n

o :

…42†

The boundary conditions in Eq. (40) are enforced as constraint conditions by introducing Lagrange

A. Barut et al. / Composite Structures 55 (2002) 277±294 …s† Kal …t†

0…s† Kzl …t†,

multiplier functions, with …a ˆ n; t; z† and de®ned along the lth boundary segment. These boundary conditions are written in integral form as Z n T o …s† …s† …s† Kl …t† Vl q…s† ^ ul dt ˆ 0; …43† …s†

Cl

…s†

where the matrix Kl contains the Lagrange multiplier functions in the form 2 …s† 3 Knl …t† 6 7 …s† Ktl …t† 6 7 …s† Kl …t† ˆ 6 7: …s† 4 5 Kzl …t† 0…s† Kzl …t† …s†

The Lagrange multiplier functions Kal …t†, with a ˆ n; 0…s† t; z, and Kzl …t† are de®ned in terms of Legendre polynomials as 

J   X  …s† 0…s† …s† 0…s† kj…al† ; kj…zl† Pj …n†; Kal …t…n††; Kzl …t…n†† ˆ jˆ0

with a ˆ n; t; z;

…44†

where Pj represents the jth-order Legendre polynomial and kj…al† with …a ˆ n; t; z† and k0j…zl† are the unknown Lagrange multipliers associated with each Legendre …s† polynomial, Pj , and boundary segment, Cl . Substituting the expressions for the Lagrange multiplier functions from Eq. (44) into Eq. (43) and rearranging the terms, the constraint equations representing the prescribed displacements can be rewritten as …s†T

…s†

kl Cl q…s†

…s†T …s†

kl f lc ˆ 0;

with l ˆ 1; . . . ; L;

…45†

where …s†T

kl

n T o …s† …s†T …s†T ˆ k1l ; k2l ; . . . ; kJl

…46†

n o …s†T …s† …s† …s† 0…s† kkl ˆ kk…nl† ; kk…tl† ; kk…zl† ; kk…zl† ; h i T …s†T …s†T …s†T Cl ˆ C…s† C2l


CJl 1l

…s†

Cjl ˆ

Z

…s†T

…s†

Cl

Pj Vl

dC

…47† …48†

…49†

and

n T o …s†T …s† …s†T …s†T f lc ˆ f 1…lc† ; f 2…lc† ; . . . ; f J …lc† with …s†T

f j…lc† ˆ

T

k…s† f …s† c ˆ 0;

…52†

where

n T o T …s† …s†T …s†T k…s† ˆ k1 ; k2 ; . . . ; kL ; h i T T …s†T …s†T C…s† ˆ C…s† C


C L 1 2

…53a† …53b†

and

n T o T …s† …s†T …s†T ˆ f 1c ; f 2c ; . . . ; f Lc : f …s† c

…53c†

The system of constraint equations in Eq. (52) is unique, provided the rank of the matrix C…s† is equal to the total number of constraint equations. Also, Eq. (52) can be treated as the potential energy of the reaction forces producing zero energy since C…s† q…s† ˆ 0, and they can be added to the expression for the total potential energy. Thus, the variation of the total potential energy will automatically result in the equations of equilibrium, as well as the constraint equations. 3.4. Equilibrium equations The derivation of the equilibrium equations and boundary conditions is based on the principal of stationary potential energy. The total potential in the repaired skin with a bonded patch can be expressed in the form pˆU

V

Vc ;

…54†

where U is the strain energy and V and Vc are the potential of the external and reaction forces, respectively. The strain energy associated with the skin, adhesive, and patch is given by X Uˆ Uq …55† qˆs;a;p

with

with

T

k…s† C…s† q…s†

283

Z …s† Cl

…s†T

ul Pj ^

…50†

in which Uq is expressed as Z T 1 Uq ˆ s…q† e…q† dA; with q ˆ s; a; p; 2 Aq

where Aq denotes the areas of the skin, adhesive, and patch. Substituting for the stress and strain resultant vectors from Eqs. (18), (26), (34), and (37) and rearranging terms yield  T   …s†  Kss Ksp 1 q…s† q Uˆ ; …57† T KTsp Kpp q…p† 2 q…p† Z Z T T T …a† …s† Kss ˆ L…s† E…s† L…s† dA ‡ L…s† …58a† a E La dA; As

dC:

…51†

The constraint equations in Eq. (45) can be assembled to form a single matrix equation combining all of them as

…56†

Aa

Z Kpp ˆ

T

Ap

Z

Ksp ˆ

T

L…p† E…p† L…p† dA ‡

Aa

T

…a† …p† L…s† a E La dA:

Z

T

Aa

…a† …p† L…p† a E La dA;

…58b† …58c†

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A. Barut et al. / Composite Structures 55 (2002) 277±294

The potential of the external forces denoted by V is obtained from Z V ˆ

C…s†

n o …s† …s† …s† …s† u…s† p p ‡ u p ‡ u m ‡ u m ‡ u dC; x y z x y x y z z;x z;y …59†

where the external boundary of the skin is denoted by C…s† , the applied tractions by px ; py , and pz , and the moments by mx and my ; their positive signs are shown in Fig. 1. Substituting for the displacement components and their derivatives from Eq. (6) permits this expression in matrix notation as T

V ˆ q…s† p…s†

…60†

in which n T o T …s†T ; p p…s† ˆ p…s† e j with p…s† e

ˆ

p…s† j ˆ

Z C…s†

Z C…s†

…61†

n o …s† V…s† x px ‡ Vy py dC; n o …s† …s† V…s† z pz ‡ Vz;x mx ‡ Vz;y my dC:

…62a† …62b†

The potential energy of the reaction forces can be written in the form T

Vc ˆ k…s† C…s† q…s†

T

k…s† f …s† c :

…63†

Combining the strain energy associated with each of these regions and the potential of the external and reaction forces leads to the total potential of the bonded patch repair. The resulting expression for the total potential energy becomes  T   …s†  1 q…s† Kss Ksp q pˆ T KTsp Kpp q…p† 2 q…p†  T  …s†  T T p q…s† …64† k…s† C…s† q…s† ‡ k…s† f …s† c : …p†T 0 q Enforcing the ®rst variation of p to vanish leads to the matrix equilibrium equations including the constraint conditions as 2 38 9 8 9 T Kss Ksp C…s† < q…s† = < p…s† = 6 T 7 …65† 0 : 0 5 q…p† ˆ 4 Ksp Kpp : …s† ; : …s† ; …s† f k C 0 0 c The solution to this equation permits the recovery of the displacement, strain, and stress components in the skin, patch, and adhesive through Eqs. (6), (18), (26), (34), and (37).

4. Results and discussion The present analysis results are validated against the experimental measurements and FE analysis. Two patch-repaired aluminum specimens of di€erent thicknesses, with a central circular cutout under uniaxial tension, are considered. The planar geometries of the specimens are identical, as shown in Fig. 3(a). The skin has a rectangular geometry, with length and width dimensions speci®ed by Ls ˆ 10 in. and Ws ˆ 4 in., respectively. The skin has a central circular cutout, with hole diameter speci®ed as d ˆ 0:75 in. As shown in Fig. 3(a), the aluminum patch is located centrally over the circular cutout, and it has a square geometry, with the panel length speci®ed as Wp ˆ 1:125 in. The Young's modulus and Poisson's ratio used for the skin and the patch are, respectively, E ˆ 10:2  106 psi and m ˆ 0:33. Also, the adhesive bond between the skin and the patch has a shear modulus and Poisson's ratio speci®ed by Ga ˆ 60  106 psi and m ˆ 0:34, respectively. 4.1. Experimental measurements The two test specimens, whose dimensions and material properties are speci®ed above, were prepared with either a thick skin with a thin patch, referred to as Repair-I, or a thin skin with a thick patch, named as Repair-II. As depicted in Fig. 3(b), the specimens Repair-I and Repair-II have skin thicknesses equal to hs ˆ 0:088 and 0.0635 in., respectively. The patch thicknesses for Repair-I and Repair-II specimens are speci®ed as hp ˆ 0:024 and 0.049 in., respectively. Hence, the skinto-patch thickness ratios for Repair-I and Repair-II specimens are, respectively, 3.67 and 1.30. In both specimens, the nominal thickness of the adhesive bond (3M Scotch±Welde structural adhesive ®lm, AF-1632U) is measured as ha ˆ 0:0025 in. The skin and patch were prepared for bonding in accordance with industry standards. The skin/adhesive/ patch assembly was then placed in a vacuum bag under a vacuum of approximately 10 in. of mercury. Following the manufacturer's instructions, the adhesive was cured in an oven for 60 min at a temperature of 275°. Strain gages were placed at several locations on the specimen, as shown in Fig. 4(a). The global coordinates of these strain gages are given in Table 1. As shown in Fig. 4(b), the tensile tests were performed on a Testworks-controlled MTS/Sintech servo-mechanical load frame, and the strain gage readings were acquired using Labview software and National Instruments data-acquisition hardware. The out-of-plane de¯ection of the specimen arising from the eccentricity between the mid-surface of the skin and patch was also measured by using a digital dial gage at two points, labeled as A1 and B1 (Fig. 4(a)). Point A1

A. Barut et al. / Composite Structures 55 (2002) 277±294

285

(b)

(a)

Fig. 3. Geometric description of patch-repaired aluminum specimens: (a) top view and (b) side view.

Fig. 4. Test setup: (a) gage placement and (b) a specimen under uniaxial loading.

is on the side of the skin without the patch and located 0.34375 in. away from the edge of the hole along the horizontal centerline. Point B1 is on top of the patch

along the horizontal centerline and 0.15625 in. away from the vertical centerline. Also, the maximum load applied was 800 lb at a loading rate of 0.05 in./min.

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A. Barut et al. / Composite Structures 55 (2002) 277±294

and vertical centerlines of these three sections, symmetry conditions were applied as displacement boundary conditions. These symmetry conditions are expressed as

Table 1 Global coordinates of the strain gages Gage label

Coordinate (in.)

Specimens Repair-I

Repair-II

Hole (left)

x y z

1.53125 5.00000 0.00000

1.53125 5.00000 0.00000

Hole (right)

x y z

2.46875 5.00000 0.00000

2.46875 5.00000 0.00000

Patch

x y z

2.00000 5.00000 0.11450

2.00000 5.00000 0.11500

Far ®eld

x y z

2.93750 6.93750 0.08800

2.93750 6.93750 0.06350

4.2. Finite element analysis In order to compare the stress distribution in the skin, adhesive, and the patch, in addition to the experimental strain and displacement measurements at several locations, a three-dimensional FE analysis was conducted. Utilizing ANSYS, a commercial FEA program, a global model and sub-model of the patch-repaired aluminum specimen were modeled with brick elements (SOLID45). For the global model, the upper right quarter of the entire patch-repaired skin was discretized (taking advantage of symmetry conditions) by 3008 brick elements, of which 1408 were used for the skin and 640 and 960 for the adhesive and the patch sections, respectively. Also, the thickness of each section was discretized into one division, which was sucient for the coarse global model. The FE model consisted of 4968 nodes with 14,501 active degrees of freedom. Along the horizontal

ux …0; y† ˆ uy …x; 0† ˆ uz;x …0; y† ˆ uz;y …x; 0† ˆ 0: The skin is clamped along the horizontal edges and free along the vertical edges. These conditions are applied to the FE model as uy …x; Ls =2† ˆ dy =2

and

uz …x; Ls =2† ˆ 0

in which dy denotes the uniform stretching of the skin along the horizontal edges and provides clamped boundary conditions. In order to increase the accuracy around the circular cutout, a re®ned sub-model was constructed by cutting out a square portion of the original model. The results obtained from the global model were then used as displacement boundary conditions on the cut boundaries around the re®ned sub-model. The length of the square sub-model was chosen to be W ˆ 0:8 in., which is about twice as large as the hole radius. The sub-FE model consists of 20,000 elements (4000 used for the skin, 8000 for the adhesive, and 8000 for the patch sections) with 24,849 nodal points and 72,240 active degrees of freedom. Also, in order to increase the accuracy of stresses in the thickness direction around the hole cutout, the skin and patch sections were discretized into two divisions and the adhesive thickness was divided into four equal-size intervals. As in the global model, the symmetry conditions were imposed along the horizontal and vertical centerlines of the sub-model. Along the horizontal and vertical cut boundaries, however, the displacements calculated from the coarse (global) FE model were interpolated over the nodes, thus applying the prescribed displacement boundary conditions.

Fig. 5. The magni®ed deformed con®guration of a specimen after loading.

A. Barut et al. / Composite Structures 55 (2002) 277±294

4.3. Present analysis Instead of applying symmetry conditions along the centerlines, as in the FE analysis, the entire geometry of the aluminum specimens was considered in the calculation of the system sti€ness matrix and the right-hand side load vector. However, advantage was taken of the symmetry conditions in order to reduce the number of terms in the Chebyshev polynomials by rede®ning the displacement components as …q†

ux ˆ …q†

uy ˆ …q†

uz ˆ

M q ‡1 X

m 1 X

mˆ1;3;5;7;... nˆ0;2;4;6;... Mq X

m‡1 X

mˆ0;2;4;6;... nˆ1;3;5;7;... Mq X

m X

mˆ0;2;4;6;... nˆ0;2;4;6;...

…q† cxmn Tm …x†Tn …y†;

287

of the skin and patch and the sti€ness of the adhesive bonding between these components. Comparisons of strain component eyy measured at critical locations for both specimens are plotted in Fig. 6(a) and (b), respectively. The comparison shows remarkable agreement between the experimental measurements and the strains calculated from the present and the FE analyses. Comparing the strain plots in both ®gures, it can be observed that the far-®eld strain and the strain measured at the center of the patch in specimen Repair-I are nearly the same. On the other hand, the strains measured at these two locations di€er signi®cantly in Repair-II specimen. Also, the strains measured near the hole in Repair-II

…q† cymn Tm …x†Tn …y†;

…q† czmn Tm …x†Tn …y†

…q†

in which ux is asymmetric in the x-direction and sym…q† metric in the y-direction; uy is symmetric in the x-di…q† rection and asymmetric in the y-direction; and uz is symmetric in both directions. These approximations automatically satisfy the symmetry conditions along the centerlines and reduce the number of unknowns considerably. In the skin, both in-plane and transverse displacement ®elds were approximated by complex potential functions of order 3 (i.e., Ns2 ˆ Ns1 ˆ 3, resulting in three positive and three negative powers of the mapping function) and Chebyshev polynomials of order Ms ˆ 24, thus leading to a total of 321 generalized coordinates (unknowns). Due to the absence of a hole cutout in the patch, the displacement ®elds were represented by only Chebyshev polynomials of order Mp ˆ 24, thus introducing an additional 273 generalized coordinates and increasing the total number of unknowns to 594. In order to apply clamped boundary conditions along the upper and lower vertical edges, the reaction forces associated with the transverse de¯ection and the slope normal to each boundary line were represented by 10term Legendre polynomials, thus resulting in a total of 40 constraint equations. Additional constraint equations were also used to suppress the rigid-body motion of the skin and obtain single-valuedness of the transverse de¯ection ®eld associated with the complex functions. A typical deformed con®guration, obtained from the present analysis, representative of both specimens is shown in Fig. 5. In both specimens, due to the eccentric position of the patch with respect to the plane of the skin, the stretching and bending actions of both the skin and patch are coupled through the interaction with the adhesive, thus causing both the skin and patch to bend when axial load is applied in the plane of the skin. The severity of coupling depends on the sti€ness properties

Fig. 6. Comparison of present analysis against FE analysis and experimental measurements for strains measured at critical locations in (a) Repair-I and (b) Repair-II specimens.

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A. Barut et al. / Composite Structures 55 (2002) 277±294

specimen are slightly higher than those measured in Repair-I specimen. However, as observed in Fig. 6, the di€erence in patch strains between these two specimens is considerable. The signi®cant di€erence in the in-plane strain measurements for these specimens is basically due to the di€erence in their bending deformations. The bending deformations caused by the coupling action between skin and patch always tend to increase the normal strain at the bottom of the skin and decrease at the top of the patch. Although the overall thickness and patch eccentricity of both specimens are about the same, the RepairI specimen has a thicker skin than the Repair-II specimen, thus yielding higher bending sti€ness and lower bending deformations than the Repair-II specimen. This is also veri®ed through a comparison of transverse displacements as given in Table 2, which gives the out-ofTable 2 Comparison of out-of-plane displacements at measurement points Specimens

Locations

Experiment (in.)

Analytical (in.)

FEM (in.)

Repair-I

Point A1 Point B1

0.00520 0.00613

0.003902 0.004013

0.004088 0.004253

Repair-II

Point A1 Point B1

0.01125 0.01357

0.010365 0.010328

0.01021 0.01025

plane displacements measured and calculated at points along the vertical centerlines of the skin and patch. The comparison of the displacements between experimental measurements and those evaluated from the present and FE analyses are in favorable agreement. As observed in this table, the specimen with the thinner skin component (i.e., Repair-II) undergoes a transverse de¯ection whose order of magnitude is equal to 0.01 in., whereas the specimen with the thicker skin component (Repair-I) undergoes a lower transverse de¯ection with an order of magnitude around 0.005 in. The three-dimensional contour plots of the in-plane stress resultant, Nyy , in the skin and patch of both specimens for a uniform loading of r0 ˆ 1 lb=in:2 are shown in Fig. 7(a) and (b), respectively. These plots demonstrate the capability of the present analysis to capture stress concentrations near the circular cutouts. Also, a comparison between the FE and present analyses for the variation of the in-plane stress resultant, Nyy , along the horizontal centerline of the skin are shown in Fig. 8. Excellent agreement exists between the present and FE analyses. As observed in Fig. 8, the patch reduces the stress intensi®cation by about 30% from the well-known stress concentration of 3r0 . From Fig. 8(a) and (b), it is observed that the reduction of stress intensi®cation is about the same in both specimens. However, it must be kept in mind that the point-wise

(a)

(b) Fig. 7. Three-dimensional contour plots of in-plane stress resultant, Nyy , in (a) the skin and (b) patch sections of both Repair-I and Repair-II specimens.

A. Barut et al. / Composite Structures 55 (2002) 277±294

(a)

(b)

Fig. 8. Comparison of present analysis against FEM for the variation of the in-plane stress resultant along the horizontal centerline of (a) Repair-I and (b) Repair-II specimens.

stresses vary through the thickness because of the bending deformations. For this reason, the comparison of the strain measurements in Fig. 6 reveals that the Repair-I specimen (the skin-to-patch thickness ratio is 3.67) is preferable for improving the skin's in-plane and bending behavior as compared to the poor bending performance of the Repair-II specimen. The three-dimensional contour plots of the transverse shear and peeling stress distributions in the adhesive for the Repair-I and Repair-II specimens are shown in Fig. 9. The transverse shear stress components are asymmetrically distributed, whereas the peeling stress distribution is symmetric in both specimens. Around the hole boundary, both rxz and ryz reach peak values around 45° and 135°. The peeling

289

stress, rzz , reaches the local peak value at 90° and 270° around the hole boundary. Since the direction of the loading is parallel to y-axis, the response of ryz along the horizontal edges is higher than rxz along the vertical edges of the patch, as expected. Similarly, the peeling stress distribution along the horizontal edges is higher than that along the vertical edges. Comparisons of the transverse shear, ryz , and peeling, rzz , stresses between the present and FE analyses along one half of the upper horizontal edge for both specimens are given in Fig. 10(a) and (b), respectively. Favorable agreement is observed between both analyses for the comparison of transverse shear, ryz , stress variation. For the peeling stress variation, however, the di€erence between the present and FE analyses may be attributed to the assumptions used in describing the kinematics of the plates, that is, the Kirchho€ versus the three-dimensional continuum models. As observed in Fig. 10, stress concentrations occur in both transverse shear and peeling stresses near the corner. The adhesive stresses around one quarter of the hole boundary computed from the re®ned FE sub-model are compared against the present analysis for both specimens in Fig. 11. In the FE analysis, the adhesive stresses are computed at ®ve equally spaced points through the thickness of the adhesive. As illustrated, through-thethickness variations of transverse shear stresses in both specimens change slightly (i.e., close to uniform), whereas non-uniform variations of peeling stresses are observed through the thickness of the adhesive (Fig. 11(c)). The comparison plots for transverse shear stresses indicate close agreement between the present and FE analyses. However, the present and FE analyses do not lead to similar predictions of the peeling stress because the present analysis is based on the assumption of uniform transverse normal stress variation through the adhesive thickness. Therefore, higher-order variations of transverse displacements through adhesive thickness must be included in order to obtain a non-uniform variation of peeling stress within the present formulation. 5. Conclusions In this study, an analytical approach for determining the stress and displacement ®elds in a patch-repaired skin with a circular cutout under in-plane loading has been presented. The analytical approach utilizes a complete set of complex (local) functions superimposed with Chebyshev (global) polynomials to approximate the kinematic ®eld. The matrix form of equilibrium equations has been obtained via the principle of minimum potential energy. The present approach has been validated against both experimental measurements and FE analysis predictions. The comparison of the strain

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A. Barut et al. / Composite Structures 55 (2002) 277±294

(a)

(b)

(c) Fig. 9. Three-dimensional contour plots of adhesive stresses, (a) ryz , (b) rxz , and (c) rzz , in Repair-I and Repair-II specimens.

measurements between the present analysis and the experiment indicates close agreement. The agreement between the present and FE analyses for in-plane stress calculations is remarkable. Although the reductions in stress intensi®cation around the hole boundary for both specimens are nearly the same, the strain and in-plane stress calculations indicate that the specimen with the higher skin-to-patch thickness ratio (i.e., Repair-I) provides higher bending resistance and lower transverse de¯ection than the other specimen (Repair-II). Thus, the performance of the Repair-I specimen appears to be superior to that of the Repair-II specimen, which is poor in bending behavior. The comparison of the adhesive stresses evaluated by the present and FE analyses shows favorable agreement for transverse shear stresses. However, the non-uniform variation of the peeling stress component through the thicknesses of the specimens as obtained from the FE analysis suggests that further

improvements in the present model may still be achieved by incorporating a higher-order theory with transverse shear deformations into the present analytical formulation.

Appendix A …q†

…q†

The complex potential functions Unk …zek †, with q ˆ s; p, appearing in Eqs. (2a)±(2c) are de®ned as  n …s† …s† …s† Unk …zek † ˆ nek ; …A:1a†  n …p† …p† …p† Unk …zek † ˆ zek :

…A:1b† …q†

The mapping functions nek , ®rst introduced by Bowie [15], map a cutout onto a unit circle. In this analysis, the

A. Barut et al. / Composite Structures 55 (2002) 277±294

291

(a)

(b) Fig. 10. Comparison of present analysis against FE analysis for transverse shear, (a) ryz and (b) rxz , and peeling, (c) rzz , stresses along the upper horizontal edge of the adhesive section in Repair-I and Repair-II specimens.

mapping functions for an elliptical cutout, introduced by Lekhnitskii [16], are employed in the form

…q† nek

ˆ

…q† zek



r     …q†

zek

a

2

…q†

a2 lek  2 …q† i lek b

2

b2

…k ˆ 1; 2†

0…q†

…A:2† …q†

…q†

in which zek ˆ xq ‡ lek yq , with (q ˆ s; p), and a and b are the major p and minor axes of the elliptical cutout with i ˆ 1. The sign of the square root term is chosen …q† so that jnek j P 1. …q† …q† Inverting the mapping function provides xek …nek † as   sek …q† …q† …q† …q† zek ˆ xek nek ˆ rek nek …A:3† …q† nek in which …q†

rek ˆ

1 a 2

 …q† ilek b ;

…q†

sek ˆ

 1 …q† a ‡ ilek b : 2

…q†

…A:4†

…q†

The unknown complex constants le1 and le2 , with …q† …q† …q ˆ s; p†, and their complex conjugates, i.e., le3 ˆ le3 …q† …q† and le4 ˆ le2 , are the roots of the characteristic equation obtained from the in-plane compatibility condition a11



…q†

lek

4

0…q† …q†

0…q†

2a16



…q†

lek

3

  2 0…q† 0…q† …q† ‡ 2a26 ‡ a66 lek

0…q†

2a26 lek ‡ a22 ˆ 0

…A:5† 0…q†

in which the coecients aij are components of the ¯exibility matrix a0…q† , which is the inverse of the in-plane sti€ness matrix A0…q† . Both the ¯exibility and the sti€ness matrices, a0…q† and A0…q† , are measured with respect to the local coordinate system …xq ; yq †. The angle wq represents the orientation of the local coordinate system with respect to the global coordinate system. Thus, the components of A0…q† can be directly obtained by transforming the components of the in-plane sti€ness matrix, A…q† , de®ned in the global system through the orientation angle. This transformation relation is available in any

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A. Barut et al. / Composite Structures 55 (2002) 277±294

(a)

(b)

(c) Fig. 11. Comparison of present against FE analyses for the variation of transverse, (a) ryz and (b) rxz , and peeling, (c) rzz , stresses along the hole boundary in Repair-I and Repair-II specimens.

textbook on composite materials. Also, the com…q† …q† plex constants pk and qk in Eqs. (3a) and (3b) are de®ned as  2 …q† 0…q† …q† 0…q† 0…q† …q† ‡ a12 a16 lek ; …A:6a† pk ˆ a11 lek

…q†

0…q† …q†

0…q†

…q†

qk ˆ a12 lek ‡ a22 =lek

0…q†

a26 : …q†

…A:6b† …q†

The complex functions Fnk …zjk † appearing in the expression for the local functions, uz…q† , with q ˆ s; p, Eq. (2c) are de®ned as

A. Barut et al. / Composite Structures 55 (2002) 277±294

8 …q†  n n 2 …q†  r sjk …q† …q† > > njk ; > jkn njk n 2 > > > > …q† …q† 2 > rjk …njk † …q† …q† > > > <  2 n sjk ln njk ; …q† …q† …q† Fnk …zjk † ˆ zjk ; > > 2 > …q† …q† > sjk …njk † …q† …q† > > r ln n ‡ ; > jk jk > > r…q†  n‡1 s2…q†  n 1 > > …q† …q† jk jk : njk njk ; n‡1 n 1

with

n P 3;

…q†T

Vz…n† ˆ

n ˆ 2; n ˆ 0; 1; nˆ

1;

n6

2;

…q†

…q†

…q† n

with n P 0

…A:7b† …q†

in which the expressions for the mapping function, njk , …q† …q† and the constants, rjk and sjk , are, respectively, in the …q† …q† …q† same form as the expressions for nek , rjk , and sjk , except that the subscript e is replaced by j. …q† In Eqs. (A.7a) and (A.7b), the complex variables zjk , with q ˆ s; p, are de®ned in the form …q† zjk

ˆ xq ‡

…q† ljk yq

…A:8† …q†

…q†

in which the unknown complex constants lj1 and lj2 and their conjugates are obtained from the bending equilibrium equations for an unpatched plate,  4  3 0…q† …q† 0…q† …q† D22 ljk ‡ 4D26 ljk   2 0…q† 0…q† …q† ‡ 2D12 ‡ 4D66 ljk 0…q† …q†

0…q†

‡ 4D16 ljk ‡ D11 ˆ 0;

…A:9†

0…q†

where Dij are the components of the bending sti€ness matrix D0…q† , which is de®ned with respect to the local coordinate system …xq ; yq †. The bending sti€ness matrix, D0…q† , can be directly obtained from the transformation of matrix D…q† de®ned in the global coordinates. This transformation relation is available in any textbook on composite materials. The local vector of complex interpolation functions …q† Va , with …a ˆ x; y; z†, appearing in Eqs. (5a) and (5b) is de®ned as   …q†T …q†T …q†T …q†T …q†T Va ˆ Va…Nq1 † ; Va…Nq1 ‡1† ; . . . ; Van ; . . . ; Va…Nq2 † …A:10† with …q†T Va…n†

 ˆ

…q†T …q†T Va…n1† ; Va…n2†



in which n h i …q†T …q† …q† Va…nk† ˆ 2Re dak Unk ;

…q ˆ s; p; a ˆ x; y†

…A:11†

h io …q† …q† 2Im dak Unk

…q ˆ s; p; a ˆ x; y; k ˆ 1; 2† and …q†T Vz

 ˆ

…q†T …q†T …q†T …q†T Vz…Nq1 † ; Vz…Nq1 ‡1† ; . . . ; Vz…n† ; . . . ; Vz…Nq2 †

…q†T

…q†T



Vz…n1† ; Vz…n2†

…q ˆ s; p†

…A:14†

in which n h i …q†T …q† Vz…nk† ˆ 2Re Fnk ;

h io …q† 2Im Fnk

…q ˆ s; p; k ˆ 1; 2†: …A:15†

…A:7a† Fnk …zjk † ˆ …zjk †



293

…A:12†  …A:13†

The corresponding vectors of unknown coecients a…q† and b…q† are de®ned in the form n T o T T …q† …q†T …q†T a…q† ˆ aNq1 ; aNq1 ‡1 ; . . . ; a…q† …A:16† n ; . . . ; aNq2 in which n T o T …q† …q†T a…q† ˆ a ; a n1 n2 n

…A:17†

with

n h i h io …q†T …q† …q† ank ˆ Re ank ; Im ank

…A:18†

and

n T o T T …q† …q†T …q†T ; . . . ; b b…q† ˆ bNq1 ; bNq1 ‡1 ; . . . ; b…q† Nq2 n

…A:19†

in which n T o T …q† …q†T ˆ bn1 ; bn2 b…q† n

…A:20†

with

n h i h io …q†T …q† …q† bnk ˆ Re bnk ; Im bnk :

…A:21† …q†

Also, the vector of global interpolation functions Va , with …a ˆ x; y; z†, and the associated unknown vectors c…q† a in Eqs. (5a) and (5b) are de®ned as  T V…q† ˆ T00 …x; y†; T10 …x; y†; T01 …x; y†; T20 …x; y†; T11 …x; y†; a T02 …x; y†; . . . ; T0Mq …x; y†g in which Tij …x; y† ˆ Ti …x†Tj …y† and  cTa ˆ ca00 ; ca10 ; ca01 ; ca20 ; ca11 ; ca02 ; . . . ; ca0Mq :

…A:22† …A:23†

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