- Email: [email protected]
Thermal stresses in one-sided bond repair: geometrically nonlinear analysis
Theoretical and Applied Fracture Mechanics 40 (2003) 197–209 www.elsevier.com/locate/tafmec
Thermal stresses in one-sided bond repair: geometrically nonlinear analysis C.N. Duong *, J. Yu The Boeing Company, 5301 Bolsa Avenue, MC H013-A316, Huntington Beach, CA 92647-2099, USA
Abstract The problem of thermal stresses in an infinite isotropic plate rigidly bonded with an unsymmetrical, polygon-shaped reinforcement is analyzed within large deflection plate theory, using the equivalent inclusion method [J. Thermal Stresses 26 (2003) 457]. For simplicity, the plate assumes to be cooled down uniformly from a stress free temperature. The earlier approach proposed in [J. Thermal Stresses 26 (2003) 457] has been extended in the present paper to include geometric nonlinearity. Extension of the method to address residual thermal stresses associated with curing of the adhesive is also discussed. 2003 Elsevier Ltd. All rights reserved.
1. Introduction An infinite isotropic plate with a finite reinforcement made of a different material will develop thermal stresses during uniform temperature excursions. When the plate is bonded unsymmetrically with a reinforcement on one of its two sides, the thermal load will induce both in-plane deformations and the out-of-plane deflection in the plate. This is because the plate and the reinforcement inside the bonded region can be treated as an asymmetric laminate with its bending and extensional behaviors being coupled. Analytical solutions for this particular and related thermal stress problems existed for elliptical and polygonal reinforcements [1,2]. However, these solutions were derived within
*
Corresponding author. Tel.: +1-714-896-6204; fax: +1-714896-6205. E-mail address: [email protected] (C.N. Duong).
a small deflection plate theory. Unfortunately, it was shown in [3] that the effect of large deflection might be significant on the thermal stresses in a onesided reinforced plate for a large temperature excursion. This necessitates a geometrically nonlinear analysis of a bonded plate with an unsymmetrical reinforcement under uniform cooling. As shown earlier in [4] and later in [1], the problem of an isotropic plate with an inhomogeneity or reinforcement under thermal loading can be solved more conveniently by the equivalent inclusion method. This is because analytical solutions for elliptical and polygon-shaped inclusions with uniform eigenstrains [5,6] and uniform eigencurvatures [7] are available. An analytical solution procedure based on an engineering approach is therefore introduced to solve for a large deflection problem of a bonded plate under uniform cooling, also using the equivalent inclusion method. Extension of the method to address thermal stresses associated with curing of the adhesive will also be
0167-8442/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0167-8442(03)00046-6
198
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
discussed. It is worthwhile to note that an analytical solution of the present problem for thermal stresses has been obtained previously using the equivalent inclusion method in [8] for a case of plane stress and in [1] for a case of linear bending as mentioned earlier. The solution of the problem treated here is of practical interest since a plate with a bonded reinforcement has been found in many engineering applications. A particular example in aerospace application is a bonded composite repair over a cracked metallic structure [9]. A repair method using composite patches to reinforce the cracked structure has been shown to very promising owing to the light weight, high stiffness and strength of the composite [9]. Although a symmetrical repair is the most effective reinforcement, unsymmetrical repairs provide a clear advantage when it is difficult or not possible to access both sides of a structure. Analysis of a bonded repair is usually carried out in two-stages following RoseÕs approach [10]. In the first stage, the thermomechanical response of an uncracked, reinforced plate is determined. This is then followed by a fracture analysis of a cracked reinforced plate with known traction acting on the crackÕs surfaces [11]. The solution method presented here therefore can be used to estimate thermal stresses resulting from curing or due to low operating temperature in the first stage analysis of RoseÕs approach for an unsymmetrical repair. It should be noted that the solution method presented here does not account for the flexibility of the adhesive as in a realistic bonded repair. In other words, it is assumed that the cured adhesive bond does not allow any relative displacement between the plate and the reinforcement. This rigid-bond assumption ignores the finite width of the load transfer zone around the boundary of the reinforcement, which is valid when the actual width of the load transfer zone is small relative to the in-plane dimensions of the reinforcement [2].
2. Preliminary analysis Consider a reinforcement that spans across the entire width of the plate, so that the stresses in the
reinforcement and the plate are uniform in the width direction everywhere within the reinforced portion as shown in Fig. 1. The reinforcement and plate are assumed to be fused together with no relative sliding at the interface. The whole reinforced plate is subjected to a uniform temperature excursion. This problem will be solved exactly within a large deflection theory. In the absence of the transverse load, the equilibrium equations based on the von Karman theory are given by [12]: N22;2 ¼ 0; ð1Þ M22;22 þ N22 w;22 ¼ 0; ð2Þ where N22 and M22 are the stress and moment resultant defined explicitly in the next section, and a comma indicates partial differentiation. It then follows from Eq. (1) upon integration that: N22 ¼ const ¼ C:
ð3Þ
On the other hand, the kinematics and constitutive relations for a laminated plate is given by [13]: e22 ¼ v;2 þ eNL 22 þ j22 ðz h0 Þ; 1 2 eNL 22 ¼ 2w;2 ;
N22 ¼ A22 ðv;2 þ 12w2;2 a22 DT Þ B22 w;22 ; M22 ¼ B22 ðv;2 þ 12w2;2 a22 DT Þ D22 w;22 ;
ð4Þ
ð5Þ
where A22 , B22 , and D22 are extensional, coupling, and bending stiffnesses (in Voight notation) that are also defined explicitly in the next section; v and w are the longitudinal and transverse displacement, respectively; a22 is the thermal expansion coefficient; eNL 22 is the nonlinear strain; h0 is the zcoordinate of the reference plane; and j22 is the curvature defined by j22 ¼ o2 w=ox22 . The reference plane is chosen to be at the mid-plane of the t base plate, i.e., h0 ¼ 2p , for both regions inside and outside the overlap. Applying Eqs. (2), (3) and (5) to the region outside the overlap (see Fig. 1(c)) and enforcing the boundary condition at infinity, i.e., 0 N122 ¼ 0, yields: 0 N22 ¼ 0; 0 M22;22 ¼ 0; 0 0 N22 ¼ A022 ðv0;2 þ 12w02 ;2 a22 DT Þ ¼ 0; 0 M22 ¼ D022 w0;22 ;
ð6Þ
ð7Þ
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
N 0∞ 22
199
N 0∞ 22
N ∞0 22
N 022 = N 0∞ 22 2.
2.
N I22 = N 022 = N 0∞ 22 ∞
γ
N I22 ∆T ≠ 0
M due to offset = N I22 . γ N 0∞ 22
N 0∞ 22 (a)
(b)
(c)
Fig. 1. Plate reinforced with a patch spanning across the entire widthÕs plate. (a) Plan view, (b) side view, (c) Free-body diagram used 0 to explain the effects of shifting the reference planeÕs position. Note that for the case of purely thermal loading, N122 ¼ 0.
where B022 has been set equal to zero since the plate outside the overlap is a single layer of homogeneous material and thus symmetric with respect to the reference plane, and the superscript 0 denotes the region outside the overlap. Substituting the 0 second equation of Eq. (7) for M22 into the second equation of Eq. (6) gives: w0;2222 ¼ 0:
ð8Þ
By integrating Eq. (8) and enforcing the simply 0 supported conditions at the ends, i.e., M22 ðl1 Þ ¼ 0 0 0 w ðl1 Þ ¼ Q22 ðl1 Þ ¼ 0 where Q22 is the transverse shear force defined as Q022 ¼ dM22 =dx2 , the transverse displacement outside the overlap is then given by: w0 ðx2 Þ ¼ C1 ðl1 x2 Þ;
ð9Þ
where C1 is an unknown constant which must be determined from the displacement, slope, and traction continuity at end of the overlap. In contrast, the first equation of Eq. (7) can be rewritten as: 0 ðv0;2 þ 12w02 ;2 Þ ¼ a22 DT :
ð10Þ
Similarly, the region inside the overlap also has I ¼ const ¼ 0 from consideration of the force N22 equilibrium. With that, applying Eqs. (2), (3) and (5) to the region inside the overlap yields: I N22 ¼ 0; ð11Þ I M22;22 ¼ 0; I I I I ¼ AI22 ðvI;2 þ 12wI2 N22 ;2 a22 DT Þ B22 w;22 ¼ 0; I I I I M22 ¼ BI22 ðvI;2 þ 12wI2 ;2 a22 DT Þ D22 w;22 ;
ð12Þ
200
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
where the superscript I indicates the region inside the overlap. It is worthy to note that AI22 , BI22 , DI22 , and aI22 are stiffnesses and thermal expansion coefficient of a laminated plate composing of the reinforcement and the base plate. Eq. (12) can also be rewritten as: BI22 I I ðvI;2 þ 12wI2 ;2 Þ ¼ I w;22 þ a22 DT ; A22 I2 B22 I I I ¼ M22 D 22 w;22 : AI22
ð13Þ
Substituting the second equation of Eq. (13) into the second equation of Eq. (11) gives: wI;2222
¼ 0:
ð14Þ
Upon integrating Eq. (14) and enforcing the symmetry conditions at the center of the reinforced plate, one finally obtains: j wI ðx2 Þ ¼ C2 þ x22 ; 2
ð15Þ
where C2 and j are constants yet to be determined. From which, wI;2 ðx2 Þ ¼ j x2 ; wI;22 ðx2 Þ ¼ j ¼ const:
ð16Þ
Thus, the constant j is the curvature of the plate inside the overlap, i.e., j22 ¼ j. Substituting Eq. (16) into the first equation of Eq. (13) yields: vI;2 ðx2 Þ ¼
BI22 j2 2 I x: j þ a DT 22 AI22 2 2
ð17Þ
The fist two conditions for determining the three constants C1 , C2 and j, are the displacement and slope continuity conditions at the end of the overlap, i.e., w0 ð‘Þ ¼ wI ð‘Þ; w0;2 ð‘Þ ¼ wI;2 ð‘Þ:
ð18Þ
The third condition is derived from the continuity of the normal traction acting on the base plate at the inhomogeneity interface, i.e., IðpÞ
0 N22 ð‘Þ ¼ A022 ðvI;2 ð‘Þ þ 12wI2 ;2 ð‘Þ a22 DT Þ 0 ð‘Þ ¼ 0; ¼ N22
ð19Þ
IðpÞ
where N22 is the stress resultant in the base plate of the patch-plate combination. From which, vI;2 ð‘Þ ¼ a022 DT
‘2 2 j ; 2
ð20Þ
where the result from Eq. (16) for wI;2 ð‘Þ has been utilized in the derivation of Eq. (20). However, vI;2 ð‘Þ can also be obtained by evaluating Eq. (17) at x2 ¼ ‘ as: vI;2 ð‘Þ ¼
BI22 ‘2 2 I j : j þ a DT 22 AI22 2
ð21Þ
Thus, the constant j can be determined from Eqs. (20) and (21) as: j¼
AI22 0 a22 aI22 DT : I B22
ð22Þ
Once j is found, then C1 and C2 can be determined from the conditions given by Eq. (18). For future discussion, the average nonlinear strain inside the overlap, i.e., ‘ 6 x2 6 ‘, is also calculated to be: R ‘ 1 I2 R‘ 1 2 w dx2 ðjx2 Þ dx2 1 2 2 ‘ 2 ;2 NL e22 ¼ ¼ ‘ 2 ¼ j‘: 6 2‘ 2‘ ð23Þ In summary, under purely thermal loading, this preliminary analysis predicts that the curvature of the reinforced plate is constant inside the overlap and equals to zero outside the overlap. Furthermore, since the curvature of the plate is zero outside the overlap, the term N22 w;22 associated with the large transverse deflection in the moment equilibrium equation also equals to zero in that region. On the other hand, when the reinforced plate is subjected to the remote stress, i.e., 0 N122 6¼ 0, the curvature of the plate inside the overlap is not necessarily constant and zero respectively for regions inside and outside the over0 lap. However, if N122 is sufficiently small, the trends for the curvature and N22 w;22 as predicted by the above analysis are probably still true within 0 the approximate sense for the case N122 6¼ 0. In particular, the curvature is still approximately constant and zero inside and outside the overlap while N22 w;22 can be assumed to be zero outside the overlap. In addition, the average nonlinear strain also takes approximately a same form as Eq. (23).
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
In that case, the average moment inside the overlap is derived from Eq. (2) as follows. I 0 þ N122 j ¼ 0; M22;22
N 0 j x22 I ; ¼ 122 M22 2 R‘ MI 1 I 0 M 22 ¼ ‘ 22 ¼ j‘2 N122 ; 6 2‘
ð24Þ
I 0 0 ¼ N22 ¼ N122 , and wI;22 ðx2 Þ j via Eq. since N22 (16). In contrast, a linear analysis of the latter thermo-mechanical problem will significantly over-predict the bending stresses inside the overlap. This is because the linear analysis ignores the nonlinear term N22 w;22 in the moment equilibrium equation and the nonlinear strain term eNL ij in the kinematics relation. However, if in the linear analysis the reference plane of the overlap is purportedly shifted from z ¼ h0 ¼ tp =2 to z ¼ h0 c ¼ tp c where c ¼ 16j‘2 and j is the sought curvature, 2 then the effect of the geometrically nonlinearity will be accounted for approximately in the average sense as explained immediately below. In that case, the kinematics relation for regions inside and outside the overlap are given by:
e22 ¼ v;2 þ j22 ðz h0 þ cÞ
jx2 j 6 ‘;
e22 ¼ v;2 þ j22 ðz h0 Þ jx2 j > ‘:
ð25Þ
By comparing the first equation of Eq. (25) with that of Eq. (4), it becomes clear that the effect of the shift of the reference planeÕs position inside the overlap is to raise the linear strains there by an amount equal to the average nonlinear strain predicted by the corresponding nonlinear analysis, 2 2 1 2 2 since j22 c ¼ j6 ‘ and eNL 22 ¼ 6j ‘ via Eq. (23). On the other hand, from Fig. 1(c), this positional shift of the reference plane will also induce a constant 0 0 moment of N122 c ¼ 16j‘2 N122 that equals to the average moment associated with nonlinear term N22 w;22 in the nonlinear analysis via Eq. (24). In addition, both the present linear analysis with the mentioned shift of the reference planeÕs position and the approximated nonlinear analysis described in the preceding paragraph neglect the nonlinear term N22 w;22 in the formulation for the region outside the overlap. In other words, it appears that the present geometrically nonlinear problem can be
201
solved approximately by the geometrically linear analysis but with the appropriate shift of the reference planeÕs position inside the overlap, providing that the stress resultant remains small everywhere. Even though this ‘‘geometrically linear analysis’’ approach may lack the scientific rigor, however, it appears technically sound from the engineering viewpoint. This engineering approach is therefore proposed for the analysis of thermal stresses in a polygonal patch considered in the next section.
3. Formulation for thermal stresses in a polygonal patch Consider an infinite isotropic plate bonded rigidly with a polygon-shaped reinforcement X. The reinforcement is either isotropic or orthotropic with its material principal axes parallel to the global coordinate axes. Let us also assume that the reinforcement when it is orthotropic will have a larger mismatch in the thermal expansion coefficient with the base plate along the x2 -direction. The bonded plate assumes to be subjected to a uniform temperature change DT (see Fig. 2). Extension of the present approach to address residual thermal stresses associated with curing will be discussed in a latter part of the next section. For simplicity, all shear components are also assumed to be zero. Based on results from the finite element analysis, this assumption is found to be valid for x2
Polygon-shaped reinforcement
2L
x1
2W
Fig. 2. Geometry of a bonded plate with a polygon-shaped reinforcement.
202
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
the present problem. Since the reinforcement is rigidly bonded to a plate, the reinforced region of the plate is treated as an inhogomogeneity. It is rather unwieldy to solve the stated problem using a rigorous geometrically nonlinear analysis. We will therefore present here an approximate solution based on an engineering approach similar to that outlined in Section 2 above. The approach proposed in that section will be extended in this section with details shown below, and its results will be compared with the FE solutions for a wide range of reinforcementÕs configurations. In contrast to the results of patching across the entire width of the plate, the stress resultants inside the plate reinforced with a polygonal patch due to a uniform temperature change will not necessarily equal to zero even without remote applied stresses. It should remember that except at infinity, only the average stress resultant across the plate width is required to equal to zero. However, if these stress resultants are assumed to be small, then the thermal stresses in the latter problem can be solved approximately by a geometrically linear analysis with the reference plane of the reinforced region being shifted purportedly from z ¼ h0 ¼ tp =2 to t z ¼ h0 c ¼ 2p c as mentioned in Section 2. Thus, the considered bonded plate is divided into two regions; each will be modeled using different kinematics and constitutive relations. The region outside of the reinforcement is a homogeneous infinite isotropic plate. On the other hand, the inhomogeneous region is a finite asymmetric laminated plate that composes of two layers corresponding to the base plate and the reinforcement. The kinematics relations used to describe the inhomogeneity and the surrounding plate are summarized below, which are the 2-D generalization of Eq. (25). For the inhomogeneity, eIij ¼ eIij þ ðz h0 cÞjIij ; 1 I eIij ¼ ui;j þ uIj;i ; 2
ð26Þ
c ¼ 16j0 ‘2 ; jIij ¼
o2 wI ; oxi oxj
ð27Þ
while for the surrounding isotropic plate outside the inhomogeneity, epij ¼ epij þ ðz h0 Þjpij ; epij ¼
1 p ui;j þ upj;i ; 2
jpij ¼
ð28Þ
o2 wp ; oxi oxj
where j0 is a representative curvature in the x2 direction inside region X, ‘ is the longest halflength of the polygonal reinforcement measured from its center, and the rest are previously defined. This conjecture will then be tested for its validity for a wide range of reinforcementÕs configurations. As shown later in Section 5, the above conjecture will yield sufficiently accurate results for stresses in the base plate near the center of the reinforcement and thus at the prospective crack location in the bonded repair when these results are compared with FE solutions. Once the kinematics relations are established, the constitutive relations for the inhomogeneity and the surrounding plate can be derived in a straightforward manner using the classical plate theory [14]. Following the work in [1], the constitutive relations for the inhomogeneity and for the surrounding plate are given by: For the inhomogeneity, ðTÞ
NijI ¼ AIijkl eIkl þ BIijkl jIkl Nij ; ðTÞ
ð29Þ
MijI ¼ BIijkl eIkl þ DIijkl jIkl Mij
and for the surrounding isotropic plate, Nijp ¼ Apijkl epkl ; Mijp ¼ Dpijkl jPkl ;
ð30Þ
where Aijkl , Bijkl and Dijkl are the extensional, coupling, and bending stiffness tensors, respectively; Nij and Mij are the stress and moment reðTÞ ðTÞ sultant. In Eq. (29), Nij and Mij are the stress and moment resultant associated with the thermal strains and their explicit formula for the present problem will be derived in the next paragraph while Aijkl , Bijkl , Dijkl , Nij and Mij are defined as [14]:
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
Aijkl ¼
Z
t
Cijkl dz; 0
Bijkl ¼
Z
t
ð31Þ
Cijkl ðz hÞdz;
0
Z
Dijkl ¼
t 2
Cijkl ðz hÞ dz;
0
Nij ¼
Z
t
0
Mij ¼
Z
t
ðTÞ Cijkl ekl eij dz;
Cijkl ekl
ðTÞ eij
ðz hÞ dz;
ð32Þ
0
where Cijkl is the regular elasticity tensor, t is the total thickness of an inhomogeneity or of a surrounding base plate, and h is equal to h0 for a surrounding base plate and ðh0 cÞ for an inhomogeneity. It was shown in [1,8] that the problem of a bonded plate under uniform cooling can be formulated as an initial strain problem with an initial strain prescribed in the bonded patch and that initial strain is given by: ðTÞðRÞ
eij
¼ Daij DT ;
ð33Þ
P where Daij ¼ aR ij aij , aij is the thermal expansion coefficient tensor; and the superscript R signifies the reinforcement. It then follows from [1,14] that ðTÞ ðTÞ Nij and Mij in Eq. (29) are given by: R R E11 Da11 mR ðTÞ 12 E22 Da22 N11 ¼ þ DT tR ; R R 1 mR 1 mR 12 m21 12 m21 R R R m21 E11 Da11 E22 Da22 ðTÞ N22 ¼ þ DT tR ; R R 1 mR 1 mR 12 m21 12 m21 ð34Þ
uniform cooling was formulated as an inhomogeneity problem with the kinematics and constitutive relations prescribed by Eqs. (26), (28), (29), (30), respectively. This inhomogeneity problem then will be solved by the equivalent inclusion method. For the purpose of clarity, this method will be demonstrated first for a case when c in Eq. (27) is prescribed as a known constant. In the equivalent inclusion method, the stress, in-plane reference strain and curvature fields induced by an inhomogeneity occupied region X will be the same as those induced by eigenstrain field eij and eigencurvature field jij in the same region of a homop geneous material of Cijkl when eij and jij are selected appropriately as shown in Fig. 3. The latter homogeneous problem is called an inclusion problem with the constitutive relation given by: 8 < AP eH e for x inside X; ijkl kl kl NijH ¼ : AP eH for x outside X; ijkl kl ð35Þ ( H P D j j for x inside X; ijkl kl kl MijH ¼ for x outside X: DPijkl jH kl In the above equation, the superscript H denotes a homogeneous problem. It should be reminded that Apijkl and Dpijkl are defined earlier in Section 3 by Eq. (31) with h ¼ h0 . For a constant (or uniform) eij and jij , from [6,7], the elastic solutions of the homogeneous problem can be expressed as: eH ij ðxÞ ¼ Sijkl ðxÞekl ;
ðTÞ
P C ijkl
2
ðTÞ
ð36Þ
jH ij ðxÞ ¼ Kijkl ðxÞjkl ;
N12 ¼ 0; Mij ¼
203
2
ðtP þ tR h0 Þ ðtP h0 Þ ðTÞ Nij : 2 tR
With the stated problem completely formulated, it remains now to outline its solution procedure and that will be the topic of the next section.
ε*ij
∆T
κ *ij
Ω
Ω
I C ijkl
4. Solution method In Section 3, a problem of an infinite isotropic plate with a polygon-shaped reinforcement under
(I)
(II)
Fig. 3. An illustration of the equivalent inclusion method. (I) an inhomogeneity problem and (II) an inclusion problem with a uniform eigenstrain and a uniform eigencurvature.
204
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
where Sijkl and Kijkl are Eshelby and Eshelby-type tensors for eigenstrain and eigencurvature, respectively. Algorithms for evaluating these tensors H are detailed in [6,7]. eH ij and jij are in general not uniform inside X except for an elliptical region. The equivalency condition between the inhomogeneity problem and the homogeneous problem requires that the in-plane strain and curvature field inside X as well as the stress and moment resultant of the two problems must be the same, i.e., eIij
¼
¼
ð37Þ
NijH ;
MijI ¼ MijH :
eIij
)
" ¼
AIijkl
DAaa11 ðS1111 ð0Þe11 þ S1122 ð0Þe22 Þ þ DAaa22 S2211 ð0Þe11 þ S2222 ð0Þe22 þ APaa11 e11 þ APaa22 e22 þ BIaa11 K1111 ð0Þj11 þ K1122 ð0Þj22 þ BIaa22 K2211 ð0Þj11 ðTÞ þ K2222 ð0Þj22 ¼ Naa ;
BIijkl
#1 ("
BIijkl DIijkl ( ) eH mn emn jH mn ðTÞðRÞ
H Since eH ij and jij are in general not uniform inside X as mentioned above, the equivalency condition given by Eq. (37) can only be satisfied approximately when eij and jij are approximated by a (unknown) constant tensor. By enforcing condition (37) at point ð0; 0Þ, i.e., the origin of the coordinate system, and by substituting results from Eqs. (26), (29) and (36) into (37), one finally obtains the following linear equations for the unknown eij and jij (without summation on the subscript a):
jmn ðTÞ
ð38Þ
DAijkl ¼ AIijkl APijkl ; for a ¼ 1; 2:
Once eij and jij are determined, the in-plane strain and curvature fields of the homogeneous problem can be calculated from Eqs. (36) while the
APklmn 0
( þ
0
#
DPklmn )) ðTÞ
Nkl
ðTÞ
;
ð40Þ
jkl ðTÞ
where Cijkl , ekl , Nkl and Mkl are defined earlier in Section 3. So far we have assumed that c is known a priori. However, in reality, it is also part of the sought solutions since it depends on the x2 -component of the curvature inside X via Eq. (27). Since the curvature field inside X in general is not uniform, the curvature at any point inside X can be used as j0 in the evaluation of c. However, it is found that the following definition of j0 will yield sufficiently accurate solutions for a bonded plate with reinforcements of different shapeÕs aspect ratios: j0 ¼ jI22 ð0; y Þ; y ¼ minimum of f‘; W g:
BIaa11 S1111 ð0Þe11 þ S1122 ð0Þe22 þ BIaa22 S2211 ð0Þe11 þ S2222 ð0Þe22 þ DDaa11 K1111 ð0Þj11 þ K1122 ð0Þj22 þ DDaa22 K2211 ð0Þj11 ðTÞ ; þ K2222 ð0Þj22 þ DPaa11 j11 þ DPaa22 j22 ¼ Maa
DDijkl ¼ DIijkl DPijkl
(
jIij
eH ij ;
jIij ¼ jH ij ; NijI
stress and moment resultant are evaluated using Eq. (35). Similar to those given in [1], the stresses in the base plate and in the reinforcement inside X are finally given by: I P ekl þ ðz h0 þ cÞjIkl ; rPij ¼ Cijkl ð39Þ ðTÞðRÞ R I þ ðz h0 þ cÞjIkl ; rR ij ¼ Cijkl ekl ekl
ð41Þ
In the above equation, the notation jI22 ð0; y Þ means that jI22 is evaluated at a point ð0; y Þ on the x2 -axis, where y is either the half-length ‘ or the half-width W of the reinforcement whichever is smaller. With the solution procedure for a fixed value of c described and j0 defined, the solution of the geometrically nonlinear problem can be obtained by the following simple iterative procedure. First, j0 and thus c are assumed to be zero, and the elastic solutions for stress, strain and curvature field are obtained in a usual manner using the given value of c as described earlier. Second, the condition imposed by Eq. (41) is verified. If that condition is satisfied, then the just obtained solution has converged to the true solution of the problem. Otherwise, another value of j0 , thus c, will be assumed and the above steps are repeated until convergence. A new trial value for j0 will be
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
calculated using the following recursive formula [15]: ðiþ1Þ
j0
ðiÞ
IðiÞ
¼ ð1 kÞj0 þ kj22 ð0; y Þ; ðiþ1Þ
ð42Þ
ðiÞ
where j0 and j0 are values of j0 in the current IðiÞ and previous iterative cycle, respectively; jij is the elastic solution for the curvature field obtained from the previous iterative cycle that is based on ðiÞ c ¼ 16j0 ‘2 ; and k is a scalar factor which takes values between 0 and 1. It should be noted that a normal direct substitution method corresponds to a case of k ¼ 1. It remains now to discuss the extension of the present method to address residual thermal stresses associated with curing of the adhesive. While the uniform temperature model considered here truly represents the thermal loading during high altitude cruising, it may appear unrealistic for simulating the curing process of the adhesive. However, by using simple curing models presented in [10,16,17], the curing process of the adhesive can be simulated as a uniform cooling process of a whole bonded structure with the base plateÕs thermal expansion coefficient being approximated by an (lower) ‘‘effective’’ value. In a similar context, the residual thermal stress problem associated with curing was formulated as an initial strain problem in [8]. As shown in [8], the only difference between the curing problem and the present uniform cooling problem is that the initial strain prescribed in the patch for the former problem is given by the following expression instead of Eq. (33): ðTÞðRÞ
eij
0 ¼ aR ij DTcuring eij ;
ð43Þ
where DTcuring ¼ Tambient Tcuring and in general Tcuring > Tambient ; e0ij is the strain solution of an inp clusion problem of a homogeneous material Cijkl with a uniform eigenstrain field ap DTcuring dij prescribed inside region P; ap is the thermal expansion coefficient of the base plate; and the inclusion region P assumes the same shape as that of the heating blanket. Thus, the thermal effect due to curing can be addressed in a similar manner as that considered here if any curing model mentioned above is employed. It is important to note that other advanced curing models such as the one
205
considered in [2] for a geometrically linear analysis of residual thermal stresses can also be extended to include the effect of geometric nonlinearity using the present results. For instance, by shifting the neutral plane of the curing model presented in [2] by an amount of c as postulated by Eq. (27) and iteratively solving for c as described earlier, perhaps one can obtain approximate nonlinear solutions of the curing problem with various degrees of thermal constraints. It would be difficult to supply a precise error analysis for the present approximate method: its accuracy is most readily assessed by comparison with finite element results for particular cases. This is what is depicted in Section 5.
5. Results and discussion To illustrate the present method, particular cases corresponding to typical repair configurations are considered in this section. All computations in this section unless noted otherwise are carried out for a temperature excursion of )75 C, corresponding to a typical temperature change experienced by the aircraft during high altitude cruising. The first case is an infinite thin plate repaired with a square patch ð‘=W ¼ 1Þ. A patch is bonded rigidly to a plate. The material properties and thickness of the base plate and the patch are given below: Base plate: Aluminum, Ep ¼ 72:4 GPa, mp ¼ 0:33, ap ¼ 22:5E)06/C, tP ¼ 1:6 mm. Patch: Boron/Epoxy, Ex ¼ 18:7 GPa, Ey ¼ 193:6 GPa, myx ¼ 0:21, Gxy ¼ 5:5 GPa, ax ¼ 21:4E)06/ C, ay ¼ 4:3E)06/C, tR ¼ 0:66 mm. The patch stiffness ratio ER tR =Ep tp (S) is 1.1. The sensitivity of the length of the patch relative to the base plate thickness is studied. Four patch lengths ð2 ‘Þ of 10.16, 20.32, 30.48, and 40.64 cm are considered in the analysis. Since the thermal stresses in the patch are compressive and they will be negated under far field tensile stress [1,2], we therefore concentrate our effort to the plateÕs thermal stresses. Thermal stresses in the base plate at a patchÕs center in the x2 -component are of special interest,
206
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
as they are needed for fracture analysis using Wang–RoseÕs crack bridging model [11]. Since the stress distributions in the plate and patch are linear through their thickness, it will be more meaningful to report these stresses in terms of the mean and bending values. These values of thermal stresses are plotted in Fig. 4 as the functions of the normalized half patch length ‘=tp where tp is a plateÕs thickness. The mean and bending stresses are defined as half of the sum and difference of stresses near the top and bottom surfaces of the base plate. All analytical solutions are obtained using k in the recursive formula given in Eq. (42) equal to 0.1, and they converge within about 30 iterations. To assess the accuracy of the analytical method, results from the finite element analysis are also obtained and compared with the analytical predictions in Fig. 4. Finite element analyses are
Mean stress σm(MPa)
L/W=1, S=1.1 50 45 40
3-D F.E. Analytical
35 30 25 20 0
25
50
75
100
125
carried out using ABAQUS with the base plate and each ply of the patch modeled separately. The base plate is modeled as three layers of 20-node solid elements while each ply of the patch is by one layer of solid elements. To avoid the adverse effect of the small thickness-to-length ratio on the solution, stiffness matrices of all solid elements are evaluated using reduced integration. In the FE model, the base plate was restrained from the outof-plane deflection along its periphery. A typical mesh used in the FE analyses is given in Fig. 5. Similar results but for different patchÕs aspect ratios are presented in Fig. 6 for ‘=W ¼ 2 and in Fig. 7 for ‘=W ¼ 1=2. In contrast to the planestress and linear bending results [1,8], from Figs. 4, 6, 7, thermal stresses in the base plate are affected by the patchÕs size. For a same patchÕs aspect ratio, the mean components of the thermal stresses in the plate increase with patchÕs sizes while the bending components show a reverse trend. These bending stresses will eventually vanish for large values of ‘=tp as expected. In general, analytical predictions are in quite good agreement with the FE results (within 12%) for all aspect ratios ‘=W considered. To show the effect of a patchÕs thickness on the thermal stresses, analyses of a few selective patch configurations above but with a thickness of 1.057 mm are performed and their results are summarized in Table 1. For reference, corresponding results previously presented in Figs. 4, 6 and 7 are
150
Normalized half patch length / tp
(a)
Bending stress σb(MPa)
L/W=1, S=1.1
(b)
50 45 40 35 30 25 20 15 10 5 0
3-D F.E. Analytical
0
25
50
75
100
125
150
Normalized half patch length / tp
Fig. 4. Thermal stresses in a middle of a plate reinforced with a square patch for different normalized half patch lengths. (a) The mean stresses, (b) the bending stresses.
Fig. 5. A typical mesh used in the FE analysis.
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
L/W=1/2, S=1.1
L/W=2, S=1.1 45
50 45
Mean stress σm(MPa)
Mean stress σm(MPa)
55
3-D F.E. Analytical
40 35 30 25 0
50
100
150
200
250
40
35
3-D F.E. Analytical
30
25 50
300
Normalized half patch length / tp
(a)
75
100
125
150
Normalized half patch length / tp
(a)
L/W=2, S=1.1
L/W=1/2, S=1.1
35
45
Bending stress σb(MPa)
Bending stress σb(MPa)
207
3-D F.E.
30
Analytical
25 20 15 10 5 0 0
50
100
150
200
250
3-D F.E. Analytical
35 30 25 20 50
300
Normalized half patch length / tp
(b)
40
(b)
75
100
125
150
Normalized half patch length / tp
Fig. 6. Thermal stresses in a middle of a plate reinforced with a rectangular patch ðL=W ¼ 2Þ for different normalized half patch lengths. (a) The mean stresses, (b) the bending stresses.
Fig. 7. Thermal stresses in a middle of a plate reinforced with a rectangular patch ðL=W ¼ 1=2Þ for different normalized half patch lengths. (a) The mean stresses, (b) the bending stresses.
repeated in the table. It follows from Table 1 that a thicker patch will induce higher thermal stresses in the plate as expected and a same good agreement is observed between the analytical and the FE results. Additional results but not shown here indicate that bonded plates with a same patch stiffness ratio ER tR =Ep tp , same aspect ratio ‘=W and same parameter ‘=tp will have the same thermal stresses.
Thus, for a given temperature change, the nonlinear thermal stresses can be characterized in terms of three nondimensional parameters ER tR = Ep tp , ‘=W and ‘=tp . So far only patches of rectangular or square shapes have been considered. Another common shape of the patch is therefore selected for presentation. An octagonal patch sometimes is preferred
Table 1 Comparison of thermal stresses in a middle of a patched plate with different patchÕs thicknesses L=tp
L=W
tR =tp
128 128 128 128
1 1 2 2
0.416 0.666 0.416 0.666
Analytical method
FE method
rm (MPa)
rb (MPa)
rm (MPa)
rb (MPa)
45.6 49.8 44.4 49.9
17.6 20.1 18.4 20.2
45.1 49.3 47.3 53.0
15.7 18.5 17.8 20.0
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
45˚
8.89 cm
Mean stress σm(MPa)
208
x2
17.78 cm
x1
L/W = 1, S=1.1 100 90 80 70 60 50 40 30 20 10 0 0
(a)
F.E. Analytical
25 50 75 100 125 Temperature change (°C)
150
6.35 cm 12.7 cm
Fig. 8. Geometry of the example problem with an octagonal patch.
Bending stress σb(MPa)
L/W = 1, S=1.1 28 F.E. Analytical
24 20 16 12 8 4 0 0
25
(b)
• Analytical predictions, rm ¼ 32:1 MPa, rb ¼ 36:4 MPa. • FE results, rm ¼ 35:1 MPa, rb ¼ 34:7 MPa. Again the analytical method predicts well the nonlinear thermal stresses. As a final example, an infinite plate rigidly bonded with a square boron/epoxy patch (‘=W ¼ 1, S ¼ 1:1, ‘=tp ¼ 128) under a wide range of temperature excursions is considered. The mean and bending components of the thermal stresses in a middle of the plate are plotted in Fig. 9(a) and (b), respectively, for different cooling temperature ranges jDT j. For a future discussion, the normalized bending stresses ðrb =rm Þ are also plotted versus jDT j in Fig. 10. From Fig. 9, it seems that
150
Fig. 9. Thermal stresses in a middle of a plate reinforced with a square patch for different temperature excursions. (a) The mean stresses, (b) the bending stresses.
L/W = 1, S=1.1 Normalized bending stress σb/σm
over a rectangular patch because of its superior performance in preventing peeling near the patchÕs corner. An analysis of an octagonal patch therefore will be demonstrated next. The geometry of the octagonal patch is shown in Fig. 8. Material properties and thickness of the patch and the base plate are identical to those given earlier in the beginning of this section. Analytical predictions as well as FE results for the thermal stresses in the base plate are given below.
50 75 100 125 Temperature change (°C)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
F.E. Analytical
0
25
50 75 100 125 Temperature change (°C)
150
Fig. 10. Normalized bending stresses in a middle of a reinforced plate for different temperature excursions.
analytical predictions for the mean stresses are in excellent agreement with FE results for all jDT j while the analytical predictions for the bending stresses show a larger discrepancy with FE solutions. The deviation between analytical and FE results for the bending stresses may exceed 20% for jDT j > 100 C and attains a largest value of 30% when jDT j ¼ 138 C. However, since the contribution of the bending component to the total
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
stress is significantly smaller for jDT j > 100 C as illustrated in Fig. 10 via rb =rm , this discrepancy may not be significant as it appears in Fig. 9. For example, the analytical method predicts that the bending stress is 21% of the mean stresses when jDT j ¼ 138 C while the FE method yields a ratio of 28%. Thus, the agreement between two methods is considered to be satisfactory, in view of the approximate nature of the present analytical model.
6. Conclusion An analytical method to estimate thermal stresses in an infinite isotropic plate bonded unsymmetrically with a polygon-shaped reinforcement is presented within a geometrically nonlinear analysis. The method is based on an engineering approach. By postulating certain simplifications or assumptions in the formulation, the geometrically nonlinear problem can be solved approximately in a similar manner as that for a linear analysis. Even though these assumptions appear to be rather significant assumptions from the outset, they are justified as reasonable for the present thermal stress problem by checking the ability of the present method in predicting results in reasonably close agreement with FE solutions for numerous practical cases.
Acknowledgements The authors take this opportunity to gratefully acknowledge the support provided for this research by the USAF, WPAFB Ohio, as part of the composite repair of aircraft structures (CRAS) program under the management of Capt. Williams Shipman. The contract number is F33615-97C3219.
209
References [1] C.N. Duong, J. Yu, Thermal stresses in a one-sided bonded repair by a plate inclusion model, Journal of Thermal Stresses 26 (2003) 457–464. [2] C.H. Wang, D. Erjavec, Geometrically linear analysis of the thermal stresses in one-sided composite repairs, Journal of Thermal Stresses 23 (2000) 833–851. [3] J.C. Klug, C.T. Sun, Large deflection effects of cracked aluminum plates repaired with bonded composite patches, Composite Structures 42 (1998) 291–296. [4] H.G. Beom, Y.Y. Earmme, The elastic field of an elliptic cylindrical inclusion in a laminate with multiple isotropic layers, Journal of Applied Mechanics 66 (1999) 165–171. [5] T. Mura, Mechanics of Defects in Solids, Martinus Nijhoff, 1987. [6] G.J. Rodin, EshelbyÕs inclusion problem for polygons and polyhedra, Journal of Mechanics and Physics of Solids 44 (1996) 1977–1995. [7] C.N. Duong, J. Yu, Analysis of a plate containing a polygon-shape inclusion with a uniform eigencurvature, Journal of Applied Mechanics, in press. [8] C.N. Duong, J. Yu, An analytical estimate of thermal effects in a composite bonded repair: Plane stress analysis, International Journal of Solids and Structures 39 (2002) 1003–1014. [9] A.A. Baker, Crack patching: Experimental studies, practical applications, in: A.A. Baker, R. Jones (Eds.), Bonded Repair of Aircraft Structures, Martinus Nijhoff, 1988. [10] L.R.F. Rose, Theoretical analysis of crack patching, in: A.A. Baker, R. Jones (Eds.), Bonded Repair of Aircraft Structures, Martinus Nijhoff, 1988. [11] C.H. Wang, L.R.F. Rose, A crack bridging model for bonded plates subjected to tension and bending, International Journal of Solids and Structures 36 (1999) 1985–2014. [12] S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959. [13] C.T. Sun, H. Chin, Analysis of asymmetric composite laminates, AIAA Journal 26 (1988) 714–718. [14] R.M. Jones, Mechanics of Composite Materials, McGrawHill, 1975. [15] B. Carnahan et al., Applied Numerical Method, John Wiley and Sons, 1969. [16] W.M.Th. Van Barneveld, Theoretical Analysis of In-Field Bonded Repair of Cracked, Orthotropic Aircraft Structures, Final Thesis, Delft Technical University, Netherlands, 1994. [17] R.S. Freddel, Damage Tolerant Repair Techniques for Pressurized Aircraft Fuselage, Ph.D. Thesis, Delft Technical University, Netherlands, 1994.
Thermal stresses in one-sided bond repair: geometrically nonlinear analysis C.N. Duong *, J. Yu The Boeing Company, 5301 Bolsa Avenue, MC H013-A316, Huntington Beach, CA 92647-2099, USA
Abstract The problem of thermal stresses in an infinite isotropic plate rigidly bonded with an unsymmetrical, polygon-shaped reinforcement is analyzed within large deflection plate theory, using the equivalent inclusion method [J. Thermal Stresses 26 (2003) 457]. For simplicity, the plate assumes to be cooled down uniformly from a stress free temperature. The earlier approach proposed in [J. Thermal Stresses 26 (2003) 457] has been extended in the present paper to include geometric nonlinearity. Extension of the method to address residual thermal stresses associated with curing of the adhesive is also discussed. 2003 Elsevier Ltd. All rights reserved.
1. Introduction An infinite isotropic plate with a finite reinforcement made of a different material will develop thermal stresses during uniform temperature excursions. When the plate is bonded unsymmetrically with a reinforcement on one of its two sides, the thermal load will induce both in-plane deformations and the out-of-plane deflection in the plate. This is because the plate and the reinforcement inside the bonded region can be treated as an asymmetric laminate with its bending and extensional behaviors being coupled. Analytical solutions for this particular and related thermal stress problems existed for elliptical and polygonal reinforcements [1,2]. However, these solutions were derived within
*
Corresponding author. Tel.: +1-714-896-6204; fax: +1-714896-6205. E-mail address: [email protected] (C.N. Duong).
a small deflection plate theory. Unfortunately, it was shown in [3] that the effect of large deflection might be significant on the thermal stresses in a onesided reinforced plate for a large temperature excursion. This necessitates a geometrically nonlinear analysis of a bonded plate with an unsymmetrical reinforcement under uniform cooling. As shown earlier in [4] and later in [1], the problem of an isotropic plate with an inhomogeneity or reinforcement under thermal loading can be solved more conveniently by the equivalent inclusion method. This is because analytical solutions for elliptical and polygon-shaped inclusions with uniform eigenstrains [5,6] and uniform eigencurvatures [7] are available. An analytical solution procedure based on an engineering approach is therefore introduced to solve for a large deflection problem of a bonded plate under uniform cooling, also using the equivalent inclusion method. Extension of the method to address thermal stresses associated with curing of the adhesive will also be
0167-8442/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0167-8442(03)00046-6
198
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
discussed. It is worthwhile to note that an analytical solution of the present problem for thermal stresses has been obtained previously using the equivalent inclusion method in [8] for a case of plane stress and in [1] for a case of linear bending as mentioned earlier. The solution of the problem treated here is of practical interest since a plate with a bonded reinforcement has been found in many engineering applications. A particular example in aerospace application is a bonded composite repair over a cracked metallic structure [9]. A repair method using composite patches to reinforce the cracked structure has been shown to very promising owing to the light weight, high stiffness and strength of the composite [9]. Although a symmetrical repair is the most effective reinforcement, unsymmetrical repairs provide a clear advantage when it is difficult or not possible to access both sides of a structure. Analysis of a bonded repair is usually carried out in two-stages following RoseÕs approach [10]. In the first stage, the thermomechanical response of an uncracked, reinforced plate is determined. This is then followed by a fracture analysis of a cracked reinforced plate with known traction acting on the crackÕs surfaces [11]. The solution method presented here therefore can be used to estimate thermal stresses resulting from curing or due to low operating temperature in the first stage analysis of RoseÕs approach for an unsymmetrical repair. It should be noted that the solution method presented here does not account for the flexibility of the adhesive as in a realistic bonded repair. In other words, it is assumed that the cured adhesive bond does not allow any relative displacement between the plate and the reinforcement. This rigid-bond assumption ignores the finite width of the load transfer zone around the boundary of the reinforcement, which is valid when the actual width of the load transfer zone is small relative to the in-plane dimensions of the reinforcement [2].
2. Preliminary analysis Consider a reinforcement that spans across the entire width of the plate, so that the stresses in the
reinforcement and the plate are uniform in the width direction everywhere within the reinforced portion as shown in Fig. 1. The reinforcement and plate are assumed to be fused together with no relative sliding at the interface. The whole reinforced plate is subjected to a uniform temperature excursion. This problem will be solved exactly within a large deflection theory. In the absence of the transverse load, the equilibrium equations based on the von Karman theory are given by [12]: N22;2 ¼ 0; ð1Þ M22;22 þ N22 w;22 ¼ 0; ð2Þ where N22 and M22 are the stress and moment resultant defined explicitly in the next section, and a comma indicates partial differentiation. It then follows from Eq. (1) upon integration that: N22 ¼ const ¼ C:
ð3Þ
On the other hand, the kinematics and constitutive relations for a laminated plate is given by [13]: e22 ¼ v;2 þ eNL 22 þ j22 ðz h0 Þ; 1 2 eNL 22 ¼ 2w;2 ;
N22 ¼ A22 ðv;2 þ 12w2;2 a22 DT Þ B22 w;22 ; M22 ¼ B22 ðv;2 þ 12w2;2 a22 DT Þ D22 w;22 ;
ð4Þ
ð5Þ
where A22 , B22 , and D22 are extensional, coupling, and bending stiffnesses (in Voight notation) that are also defined explicitly in the next section; v and w are the longitudinal and transverse displacement, respectively; a22 is the thermal expansion coefficient; eNL 22 is the nonlinear strain; h0 is the zcoordinate of the reference plane; and j22 is the curvature defined by j22 ¼ o2 w=ox22 . The reference plane is chosen to be at the mid-plane of the t base plate, i.e., h0 ¼ 2p , for both regions inside and outside the overlap. Applying Eqs. (2), (3) and (5) to the region outside the overlap (see Fig. 1(c)) and enforcing the boundary condition at infinity, i.e., 0 N122 ¼ 0, yields: 0 N22 ¼ 0; 0 M22;22 ¼ 0; 0 0 N22 ¼ A022 ðv0;2 þ 12w02 ;2 a22 DT Þ ¼ 0; 0 M22 ¼ D022 w0;22 ;
ð6Þ
ð7Þ
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
N 0∞ 22
199
N 0∞ 22
N ∞0 22
N 022 = N 0∞ 22 2.
2.
N I22 = N 022 = N 0∞ 22 ∞
γ
N I22 ∆T ≠ 0
M due to offset = N I22 . γ N 0∞ 22
N 0∞ 22 (a)
(b)
(c)
Fig. 1. Plate reinforced with a patch spanning across the entire widthÕs plate. (a) Plan view, (b) side view, (c) Free-body diagram used 0 to explain the effects of shifting the reference planeÕs position. Note that for the case of purely thermal loading, N122 ¼ 0.
where B022 has been set equal to zero since the plate outside the overlap is a single layer of homogeneous material and thus symmetric with respect to the reference plane, and the superscript 0 denotes the region outside the overlap. Substituting the 0 second equation of Eq. (7) for M22 into the second equation of Eq. (6) gives: w0;2222 ¼ 0:
ð8Þ
By integrating Eq. (8) and enforcing the simply 0 supported conditions at the ends, i.e., M22 ðl1 Þ ¼ 0 0 0 w ðl1 Þ ¼ Q22 ðl1 Þ ¼ 0 where Q22 is the transverse shear force defined as Q022 ¼ dM22 =dx2 , the transverse displacement outside the overlap is then given by: w0 ðx2 Þ ¼ C1 ðl1 x2 Þ;
ð9Þ
where C1 is an unknown constant which must be determined from the displacement, slope, and traction continuity at end of the overlap. In contrast, the first equation of Eq. (7) can be rewritten as: 0 ðv0;2 þ 12w02 ;2 Þ ¼ a22 DT :
ð10Þ
Similarly, the region inside the overlap also has I ¼ const ¼ 0 from consideration of the force N22 equilibrium. With that, applying Eqs. (2), (3) and (5) to the region inside the overlap yields: I N22 ¼ 0; ð11Þ I M22;22 ¼ 0; I I I I ¼ AI22 ðvI;2 þ 12wI2 N22 ;2 a22 DT Þ B22 w;22 ¼ 0; I I I I M22 ¼ BI22 ðvI;2 þ 12wI2 ;2 a22 DT Þ D22 w;22 ;
ð12Þ
200
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
where the superscript I indicates the region inside the overlap. It is worthy to note that AI22 , BI22 , DI22 , and aI22 are stiffnesses and thermal expansion coefficient of a laminated plate composing of the reinforcement and the base plate. Eq. (12) can also be rewritten as: BI22 I I ðvI;2 þ 12wI2 ;2 Þ ¼ I w;22 þ a22 DT ; A22 I2 B22 I I I ¼ M22 D 22 w;22 : AI22
ð13Þ
Substituting the second equation of Eq. (13) into the second equation of Eq. (11) gives: wI;2222
¼ 0:
ð14Þ
Upon integrating Eq. (14) and enforcing the symmetry conditions at the center of the reinforced plate, one finally obtains: j wI ðx2 Þ ¼ C2 þ x22 ; 2
ð15Þ
where C2 and j are constants yet to be determined. From which, wI;2 ðx2 Þ ¼ j x2 ; wI;22 ðx2 Þ ¼ j ¼ const:
ð16Þ
Thus, the constant j is the curvature of the plate inside the overlap, i.e., j22 ¼ j. Substituting Eq. (16) into the first equation of Eq. (13) yields: vI;2 ðx2 Þ ¼
BI22 j2 2 I x: j þ a DT 22 AI22 2 2
ð17Þ
The fist two conditions for determining the three constants C1 , C2 and j, are the displacement and slope continuity conditions at the end of the overlap, i.e., w0 ð‘Þ ¼ wI ð‘Þ; w0;2 ð‘Þ ¼ wI;2 ð‘Þ:
ð18Þ
The third condition is derived from the continuity of the normal traction acting on the base plate at the inhomogeneity interface, i.e., IðpÞ
0 N22 ð‘Þ ¼ A022 ðvI;2 ð‘Þ þ 12wI2 ;2 ð‘Þ a22 DT Þ 0 ð‘Þ ¼ 0; ¼ N22
ð19Þ
IðpÞ
where N22 is the stress resultant in the base plate of the patch-plate combination. From which, vI;2 ð‘Þ ¼ a022 DT
‘2 2 j ; 2
ð20Þ
where the result from Eq. (16) for wI;2 ð‘Þ has been utilized in the derivation of Eq. (20). However, vI;2 ð‘Þ can also be obtained by evaluating Eq. (17) at x2 ¼ ‘ as: vI;2 ð‘Þ ¼
BI22 ‘2 2 I j : j þ a DT 22 AI22 2
ð21Þ
Thus, the constant j can be determined from Eqs. (20) and (21) as: j¼
AI22 0 a22 aI22 DT : I B22
ð22Þ
Once j is found, then C1 and C2 can be determined from the conditions given by Eq. (18). For future discussion, the average nonlinear strain inside the overlap, i.e., ‘ 6 x2 6 ‘, is also calculated to be: R ‘ 1 I2 R‘ 1 2 w dx2 ðjx2 Þ dx2 1 2 2 ‘ 2 ;2 NL e22 ¼ ¼ ‘ 2 ¼ j‘: 6 2‘ 2‘ ð23Þ In summary, under purely thermal loading, this preliminary analysis predicts that the curvature of the reinforced plate is constant inside the overlap and equals to zero outside the overlap. Furthermore, since the curvature of the plate is zero outside the overlap, the term N22 w;22 associated with the large transverse deflection in the moment equilibrium equation also equals to zero in that region. On the other hand, when the reinforced plate is subjected to the remote stress, i.e., 0 N122 6¼ 0, the curvature of the plate inside the overlap is not necessarily constant and zero respectively for regions inside and outside the over0 lap. However, if N122 is sufficiently small, the trends for the curvature and N22 w;22 as predicted by the above analysis are probably still true within 0 the approximate sense for the case N122 6¼ 0. In particular, the curvature is still approximately constant and zero inside and outside the overlap while N22 w;22 can be assumed to be zero outside the overlap. In addition, the average nonlinear strain also takes approximately a same form as Eq. (23).
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
In that case, the average moment inside the overlap is derived from Eq. (2) as follows. I 0 þ N122 j ¼ 0; M22;22
N 0 j x22 I ; ¼ 122 M22 2 R‘ MI 1 I 0 M 22 ¼ ‘ 22 ¼ j‘2 N122 ; 6 2‘
ð24Þ
I 0 0 ¼ N22 ¼ N122 , and wI;22 ðx2 Þ j via Eq. since N22 (16). In contrast, a linear analysis of the latter thermo-mechanical problem will significantly over-predict the bending stresses inside the overlap. This is because the linear analysis ignores the nonlinear term N22 w;22 in the moment equilibrium equation and the nonlinear strain term eNL ij in the kinematics relation. However, if in the linear analysis the reference plane of the overlap is purportedly shifted from z ¼ h0 ¼ tp =2 to z ¼ h0 c ¼ tp c where c ¼ 16j‘2 and j is the sought curvature, 2 then the effect of the geometrically nonlinearity will be accounted for approximately in the average sense as explained immediately below. In that case, the kinematics relation for regions inside and outside the overlap are given by:
e22 ¼ v;2 þ j22 ðz h0 þ cÞ
jx2 j 6 ‘;
e22 ¼ v;2 þ j22 ðz h0 Þ jx2 j > ‘:
ð25Þ
By comparing the first equation of Eq. (25) with that of Eq. (4), it becomes clear that the effect of the shift of the reference planeÕs position inside the overlap is to raise the linear strains there by an amount equal to the average nonlinear strain predicted by the corresponding nonlinear analysis, 2 2 1 2 2 since j22 c ¼ j6 ‘ and eNL 22 ¼ 6j ‘ via Eq. (23). On the other hand, from Fig. 1(c), this positional shift of the reference plane will also induce a constant 0 0 moment of N122 c ¼ 16j‘2 N122 that equals to the average moment associated with nonlinear term N22 w;22 in the nonlinear analysis via Eq. (24). In addition, both the present linear analysis with the mentioned shift of the reference planeÕs position and the approximated nonlinear analysis described in the preceding paragraph neglect the nonlinear term N22 w;22 in the formulation for the region outside the overlap. In other words, it appears that the present geometrically nonlinear problem can be
201
solved approximately by the geometrically linear analysis but with the appropriate shift of the reference planeÕs position inside the overlap, providing that the stress resultant remains small everywhere. Even though this ‘‘geometrically linear analysis’’ approach may lack the scientific rigor, however, it appears technically sound from the engineering viewpoint. This engineering approach is therefore proposed for the analysis of thermal stresses in a polygonal patch considered in the next section.
3. Formulation for thermal stresses in a polygonal patch Consider an infinite isotropic plate bonded rigidly with a polygon-shaped reinforcement X. The reinforcement is either isotropic or orthotropic with its material principal axes parallel to the global coordinate axes. Let us also assume that the reinforcement when it is orthotropic will have a larger mismatch in the thermal expansion coefficient with the base plate along the x2 -direction. The bonded plate assumes to be subjected to a uniform temperature change DT (see Fig. 2). Extension of the present approach to address residual thermal stresses associated with curing will be discussed in a latter part of the next section. For simplicity, all shear components are also assumed to be zero. Based on results from the finite element analysis, this assumption is found to be valid for x2
Polygon-shaped reinforcement
2L
x1
2W
Fig. 2. Geometry of a bonded plate with a polygon-shaped reinforcement.
202
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
the present problem. Since the reinforcement is rigidly bonded to a plate, the reinforced region of the plate is treated as an inhogomogeneity. It is rather unwieldy to solve the stated problem using a rigorous geometrically nonlinear analysis. We will therefore present here an approximate solution based on an engineering approach similar to that outlined in Section 2 above. The approach proposed in that section will be extended in this section with details shown below, and its results will be compared with the FE solutions for a wide range of reinforcementÕs configurations. In contrast to the results of patching across the entire width of the plate, the stress resultants inside the plate reinforced with a polygonal patch due to a uniform temperature change will not necessarily equal to zero even without remote applied stresses. It should remember that except at infinity, only the average stress resultant across the plate width is required to equal to zero. However, if these stress resultants are assumed to be small, then the thermal stresses in the latter problem can be solved approximately by a geometrically linear analysis with the reference plane of the reinforced region being shifted purportedly from z ¼ h0 ¼ tp =2 to t z ¼ h0 c ¼ 2p c as mentioned in Section 2. Thus, the considered bonded plate is divided into two regions; each will be modeled using different kinematics and constitutive relations. The region outside of the reinforcement is a homogeneous infinite isotropic plate. On the other hand, the inhomogeneous region is a finite asymmetric laminated plate that composes of two layers corresponding to the base plate and the reinforcement. The kinematics relations used to describe the inhomogeneity and the surrounding plate are summarized below, which are the 2-D generalization of Eq. (25). For the inhomogeneity, eIij ¼ eIij þ ðz h0 cÞjIij ; 1 I eIij ¼ ui;j þ uIj;i ; 2
ð26Þ
c ¼ 16j0 ‘2 ; jIij ¼
o2 wI ; oxi oxj
ð27Þ
while for the surrounding isotropic plate outside the inhomogeneity, epij ¼ epij þ ðz h0 Þjpij ; epij ¼
1 p ui;j þ upj;i ; 2
jpij ¼
ð28Þ
o2 wp ; oxi oxj
where j0 is a representative curvature in the x2 direction inside region X, ‘ is the longest halflength of the polygonal reinforcement measured from its center, and the rest are previously defined. This conjecture will then be tested for its validity for a wide range of reinforcementÕs configurations. As shown later in Section 5, the above conjecture will yield sufficiently accurate results for stresses in the base plate near the center of the reinforcement and thus at the prospective crack location in the bonded repair when these results are compared with FE solutions. Once the kinematics relations are established, the constitutive relations for the inhomogeneity and the surrounding plate can be derived in a straightforward manner using the classical plate theory [14]. Following the work in [1], the constitutive relations for the inhomogeneity and for the surrounding plate are given by: For the inhomogeneity, ðTÞ
NijI ¼ AIijkl eIkl þ BIijkl jIkl Nij ; ðTÞ
ð29Þ
MijI ¼ BIijkl eIkl þ DIijkl jIkl Mij
and for the surrounding isotropic plate, Nijp ¼ Apijkl epkl ; Mijp ¼ Dpijkl jPkl ;
ð30Þ
where Aijkl , Bijkl and Dijkl are the extensional, coupling, and bending stiffness tensors, respectively; Nij and Mij are the stress and moment reðTÞ ðTÞ sultant. In Eq. (29), Nij and Mij are the stress and moment resultant associated with the thermal strains and their explicit formula for the present problem will be derived in the next paragraph while Aijkl , Bijkl , Dijkl , Nij and Mij are defined as [14]:
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
Aijkl ¼
Z
t
Cijkl dz; 0
Bijkl ¼
Z
t
ð31Þ
Cijkl ðz hÞdz;
0
Z
Dijkl ¼
t 2
Cijkl ðz hÞ dz;
0
Nij ¼
Z
t
0
Mij ¼
Z
t
ðTÞ Cijkl ekl eij dz;
Cijkl ekl
ðTÞ eij
ðz hÞ dz;
ð32Þ
0
where Cijkl is the regular elasticity tensor, t is the total thickness of an inhomogeneity or of a surrounding base plate, and h is equal to h0 for a surrounding base plate and ðh0 cÞ for an inhomogeneity. It was shown in [1,8] that the problem of a bonded plate under uniform cooling can be formulated as an initial strain problem with an initial strain prescribed in the bonded patch and that initial strain is given by: ðTÞðRÞ
eij
¼ Daij DT ;
ð33Þ
P where Daij ¼ aR ij aij , aij is the thermal expansion coefficient tensor; and the superscript R signifies the reinforcement. It then follows from [1,14] that ðTÞ ðTÞ Nij and Mij in Eq. (29) are given by: R R E11 Da11 mR ðTÞ 12 E22 Da22 N11 ¼ þ DT tR ; R R 1 mR 1 mR 12 m21 12 m21 R R R m21 E11 Da11 E22 Da22 ðTÞ N22 ¼ þ DT tR ; R R 1 mR 1 mR 12 m21 12 m21 ð34Þ
uniform cooling was formulated as an inhomogeneity problem with the kinematics and constitutive relations prescribed by Eqs. (26), (28), (29), (30), respectively. This inhomogeneity problem then will be solved by the equivalent inclusion method. For the purpose of clarity, this method will be demonstrated first for a case when c in Eq. (27) is prescribed as a known constant. In the equivalent inclusion method, the stress, in-plane reference strain and curvature fields induced by an inhomogeneity occupied region X will be the same as those induced by eigenstrain field eij and eigencurvature field jij in the same region of a homop geneous material of Cijkl when eij and jij are selected appropriately as shown in Fig. 3. The latter homogeneous problem is called an inclusion problem with the constitutive relation given by: 8 < AP eH e for x inside X; ijkl kl kl NijH ¼ : AP eH for x outside X; ijkl kl ð35Þ ( H P D j j for x inside X; ijkl kl kl MijH ¼ for x outside X: DPijkl jH kl In the above equation, the superscript H denotes a homogeneous problem. It should be reminded that Apijkl and Dpijkl are defined earlier in Section 3 by Eq. (31) with h ¼ h0 . For a constant (or uniform) eij and jij , from [6,7], the elastic solutions of the homogeneous problem can be expressed as: eH ij ðxÞ ¼ Sijkl ðxÞekl ;
ðTÞ
P C ijkl
2
ðTÞ
ð36Þ
jH ij ðxÞ ¼ Kijkl ðxÞjkl ;
N12 ¼ 0; Mij ¼
203
2
ðtP þ tR h0 Þ ðtP h0 Þ ðTÞ Nij : 2 tR
With the stated problem completely formulated, it remains now to outline its solution procedure and that will be the topic of the next section.
ε*ij
∆T
κ *ij
Ω
Ω
I C ijkl
4. Solution method In Section 3, a problem of an infinite isotropic plate with a polygon-shaped reinforcement under
(I)
(II)
Fig. 3. An illustration of the equivalent inclusion method. (I) an inhomogeneity problem and (II) an inclusion problem with a uniform eigenstrain and a uniform eigencurvature.
204
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
where Sijkl and Kijkl are Eshelby and Eshelby-type tensors for eigenstrain and eigencurvature, respectively. Algorithms for evaluating these tensors H are detailed in [6,7]. eH ij and jij are in general not uniform inside X except for an elliptical region. The equivalency condition between the inhomogeneity problem and the homogeneous problem requires that the in-plane strain and curvature field inside X as well as the stress and moment resultant of the two problems must be the same, i.e., eIij
¼
¼
ð37Þ
NijH ;
MijI ¼ MijH :
eIij
)
" ¼
AIijkl
DAaa11 ðS1111 ð0Þe11 þ S1122 ð0Þe22 Þ þ DAaa22 S2211 ð0Þe11 þ S2222 ð0Þe22 þ APaa11 e11 þ APaa22 e22 þ BIaa11 K1111 ð0Þj11 þ K1122 ð0Þj22 þ BIaa22 K2211 ð0Þj11 ðTÞ þ K2222 ð0Þj22 ¼ Naa ;
BIijkl
#1 ("
BIijkl DIijkl ( ) eH mn emn jH mn ðTÞðRÞ
H Since eH ij and jij are in general not uniform inside X as mentioned above, the equivalency condition given by Eq. (37) can only be satisfied approximately when eij and jij are approximated by a (unknown) constant tensor. By enforcing condition (37) at point ð0; 0Þ, i.e., the origin of the coordinate system, and by substituting results from Eqs. (26), (29) and (36) into (37), one finally obtains the following linear equations for the unknown eij and jij (without summation on the subscript a):
jmn ðTÞ
ð38Þ
DAijkl ¼ AIijkl APijkl ; for a ¼ 1; 2:
Once eij and jij are determined, the in-plane strain and curvature fields of the homogeneous problem can be calculated from Eqs. (36) while the
APklmn 0
( þ
0
#
DPklmn )) ðTÞ
Nkl
ðTÞ
;
ð40Þ
jkl ðTÞ
where Cijkl , ekl , Nkl and Mkl are defined earlier in Section 3. So far we have assumed that c is known a priori. However, in reality, it is also part of the sought solutions since it depends on the x2 -component of the curvature inside X via Eq. (27). Since the curvature field inside X in general is not uniform, the curvature at any point inside X can be used as j0 in the evaluation of c. However, it is found that the following definition of j0 will yield sufficiently accurate solutions for a bonded plate with reinforcements of different shapeÕs aspect ratios: j0 ¼ jI22 ð0; y Þ; y ¼ minimum of f‘; W g:
BIaa11 S1111 ð0Þe11 þ S1122 ð0Þe22 þ BIaa22 S2211 ð0Þe11 þ S2222 ð0Þe22 þ DDaa11 K1111 ð0Þj11 þ K1122 ð0Þj22 þ DDaa22 K2211 ð0Þj11 ðTÞ ; þ K2222 ð0Þj22 þ DPaa11 j11 þ DPaa22 j22 ¼ Maa
DDijkl ¼ DIijkl DPijkl
(
jIij
eH ij ;
jIij ¼ jH ij ; NijI
stress and moment resultant are evaluated using Eq. (35). Similar to those given in [1], the stresses in the base plate and in the reinforcement inside X are finally given by: I P ekl þ ðz h0 þ cÞjIkl ; rPij ¼ Cijkl ð39Þ ðTÞðRÞ R I þ ðz h0 þ cÞjIkl ; rR ij ¼ Cijkl ekl ekl
ð41Þ
In the above equation, the notation jI22 ð0; y Þ means that jI22 is evaluated at a point ð0; y Þ on the x2 -axis, where y is either the half-length ‘ or the half-width W of the reinforcement whichever is smaller. With the solution procedure for a fixed value of c described and j0 defined, the solution of the geometrically nonlinear problem can be obtained by the following simple iterative procedure. First, j0 and thus c are assumed to be zero, and the elastic solutions for stress, strain and curvature field are obtained in a usual manner using the given value of c as described earlier. Second, the condition imposed by Eq. (41) is verified. If that condition is satisfied, then the just obtained solution has converged to the true solution of the problem. Otherwise, another value of j0 , thus c, will be assumed and the above steps are repeated until convergence. A new trial value for j0 will be
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
calculated using the following recursive formula [15]: ðiþ1Þ
j0
ðiÞ
IðiÞ
¼ ð1 kÞj0 þ kj22 ð0; y Þ; ðiþ1Þ
ð42Þ
ðiÞ
where j0 and j0 are values of j0 in the current IðiÞ and previous iterative cycle, respectively; jij is the elastic solution for the curvature field obtained from the previous iterative cycle that is based on ðiÞ c ¼ 16j0 ‘2 ; and k is a scalar factor which takes values between 0 and 1. It should be noted that a normal direct substitution method corresponds to a case of k ¼ 1. It remains now to discuss the extension of the present method to address residual thermal stresses associated with curing of the adhesive. While the uniform temperature model considered here truly represents the thermal loading during high altitude cruising, it may appear unrealistic for simulating the curing process of the adhesive. However, by using simple curing models presented in [10,16,17], the curing process of the adhesive can be simulated as a uniform cooling process of a whole bonded structure with the base plateÕs thermal expansion coefficient being approximated by an (lower) ‘‘effective’’ value. In a similar context, the residual thermal stress problem associated with curing was formulated as an initial strain problem in [8]. As shown in [8], the only difference between the curing problem and the present uniform cooling problem is that the initial strain prescribed in the patch for the former problem is given by the following expression instead of Eq. (33): ðTÞðRÞ
eij
0 ¼ aR ij DTcuring eij ;
ð43Þ
where DTcuring ¼ Tambient Tcuring and in general Tcuring > Tambient ; e0ij is the strain solution of an inp clusion problem of a homogeneous material Cijkl with a uniform eigenstrain field ap DTcuring dij prescribed inside region P; ap is the thermal expansion coefficient of the base plate; and the inclusion region P assumes the same shape as that of the heating blanket. Thus, the thermal effect due to curing can be addressed in a similar manner as that considered here if any curing model mentioned above is employed. It is important to note that other advanced curing models such as the one
205
considered in [2] for a geometrically linear analysis of residual thermal stresses can also be extended to include the effect of geometric nonlinearity using the present results. For instance, by shifting the neutral plane of the curing model presented in [2] by an amount of c as postulated by Eq. (27) and iteratively solving for c as described earlier, perhaps one can obtain approximate nonlinear solutions of the curing problem with various degrees of thermal constraints. It would be difficult to supply a precise error analysis for the present approximate method: its accuracy is most readily assessed by comparison with finite element results for particular cases. This is what is depicted in Section 5.
5. Results and discussion To illustrate the present method, particular cases corresponding to typical repair configurations are considered in this section. All computations in this section unless noted otherwise are carried out for a temperature excursion of )75 C, corresponding to a typical temperature change experienced by the aircraft during high altitude cruising. The first case is an infinite thin plate repaired with a square patch ð‘=W ¼ 1Þ. A patch is bonded rigidly to a plate. The material properties and thickness of the base plate and the patch are given below: Base plate: Aluminum, Ep ¼ 72:4 GPa, mp ¼ 0:33, ap ¼ 22:5E)06/C, tP ¼ 1:6 mm. Patch: Boron/Epoxy, Ex ¼ 18:7 GPa, Ey ¼ 193:6 GPa, myx ¼ 0:21, Gxy ¼ 5:5 GPa, ax ¼ 21:4E)06/ C, ay ¼ 4:3E)06/C, tR ¼ 0:66 mm. The patch stiffness ratio ER tR =Ep tp (S) is 1.1. The sensitivity of the length of the patch relative to the base plate thickness is studied. Four patch lengths ð2 ‘Þ of 10.16, 20.32, 30.48, and 40.64 cm are considered in the analysis. Since the thermal stresses in the patch are compressive and they will be negated under far field tensile stress [1,2], we therefore concentrate our effort to the plateÕs thermal stresses. Thermal stresses in the base plate at a patchÕs center in the x2 -component are of special interest,
206
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
as they are needed for fracture analysis using Wang–RoseÕs crack bridging model [11]. Since the stress distributions in the plate and patch are linear through their thickness, it will be more meaningful to report these stresses in terms of the mean and bending values. These values of thermal stresses are plotted in Fig. 4 as the functions of the normalized half patch length ‘=tp where tp is a plateÕs thickness. The mean and bending stresses are defined as half of the sum and difference of stresses near the top and bottom surfaces of the base plate. All analytical solutions are obtained using k in the recursive formula given in Eq. (42) equal to 0.1, and they converge within about 30 iterations. To assess the accuracy of the analytical method, results from the finite element analysis are also obtained and compared with the analytical predictions in Fig. 4. Finite element analyses are
Mean stress σm(MPa)
L/W=1, S=1.1 50 45 40
3-D F.E. Analytical
35 30 25 20 0
25
50
75
100
125
carried out using ABAQUS with the base plate and each ply of the patch modeled separately. The base plate is modeled as three layers of 20-node solid elements while each ply of the patch is by one layer of solid elements. To avoid the adverse effect of the small thickness-to-length ratio on the solution, stiffness matrices of all solid elements are evaluated using reduced integration. In the FE model, the base plate was restrained from the outof-plane deflection along its periphery. A typical mesh used in the FE analyses is given in Fig. 5. Similar results but for different patchÕs aspect ratios are presented in Fig. 6 for ‘=W ¼ 2 and in Fig. 7 for ‘=W ¼ 1=2. In contrast to the planestress and linear bending results [1,8], from Figs. 4, 6, 7, thermal stresses in the base plate are affected by the patchÕs size. For a same patchÕs aspect ratio, the mean components of the thermal stresses in the plate increase with patchÕs sizes while the bending components show a reverse trend. These bending stresses will eventually vanish for large values of ‘=tp as expected. In general, analytical predictions are in quite good agreement with the FE results (within 12%) for all aspect ratios ‘=W considered. To show the effect of a patchÕs thickness on the thermal stresses, analyses of a few selective patch configurations above but with a thickness of 1.057 mm are performed and their results are summarized in Table 1. For reference, corresponding results previously presented in Figs. 4, 6 and 7 are
150
Normalized half patch length / tp
(a)
Bending stress σb(MPa)
L/W=1, S=1.1
(b)
50 45 40 35 30 25 20 15 10 5 0
3-D F.E. Analytical
0
25
50
75
100
125
150
Normalized half patch length / tp
Fig. 4. Thermal stresses in a middle of a plate reinforced with a square patch for different normalized half patch lengths. (a) The mean stresses, (b) the bending stresses.
Fig. 5. A typical mesh used in the FE analysis.
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
L/W=1/2, S=1.1
L/W=2, S=1.1 45
50 45
Mean stress σm(MPa)
Mean stress σm(MPa)
55
3-D F.E. Analytical
40 35 30 25 0
50
100
150
200
250
40
35
3-D F.E. Analytical
30
25 50
300
Normalized half patch length / tp
(a)
75
100
125
150
Normalized half patch length / tp
(a)
L/W=2, S=1.1
L/W=1/2, S=1.1
35
45
Bending stress σb(MPa)
Bending stress σb(MPa)
207
3-D F.E.
30
Analytical
25 20 15 10 5 0 0
50
100
150
200
250
3-D F.E. Analytical
35 30 25 20 50
300
Normalized half patch length / tp
(b)
40
(b)
75
100
125
150
Normalized half patch length / tp
Fig. 6. Thermal stresses in a middle of a plate reinforced with a rectangular patch ðL=W ¼ 2Þ for different normalized half patch lengths. (a) The mean stresses, (b) the bending stresses.
Fig. 7. Thermal stresses in a middle of a plate reinforced with a rectangular patch ðL=W ¼ 1=2Þ for different normalized half patch lengths. (a) The mean stresses, (b) the bending stresses.
repeated in the table. It follows from Table 1 that a thicker patch will induce higher thermal stresses in the plate as expected and a same good agreement is observed between the analytical and the FE results. Additional results but not shown here indicate that bonded plates with a same patch stiffness ratio ER tR =Ep tp , same aspect ratio ‘=W and same parameter ‘=tp will have the same thermal stresses.
Thus, for a given temperature change, the nonlinear thermal stresses can be characterized in terms of three nondimensional parameters ER tR = Ep tp , ‘=W and ‘=tp . So far only patches of rectangular or square shapes have been considered. Another common shape of the patch is therefore selected for presentation. An octagonal patch sometimes is preferred
Table 1 Comparison of thermal stresses in a middle of a patched plate with different patchÕs thicknesses L=tp
L=W
tR =tp
128 128 128 128
1 1 2 2
0.416 0.666 0.416 0.666
Analytical method
FE method
rm (MPa)
rb (MPa)
rm (MPa)
rb (MPa)
45.6 49.8 44.4 49.9
17.6 20.1 18.4 20.2
45.1 49.3 47.3 53.0
15.7 18.5 17.8 20.0
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
45˚
8.89 cm
Mean stress σm(MPa)
208
x2
17.78 cm
x1
L/W = 1, S=1.1 100 90 80 70 60 50 40 30 20 10 0 0
(a)
F.E. Analytical
25 50 75 100 125 Temperature change (°C)
150
6.35 cm 12.7 cm
Fig. 8. Geometry of the example problem with an octagonal patch.
Bending stress σb(MPa)
L/W = 1, S=1.1 28 F.E. Analytical
24 20 16 12 8 4 0 0
25
(b)
• Analytical predictions, rm ¼ 32:1 MPa, rb ¼ 36:4 MPa. • FE results, rm ¼ 35:1 MPa, rb ¼ 34:7 MPa. Again the analytical method predicts well the nonlinear thermal stresses. As a final example, an infinite plate rigidly bonded with a square boron/epoxy patch (‘=W ¼ 1, S ¼ 1:1, ‘=tp ¼ 128) under a wide range of temperature excursions is considered. The mean and bending components of the thermal stresses in a middle of the plate are plotted in Fig. 9(a) and (b), respectively, for different cooling temperature ranges jDT j. For a future discussion, the normalized bending stresses ðrb =rm Þ are also plotted versus jDT j in Fig. 10. From Fig. 9, it seems that
150
Fig. 9. Thermal stresses in a middle of a plate reinforced with a square patch for different temperature excursions. (a) The mean stresses, (b) the bending stresses.
L/W = 1, S=1.1 Normalized bending stress σb/σm
over a rectangular patch because of its superior performance in preventing peeling near the patchÕs corner. An analysis of an octagonal patch therefore will be demonstrated next. The geometry of the octagonal patch is shown in Fig. 8. Material properties and thickness of the patch and the base plate are identical to those given earlier in the beginning of this section. Analytical predictions as well as FE results for the thermal stresses in the base plate are given below.
50 75 100 125 Temperature change (°C)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
F.E. Analytical
0
25
50 75 100 125 Temperature change (°C)
150
Fig. 10. Normalized bending stresses in a middle of a reinforced plate for different temperature excursions.
analytical predictions for the mean stresses are in excellent agreement with FE results for all jDT j while the analytical predictions for the bending stresses show a larger discrepancy with FE solutions. The deviation between analytical and FE results for the bending stresses may exceed 20% for jDT j > 100 C and attains a largest value of 30% when jDT j ¼ 138 C. However, since the contribution of the bending component to the total
C.N. Duong, J. Yu / Theoretical and Applied Fracture Mechanics 40 (2003) 197–209
stress is significantly smaller for jDT j > 100 C as illustrated in Fig. 10 via rb =rm , this discrepancy may not be significant as it appears in Fig. 9. For example, the analytical method predicts that the bending stress is 21% of the mean stresses when jDT j ¼ 138 C while the FE method yields a ratio of 28%. Thus, the agreement between two methods is considered to be satisfactory, in view of the approximate nature of the present analytical model.
6. Conclusion An analytical method to estimate thermal stresses in an infinite isotropic plate bonded unsymmetrically with a polygon-shaped reinforcement is presented within a geometrically nonlinear analysis. The method is based on an engineering approach. By postulating certain simplifications or assumptions in the formulation, the geometrically nonlinear problem can be solved approximately in a similar manner as that for a linear analysis. Even though these assumptions appear to be rather significant assumptions from the outset, they are justified as reasonable for the present thermal stress problem by checking the ability of the present method in predicting results in reasonably close agreement with FE solutions for numerous practical cases.
Acknowledgements The authors take this opportunity to gratefully acknowledge the support provided for this research by the USAF, WPAFB Ohio, as part of the composite repair of aircraft structures (CRAS) program under the management of Capt. Williams Shipman. The contract number is F33615-97C3219.
209
References [1] C.N. Duong, J. Yu, Thermal stresses in a one-sided bonded repair by a plate inclusion model, Journal of Thermal Stresses 26 (2003) 457–464. [2] C.H. Wang, D. Erjavec, Geometrically linear analysis of the thermal stresses in one-sided composite repairs, Journal of Thermal Stresses 23 (2000) 833–851. [3] J.C. Klug, C.T. Sun, Large deflection effects of cracked aluminum plates repaired with bonded composite patches, Composite Structures 42 (1998) 291–296. [4] H.G. Beom, Y.Y. Earmme, The elastic field of an elliptic cylindrical inclusion in a laminate with multiple isotropic layers, Journal of Applied Mechanics 66 (1999) 165–171. [5] T. Mura, Mechanics of Defects in Solids, Martinus Nijhoff, 1987. [6] G.J. Rodin, EshelbyÕs inclusion problem for polygons and polyhedra, Journal of Mechanics and Physics of Solids 44 (1996) 1977–1995. [7] C.N. Duong, J. Yu, Analysis of a plate containing a polygon-shape inclusion with a uniform eigencurvature, Journal of Applied Mechanics, in press. [8] C.N. Duong, J. Yu, An analytical estimate of thermal effects in a composite bonded repair: Plane stress analysis, International Journal of Solids and Structures 39 (2002) 1003–1014. [9] A.A. Baker, Crack patching: Experimental studies, practical applications, in: A.A. Baker, R. Jones (Eds.), Bonded Repair of Aircraft Structures, Martinus Nijhoff, 1988. [10] L.R.F. Rose, Theoretical analysis of crack patching, in: A.A. Baker, R. Jones (Eds.), Bonded Repair of Aircraft Structures, Martinus Nijhoff, 1988. [11] C.H. Wang, L.R.F. Rose, A crack bridging model for bonded plates subjected to tension and bending, International Journal of Solids and Structures 36 (1999) 1985–2014. [12] S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959. [13] C.T. Sun, H. Chin, Analysis of asymmetric composite laminates, AIAA Journal 26 (1988) 714–718. [14] R.M. Jones, Mechanics of Composite Materials, McGrawHill, 1975. [15] B. Carnahan et al., Applied Numerical Method, John Wiley and Sons, 1969. [16] W.M.Th. Van Barneveld, Theoretical Analysis of In-Field Bonded Repair of Cracked, Orthotropic Aircraft Structures, Final Thesis, Delft Technical University, Netherlands, 1994. [17] R.S. Freddel, Damage Tolerant Repair Techniques for Pressurized Aircraft Fuselage, Ph.D. Thesis, Delft Technical University, Netherlands, 1994.