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On the stress singularity of dissimilar anisotropic wedges and junctions in antiplane shear
Composite Structures 73 (2006) 432–442 www.elsevier.com/locate/compstruct
On the stress singularity of dissimilar anisotropic wedges and junctions in antiplane shear Chuan-I Liu a
a,*
, Ching-Hwei Chue
b
Department of Structure Analysis, Aerospace Industrial Development Corporation, Taichung, Taiwan, ROC b Department of Mechanical Engineering, National Cheng Kung University, Tainan, ROC Available online 29 March 2005
Abstract In this paper, rk1 type antiplane stress singularities in dissimilar anisotropic wedges and junctions are studied. For some special wedge angles, the eigenequations derived are a function of fiber orientations and the elastic constants in the principal material coordinates. The boundary surfaces of the wedges can be subjected to combinations of free and clamped boundary conditions. The repeated occurrence of stress singularity orders is examined. By properly selecting the fiber orientations, the degree of the stress singularities can be diminished or even disappeared. The singularities of the materials reduced to isotropic are discussed as well. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Stress singularity; Wedge and junction; Antiplane; Anisotropic material
1. Introduction In composite structures, failures often commence at the apex of the wedge or junction where the stresses are singular. It is known that stress singularity orders depend on the wedge angles, edge boundary conditions, composite material properties, and fiber orientations. If fiber orientations and material constants, defined in the principal material coordinates are directly included in the eigenequations, certain useful information for structural design can be obtained. If x–y is a symmetric plane for anisotropic material, the in-plane and antiplane stress fields of generalized plane deformation are decoupled. Bogy [1], Kuo and Bogy [2] employed a complex function representation of the solution in conjunction with the Mellin transform method [3] to analyze the in-plane stress singularities in an anisotropic wedge. Ma and Hour [4] used the Mellin transform to study the antiplane stress singularities of *
Corresponding author. Tel.: +886 4 27070001 503251; fax: +886 4 22842824. E-mail address: [email protected] (C.-I. Liu). 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.02.015
the same wedge problem. Ma and Hour [4] found that the order of stress singularity is real for the general anisotropic bimaterial wedge of the antiplane problem subjected to various boundary conditions (i.e. free–free, clamped–clamped and free–clamped). Recently, Xie and Chaudhuri [5] proposed an eigenfunction approach for analyzing the three-dimensional asymptotic stress field of a symmetric bimaterial wedge under antiplane shear loading. A similar problem with an asymmetric bimaterial pie-shaped wedge has been solved by Chiu and Chaudhuri [6]. The problems of cracking are also extensively investigated. For example, Ma and Hour [7] studied antiplane problems in anisotropic materials with an inclined crack terminating at a bimaterial interface. Shahani [8] investigated the anisotropic finite wedge under antiplane deformation. Pageau et al. [9] obtained the numerical antiplane singular stress field of anisotropic multi-material wedges and junctions. Using the contours of stress singularity order, Chue and Liu [10,11] studied the bimaterial wedge problem with arbitrary fiber orientation and obtained the disappearance conditions of stress singularities in composite laminates.
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
The antiplane wedge and junction problems are of major concern in this article. However, the article will not focus on deriving the general antiplane stress singularity of the anisotropic wedge problem. Certain particular wedge angles possessing similar natural characteristics are selected. Also, several practical applications for engineers to design the fiber orientation of composite material are found. Based on LekhnitskiiÕs formulations [12], the eigenequations, which govern the antiplane stress singularity orders for 90°–90°, 90°–180°, 180°–180°, 90°–270° anisotropic wedges and 90°–270° junction, are derived. Consider an anisotropic wedge (Fig. 1(a)) with free–free boundary conditions and a1 = a2 = 90°, that the ^xk -axis (k = 1, 2) of the material principal axes lies in the xyplane and makes an angle gk (90° 6 gk 6 90°) with the positive x-axis. If these two wedges are the same material but with different orientations (g1 5 g2), the antiplane stress singularity orders are listed in Table 1. It is shown that the stress singularity is unvaried as the gk (k = 1 or 2) is changed to 90° gk. The repeated appearance of singularity orders will be discussed analytically in Section 3. In addition, the disappearance conditions of the stress singularity can be obtained from the derived eigenequations. This theoretical conclusion for vanishing singular stress fields can be used to improve the safety of structures.
2. Formulations of a two-bonded anisotropic wedge in antiplane fields Fig. 1(a) and (b) shows the configurations of twobonded composite wedge and junction, respectively. Oxyz is the global coordinate system. Let ^xk , ^y k and ^zk denote the principal material coordinates of composite wedge Xk (k = 1, 2). The ^zk -axis coincides with z-axis and ^xk -axis makes an angle gk (90° 6 gk 6 90°) with the positive x-axis. Therefore, the x–y plane is an elastic
433
Table 1 The antiplane stress singularity order (k 1) of the 90°–90° wedge with ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ free–free boundary conditions. (2^s44 ¼ ^s55 , 2^s44 ¼ ^s55 and ^s55 ¼ ^s55 ) (NS: no singularity)
g1 = 90° g1 = 75° g1 = 60° g1 = 45° g1 = 30° g1 = 15° g1 = 0° g1 = 15° g1 = 30° g1 = 45° g1 = 60° g1 = 75° g1 = 90°
g2 = 20°
g2 = 70°
0.0664073 0.112551 0.142152 0.152042 0.142152 0.112551 0.0664073 0.0152018 0.0240091 (NS) 0.0384675 (NS) 0.0240091 (NS) 0.0152018 0.0664073
0.0664073 0.112551 0.142152 0.152042 0.142152 0.112551 0.0664073 0.0152018 0.0240091 (NS) 0.0384675 (NS) 0.0240091 (NS) 0.0152018 0.0664073
symmetric surface. The in-plane and antiplane stress fields are then decoupled. For the antiplane field, the (k) ðkÞ exist. The relastresses sðkÞ yz , sxz and displacement w ðkÞ ðkÞ tions between strains cyz , cxz and stresses in the global coordinate system are: ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ ðkÞ cðkÞ yz ¼ s44 syz þ s45 sxz ;
ð1Þ
ðkÞ ðkÞ cðkÞ xz ¼ s45 syz þ s55 sxz ; ðkÞ
where sij are the components of elastic compliance matrix in the global coordinate system. Consider the orthotropic composite materials, which the elastic constants ðkÞ ðkÞ of ^s44 and ^s55 defined in ^xk ^y k^zk -coordinate system exist ðkÞ ðkÞ but ^s45 vanish. Therefore, sij can be computed through the following coordinate transformation ðkÞ
ðkÞ
ðkÞ
s44 ¼ ^s44 cos2 ðgk Þ þ ^s55 sin2 ðgk Þ; ðkÞ ðkÞ ðkÞ s45 ¼ ^s44 ^s55 cosðgk Þ sinðgk Þ; ðkÞ s55
¼
ðkÞ ^s55 cos2 ðgk Þ
ðk ¼ 1; 2Þ:
ð2Þ
ðkÞ þ ^s44 sin2 ðgk Þ;
The stresses and displacements can be expressed in terms of complex function /k(zk) as follows [12]
Fig. 1. Geometry of two dissimilar anisotropic materials for (a) wedge, (b) junction.
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0 sðkÞ xz ¼ 2Re½lk /k ðzk Þ;
ðkÞ
w
¼
2Re½/0k ðzk Þ;
ð3Þ
150
zk ¼ x þ lk y ¼ rðcos h þ lk sin hÞ rfk ðhÞ; ðkÞ s45
qffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ s ðkÞ ðkÞ 44 ¼ i ^s44 ^s55 : lk
ð4Þ 120
ð5Þ
Constant lk is the root of the following equation with positive imaginary part ðkÞ
100 10 1
mk
¼ 2Re½tk /k ðzk Þ;
where
tk ¼
0.1 0.01
180
ðkÞ
90
60
ðkÞ
s55 l2k 2s45 lk þ s44 ¼ 0:
ð6Þ
Substituting Eq. (2) into Eq. (6) and solving for lk, it gives pffiffiffiffiffiffi mk cos gk þ i sin gk lk ¼ pffiffiffiffiffiffi ; ð7Þ mk sin gk þ i cos gk ðkÞ
Θ k (Degree)
sðkÞ yz
ðkÞ
where mk ¼ ^s44 =^s55 . In order to obtain the stress singularity orders, the complex stress potential, /k(zk), is expressed in the following form:
/k ðzk Þ ¼ c1k zkk þ c2k zkk
ðk ¼ 1; 2Þ;
ð8Þ
where c1k and c2k are unknown complex constants. k is the complex eigenvalue to be determined. The eigenvalues in the range 0 < Re[k] 6 1 are of interest in the studying of singular stress fields. Substituting Eqs. (4) and (8) into Eq. (3), the stresses and displacements can be rearranged in the following forms h i k1 k1 k fk sðkÞ c1k lk fk1 þ c2k l ; k xz ¼ 2Re kr h i k1 k1 c1k fk1 þ c2k fk ð9Þ ; sðkÞ k yz ¼ 2Re kr h i k wðkÞ ¼ 2Re rk c1k tk fkk þ c2ktk fk ; where (k1) is the stress singularity order and a smaller value of k stands for a stronger stress singularity. If the absolute value and principal argument of lk are defined as Rk and Hk, respectively, then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk cos2 gk þ sin2 gk ; ð10Þ Rk jlk j ¼ mk sin2 gk þ cos2 gk pffiffiffiffiffiffi 2 mk : tanHk tanðArg lk Þ ¼ ð1 mk Þ sinð2gk Þ
ð11Þ
The variations of principal argument Hk with gk for different mk are shown in Fig. 2. It shows that the range of Hk is 0° < Hk < 180°. When 0° < gk < 90°, Hk is closer to 180° for larger mk. Also when 0° < gk < 90°, Hk is closer to 0° for smaller mk. When mk 1 (i.e. near isotropic material), Hk is equal to 90° for all fiber orientations, gk.
30
0 -90
-60
-30
0
η k (Degree)
30
60
90
Fig. 2. The variation of Hk with principal axis orientation gk for mk = 100, 10, 1, 0.1 and 0.01.
3. Examples and discussions Owing to geometric and material discontinuities, the antiplane stress singularities for anisotropic bimaterial wedges (with wedge angles 90°–90°, 90°–180°, 180°– 180°, 90°–270°) and junctions (with angles 90°–270°) are investigated. The reason for selecting these wedge angles is due to the similar rule for repeating occurrences of stress singularities. Also, they are very common structures in engineering applications. The boundary surfaces of the wedges can be combinations of free and clamped boundary conditions. 3.1. 90°–90° wedges Consider an anisotropic 90°–90° wedge perfectly bonded along the positive x-axis as shown in Table 3. p From Eq. (4), it can be seen that f ¼ l h ¼ 1 1, 2 p f2 h ¼ 2 ¼ l2 on the boundaries and f1(h = 0) = f2(h = 0) = 1 on the bonding surface, in which l1 and l2 are defined in Eq. (7). Based on Eq. (9), the continuity ð2Þ conditions of the stresses ðsð1Þ yz ðh ¼ 0Þ ¼ syz ðh ¼ 0ÞÞ and (1) (2) the displacements (w (h = 0) = w (h = 0)) along the interface will result in the following relations c11 þ c21 c12 c22 ¼ 0; ð12Þ c11 c21 xc12 þ xc22 ¼ 0; where x is the ratio of the elastic constants ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð2Þ ð2Þ ð1Þ ð1Þ x ¼ ^s44 ^s55 ^s44 ^s55 :
ð13Þ
ð14Þ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
3.1.1. Free–free boundary conditions p The free–free boundary conditions (i.e. sð1Þ xz h ¼ 2 ¼ p 0 and sð2Þ xz h ¼ 2 ¼ 0) can be reduced to the relations k1 ¼ 0; c11 lk1 þ c21 l
ð15Þ
k
k
l2 Þ ¼ 0: c12 ðl2 Þ þ c22 ð
ð16Þ
Eqs. (12), (13), (15) and (16) yield four linear homogeneous equations with unknowns c11, c21 , c12 and c22 . They can be rearranged in the following matrix form 2 k 32 3 c11 k1 l1 l 0 0 61 7 6 1 1 1 76 c21 7 6 7 ð17Þ 6 76 7 ¼ 0: 4 1 1 5 4 c12 5 x x 0
0
ðl2 Þ
k
ð l2 Þ
k
c22
For non-trivial solutions of cij (i, j = 1, 2), the determinant of the matrix must vanish. Only eigenvalues with 0 < Re[k] 6 1, which result in singular stress fields, are of interest. Using the definitions of Eqs. (7), (10) and (11), the eigenequation of Eq. (17) can be simplified as tan½kðH2 pÞ cotðkH1 Þ ¼ x:
ð18Þ
The stress singularity depends on Hk (the principal argument of lk) and x (the ratio of the elastic constants). While the fiber orientations (g1, g2) are changed to 90° g1 and 90° g2 respectively, a new parameter l 0 k, similar to Eq. (7), is defined as pffiffiffiffiffiffi mk sin gk þ i cos gk l0k ¼ pffiffiffiffiffiffi : ð19Þ mk cos gk þ i sin gk The absolute value R 0 k and principal argument H 0 k of l 0 k are generated as follows sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk sin2 gk þ cos2 gk 1 0 0 Rk jlk j ¼ ð20Þ ¼ ; mk cos2 gk þ sin2 gk Rk tanH0k
tanðArgl0k Þ
pffiffiffiffiffiffi 2 mk ¼ tanHk : ¼ ð1 mk Þ sinð2gk Þ ð21Þ
It is indicated that the principal arguments are unvaried when the directions are changed from gk to 90° gk. Therefore, the orders of stress singularities are the same when the directions of principal axes ^xk are changed from (g1, g2) to (90° g1, 90° g2). The same conclusion can also be reached when the directions of the principal axis ^xk is change from (g1, g2) to (g1, 90° g2) or (90° g1, g2). The numerical results listed in Table 1 ð1Þ ð1Þ ð2Þ ð2Þ are a typical example for 2^s44 ¼ ^s55 , 2^s44 ¼ ^s55 and ð1Þ ð2Þ ^s55 ¼ ^s55 . In most engineering applications, the bonded wedges are formed by using same materials with different fiber ð1Þ ð2Þ ð1Þ ð2Þ orientations (i.e. ^s44 ¼ ^s44 , ^s55 ¼ ^s55 , g1 5 g2). It leads
435
to x = 1 and l1 5 l2. Substituting x = 1 into Eq. (18), the eigenequation can be simplified as sin½ðH2 H1 pÞk ¼ 0:
ð22Þ
The solution is given by k¼
np H2 H1 p
ðn ¼ integerÞ:
ð23Þ
The values of (H2 H1), which ranges from p to p, can be obtained from Fig. 2 while mk and gk are given. The values of interest for (k 1) are in the range 1 < k 1 < 0. Fig. 3(a) plots the variations of (k 1) with H2 H1. As H2 < H1, the stress field is singular (Fig. 3(a)). However, as H2 P H1, there is no singular stress field. The strongest stress singularity occurs when H2 H1 = p and its order is k 1 = 0.5. Consider the case that both materials are isotropic, ð1Þ ð1Þ ð2Þ ð2Þ i.e. ^s44 ¼ ^s55 and ^s44 ¼ ^s55 . From Fig. 2, it gives H1 = H2 = p/2 and k = n by Eq. (18). It means that there is no singular stress field for any dissimilar isotropic 90°–90° wedge with free–free boundary conditions. 3.1.2. Clamped–clamped boundary conditions The clamped–clamped conditions (i.e. boundary wð1Þ h ¼ p2 ¼ 0 and wð2Þ h ¼ p2 ¼ 0) can be expressed the following forms: k1 ¼ 0; c11 lk1 c21 l k
ð24Þ k
c12 ðl2 Þ c22 ð l2 Þ ¼ 0:
ð25Þ
Eqs. (12), (13), (24) and (25) lead to the characteristic matrix. The eigenequation can be simplified to the following form tan½kðH2 pÞ cotðkH1 Þ ¼ 1=x:
ð26Þ
Consider the case that the materials of the two wedges are the same, x = 1. The eigenequation of Eq. (26) is the same with Eq. (18) by taking x = 1. This solution is given by Eq. (23). 3.1.3. Clamped–free boundary conditions Consider the case that the boundary surfaces at h = p/ 2 and h = p/2 are clamped and free respectively. The characteristic matrix is given by combining Eqs. (12), (13), (16) and (24) and the eigenequation yields tan½kðH2 pÞ tanðkH1 Þ ¼ x:
ð27Þ
If the boundary conditions are reversed (i.e. free– clamped), Eqs. (12), (13), (15), and (25) are used to form the characteristic matrix. The eigenequation becomes tan½kðH2 pÞ tanðkH1 Þ ¼ 1=x:
ð28Þ
If the same materials are used (x = 1) with different fiber orientations, both Eqs. (27) and (28) are reduced to cos½ðH2 H1 pÞk ¼ 0:
ð29Þ
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Fig. 3. The variations of stress singularity k 1 with H2 H1 or H1(x = 1). (a) 90°–90° wedges with free–free or clamped–clamped, (b) 90°–90° wedges with clamped–free or free–clamped, (c) 90°–180° wedges with free–free or clamped–clamped, (d) 90°–180° wedges with clamped–free or free– clamped, (e) 90°–270° wedges with free–free or clamped–clamped, (f) 90°–270° wedges with clamped–free or free–clamped B.C.
The solution of Eq. (29) is k¼
ð2n þ 1Þp 2ðH2 H1 pÞ
ðn ¼ integerÞ:
ð30Þ
Fig. 3(b) plots the variations of (k 1) with H2 H1. The dotted line represents the second root of (k 1). No singular stress field is found when p/2 < H2 H1 < p. If the two wedges have the same orientations of principal axes (i.e. H2 H1 = 0), the structure reduces to one material wedge with 180° angle and free–clamped boundary conditions. The stress singularity order k 1
is equal to 0.5 and independent of mk. The strongest singularity order is k 1 = 0.75 as H2 H1 = p. There are two roots for p < H2 H1 < p/2. It is worth considering the case that the problem is reduced to dissimilar isotropic bonded wedges. For isotropic materials, lk = i and H1 = H2 = p/2, the eigenequation obtained from Eq. (27) can be simplified as following cosðkpÞ ¼
1x ; 1þx
ð31Þ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
which is exactly the same as Eq. (28) in Ref. [4] for both of the wedge angles equal to p/2. Comparing Eqs. (27) and (28), the singularity orders are the same for free–clamped and clamped–free boundary conditions if the materials of two isotropic wedges are interchanged. This conclusion is in agreement with Chiu and Chaudhuri [6].
Consider the case of an anisotropic 90°–180° wedge perfectly bondedalong the x-axis (Table 3). From Eq. (4), there are f1 h ¼ p2 ¼ l1 and f2(h = p) = epi on the boundaries. Eqs. (12) and (13) are the continuity conditions of the stresses and displacements along the interface. 3.2.1. Free–free boundary conditions ð2Þ The condition of stress syz ðh ¼ pÞ ¼ 0 at free surface can be reduced to the following equation c12 e
ðk1Þpi
þ c22 e
ðk1Þpi
Eq. (36) with x ¼ ^sð2Þ =^sð1Þ . After mathematical operations, Eq. (36) can be reduced to Eq. (23) in Ref. [4] for a = p/2 and b = p. 3.2.2. Clamped–clamped boundary conditions The boundary condition at the clamped surface is w(2)(h = p) = 0. Using Eq. (9), the result is shown: c12 ekpi c22 ekpi ¼ 0:
3.2. 90°–180° wedges
¼ 0:
ð32Þ
The determinant of the matrix that governs the eigenvalues is obtained by combining Eqs. (12), (13), (15) and (32). The result can be simplified to the following eigenequation tanðkpÞ cotðkH1 Þ ¼ x:
ð33Þ ð1Þ
ð1Þ
Notethat the eigenequation depends only on ^s44 , ^s55 , ð2Þ ð2Þ g1 and ^s44 ^s55 . The directions of the principal axes of region X2 will not affect the stress singularity. This conclusion is also true for the case that a wedge with arbitrary angle is bonded to a half surface subjected to arbitrary boundary conditions. Again, it can be proved that the stress singularity orders remain the same when the direction of the principal axis g1 is changed to 90° g1. As the material properties of two regions are identical (x = 1), the eigenequation becomes sin½ðH1 þ pÞk ¼ 0:
ð34Þ
The solution k is np ðn ¼ integerÞ: k¼ H1 þ p
ð35Þ
Fig. 3(c) shows the variations of stress singularity with H1. The strongest stress singularity occurs at H1 = p and k 1 = 0.5. However, weak stress singularity is found as H1 is close to 0. The value H1 can be obtained from Fig. 2 or Eq. (11). ð1Þ ð1Þ When material 1 is isotropic (i.e. ^s44 ¼ ^s55 ¼ ^sð1Þ ), the eigenequation becomes ½1 þ ð1 þ xÞq cosðkpÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi sinðkp=2Þ ¼ 0;
ð2Þ ð2Þ where x ¼ ^s44 ^s55 =^sð1Þ . If ð2Þ ð2Þ (i.e. ^s44 ¼ ^s55 ¼ ^sð2Þ ), the
ð36Þ
material 2 is also isotropic
eigenequation is equal to
437
ð37Þ
Again, the determinant of the matrix is also obtained by combining Eqs. (12), (13), (24) and (37). Thus, tanðkpÞ cotðkH1 Þ ¼ 1=x:
ð38Þ
The singularity order is independent of g2. If the materials of the two wedges are the same (x = 1), the eigenequation is equal to Eq. (34) and the solution is in Eq. (35). When material 1 becomes isotropic, the eigenequation of Eq. (38), regardless of g2, becomes ½1 þ ð1 þ 1=xÞ cosðkpÞ sinðkp=2Þ ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffi ffi
ð39Þ
ð2Þ ð2Þ where x ¼ ^s44 ^s55 =^sð1Þ . If the material 2 is ð2Þ ð2Þ pic (i.e. ^s44 ¼ ^s55 ¼ ^sð2Þ ), the eigenequation ð2Þ ð1Þ
also isotrois equal to Eq. (39) with x ¼ ^s =^s . After simplification, it is the same as Eq. (38) in Ref. [4] for a = p/2 and b = p. Comparing Eqs. (36) and (39), the singularity orders of the two cases with free–free and clamped–clamped boundary conditions are the same if the materials of two isotropic wedges are interchanged. It is coincident with Chiu and Chaudhuri [6]. 3.2.3. Clamped–free boundary conditions Consider the case that the boundary surfaces at h = p/2 and h = p are clamped and free respectively. Using Eqs. (12), (13), (24) and (32), the determinant of the matrix can be simplified to the following eigenequation tanðkpÞ tanðkH1 Þ ¼ x:
ð40Þ
It is independent of g2. Again, it is easy to prove that the stress singularity order remains the same when the direction of the principal axis g1 is changed to 90° g1. As the material properties of two regions are identical (x = 1), the eigenequation is cos½ðH1 þ pÞk ¼ 0:
ð41Þ
The roots for k are k¼
ð2n þ 1Þp 2ðH1 þ pÞ
ðn ¼ integerÞ:
ð42Þ
Fig. 3(d) plots the variations of (k 1) with H1. There are two roots as H1 > p/2. The strongest stress singularity is k 1 = 0.75 for H1 = p and the weakest singularity is k 1 = 0.5 for H1 = 0. The range of the first stress singularity order is 0.75 6 k 1 6 0.5. When the material 1 with wedge angle p/2 is isotropic, Eq. (40) can be arranged as follows
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½1 þ ð1 þ xÞ cosðkpÞ qffiffiffiffiffiffiffiffiffiffiffiffi ffi cosðkp=2Þ ¼ 0;
ð43Þ
ð2Þ ð2Þ ¼ ^s44 ^s55 =^sð1Þ . ð2Þ ð1Þ
If both materials are isotropic where x (i.e. x ¼ ^s =^s ), Eq. (43) is exactly the same as Eq. (28) in Ref. [4] for a = p/2 and b = p. 3.2.4. Free–clamped boundary conditions Eqs. (12), (13), (15) and (37) lead to the characteristic matrix for this case. The eigenequation becomes tanðkpÞ tanðkH1 Þ ¼ 1=x:
ð44Þ
It is also independent of g2. If the same materials are used (x = 1), the eigenequation is again given by Eq. (41). If material 1 is isotropic, Eq. (44) can be rewritten as follows ½1 þ ð1 þ 1=xÞ cosðkpÞ cosðkp=2Þ ¼ 0:
ð45Þ
Comparing Eqs. (43) and (45), the singularity order is the same for free–clamped and clamped–free boundary conditions if the materials of two isotropic wedges are interchanged. From the previous results of Section 3.2, certain conclusions can be made: (1) The antiplane stress singularities for 90°–180° wedges with arbitrary boundary ð1Þ ð1Þ ð2Þ ð2Þ conditions depend on ^s44 , ^s55 , g1 and ^s44 ^s55 ; (2) the stress singularity order remains unchanged when the direction of the principal axis ^x1 is changed from g1 to 90° g1; (3) as the material properties of regions 1 and 2 are identical (x = 1; g1, g2 = arbitrary), the stress singularity orders are equal for free–free and clamped– clamped boundary conditions. This is also true for clamped–free and free–clamped boundary conditions; (4) if two isotropic materials are interchanged for cases of free–free and clamped–clamped boundary conditions, the singularity orders are the same. This is also true for clamped–free and free–clamped boundary conditions. It agrees with Chiu and Chaudhuri [6]. 3.3. 180°–180° wedges Consider a semi-infinite interface crack existing in two dissimilar anisotropic materials as shown in Table 3. From Eq. (4), it can be seen that f1(h = p) = epi and f2(h = p) = epi on the boundaries. The continuity conditions of the stresses and displacements along the interface are defined in Eqs. (12) and (13).
The root of Eq. (47) is k = 0.5. It is independent of material properties. 3.3.2. Clamped–clamped boundary conditions The displacement w(1)(h = p) at clamped surface is shown: c11 ekpi c21 ekpi ¼ 0:
ð48Þ
The eigenequation, given by Eqs. (12), (13), (37) and (48), is identical to Eq. (47). 3.3.3. Clamped–free boundary conditions Consider the interface crack at h = p and h = p with clamped and free boundary conditions, respectively. Then, Eqs. (12), (13), (32) and (48) lead to the eigenequation: 1x cosð2kpÞ ¼ : ð49Þ 1þx When the material properties of regions 1 and 2 are identical (i.e. x = 1), the roots of Eq. (49) are k = 0.25 and 0.75. 3.4. 90°–270° wedges Consider an anisotropic 90°–270° wedge perfectly bonded along the line h = p/2 shown in Table 3. This wedge is called as ‘‘debonded junction’’. The continuity ð1Þ ð2Þ conditions of the stresses (i.e. sxz ðh ¼ p2Þ ¼ sxz ðh ¼ p2Þ) ð1Þ p ð2Þ p and displacements (i.e. w ðh ¼ 2Þ ¼ w ðh ¼ 2Þ) along the interface are written as followings: k1 c12 lk21 c22 l k2 ¼ 0; c11 lk1 þ c21 l
ð50Þ
k1 xc12 lk2 þ xc22 l k2 ¼ 0: c11 lk1 c21 l
ð51Þ
3.4.1. Free–free boundary conditions The stresses at h = p and h = p are traction free. Connecting with Eqs. (32), (46), (50) and (51), the eigenequation for this problem is as follows sin½ðH2 H1 þ 2pÞk x 1 : ¼ sin½ðH2 þ H1 Þk xþ1
ð52Þ
3.3.1. Free–free boundary conditions ð1Þ The stress syz ðh ¼ pÞ ¼ 0 at free surface is given by
The eigenvalues depend on H1, H2 and x. Following the same procedure as in Section 3.1.1, the principal arguments Hk will be unvaried when the direction of the principal axis ^xk is changed from gk to 90° gk. Therefore, the singularity orders are the same when the direction of the ^xk -axis is changed from (g1, g2) to (90° g1, 90° g2), (g1, 90° g2), or (90° g1, g2). If the same materials are used (x = 1), Eq. (52) becomes
c11 eðk1Þpi þ c21 eðk1Þpi ¼ 0:
sin½ðH2 H1 þ 2pÞk ¼ 0:
ð46Þ
The eigenequation can be determined by Eqs. (12), (13), (32) and (46), therefore sinð2kpÞ ¼ 0:
ð47Þ
ð53Þ
Solving above equation, the root is k¼
np ðH2 H1 þ 2pÞ
ðn ¼ integerÞ:
ð54Þ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
½x 1 2ðx þ 1Þ cosðkpÞ sinðkpÞ ¼ 0:
ð55Þ
-0.4
-0.3333 -0.3
λ−1
Fig. 3(e) plots the variations of (k 1) with H2 H1. The strongest stress singularity is k 1 = 2/3 for H2 H1 = p. If the fiber directions of two wedges are the same (H2 H1 = 0), it is reduced to the case that a crack exists in an anisotropic body and k = 0.5. When material 1 and 2 are two different isotropic materials (x 5 1), the eigenequation of Eq. (52) is rewritten as follows
439
N.S.
-0.2
It is the same with Eq. (23) in Ref. [4] for a = p/2 and b = 3p/2. -0.1
-0.4
mk
0
100 10 1.1 0.1 0.01
-0.3
0
30
60
90
120
150
180
Θ1 (Degree)
Re[λ-1]
Fig. 5. The variations of stress singularity (k 1) with H1 (x = 1 and g1 = g2).
3.4.2. Clamped–clamped boundary conditions The boundaries at h = p and h = p are clamped. Using Eqs. (37), (48), (50) and (51), the following eigenequation can be obtained
-0.2
sin½ðH2 H1 þ 2pÞk 1=x 1 : ¼ sin½ðH2 þ H1 Þk 1=x þ 1
-0.1
ð56Þ
Eq. (53) is valid for this case while x = 1. For dissimilar isotropic materials, the eigenequation is as follows 0 -90
-60
-30
0
30
60
½1=x 1 2ð1=x þ 1Þ cosðkpÞ sinðkpÞ ¼ 0:
90
η1 (Degree)
(a)
It is the same with Eq. (38) in Ref. [4] for a = p/2 and b = 3p/2. Comparing Eqs. (55) and (57), the singularity orders are the same for free–free and clamped–clamped boundary surfaces if the two isotropic materials are interchanged. This conclusion is identical to Ref. [6].
mk
0.6
100 10 1.1 0.1 0.01
0.5
0.3
3.4.3. Clamped–free boundary conditions The boundaries at h = p and h = p are clamped and traction free, respectively. By rearranging Eqs. (32), (48), (50) and (51), the eigenequation can be expressed in the following form:
0.2
cos½ðH2 H1 þ 2pÞk 1 x : ¼ cos½ðH2 þ H1 Þk 1þx
0.4
Im[λ-1]
ð57Þ
Again, it can be proved that the stress singularity orders remain the same when the direction of the principal axis gk is changed to 90° gk. If x = 1, the equation becomes
0.1
cos½ðH2 H1 þ 2pÞk ¼ 0:
0 -90
(b)
ð58Þ
-60
-30
0
30
60
90
η1 (Degree)
Fig. 4. The variations of stress singularity order (k 1) with g1 for g2 = 0° and mk = 100, 10, 1.1, 0.1 and 0.01. (a) Re[k 1], (b) Im[k 1].
ð59Þ
The root of this equation is k¼
ð2n þ 1Þp 2ðH2 H1 þ 2pÞ
ðn ¼ integerÞ:
ð60Þ
440
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
Table 2 The eigenequations of antiplane anisotropic wedges and junction Free–free
Clamped–clamped
Clamped–free
Free–clamped
90°–90° Wedge
tan½kðH2 pÞ cotðkH1 Þ ¼x
tan½kðH2 pÞ cotðkH1 Þ ¼ 1=x
tan½kðH2 pÞ tanðkH1 Þ ¼ x
tan½kðH2 pÞ tanðkH1 Þ ¼ 1=x
90°–180° Wedge
tanðkpÞ cotðkH1 Þ ¼ x
tanðkpÞ cotðkH1 Þ ¼ 1=x
tanðkpÞ tanðkH1 Þ ¼ x
tanðkpÞ tanðkH1 Þ ¼ 1=x
cosð2kpÞ ¼ 1x 1þx
cosð2kpÞ ¼ x1 xþ1
180°–180° Wedge 90°–270° Wedge 90°–270° Junction
sinð2kpÞ ¼ 0; k 1 ¼ 0:5
sin½ðH2 H1 þ 2pÞk x 1 sin½ðH2 H1 þ 2pÞk 1 x cos½ðH2 H1 þ 2pÞk 1 x cos½ðH2 H1 þ 2pÞk x 1 ¼ ¼ ¼ ¼ sin½ðH2 þ H1 Þk xþ1 sin½ðH2 þ H1 Þk 1þx cos½ðH2 þ H1 Þk 1þx cos½ðH2 þ H1 Þk xþ1
2 k k R2 2x cos½ðH2 H1 þ 2pÞk 1x cos½ðH1 þ H2 Þk ¼ ð1þxÞ þ RR21 2 R1 1þx
Fig. 3(f) plots the variations of k 1 with H2 H1. There are three roots for H2 H1 > p/2. The strongest stress singularity is k 1 = 5/6 for H2 H1 = p and the weakest singularity of the first root is k 1 = 0.5 for H2 H1 = p. If the fiber orientations of the two wedges are the same (H2 = H1), it is reduced to the case that a crack exists in an anisotropic body with clamped– free boundary surfaces and k 1 = 0.75 and 0.25. For two different isotropic materials, Eq. (58) becomes ð1 xÞ cosðkpÞ ð1 þ xÞ cosð2kpÞ ¼ 0:
ð61Þ
It is the same with Eq. (28) in Ref. [4] for a = p/2 and b = 3p/2. 3.4.4. Free–clamped boundary conditions The boundaries at h = p and h = p are traction free and clamped, respectively. Using Eqs. (37), (46), (50) and (51), the eigenequation can be simplified as cos½ðH2 H1 þ 2pÞk 1 1=x : ¼ cos½ðH2 þ H1 Þk 1 þ 1=x
ð62Þ
Eq. (59) is also valid in this case for x = 1. As the materials are isotropic, the eigenequation is rewritten as follows ð1 1=xÞ cosðkpÞ ð1 þ 1=xÞ cosð2kpÞ ¼ 0:
ð63Þ
Eqs. (61) and (63) are the same if the materials of two isotropic wedges are interchanged. Important conclusions drawn from Section 3.4 can be listed as follows: (1) The antiplane stress singularities for 90°–270° wedges with arbitrary boundary conditions remain unchanged when the directions of the ^xk -axis are changed from (g1, g2) to (90° g1, 90° g2), (g1, 90° g2) or (90° g1, g2). (2) As the material properties of regions 1 and 2 are identical (x = 1; g1, g2 = arbitrary), the stress singularity orders are equal for free– free and clamped–clamped boundary conditions. This is also true for clamped–free and free–clamped boundary conditions. (3) If two isotropic materials are interchanged for cases of free–free and clamped–clamped boundary conditions, the singularity orders are the
same. This is also true for clamped–free and free– clamped boundary conditions. This conclusion is in agreement with Chiu and Chaudhuri [6]. 3.5. 90°–270° junction Consider an anisotropic 90°–270° junction (Fig. 1(b)) perfectly bonded along the line h = p/2 and h = p. The regions of wedges with angle 90° and 270° are denoted as X1 and X2, respectively. The continuity conditions ð1Þ ð2Þ of the stresses (i.e. syz ðh ¼ pÞ ¼ syz ðh ¼ pÞ) and dis(1) (2) placements (i.e. w (h = p) = w (h = p)) along the interface h = p are in the forms given below c11 eðk1Þpi þ c21 eðk1Þpi c12 eðk1Þpi c22 eðk1Þpi ¼ 0; ð64Þ c11 ekpi c21 ekpi xc12 ekpi þ xc22 ekpi ¼ 0:
ð65Þ
Eqs. (50), (51), (64) and (65) form the eigenequation for this problem. Using the definition in Eqs. (11) and (12), the eigenequation leads to 2 1x cos½ðH2 H1 þ 2pÞk cos½ðH1 þ H2 Þk 1þx # " k k 2x R2 R2 þ ¼ : ð66Þ 2 R1 R1 ð1 þ xÞ The eigenvalues depend on Hk, Rk and x. Here, k may be a complex number. If the fiber directions of Eq. (66) are changed from (g1, g2) to (90° g1, 90° g2) simultaneously, the result shows that Eq. (66) is unvaried if the relations of Eqs. (20) and (21) are used. If x = 1, Eq. (66) can be rewritten as follows k iðH þ2pÞk R2 e 2 Rk1 eiH1 k Rk1 eiðH1 2pÞk Rk2 eiH2 k ¼ 0: ð67Þ In view of Eq. (67), the stress singularity k should be a complex number. The solution of Eq. (67) becomes h i 2np ðH2 H1 þ 2pÞ i ln RR21 k¼ ðn ¼ integerÞ: ð68Þ 2 2 R2 ln R1 þ ðH2 H1 þ 2pÞ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
441
Table 3 The eigenequations of antiplane anisotropic wedges and junction as x = 1 Free–free or clamped–clamped
Clamped–free or free–clamped
90°–90° Wedge
np (Fig. 3(a)) k¼ H2 H1 p
k¼
ð2n þ 1Þp (Fig. 3(b)) 2ðH2 H1 pÞ
90°–180° Wedge
k¼
k¼
ð2n þ 1Þp (Fig. 3(d)) 2ðH1 þ pÞ
180°–180° Wedge
sinð2kpÞ ¼ 0, k1 = 0.5
90°–270° Wedge
k¼
90°–270° Junction
np (Fig. 3(c)) H1 þ p
np (Fig. 3(e)) ðH2 H1 þ 2pÞ
cosð2kpÞ ¼ 0, k1 = 0.75, 0.25
k¼
ð2n þ 1Þp (Fig. 3(f)) 2ðH2 H1 þ 2pÞ
h i 2np ðH2 H1 þ 2pÞ i ln RR21 k¼ 2 ln RR21 þ ðH2 H1 þ 2pÞ2
442
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
Fig. 4(a) and (b) show the variations of stress singularity order (k 1) with g1 for g2 = 0° and mk = 100, 10, 1.1, 0.1 and 0.01. Consider the special case of g1 = g2, that the relations of R2 = R1 and H2 = p H1 are obtained by the definition in Eqs. (10) and (11). Therefore, the stress singularities of Eq. (68) are real and double roots and given by 2np k¼ ðn ¼ integerÞ: ð69Þ 3p 2H1
singular stress fields can be used to improve the safety of structures.
Fig. 5 plots the variations of (k 1) with H1. No singular stress field is found when p/2 6 H1 6 p. The strongest stress singularity is k 1 = 1/3 for H1 = 0 (i.e. H2 = p). Dempsey and Sinclair [13] found that the existence conditions of logarithmic type singularity are d4m Rank[D(k)] = m < 4 and dk 4m det½DðkÞ ¼ 0 where D[k] is the eigenmatrix. It means that, for double roots, the singularity is logarithmic type as m = 3. In this case, however, the roots k = 2p/(3p 2H1) are double roots dD(k)/dk = 0 but the rank is 2. Therefore, the stress singularity is still rk 1-type and not logarithmic type.
References
4. Conclusions This paper has derived the general closed-form solution of eigenequation for wedges and junctions bonded at special angles under antiplane shear. The repeated occurrence of stress singularity orders is examined analytically. Taking a typical case of a 90°–90° wedge with various boundary conditions (i.e. free–free, clamped– clamped or clamped–free) as an example, the singularity orders are unchanged while the fiber orientations gk (k = 1 or 2) are changed to 90° gk. The eigenequations for dissimilar material wedges are also obtained and compared with existing literature in some special cases. All of these results are given in Tables 2 and 3. The disappearance conditions of the stress singularity can be found by plotting the variations of (k 1) with the fiber orientations. The theoretical conclusion for vanishing
Acknowledgement The authors are grateful to the National Science Council of the ROC for financial support under the contract NSC89-2212-E006-119.
[1] Bogy DB. The plane solution for anisotropic elastic wedges under normal and shear loading. J Appl Mech 1972;39:1103–9. [2] Kuo MC, Bogy DB. Plane solutions for traction problems on orthotropic unsymmetrical wedges and symmetrically twinned wedges. J Appl Mech 1974;41:203–8. [3] Tranter CJ. The use of Mellin transform in finding the stress distribution in an infinite wedge. Q J Mech Appl Math 1948;1: 125–30. [4] Ma CC, Hour BL. Analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation. Int J Solids Struct 1989; 25:1295–309. [5] Xie M, Chaudhuri RA. Three-dimensional asymptotic stress field at the front of a bimaterial wedge of symmetric geometry under antiplane shear loading. Compos Struct 2001;54:509–14. [6] Chiu Jack SH, Chaudhuri RA. Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial pie-shaped wedge under antiplane shear loading. Compos Struct 2002;58: 129–37. [7] Ma CC, Hour BL. Antiplane problems in composite anisotropic materials with an inclined crack terminating at a bimaterial interface. Int J Solids Struct 1990;26:1387–400. [8] Shahani AR. Analysis of an anisotropic finite wedge under antiplane deformation. J Elast 1999;56:17–32. [9] Pageau SS, Joseph PF, Biggers Jr SB. Finite element analysis of singular stress fields in anisotropic materials loaded in antiplane shear. Int J Numer Methods Eng 1995;38:81–97. [10] Chue CH, Liu CI. Stress singularities in a bimaterial anisotropic wedge with arbitrary fiber orientation. Compos Struct 2002;58: 129–37. [11] Chue CH, Liu CI. Disappearance of free-edge stress singularity in composite laminates. Compos Struct 2002;56:111–29. [12] Lekhnitskii SG. Theory of elasticity of an anisotropic elastic body. New York: Holden-Day; 1963. [13] Dempsey JP, Sinclair GB. On the stress singularities in the plane elasticity of the composite wedge. J Elast 1979;9:373–91.
On the stress singularity of dissimilar anisotropic wedges and junctions in antiplane shear Chuan-I Liu a
a,*
, Ching-Hwei Chue
b
Department of Structure Analysis, Aerospace Industrial Development Corporation, Taichung, Taiwan, ROC b Department of Mechanical Engineering, National Cheng Kung University, Tainan, ROC Available online 29 March 2005
Abstract In this paper, rk1 type antiplane stress singularities in dissimilar anisotropic wedges and junctions are studied. For some special wedge angles, the eigenequations derived are a function of fiber orientations and the elastic constants in the principal material coordinates. The boundary surfaces of the wedges can be subjected to combinations of free and clamped boundary conditions. The repeated occurrence of stress singularity orders is examined. By properly selecting the fiber orientations, the degree of the stress singularities can be diminished or even disappeared. The singularities of the materials reduced to isotropic are discussed as well. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Stress singularity; Wedge and junction; Antiplane; Anisotropic material
1. Introduction In composite structures, failures often commence at the apex of the wedge or junction where the stresses are singular. It is known that stress singularity orders depend on the wedge angles, edge boundary conditions, composite material properties, and fiber orientations. If fiber orientations and material constants, defined in the principal material coordinates are directly included in the eigenequations, certain useful information for structural design can be obtained. If x–y is a symmetric plane for anisotropic material, the in-plane and antiplane stress fields of generalized plane deformation are decoupled. Bogy [1], Kuo and Bogy [2] employed a complex function representation of the solution in conjunction with the Mellin transform method [3] to analyze the in-plane stress singularities in an anisotropic wedge. Ma and Hour [4] used the Mellin transform to study the antiplane stress singularities of *
Corresponding author. Tel.: +886 4 27070001 503251; fax: +886 4 22842824. E-mail address: [email protected] (C.-I. Liu). 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.02.015
the same wedge problem. Ma and Hour [4] found that the order of stress singularity is real for the general anisotropic bimaterial wedge of the antiplane problem subjected to various boundary conditions (i.e. free–free, clamped–clamped and free–clamped). Recently, Xie and Chaudhuri [5] proposed an eigenfunction approach for analyzing the three-dimensional asymptotic stress field of a symmetric bimaterial wedge under antiplane shear loading. A similar problem with an asymmetric bimaterial pie-shaped wedge has been solved by Chiu and Chaudhuri [6]. The problems of cracking are also extensively investigated. For example, Ma and Hour [7] studied antiplane problems in anisotropic materials with an inclined crack terminating at a bimaterial interface. Shahani [8] investigated the anisotropic finite wedge under antiplane deformation. Pageau et al. [9] obtained the numerical antiplane singular stress field of anisotropic multi-material wedges and junctions. Using the contours of stress singularity order, Chue and Liu [10,11] studied the bimaterial wedge problem with arbitrary fiber orientation and obtained the disappearance conditions of stress singularities in composite laminates.
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
The antiplane wedge and junction problems are of major concern in this article. However, the article will not focus on deriving the general antiplane stress singularity of the anisotropic wedge problem. Certain particular wedge angles possessing similar natural characteristics are selected. Also, several practical applications for engineers to design the fiber orientation of composite material are found. Based on LekhnitskiiÕs formulations [12], the eigenequations, which govern the antiplane stress singularity orders for 90°–90°, 90°–180°, 180°–180°, 90°–270° anisotropic wedges and 90°–270° junction, are derived. Consider an anisotropic wedge (Fig. 1(a)) with free–free boundary conditions and a1 = a2 = 90°, that the ^xk -axis (k = 1, 2) of the material principal axes lies in the xyplane and makes an angle gk (90° 6 gk 6 90°) with the positive x-axis. If these two wedges are the same material but with different orientations (g1 5 g2), the antiplane stress singularity orders are listed in Table 1. It is shown that the stress singularity is unvaried as the gk (k = 1 or 2) is changed to 90° gk. The repeated appearance of singularity orders will be discussed analytically in Section 3. In addition, the disappearance conditions of the stress singularity can be obtained from the derived eigenequations. This theoretical conclusion for vanishing singular stress fields can be used to improve the safety of structures.
2. Formulations of a two-bonded anisotropic wedge in antiplane fields Fig. 1(a) and (b) shows the configurations of twobonded composite wedge and junction, respectively. Oxyz is the global coordinate system. Let ^xk , ^y k and ^zk denote the principal material coordinates of composite wedge Xk (k = 1, 2). The ^zk -axis coincides with z-axis and ^xk -axis makes an angle gk (90° 6 gk 6 90°) with the positive x-axis. Therefore, the x–y plane is an elastic
433
Table 1 The antiplane stress singularity order (k 1) of the 90°–90° wedge with ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ free–free boundary conditions. (2^s44 ¼ ^s55 , 2^s44 ¼ ^s55 and ^s55 ¼ ^s55 ) (NS: no singularity)
g1 = 90° g1 = 75° g1 = 60° g1 = 45° g1 = 30° g1 = 15° g1 = 0° g1 = 15° g1 = 30° g1 = 45° g1 = 60° g1 = 75° g1 = 90°
g2 = 20°
g2 = 70°
0.0664073 0.112551 0.142152 0.152042 0.142152 0.112551 0.0664073 0.0152018 0.0240091 (NS) 0.0384675 (NS) 0.0240091 (NS) 0.0152018 0.0664073
0.0664073 0.112551 0.142152 0.152042 0.142152 0.112551 0.0664073 0.0152018 0.0240091 (NS) 0.0384675 (NS) 0.0240091 (NS) 0.0152018 0.0664073
symmetric surface. The in-plane and antiplane stress fields are then decoupled. For the antiplane field, the (k) ðkÞ exist. The relastresses sðkÞ yz , sxz and displacement w ðkÞ ðkÞ tions between strains cyz , cxz and stresses in the global coordinate system are: ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðkÞ ðkÞ cðkÞ yz ¼ s44 syz þ s45 sxz ;
ð1Þ
ðkÞ ðkÞ cðkÞ xz ¼ s45 syz þ s55 sxz ; ðkÞ
where sij are the components of elastic compliance matrix in the global coordinate system. Consider the orthotropic composite materials, which the elastic constants ðkÞ ðkÞ of ^s44 and ^s55 defined in ^xk ^y k^zk -coordinate system exist ðkÞ ðkÞ but ^s45 vanish. Therefore, sij can be computed through the following coordinate transformation ðkÞ
ðkÞ
ðkÞ
s44 ¼ ^s44 cos2 ðgk Þ þ ^s55 sin2 ðgk Þ; ðkÞ ðkÞ ðkÞ s45 ¼ ^s44 ^s55 cosðgk Þ sinðgk Þ; ðkÞ s55
¼
ðkÞ ^s55 cos2 ðgk Þ
ðk ¼ 1; 2Þ:
ð2Þ
ðkÞ þ ^s44 sin2 ðgk Þ;
The stresses and displacements can be expressed in terms of complex function /k(zk) as follows [12]
Fig. 1. Geometry of two dissimilar anisotropic materials for (a) wedge, (b) junction.
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C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
0 sðkÞ xz ¼ 2Re½lk /k ðzk Þ;
ðkÞ
w
¼
2Re½/0k ðzk Þ;
ð3Þ
150
zk ¼ x þ lk y ¼ rðcos h þ lk sin hÞ rfk ðhÞ; ðkÞ s45
qffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ s ðkÞ ðkÞ 44 ¼ i ^s44 ^s55 : lk
ð4Þ 120
ð5Þ
Constant lk is the root of the following equation with positive imaginary part ðkÞ
100 10 1
mk
¼ 2Re½tk /k ðzk Þ;
where
tk ¼
0.1 0.01
180
ðkÞ
90
60
ðkÞ
s55 l2k 2s45 lk þ s44 ¼ 0:
ð6Þ
Substituting Eq. (2) into Eq. (6) and solving for lk, it gives pffiffiffiffiffiffi mk cos gk þ i sin gk lk ¼ pffiffiffiffiffiffi ; ð7Þ mk sin gk þ i cos gk ðkÞ
Θ k (Degree)
sðkÞ yz
ðkÞ
where mk ¼ ^s44 =^s55 . In order to obtain the stress singularity orders, the complex stress potential, /k(zk), is expressed in the following form:
/k ðzk Þ ¼ c1k zkk þ c2k zkk
ðk ¼ 1; 2Þ;
ð8Þ
where c1k and c2k are unknown complex constants. k is the complex eigenvalue to be determined. The eigenvalues in the range 0 < Re[k] 6 1 are of interest in the studying of singular stress fields. Substituting Eqs. (4) and (8) into Eq. (3), the stresses and displacements can be rearranged in the following forms h i k1 k1 k fk sðkÞ c1k lk fk1 þ c2k l ; k xz ¼ 2Re kr h i k1 k1 c1k fk1 þ c2k fk ð9Þ ; sðkÞ k yz ¼ 2Re kr h i k wðkÞ ¼ 2Re rk c1k tk fkk þ c2ktk fk ; where (k1) is the stress singularity order and a smaller value of k stands for a stronger stress singularity. If the absolute value and principal argument of lk are defined as Rk and Hk, respectively, then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk cos2 gk þ sin2 gk ; ð10Þ Rk jlk j ¼ mk sin2 gk þ cos2 gk pffiffiffiffiffiffi 2 mk : tanHk tanðArg lk Þ ¼ ð1 mk Þ sinð2gk Þ
ð11Þ
The variations of principal argument Hk with gk for different mk are shown in Fig. 2. It shows that the range of Hk is 0° < Hk < 180°. When 0° < gk < 90°, Hk is closer to 180° for larger mk. Also when 0° < gk < 90°, Hk is closer to 0° for smaller mk. When mk 1 (i.e. near isotropic material), Hk is equal to 90° for all fiber orientations, gk.
30
0 -90
-60
-30
0
η k (Degree)
30
60
90
Fig. 2. The variation of Hk with principal axis orientation gk for mk = 100, 10, 1, 0.1 and 0.01.
3. Examples and discussions Owing to geometric and material discontinuities, the antiplane stress singularities for anisotropic bimaterial wedges (with wedge angles 90°–90°, 90°–180°, 180°– 180°, 90°–270°) and junctions (with angles 90°–270°) are investigated. The reason for selecting these wedge angles is due to the similar rule for repeating occurrences of stress singularities. Also, they are very common structures in engineering applications. The boundary surfaces of the wedges can be combinations of free and clamped boundary conditions. 3.1. 90°–90° wedges Consider an anisotropic 90°–90° wedge perfectly bonded along the positive x-axis as shown in Table 3. p From Eq. (4), it can be seen that f ¼ l h ¼ 1 1, 2 p f2 h ¼ 2 ¼ l2 on the boundaries and f1(h = 0) = f2(h = 0) = 1 on the bonding surface, in which l1 and l2 are defined in Eq. (7). Based on Eq. (9), the continuity ð2Þ conditions of the stresses ðsð1Þ yz ðh ¼ 0Þ ¼ syz ðh ¼ 0ÞÞ and (1) (2) the displacements (w (h = 0) = w (h = 0)) along the interface will result in the following relations c11 þ c21 c12 c22 ¼ 0; ð12Þ c11 c21 xc12 þ xc22 ¼ 0; where x is the ratio of the elastic constants ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð2Þ ð2Þ ð1Þ ð1Þ x ¼ ^s44 ^s55 ^s44 ^s55 :
ð13Þ
ð14Þ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
3.1.1. Free–free boundary conditions p The free–free boundary conditions (i.e. sð1Þ xz h ¼ 2 ¼ p 0 and sð2Þ xz h ¼ 2 ¼ 0) can be reduced to the relations k1 ¼ 0; c11 lk1 þ c21 l
ð15Þ
k
k
l2 Þ ¼ 0: c12 ðl2 Þ þ c22 ð
ð16Þ
Eqs. (12), (13), (15) and (16) yield four linear homogeneous equations with unknowns c11, c21 , c12 and c22 . They can be rearranged in the following matrix form 2 k 32 3 c11 k1 l1 l 0 0 61 7 6 1 1 1 76 c21 7 6 7 ð17Þ 6 76 7 ¼ 0: 4 1 1 5 4 c12 5 x x 0
0
ðl2 Þ
k
ð l2 Þ
k
c22
For non-trivial solutions of cij (i, j = 1, 2), the determinant of the matrix must vanish. Only eigenvalues with 0 < Re[k] 6 1, which result in singular stress fields, are of interest. Using the definitions of Eqs. (7), (10) and (11), the eigenequation of Eq. (17) can be simplified as tan½kðH2 pÞ cotðkH1 Þ ¼ x:
ð18Þ
The stress singularity depends on Hk (the principal argument of lk) and x (the ratio of the elastic constants). While the fiber orientations (g1, g2) are changed to 90° g1 and 90° g2 respectively, a new parameter l 0 k, similar to Eq. (7), is defined as pffiffiffiffiffiffi mk sin gk þ i cos gk l0k ¼ pffiffiffiffiffiffi : ð19Þ mk cos gk þ i sin gk The absolute value R 0 k and principal argument H 0 k of l 0 k are generated as follows sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mk sin2 gk þ cos2 gk 1 0 0 Rk jlk j ¼ ð20Þ ¼ ; mk cos2 gk þ sin2 gk Rk tanH0k
tanðArgl0k Þ
pffiffiffiffiffiffi 2 mk ¼ tanHk : ¼ ð1 mk Þ sinð2gk Þ ð21Þ
It is indicated that the principal arguments are unvaried when the directions are changed from gk to 90° gk. Therefore, the orders of stress singularities are the same when the directions of principal axes ^xk are changed from (g1, g2) to (90° g1, 90° g2). The same conclusion can also be reached when the directions of the principal axis ^xk is change from (g1, g2) to (g1, 90° g2) or (90° g1, g2). The numerical results listed in Table 1 ð1Þ ð1Þ ð2Þ ð2Þ are a typical example for 2^s44 ¼ ^s55 , 2^s44 ¼ ^s55 and ð1Þ ð2Þ ^s55 ¼ ^s55 . In most engineering applications, the bonded wedges are formed by using same materials with different fiber ð1Þ ð2Þ ð1Þ ð2Þ orientations (i.e. ^s44 ¼ ^s44 , ^s55 ¼ ^s55 , g1 5 g2). It leads
435
to x = 1 and l1 5 l2. Substituting x = 1 into Eq. (18), the eigenequation can be simplified as sin½ðH2 H1 pÞk ¼ 0:
ð22Þ
The solution is given by k¼
np H2 H1 p
ðn ¼ integerÞ:
ð23Þ
The values of (H2 H1), which ranges from p to p, can be obtained from Fig. 2 while mk and gk are given. The values of interest for (k 1) are in the range 1 < k 1 < 0. Fig. 3(a) plots the variations of (k 1) with H2 H1. As H2 < H1, the stress field is singular (Fig. 3(a)). However, as H2 P H1, there is no singular stress field. The strongest stress singularity occurs when H2 H1 = p and its order is k 1 = 0.5. Consider the case that both materials are isotropic, ð1Þ ð1Þ ð2Þ ð2Þ i.e. ^s44 ¼ ^s55 and ^s44 ¼ ^s55 . From Fig. 2, it gives H1 = H2 = p/2 and k = n by Eq. (18). It means that there is no singular stress field for any dissimilar isotropic 90°–90° wedge with free–free boundary conditions. 3.1.2. Clamped–clamped boundary conditions The clamped–clamped conditions (i.e. boundary wð1Þ h ¼ p2 ¼ 0 and wð2Þ h ¼ p2 ¼ 0) can be expressed the following forms: k1 ¼ 0; c11 lk1 c21 l k
ð24Þ k
c12 ðl2 Þ c22 ð l2 Þ ¼ 0:
ð25Þ
Eqs. (12), (13), (24) and (25) lead to the characteristic matrix. The eigenequation can be simplified to the following form tan½kðH2 pÞ cotðkH1 Þ ¼ 1=x:
ð26Þ
Consider the case that the materials of the two wedges are the same, x = 1. The eigenequation of Eq. (26) is the same with Eq. (18) by taking x = 1. This solution is given by Eq. (23). 3.1.3. Clamped–free boundary conditions Consider the case that the boundary surfaces at h = p/ 2 and h = p/2 are clamped and free respectively. The characteristic matrix is given by combining Eqs. (12), (13), (16) and (24) and the eigenequation yields tan½kðH2 pÞ tanðkH1 Þ ¼ x:
ð27Þ
If the boundary conditions are reversed (i.e. free– clamped), Eqs. (12), (13), (15), and (25) are used to form the characteristic matrix. The eigenequation becomes tan½kðH2 pÞ tanðkH1 Þ ¼ 1=x:
ð28Þ
If the same materials are used (x = 1) with different fiber orientations, both Eqs. (27) and (28) are reduced to cos½ðH2 H1 pÞk ¼ 0:
ð29Þ
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Fig. 3. The variations of stress singularity k 1 with H2 H1 or H1(x = 1). (a) 90°–90° wedges with free–free or clamped–clamped, (b) 90°–90° wedges with clamped–free or free–clamped, (c) 90°–180° wedges with free–free or clamped–clamped, (d) 90°–180° wedges with clamped–free or free– clamped, (e) 90°–270° wedges with free–free or clamped–clamped, (f) 90°–270° wedges with clamped–free or free–clamped B.C.
The solution of Eq. (29) is k¼
ð2n þ 1Þp 2ðH2 H1 pÞ
ðn ¼ integerÞ:
ð30Þ
Fig. 3(b) plots the variations of (k 1) with H2 H1. The dotted line represents the second root of (k 1). No singular stress field is found when p/2 < H2 H1 < p. If the two wedges have the same orientations of principal axes (i.e. H2 H1 = 0), the structure reduces to one material wedge with 180° angle and free–clamped boundary conditions. The stress singularity order k 1
is equal to 0.5 and independent of mk. The strongest singularity order is k 1 = 0.75 as H2 H1 = p. There are two roots for p < H2 H1 < p/2. It is worth considering the case that the problem is reduced to dissimilar isotropic bonded wedges. For isotropic materials, lk = i and H1 = H2 = p/2, the eigenequation obtained from Eq. (27) can be simplified as following cosðkpÞ ¼
1x ; 1þx
ð31Þ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
which is exactly the same as Eq. (28) in Ref. [4] for both of the wedge angles equal to p/2. Comparing Eqs. (27) and (28), the singularity orders are the same for free–clamped and clamped–free boundary conditions if the materials of two isotropic wedges are interchanged. This conclusion is in agreement with Chiu and Chaudhuri [6].
Consider the case of an anisotropic 90°–180° wedge perfectly bondedalong the x-axis (Table 3). From Eq. (4), there are f1 h ¼ p2 ¼ l1 and f2(h = p) = epi on the boundaries. Eqs. (12) and (13) are the continuity conditions of the stresses and displacements along the interface. 3.2.1. Free–free boundary conditions ð2Þ The condition of stress syz ðh ¼ pÞ ¼ 0 at free surface can be reduced to the following equation c12 e
ðk1Þpi
þ c22 e
ðk1Þpi
Eq. (36) with x ¼ ^sð2Þ =^sð1Þ . After mathematical operations, Eq. (36) can be reduced to Eq. (23) in Ref. [4] for a = p/2 and b = p. 3.2.2. Clamped–clamped boundary conditions The boundary condition at the clamped surface is w(2)(h = p) = 0. Using Eq. (9), the result is shown: c12 ekpi c22 ekpi ¼ 0:
3.2. 90°–180° wedges
¼ 0:
ð32Þ
The determinant of the matrix that governs the eigenvalues is obtained by combining Eqs. (12), (13), (15) and (32). The result can be simplified to the following eigenequation tanðkpÞ cotðkH1 Þ ¼ x:
ð33Þ ð1Þ
ð1Þ
Notethat the eigenequation depends only on ^s44 , ^s55 , ð2Þ ð2Þ g1 and ^s44 ^s55 . The directions of the principal axes of region X2 will not affect the stress singularity. This conclusion is also true for the case that a wedge with arbitrary angle is bonded to a half surface subjected to arbitrary boundary conditions. Again, it can be proved that the stress singularity orders remain the same when the direction of the principal axis g1 is changed to 90° g1. As the material properties of two regions are identical (x = 1), the eigenequation becomes sin½ðH1 þ pÞk ¼ 0:
ð34Þ
The solution k is np ðn ¼ integerÞ: k¼ H1 þ p
ð35Þ
Fig. 3(c) shows the variations of stress singularity with H1. The strongest stress singularity occurs at H1 = p and k 1 = 0.5. However, weak stress singularity is found as H1 is close to 0. The value H1 can be obtained from Fig. 2 or Eq. (11). ð1Þ ð1Þ When material 1 is isotropic (i.e. ^s44 ¼ ^s55 ¼ ^sð1Þ ), the eigenequation becomes ½1 þ ð1 þ xÞq cosðkpÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi sinðkp=2Þ ¼ 0;
ð2Þ ð2Þ where x ¼ ^s44 ^s55 =^sð1Þ . If ð2Þ ð2Þ (i.e. ^s44 ¼ ^s55 ¼ ^sð2Þ ), the
ð36Þ
material 2 is also isotropic
eigenequation is equal to
437
ð37Þ
Again, the determinant of the matrix is also obtained by combining Eqs. (12), (13), (24) and (37). Thus, tanðkpÞ cotðkH1 Þ ¼ 1=x:
ð38Þ
The singularity order is independent of g2. If the materials of the two wedges are the same (x = 1), the eigenequation is equal to Eq. (34) and the solution is in Eq. (35). When material 1 becomes isotropic, the eigenequation of Eq. (38), regardless of g2, becomes ½1 þ ð1 þ 1=xÞ cosðkpÞ sinðkp=2Þ ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffi ffi
ð39Þ
ð2Þ ð2Þ where x ¼ ^s44 ^s55 =^sð1Þ . If the material 2 is ð2Þ ð2Þ pic (i.e. ^s44 ¼ ^s55 ¼ ^sð2Þ ), the eigenequation ð2Þ ð1Þ
also isotrois equal to Eq. (39) with x ¼ ^s =^s . After simplification, it is the same as Eq. (38) in Ref. [4] for a = p/2 and b = p. Comparing Eqs. (36) and (39), the singularity orders of the two cases with free–free and clamped–clamped boundary conditions are the same if the materials of two isotropic wedges are interchanged. It is coincident with Chiu and Chaudhuri [6]. 3.2.3. Clamped–free boundary conditions Consider the case that the boundary surfaces at h = p/2 and h = p are clamped and free respectively. Using Eqs. (12), (13), (24) and (32), the determinant of the matrix can be simplified to the following eigenequation tanðkpÞ tanðkH1 Þ ¼ x:
ð40Þ
It is independent of g2. Again, it is easy to prove that the stress singularity order remains the same when the direction of the principal axis g1 is changed to 90° g1. As the material properties of two regions are identical (x = 1), the eigenequation is cos½ðH1 þ pÞk ¼ 0:
ð41Þ
The roots for k are k¼
ð2n þ 1Þp 2ðH1 þ pÞ
ðn ¼ integerÞ:
ð42Þ
Fig. 3(d) plots the variations of (k 1) with H1. There are two roots as H1 > p/2. The strongest stress singularity is k 1 = 0.75 for H1 = p and the weakest singularity is k 1 = 0.5 for H1 = 0. The range of the first stress singularity order is 0.75 6 k 1 6 0.5. When the material 1 with wedge angle p/2 is isotropic, Eq. (40) can be arranged as follows
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½1 þ ð1 þ xÞ cosðkpÞ qffiffiffiffiffiffiffiffiffiffiffiffi ffi cosðkp=2Þ ¼ 0;
ð43Þ
ð2Þ ð2Þ ¼ ^s44 ^s55 =^sð1Þ . ð2Þ ð1Þ
If both materials are isotropic where x (i.e. x ¼ ^s =^s ), Eq. (43) is exactly the same as Eq. (28) in Ref. [4] for a = p/2 and b = p. 3.2.4. Free–clamped boundary conditions Eqs. (12), (13), (15) and (37) lead to the characteristic matrix for this case. The eigenequation becomes tanðkpÞ tanðkH1 Þ ¼ 1=x:
ð44Þ
It is also independent of g2. If the same materials are used (x = 1), the eigenequation is again given by Eq. (41). If material 1 is isotropic, Eq. (44) can be rewritten as follows ½1 þ ð1 þ 1=xÞ cosðkpÞ cosðkp=2Þ ¼ 0:
ð45Þ
Comparing Eqs. (43) and (45), the singularity order is the same for free–clamped and clamped–free boundary conditions if the materials of two isotropic wedges are interchanged. From the previous results of Section 3.2, certain conclusions can be made: (1) The antiplane stress singularities for 90°–180° wedges with arbitrary boundary ð1Þ ð1Þ ð2Þ ð2Þ conditions depend on ^s44 , ^s55 , g1 and ^s44 ^s55 ; (2) the stress singularity order remains unchanged when the direction of the principal axis ^x1 is changed from g1 to 90° g1; (3) as the material properties of regions 1 and 2 are identical (x = 1; g1, g2 = arbitrary), the stress singularity orders are equal for free–free and clamped– clamped boundary conditions. This is also true for clamped–free and free–clamped boundary conditions; (4) if two isotropic materials are interchanged for cases of free–free and clamped–clamped boundary conditions, the singularity orders are the same. This is also true for clamped–free and free–clamped boundary conditions. It agrees with Chiu and Chaudhuri [6]. 3.3. 180°–180° wedges Consider a semi-infinite interface crack existing in two dissimilar anisotropic materials as shown in Table 3. From Eq. (4), it can be seen that f1(h = p) = epi and f2(h = p) = epi on the boundaries. The continuity conditions of the stresses and displacements along the interface are defined in Eqs. (12) and (13).
The root of Eq. (47) is k = 0.5. It is independent of material properties. 3.3.2. Clamped–clamped boundary conditions The displacement w(1)(h = p) at clamped surface is shown: c11 ekpi c21 ekpi ¼ 0:
ð48Þ
The eigenequation, given by Eqs. (12), (13), (37) and (48), is identical to Eq. (47). 3.3.3. Clamped–free boundary conditions Consider the interface crack at h = p and h = p with clamped and free boundary conditions, respectively. Then, Eqs. (12), (13), (32) and (48) lead to the eigenequation: 1x cosð2kpÞ ¼ : ð49Þ 1þx When the material properties of regions 1 and 2 are identical (i.e. x = 1), the roots of Eq. (49) are k = 0.25 and 0.75. 3.4. 90°–270° wedges Consider an anisotropic 90°–270° wedge perfectly bonded along the line h = p/2 shown in Table 3. This wedge is called as ‘‘debonded junction’’. The continuity ð1Þ ð2Þ conditions of the stresses (i.e. sxz ðh ¼ p2Þ ¼ sxz ðh ¼ p2Þ) ð1Þ p ð2Þ p and displacements (i.e. w ðh ¼ 2Þ ¼ w ðh ¼ 2Þ) along the interface are written as followings: k1 c12 lk21 c22 l k2 ¼ 0; c11 lk1 þ c21 l
ð50Þ
k1 xc12 lk2 þ xc22 l k2 ¼ 0: c11 lk1 c21 l
ð51Þ
3.4.1. Free–free boundary conditions The stresses at h = p and h = p are traction free. Connecting with Eqs. (32), (46), (50) and (51), the eigenequation for this problem is as follows sin½ðH2 H1 þ 2pÞk x 1 : ¼ sin½ðH2 þ H1 Þk xþ1
ð52Þ
3.3.1. Free–free boundary conditions ð1Þ The stress syz ðh ¼ pÞ ¼ 0 at free surface is given by
The eigenvalues depend on H1, H2 and x. Following the same procedure as in Section 3.1.1, the principal arguments Hk will be unvaried when the direction of the principal axis ^xk is changed from gk to 90° gk. Therefore, the singularity orders are the same when the direction of the ^xk -axis is changed from (g1, g2) to (90° g1, 90° g2), (g1, 90° g2), or (90° g1, g2). If the same materials are used (x = 1), Eq. (52) becomes
c11 eðk1Þpi þ c21 eðk1Þpi ¼ 0:
sin½ðH2 H1 þ 2pÞk ¼ 0:
ð46Þ
The eigenequation can be determined by Eqs. (12), (13), (32) and (46), therefore sinð2kpÞ ¼ 0:
ð47Þ
ð53Þ
Solving above equation, the root is k¼
np ðH2 H1 þ 2pÞ
ðn ¼ integerÞ:
ð54Þ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
½x 1 2ðx þ 1Þ cosðkpÞ sinðkpÞ ¼ 0:
ð55Þ
-0.4
-0.3333 -0.3
λ−1
Fig. 3(e) plots the variations of (k 1) with H2 H1. The strongest stress singularity is k 1 = 2/3 for H2 H1 = p. If the fiber directions of two wedges are the same (H2 H1 = 0), it is reduced to the case that a crack exists in an anisotropic body and k = 0.5. When material 1 and 2 are two different isotropic materials (x 5 1), the eigenequation of Eq. (52) is rewritten as follows
439
N.S.
-0.2
It is the same with Eq. (23) in Ref. [4] for a = p/2 and b = 3p/2. -0.1
-0.4
mk
0
100 10 1.1 0.1 0.01
-0.3
0
30
60
90
120
150
180
Θ1 (Degree)
Re[λ-1]
Fig. 5. The variations of stress singularity (k 1) with H1 (x = 1 and g1 = g2).
3.4.2. Clamped–clamped boundary conditions The boundaries at h = p and h = p are clamped. Using Eqs. (37), (48), (50) and (51), the following eigenequation can be obtained
-0.2
sin½ðH2 H1 þ 2pÞk 1=x 1 : ¼ sin½ðH2 þ H1 Þk 1=x þ 1
-0.1
ð56Þ
Eq. (53) is valid for this case while x = 1. For dissimilar isotropic materials, the eigenequation is as follows 0 -90
-60
-30
0
30
60
½1=x 1 2ð1=x þ 1Þ cosðkpÞ sinðkpÞ ¼ 0:
90
η1 (Degree)
(a)
It is the same with Eq. (38) in Ref. [4] for a = p/2 and b = 3p/2. Comparing Eqs. (55) and (57), the singularity orders are the same for free–free and clamped–clamped boundary surfaces if the two isotropic materials are interchanged. This conclusion is identical to Ref. [6].
mk
0.6
100 10 1.1 0.1 0.01
0.5
0.3
3.4.3. Clamped–free boundary conditions The boundaries at h = p and h = p are clamped and traction free, respectively. By rearranging Eqs. (32), (48), (50) and (51), the eigenequation can be expressed in the following form:
0.2
cos½ðH2 H1 þ 2pÞk 1 x : ¼ cos½ðH2 þ H1 Þk 1þx
0.4
Im[λ-1]
ð57Þ
Again, it can be proved that the stress singularity orders remain the same when the direction of the principal axis gk is changed to 90° gk. If x = 1, the equation becomes
0.1
cos½ðH2 H1 þ 2pÞk ¼ 0:
0 -90
(b)
ð58Þ
-60
-30
0
30
60
90
η1 (Degree)
Fig. 4. The variations of stress singularity order (k 1) with g1 for g2 = 0° and mk = 100, 10, 1.1, 0.1 and 0.01. (a) Re[k 1], (b) Im[k 1].
ð59Þ
The root of this equation is k¼
ð2n þ 1Þp 2ðH2 H1 þ 2pÞ
ðn ¼ integerÞ:
ð60Þ
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Table 2 The eigenequations of antiplane anisotropic wedges and junction Free–free
Clamped–clamped
Clamped–free
Free–clamped
90°–90° Wedge
tan½kðH2 pÞ cotðkH1 Þ ¼x
tan½kðH2 pÞ cotðkH1 Þ ¼ 1=x
tan½kðH2 pÞ tanðkH1 Þ ¼ x
tan½kðH2 pÞ tanðkH1 Þ ¼ 1=x
90°–180° Wedge
tanðkpÞ cotðkH1 Þ ¼ x
tanðkpÞ cotðkH1 Þ ¼ 1=x
tanðkpÞ tanðkH1 Þ ¼ x
tanðkpÞ tanðkH1 Þ ¼ 1=x
cosð2kpÞ ¼ 1x 1þx
cosð2kpÞ ¼ x1 xþ1
180°–180° Wedge 90°–270° Wedge 90°–270° Junction
sinð2kpÞ ¼ 0; k 1 ¼ 0:5
sin½ðH2 H1 þ 2pÞk x 1 sin½ðH2 H1 þ 2pÞk 1 x cos½ðH2 H1 þ 2pÞk 1 x cos½ðH2 H1 þ 2pÞk x 1 ¼ ¼ ¼ ¼ sin½ðH2 þ H1 Þk xþ1 sin½ðH2 þ H1 Þk 1þx cos½ðH2 þ H1 Þk 1þx cos½ðH2 þ H1 Þk xþ1
2 k k R2 2x cos½ðH2 H1 þ 2pÞk 1x cos½ðH1 þ H2 Þk ¼ ð1þxÞ þ RR21 2 R1 1þx
Fig. 3(f) plots the variations of k 1 with H2 H1. There are three roots for H2 H1 > p/2. The strongest stress singularity is k 1 = 5/6 for H2 H1 = p and the weakest singularity of the first root is k 1 = 0.5 for H2 H1 = p. If the fiber orientations of the two wedges are the same (H2 = H1), it is reduced to the case that a crack exists in an anisotropic body with clamped– free boundary surfaces and k 1 = 0.75 and 0.25. For two different isotropic materials, Eq. (58) becomes ð1 xÞ cosðkpÞ ð1 þ xÞ cosð2kpÞ ¼ 0:
ð61Þ
It is the same with Eq. (28) in Ref. [4] for a = p/2 and b = 3p/2. 3.4.4. Free–clamped boundary conditions The boundaries at h = p and h = p are traction free and clamped, respectively. Using Eqs. (37), (46), (50) and (51), the eigenequation can be simplified as cos½ðH2 H1 þ 2pÞk 1 1=x : ¼ cos½ðH2 þ H1 Þk 1 þ 1=x
ð62Þ
Eq. (59) is also valid in this case for x = 1. As the materials are isotropic, the eigenequation is rewritten as follows ð1 1=xÞ cosðkpÞ ð1 þ 1=xÞ cosð2kpÞ ¼ 0:
ð63Þ
Eqs. (61) and (63) are the same if the materials of two isotropic wedges are interchanged. Important conclusions drawn from Section 3.4 can be listed as follows: (1) The antiplane stress singularities for 90°–270° wedges with arbitrary boundary conditions remain unchanged when the directions of the ^xk -axis are changed from (g1, g2) to (90° g1, 90° g2), (g1, 90° g2) or (90° g1, g2). (2) As the material properties of regions 1 and 2 are identical (x = 1; g1, g2 = arbitrary), the stress singularity orders are equal for free– free and clamped–clamped boundary conditions. This is also true for clamped–free and free–clamped boundary conditions. (3) If two isotropic materials are interchanged for cases of free–free and clamped–clamped boundary conditions, the singularity orders are the
same. This is also true for clamped–free and free– clamped boundary conditions. This conclusion is in agreement with Chiu and Chaudhuri [6]. 3.5. 90°–270° junction Consider an anisotropic 90°–270° junction (Fig. 1(b)) perfectly bonded along the line h = p/2 and h = p. The regions of wedges with angle 90° and 270° are denoted as X1 and X2, respectively. The continuity conditions ð1Þ ð2Þ of the stresses (i.e. syz ðh ¼ pÞ ¼ syz ðh ¼ pÞ) and dis(1) (2) placements (i.e. w (h = p) = w (h = p)) along the interface h = p are in the forms given below c11 eðk1Þpi þ c21 eðk1Þpi c12 eðk1Þpi c22 eðk1Þpi ¼ 0; ð64Þ c11 ekpi c21 ekpi xc12 ekpi þ xc22 ekpi ¼ 0:
ð65Þ
Eqs. (50), (51), (64) and (65) form the eigenequation for this problem. Using the definition in Eqs. (11) and (12), the eigenequation leads to 2 1x cos½ðH2 H1 þ 2pÞk cos½ðH1 þ H2 Þk 1þx # " k k 2x R2 R2 þ ¼ : ð66Þ 2 R1 R1 ð1 þ xÞ The eigenvalues depend on Hk, Rk and x. Here, k may be a complex number. If the fiber directions of Eq. (66) are changed from (g1, g2) to (90° g1, 90° g2) simultaneously, the result shows that Eq. (66) is unvaried if the relations of Eqs. (20) and (21) are used. If x = 1, Eq. (66) can be rewritten as follows k iðH þ2pÞk R2 e 2 Rk1 eiH1 k Rk1 eiðH1 2pÞk Rk2 eiH2 k ¼ 0: ð67Þ In view of Eq. (67), the stress singularity k should be a complex number. The solution of Eq. (67) becomes h i 2np ðH2 H1 þ 2pÞ i ln RR21 k¼ ðn ¼ integerÞ: ð68Þ 2 2 R2 ln R1 þ ðH2 H1 þ 2pÞ
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
441
Table 3 The eigenequations of antiplane anisotropic wedges and junction as x = 1 Free–free or clamped–clamped
Clamped–free or free–clamped
90°–90° Wedge
np (Fig. 3(a)) k¼ H2 H1 p
k¼
ð2n þ 1Þp (Fig. 3(b)) 2ðH2 H1 pÞ
90°–180° Wedge
k¼
k¼
ð2n þ 1Þp (Fig. 3(d)) 2ðH1 þ pÞ
180°–180° Wedge
sinð2kpÞ ¼ 0, k1 = 0.5
90°–270° Wedge
k¼
90°–270° Junction
np (Fig. 3(c)) H1 þ p
np (Fig. 3(e)) ðH2 H1 þ 2pÞ
cosð2kpÞ ¼ 0, k1 = 0.75, 0.25
k¼
ð2n þ 1Þp (Fig. 3(f)) 2ðH2 H1 þ 2pÞ
h i 2np ðH2 H1 þ 2pÞ i ln RR21 k¼ 2 ln RR21 þ ðH2 H1 þ 2pÞ2
442
C.-I. Liu, C.-H. Chue / Composite Structures 73 (2006) 432–442
Fig. 4(a) and (b) show the variations of stress singularity order (k 1) with g1 for g2 = 0° and mk = 100, 10, 1.1, 0.1 and 0.01. Consider the special case of g1 = g2, that the relations of R2 = R1 and H2 = p H1 are obtained by the definition in Eqs. (10) and (11). Therefore, the stress singularities of Eq. (68) are real and double roots and given by 2np k¼ ðn ¼ integerÞ: ð69Þ 3p 2H1
singular stress fields can be used to improve the safety of structures.
Fig. 5 plots the variations of (k 1) with H1. No singular stress field is found when p/2 6 H1 6 p. The strongest stress singularity is k 1 = 1/3 for H1 = 0 (i.e. H2 = p). Dempsey and Sinclair [13] found that the existence conditions of logarithmic type singularity are d4m Rank[D(k)] = m < 4 and dk 4m det½DðkÞ ¼ 0 where D[k] is the eigenmatrix. It means that, for double roots, the singularity is logarithmic type as m = 3. In this case, however, the roots k = 2p/(3p 2H1) are double roots dD(k)/dk = 0 but the rank is 2. Therefore, the stress singularity is still rk 1-type and not logarithmic type.
References
4. Conclusions This paper has derived the general closed-form solution of eigenequation for wedges and junctions bonded at special angles under antiplane shear. The repeated occurrence of stress singularity orders is examined analytically. Taking a typical case of a 90°–90° wedge with various boundary conditions (i.e. free–free, clamped– clamped or clamped–free) as an example, the singularity orders are unchanged while the fiber orientations gk (k = 1 or 2) are changed to 90° gk. The eigenequations for dissimilar material wedges are also obtained and compared with existing literature in some special cases. All of these results are given in Tables 2 and 3. The disappearance conditions of the stress singularity can be found by plotting the variations of (k 1) with the fiber orientations. The theoretical conclusion for vanishing
Acknowledgement The authors are grateful to the National Science Council of the ROC for financial support under the contract NSC89-2212-E006-119.
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