Stresses around the bond edges of axisymmetric deformation joints

Engineering Fracture Mechanics 66 (2000) 153±170

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Stresses around the bond edges of axisymmetric deformation joints Y.L. Li a,*, 1, S.Y. Hu a, 1, Y.Y. Yang b a

Department of Applied Physics & Applied Mechanics, Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minatoku, Tokyo 106, Japan b Forschungszentrum Karlsruhe, IMF II, D-76021 Karlsruhe, Germany Received 6 April 1999; received in revised form 20 January 2000; accepted 24 January 2000

Abstract Stresses in a two-dissimilar-materials joint under an axisymmetric deformation are analysed. The similarity and the di€erence between the stresses in an axisymmetric deformation joint and in a plane strain deformation joint are pointed out. The similarity is that the stresses near the bond edges of the joints have the same singularity. The di€erence is that the stresses in an axisymmetric deformation joint cannot be determined by only two composite material parameters, which is a correct conclusion in a plane strain deformation joint under prescribed traction. Therefore, asymptotic descriptions for the stress ®elds near the bond edges of the joints must be di€erent under axisymmetric deformation and under plane strain deformation, even though the stress singularities are same. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Axisymmetric deformation; Two-dissimilar-materials joint; Stress singularity; Asymptotic analyses

1. Introduction By virtue of linear elasticity, stresses may be singular at the bond edges of dissimilar material joints subjected to loading due to the discontinuity of geometry and/or materials. The high stresses near bond edges are the main factor for causing failure of joint materials. Thus, a lot of work on the stress singularity has been done for joints, especially under 2D deformation assumptions owing to the simplicity of mathematics, for examples, see [1±12]. In practice, axisymmetric joints have wide applications as structural components and as specimens in experiments. In contrast to the 2D study, * Corresponding author. Fax: +81-3-3402-6375. E-mail address: [email protected] (Y.L. Li). 1 On leave from Lanzhou University, People's Republic of China. 0013-7944/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 0 ) 0 0 0 1 3 - 8

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Y.L. Li et al. / Engineering Fracture Mechanics 66 (2000) 153±170

there are relatively few papers concerning the stress analyses in axisymmetric joints. This may be due to the impression that the stress singularities at the bond edges of axisymmetric deformation joints are the same as those in the plane strain deformation joints [13±15]. However, the stress states in axisymmetric deformation bodies are di€erent from those in plane strain deformation bodies. One obvious di€erence is that stresses in a two-dissimilar-materials joint under plane strain deformations subjected to prescribed surface traction can be determined by two composite elastic constants, for example, the Dundurs constants P a and b [16,17]. Thereby the stresses near the bond edge can be described by Krÿo zij …j, o† or m Km rÿom zij …j, om † where o or om …o, om < 1† and zij …j, o† or zij …j, om † depend on only a and b: K or Km is an unknown constant and depends on applied loading condition besides a and b: …r, f† is the polar coordinate system originated at one point of the bond edge. But the stresses in an axisymmtric deformation joint cannot be determined by only two material constants, even under prescribed surface traction. They depend on three composite material constants like in arbitrary 3D deformation joints. Moreover, although the singular part of the stress ®eld near the bond edge of the axisymmetric deformation joint has the form of rÿo zij …j, o†, it cannot correctly describe the real stress ®eld in the vicinity of the singular point, especially when the absolute value of o is small. This can be observed from the examples presented in Section 5. Therefore, the stress terms, which may have signi®cant contributions to the stress ®eld but do not appear in plane strain deformation joints, must be found and considered. In the present work, a general solution is presented for the axisymmetric joint (shown in Fig. 2(a)) subjected to an axisymmetric deformation. The joint consists of two dissimilar elastic materials with arbitrary wedge angles j1 and and j2 : The degenerate case of j1 ˆ j2 ˆ 908 was studied in our previous paper by a method di€erent from here [18]. The general solution here is based on the Boussinesq's solution in two harmonic functions [19] and is expressed in the form of a power series in r: Thus, it can be easily applied to obtain the stress singular exponent o and the related angular distribution functions. From this solution, the similarity and the di€erence between axisymmetric deformations and plane-strain deformations can be seen clearly. For engineering application, an asymptotic description is developed for the stress ®eld around the bond edge of the axisymmetric deformation joint. Concrete examples are calculated to demonstrate the validity of the asymptotic description.

2. General solution Let fur , uz g and fsrr , syy , szz , srz g denote the non-zero displacements and stresses, respectively, in the cylindrical coordinate system …r, y, z† de®ned in Fig. 1(b). The equation of equilibrium without body force can be expressed as @ srr @srz srr ÿ syy ‡ ‡ ˆ 0, @r @z r

@ srz @szz srz ‡ ‡ ˆ 0: @r @z r

…1†

The relations between stresses and displacements are     n @ ur n ur , e‡ e‡ , syy ˆ 2m srr ˆ 2m @r r 1 ÿ 2n 1 ÿ 2n  srz ˆ m

 @ ur @ uz ‡ , @z @r

 szz ˆ 2m

 n @uz e‡ , 1 ÿ 2n @z

…2†

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155

E where e ˆ @@urr ‡ urr ‡ @@uzz , m ˆ 2…1‡n† is the shear modulus, E is the Young's modulus and n is the Poisson's ratio. Based on the solution of Boussinesq form, two harmonic functions F…r, z†, C…r, z† are introduced [19], which satisfy

r2 F ˆ

@ 2F 1 @ F @ 2F ‡ 2 ˆ 0, ‡ @r2 r @r @z

…3†

r2 C ˆ

@ 2 C 1 @C @ 2 C ‡ ‡ ˆ 0: @r2 r @r @ z2

…4†

The displacements and stresses are related to F and C as follows:     1 @F @C 1 @F @C ÿ ÿz , uz ˆ ÿ ÿz ‡ …3 ÿ 4n †C , ur ˆ 2m @r @r 2m @z @z srr ˆ ÿ

@ 2F @ 2C @C 1 @F z @ C @C , syy ˆ ÿ ÿ ‡ 2n , ÿ z 2 ‡ 2n 2 @r @r @z r @r r @r @z

srz ˆ ÿ

@ 2F @ 2C @C ÿz ‡ …1 ÿ 2n † , @r@z @ r@z @r

szz ˆ ÿ

@ 2F @ 2C @C : ÿ z 2 ‡ 2… 1 ÿ n † 2 @z @z @z

…5†

…6†

In Fig. 1(b) point S …S…r, z† ˆ …R, 0†, R is the radius of the circular interface) is such a special point where stresses may be singular for most joints, so S is used to denote the singular point. Using the singular point S as the origin, we de®ne a new coordinate system which satis®es x ˆ R ÿ r ˆ rcos j,

z ˆ rsin j:

…7†

In order to ®nd the explicit form solutions of F and C which are valid for small r, we rewrite Eqs. (3) and (4) as

Fig. 1. Geometry of an axisymmetric joint and coordinate systems.

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Y.L. Li et al. / Engineering Fracture Mechanics 66 (2000) 153±170 1  n @ 2F @ 2F 1 @F 1 @F X x ˆ ‡ 2 ˆ , 2 @x @z R ÿ x @x R @x nˆ0 R

…8†

1  n @ 2C @ 2C 1 @C 1 @C X x ˆ ‡ ˆ : @x2 @z2 R ÿ x @x R @ x nˆ0 R

…9†

The solutions of Eqs. (8) and (9) can be expressed as series [20] F ˆ F …0 † ‡

1 X 1 …1 † 1 1 …k † F ‡ 2 F…2 † ‡


ˆ F , R R Rk kˆ0

…10†

C ˆ C …0 † ‡

1 X 1 …1 † 1 1 …k † C ‡ 2 C …2 † ‡


ˆ C : k R R R kˆ0

…11†

Suppose that at point S stresses exhibit a power type of singularity, then ÿo‡kÿm‡2 k X A…k † x m …x ‡ iz†ÿo‡kÿm‡2 ‡C …k † x m …x ÿ iz †

F…k † ˆ 2m

m

mˆ0

C…k † ˆ 2m

k h X mˆ0

m

… ÿ o ‡ k ÿ m ‡ 2†

B …mk † x m …x ‡ iz†ÿo‡kÿm‡1 ‡D…mk † x m …x ÿ iz †

,

ÿo‡kÿm‡1

…12† i

,

…13†

…k† …k† …k† , B …k† in which o is an eigenvalue, Am m , C m and Dm are unknown complex constants and i is the 2 imaginary symbol of i ˆ ÿ1: By substituting Eq. (12) in Eq. (8) and equating the coecients of x m …x ‡ iz†ÿo‡kÿm‡2 , we can obtain

A…kk † ˆ

2k ÿ 1 …kÿ1 † Akÿ1 , …kr1 †, 2k

A…mk † ˆ

i h 2m ÿ 1 …kÿ1 † 1 k† Amÿ1 ‡ , …kr2, m ˆ k ÿ 1, mA…mkÿ1 † ÿ …m ‡ 1 †A…m‡1 2m 2… ÿ o ‡ k ÿ m ‡ 1 †

…14†

k ÿ 2, . . . ,1†: ÿo‡kÿm‡2 m , and also for The same relations are valid for C …k† m by comparing the coecients of x …x ÿ iz† …k† …k† …k† …k† …k† …k† B m and Dm : Hence, the only unknowns are A0 , B 0 , C 0 and D0 …k ˆ 0, 1, 2, . . .† which will be determined from the stress-free conditions at the sides of the joint and the interfacial continuity conditions. For the axisymmetric joint shown in Fig. 1(a), these conditions can be described in the coordinates …r, j†

srr1 …r, j1 †sin j1 ‡ srz1 …r, j1 †cos j1 ˆ 0, srz1 …r, j1 †sin j1 ‡ szz1 …r, j1 †cos j1 ˆ 0, srr2 …r, ÿ j2 †sin… ÿ j2 † ‡ srz2 …r, ÿ j2 †cos… ÿ j2 † ˆ 0,

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157

srz2 …r, ÿ j2 †sin… ÿ j2 † ‡ szz2 …r, ÿ j2 †cos… ÿ j2 † ˆ 0, szz1 …r, 0† ˆ szz2 …r, 0†, srz1 …r, 0† ˆ srz2 …r, 0†, ur1 …r, 0† ˆ ur2 …r, 0†, uz1 …r, 0† ˆ uz2 …r, 0†,

…15†

where the subscripts 1 and 2 refer to the materials 1 and 2 in the joint, respectively.

3. Stresses and stress singularity For applying the boundary conditions easily, we express the stresses and displacements in the polar coordinates. With coordinate transform Eq. (7), Eqs. (12) and (13) can be rewritten in the polar coordinates r and j as F…k † ˆ 2mrÿo‡k‡2 ‡ 2 †j

i

C…k † ˆ 2mrÿo‡k‡1

h cosm j a…k † cos… ÿ o ‡ k ÿ m ‡ 2 †j ‡ c…mk † sin… ÿ o ‡ k ÿ m ÿo ‡ k ÿ m ‡ 2 m mˆ0 k X

…12† 0 k X mˆ0

h i cosm j b…mk † cos… ÿ o ‡ k ÿ m ‡ 1 †j ‡ d …mk † sin… ÿ o ‡ k ÿ m ‡ 1 †j ,

…13† 0

…k† …k† …k† …k† …k† …k† …k† …k† …k† where the new constants are de®ned by a…k† m ˆ Am ‡ C m , cm ˆ i…Am ÿ C m †, bm ˆ B m ‡ Dm , d m ˆ …k† …k† i…B m ÿ Dm †: Substituting the solutions (10) and (11) into Eqs. (5) and (6), the corresponding displacements and stresses can be obtained. They can be translated in the polar coordinates r and j as    ur ˆ rÿo‡1 a…0 † cos… ÿ o ‡ 1 †j ‡ c…0 † sin… ÿ o ‡ 1 †j ‡ … ÿ o ‡ 1 †sin j b…0 † cos oj

ÿ d …0 † sin oj



‡

  rÿo‡2  …1 † a cos… ÿ o ‡ 2 †j ‡ c…1 † sin… ÿ o ‡ 2 †j ‡ … ÿ o R

  ‡ 2 †sin j b…1 † cos… ÿ o ‡ 1 †j ‡ d …1 † sin… ÿ o ‡ 1 †j ‡

 …† 1 a 0 cos… ÿ o ‡ 2 †j 2… ÿ o ‡ 2 †

 cos j  … †  sin j  … † a 0 cos… ÿ o ‡ 1 †j ‡ c…0 † sin… ÿ o ‡ 1 †j ‡ b 0 cos… ‡ c…0 † sin… ÿ o ‡ 2 †j ‡ 2 2   1  ÿ o ‡ 1 †j ‡ d …0 † sin… ÿ o ‡ 1 †j ‡ … ÿ o ‡ 1 †sin jcos j b…0 † cos oj ÿ d …0 † sin oj 2 ‡

rÿo‡3 f





g ‡





, R2

…16†

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Y.L. Li et al. / Engineering Fracture Mechanics 66 (2000) 153±170

  uz ˆ rÿo‡1 a…0 † sin… ÿ o ‡ 1 †j ÿ c…0 † cos… ÿ o ‡ 1 †j ÿ … ÿ o ‡ 1 †sin j b…0 † sin oj 

  rÿo‡2  … †  …† …0 † 0 a 1 sin… ‡ d cos oj ‡ …3 ÿ 4n † b cos… ÿ o ‡ 1 †j ‡ d sin… ÿ o ‡ 1 †j ‡ R   ÿ o ‡ 2 †j ÿ c…1 † cos… ÿ o ‡ 2 †j ‡ … ÿ o ‡ 2 †sin j b…1 † sin… ÿ o ‡ 1 †j ÿ d …1 † cos… ÿ o …0 †



 cos j  … †   a 0 sin… ÿ o ‡ 1 †j ‡ 1 †j ‡ …3 ÿ 4n † b…1 † cos… ÿ o ‡ 2 †j ‡ d …1 † sin… ÿ o ‡ 2 †j ‡ 2   …3 ÿ 4n †  1 cos j b…0 † cos… ÿ o ‡ 1 †j ‡ d …0 † sin… ÿ o ‡ 1 †j ÿ … ÿ o ÿ c…0 † cos… ÿ o ‡ 1 †j ‡ 2 2    rÿo‡3 ‡ 1 †sin jcos j b…0 † sin oj ‡ d …0 † cos oj ‡ f





g ‡





, R2

…17†

   srr ˆ rÿo … ÿ o ‡ 1 † ÿ a…0 † cos oj ‡ c…0 † sin oj ‡ 2n b…0 † sin oj ‡ d …0 † cos oj 2m   rÿo‡1 ÿ  … † …0 † 0 … ÿ o ‡ 2 † a…1 † cos… ÿ o ‡ 1 †j … † … † ‡ osin j b cos o ‡ 1 j ÿ d sin o ‡ 1 j ÿ R     ‡ c…1 † sin… ÿ o ‡ 1 †j ‡ … ÿ o ‡ 1 †sin j b…1 † cos oj ÿ d …1 † sin oj ‡ 2n b…1 † sin… ÿ o ‡ 1 †j   1 ‡ a…0 † cos… ÿ o ‡ 1 †j ‡ c…0 † sin… ÿ o ‡ 1 †j ‡ … ÿ o 2     … † ‡ 1 †cos j a 0 cos oj ÿ c…0 † sin oj ‡ … ÿ o ‡ 1 †sin j b…0 † cos oj ÿ d …0 † sin oj ÿ n… ÿ o ÿ d …1 † cos… ÿ o ‡ 1 †j




  1  ‡ 1 †cos j b…0 † sin oj ‡ d …0 † cos oj ÿ o… ÿ o ‡ 1 †cos jsin j b…0 † cos…o ‡ 1 †j 2   rÿo‡2 ÿ d …0 † sin…o ‡ 1 †j ‡ f





g ‡





, R2

…18†

 rÿo‡1 n   syy ˆ rÿo 2n… ÿ o ‡ 1 † b…0 † sin oj ‡ d …0 † cos oj ‡ 2n… ÿ o ‡ 2 † ÿ b…1 † sin… ÿ o ‡ 1 †j 2m R    ‡ d …1 † cos… ÿ o ‡ 1 †j ‡ a…0 † cos… ÿ o ‡ 1 †j ‡ c…0 † sin… ÿ o ‡ 1 †j ‡ n… ÿ o   o  ‡ 1 †cos j b…0 † sin oj ‡ d …0 † cos oj ‡ … ÿ o ‡ 1 †sin j b…0 † cos oj ÿ d …0 † sin oj ‡

rÿo‡2 f





g ‡





, R2

…19†

Y.L. Li et al. / Engineering Fracture Mechanics 66 (2000) 153±170

szz 2m

159

   ˆ rÿo … ÿ o ‡ 1 † a…0 † cos oj ÿ c…0 † sin oj ÿ osin j b…0 † cos…o ‡ 1 †j ÿ d …0 † sin…o ‡ 1 †j  ÿ  rÿo‡1  … ÿ o ‡ 2 † a…1 † cos… ÿ o ‡ 1 †j ‡ c…1 † sin… ‡ 2…1 ÿ n † b…0 † sin oj ‡ d …0 † cos oj ‡ R     …† ÿ o ‡ 1 †j ‡ … ÿ o ‡ 1 †sin j b 1 cos oj ÿ d …1 † sin oj ‡ 2…1 ÿ n † ÿ b…1 † sin… ÿ o ‡ 1 †j 


  1 ‡ … ÿ o ‡ 1 †cos j a…0 † cos oj ÿ c…0 † sin oj ‡ …1 ÿ n †… ÿ o 2  … †   o ‡ 1 †cos j b 0 sin oj ‡ d …0 † cos oj ÿ … ÿ o ‡ 1 †cos jsin j b…0 † cos…o ‡ 1 †j 2   rÿo‡2 ÿ d …0 † sin…o ‡ 1 †j ‡ f





g ‡





, R2

‡ d …1 † cos… ÿ o ‡ 1 †j

…20†

   srz ˆ rÿo … ÿ o ‡ 1 † a…0 † sin oj ‡ c…0 † cos oj ÿ …1 ÿ 2n † b…0 † cos oj ÿ d …0 † sin oj 2m  ÿ  rÿo‡1  …† …0 † 0 … ÿ o ‡ 2 † ÿ a…1 † sin… ÿ o … † … † ÿ osin j b sin o ‡ 1 j ‡ d cos o ‡ 1 j ‡ R    ‡ 1 †j ‡ c…1 † cos… ÿ o ‡ 1 †j ‡ … ÿ o ‡ 1 †sin j b…1 † sin oj ‡ d …1 † cos oj ÿ …1 ÿ 2n † 
1   ÿ a…0 † sin… ÿ o ‡ 1 †j ‡ c…0 † cos… ÿ o  b…1 † cos… ÿ o ‡ 1 †j ‡ d …1 † sin… ÿ o ‡ 1 †j ‡ 2    1  1 ‡ 1 †j ‡ … ÿ o ‡ 1 †cos j a…0 † sin oj ‡ c…0 † cos oj ‡ … ÿ o ‡ 1 †sin j b…0 † sin oj 2 2    o ‡ d …0 † cos oj ÿ … ÿ o ‡ 1 †cos jsin j b…0 † sin…o ‡ 1 †j ‡ d …0 † cos…o ‡ 1 †j 2   1 ÿ 2n 1 ÿ 2n  …0 † … ÿ o ‡ 1 †cos j b…0 † cos oj ÿ b cos… ÿ o ‡ 1 †j ‡ d …0 † sin… ÿ o ‡ 1 †j ÿ 2 2  ÿo‡2  r ÿ d …0 † sin oj ‡ f





g ‡





, R2

…21†

…k† …k†  b…k†  c…k† d …k†  d …k† for the sake of simplicity. Eqs. (14), where a…k†  a…k† 0 , b 0 , c 0 and 0 are introduced 2 2 2 1 @F @ 2F (3) and (4) in their variants r @ r ˆ ÿ @ r2 ÿ @@ zF2 and 1r @@Cr ˆ ÿ @@ rC2 ÿ @@ zC2 have been used for deriving the above expressions. Applying the conditions in Eq. (15), the following equation can be obtained:

r

ÿo

h

m…ij0 †

  h i i rÿo‡1 h …1 † i …1 † rÿo‡2 …0 † …0 † 1 0 mij X ‡ wij X ‡ X ‡ f





g ‡





ˆ 0, R R2

…22†

…k† …k† …k† …k† …k† …k† …k† T 1 and 2 of the coecients are where X…k† ˆ fa…k† ,1 c,1 b,1 d ,1 a,2 c,2 b,2 d ,2 g , …k ˆ 0, 1,


†: The subscripts …1† 1 0 Š, ‰m Š, ‰ w Š,


are 8  8 order matrixes. Their related to materials 1 and 2, respectively. ‰m…0† ij ij ij components are the known functions of o, j1 , j2 , n1 , n2 and k ˆ m2 =m1 : Because of the arbitrariness of r, Eq. (22) is equivalent to the following equations,

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Y.L. Li et al. / Engineering Fracture Mechanics 66 (2000) 153±170

h i m…ij0 † X…0 † ˆ 0,

…23†

i h i h 0 m…ij1 † X…1 † ‡ 1 wij X…0 † ˆ 0,


:

…24†

Eq. (23) is a group of homogeneous linear algebraic equations. In order to obtain non-trivial solutions of Eq. (23), the determinant of the coecient matrix ‰m…0† ij Š has to be zero which leads to a characteristic equation for o i h det m…ij0 † ˆ P… ÿ o, a, b† ˆ 0,

…25†

where a and b are the Dundurs parameters with the de®nitions of  aˆ

…1 ÿ n2 † ÿ k…1 ÿ n1 † , …1 ÿ n2 † ‡ k…1 ÿ n1 †

bˆ

   1 1 ÿ n2 ÿ k ÿ n1 2 2 : …1 ÿ n2 † ‡ k…1 ÿ n1 †

Eq. (25) is a transcendental equation for o: Its roots are examined numerically to be the same as those in a joint with the same material combination and j1 , j2 under a plane strain deformation. Generally, o should be less than 1 in order to ensure displacements bounded. For a given o, the unknown vector X…0† in Eq. (23) can be determined to be proportional to a constant which depends on the boundary conditions at the joint ends. From Eq. (24), the unknown vector X…1† can be obtained uniquely for a given eigenvalue o when …1† …1† det‰m…1† ij Š6ˆ0: It is veri®ed that det‰mij Š ˆ P…ÿo ‡ 1, a, b†: Therefore, det‰mij Š6ˆ0 means that ÿo ‡ 1 is …0† …0† not an eigenvalue of ‰mij Š: If ÿo ‡ 1 is also an eigenvalue of ‰mij Š, i.e. both det‰m…0† ij Š ˆ 0 and …1† det‰m…1† exists ij Š ˆ 0 with the substitution of the same o, it is known from Eq. (24) that a solution for X if and only if [21] h i 0 Y T 1 wij X …0 † ˆ 0,

…26†

where Y ˆ fy1 y2 y3 y4 y5 y6 y7 y8 gT is a eigenvector of ‰m…1† ij Š with the de®nition of h i Y T m…ij1 † ˆ 0

…27†

If Eq. (26) holds, a solution of X …1† exists but is not unique. However, the non-unique portion of X…1† can be ignored, because that portion is represented by the eigenfunction associated with the eigenvalue …1† does not ÿo ‡ 1: If ÿo ‡ 1 is an eigenvalue of ‰m…0† ij Š but Eq. (26) does not hold, a solution for X …1† …1† exist. In this case, the solutions for F and C cannot be given by Eqs. (12) and (13) or Eqs. (12) ' and (13) ' with k ˆ 1: Instead, we use the following modi®ed solutions: …1 †

F



h i  1 @ rÿo‡3 a…01 † cos… ÿ o ‡ 3 †j ‡ c…01 † sin… ÿ o ‡ 3 †j ÿo ‡ 3 @ o h i 1 rÿo‡3 cos j a…11 † cos… ÿ o ‡ 2 †j ‡ c…11 † sin… ÿ o ‡ 2 †j ‡ ÿo ‡ 2

ˆ 2m

…28†

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161



h  i h @ rÿo‡2 b…01 † cos… ÿ o ‡ 2 †j ‡ d …01 † sin… ÿ o ‡ 2 †j ‡ rÿo‡2 cos j b…11 † cos… ÿ o @o i …1 † ‡ 1 †j ‡ d 1 sin… ÿ o ‡ 1 †j

C…1 † ˆ 2m

…29†

…1† …1† and d…1† are now assumed to depend on o: An application of the modi®ed in which a…1† 0 , c0 , b 0 0 solution can be seen in the next section. Similar discussions as above can be applied to obtain the …k† …k† …k† …k† …k† …k† …k† T coecient vector X …k† ˆ fa…k† ,1 c,1 b,1 d ,1 a,2 c,2 b,2 d ,2 g , …kr2† in the higher order terms of the series solutions. In summary, when o is an eigenvalue of det‰m…0† ij Š ˆ 0 but all of o ‡ k …k ˆ 1, 2, . . .† are not, the corresponding stresses can be expressed as #  ÿo "  2 r r …1 † r …2 † zij …j, o † ‡


, …30† zij …j, o † ‡ zij …j, o † ‡ sij …r, j † ˆ K R R R

where K has the stress dimension and is called as stress intensity factor, zij , z…k† ij …k ˆ 1, 2, . . . ; i, j ˆ r, z, y† are known functions. From Eq. (30) we know that for each o, the stress expressions include the terms ÿo‡2 …2† zij …j, o†, . . . : It has been veri®ed that o and not only rÿo zij …j, o†, but also rÿo‡1 z…1† ij …j, o†, r zij …j, o† depend on only two composite material constants, as the same as those in a plane strain deformation joint with the same material combination and j1 , j2 by the de®nition of zzz …0, o† ˆ 1 for zij …j, o†: But z…k† ij …j, o† cannot be determined by only two material constants. They depend on the third material constant, for example, k ˆ m2 =m1 besides a and b:

4. Special eigenvalue o0 ˆ 1 and its corresponding regular stress terms It is easy to examine that o ˆ o0  1 and ÿo0 ‡ 1 …ˆ 0† are always solutions of Eq. (25). That means …1† both det‰m…0† ij Š ˆ 0 and det‰mij Š ˆ 0 with the substitution of o0 ˆ 1: Furthermore, it can be examined that Eq. (26) holds only when …n1 ÿ n2 †sin j1 sin j2 sin…j1 ÿ j2 † ˆ 0: In order to obtain the stresses associated with o0 ˆ 1 for the axisymmetric joint with arbitrary material combination and wedge angles, the modi®ed solutions given in Eqs. (28) and (29) are used. If we consider the approximate eigenfunctions with only the ®rst two terms, they can be written as following, denoted with a superscript II,  h    1r …0 † …0 † …1 † …1 † ÿ ln r a ˆ 2mr a cos j ‡ c sin j ‡ cos 2j ‡ c sin 2j ‡ e0 cos 2j ‡ g0 sin 2j FII 0 0 0 0 0 2R   i   …1 † …1 † …0 † …0 † , …31† ‡ j a0 sin 2j ÿ c0 cos 2j ‡ cos j a0 cos j ‡ c0 sin j     r ÿ ln r b…01 † cos j ‡ d …01 † sin j ‡ f0 cos j ‡ h0 sin j ‡ j b…01 † sin j R  1 ÿ d …01 † cos j ‡ b…00 † cos jg , 2

…0 † CII 0 ˆ 2mf b0 ‡

…32†

where the subscript 0 is used…1 †to refer to…1the case of o0 ˆ 1: The new constants come from the de®nitions @ a…1 † @ c…1 † @b @d † e0  @ o0 , g0  @ o0 , f0  @ o0 , h0  @ o0 : Substituting Eqs. (31) and (32) into Eqs. (5) and (6), the

162

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displacements and stresses associated with o0 ˆ 1 are obtained, where ln r term appears. Applying the boundary conditions of Eq. (15) and equating the coecients of ln r term, a…01 † ˆ d …01 † ˆ 0,

…1 ÿ 2n †b…01 † ÿ c…01 † ˆ 0

are obtained. By introducing a new constant   c…00 † 1 …0 †  ‡ nÿ b ‡ …2n ÿ 1 †f0 ‡ g0 , g0 ˆ 2 2 0 the displacements and stresses are simpli®ed as      r 3 … ÿ 2 ‡ 2n †b…01 † ln rsin j ‡ b…01 † … ÿ 1 ‡ 2n †jcos j ‡ ÿ ‡ n sin j ‡ g0 sin j u0r ˆ a…00 † ‡ R 2 h i ‡ e0 cos j ‡ a…00 † cos j ‡ …1 ÿ n †b…00 † ‡ …2 ÿ 2n †f0 sin jg,

u0z

     r 1 …1 † …1 † … ÿ 2 ‡ 2n †b0 ln rcos j ‡ b0 …1 ÿ 2n †jsin j ‡ ÿ n cos j ˆ …3 ÿ ÿ ‡ R 2  h i …0 †  … † … † … † ÿ g0 cos j ‡ e0 sin j ‡ 2 ÿ 4n h0 sin j ‡ 1 ÿ n b0 ‡ 2 ÿ 2n f0 cos j , 4n †b…00 †

c…00 †

…33†

    s0rr 1 1 …0 † …1 † ˆ ÿ a0 ‡ j ‡ sin 2j b0 ÿ e0 ‡ 2nh0 , 2m R 2 s0zz 1 ˆ R 2m s0rz 1 ˆ 2m R

   1 …1 † … † j ÿ sin 2j b0 ‡ e0 ‡ 2 ÿ 2n h0 , 2 

  1 n ÿ 1 ‡ cos 2j b…01 † ‡ g0 , 2

h i s0yy 1 …0 † ˆ a0 ‡ 2n jb…01 † ‡ 2nh0 : 2m R

…34†

…0† …0† In Eq. (33), …3 ÿ 4n†b…0† 0 ÿ c0 represents a rigid translation of the joint in z-direction and …1 ÿ n†b0 ‡ …0† …2 ÿ 2n†f0 means a rigid rotation, so they do not produce any stress. However, a0 ˆ u0r jrˆ0  ur0 , which is the r-direction displacement at point S and produces stresses. With a given ur0 , the constants  b…1† 0 , e0 , g0 and h0 can be uniquely determined from the conditions in Eq. (15). They are all proportional to ur0 : Therefore, the stresses in Eq. (34) are proportional to ur0 as well. They can be expressed as

s0ij …j † ˆ E1

ur0 z …j †, R ij0

…35†

where E1 is the Young's modulus of material 1. zij0 …ij ˆ rr, yy, zz, rz† is a known function of j and depends on the material properties and the wedge angles of the joint. When …n1 ÿ n2 †sin j1 sin j2 sin…j1 ÿ j2 † ˆ 0, zij0 is a constant. It can be seen that the stresses given in Eq. (35) are independent of r: They

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163

depend on the radial direction displacement ur0 of the singular point, i.e. the bond edge point. This is di€erent from the situation in plane deformation problem where the displacements themselves at the singular point do not produce any stress. The regular stress terms of Eq. (35) cannot be obtained by simply letting o ˆ 0 from Eq. (30). Their meanings are di€erent. It has been demonstrated that the regular stress terms in Eq. (35) must be considered for developing an asymptotic description to the stress ®eld near the bond edge point [18]. In the case that j1 ˆ j2 ˆ 908, s0ij …j† are the same as those given in the paper [18] obtained by a di€erent method.

5. Asymptotic description and numerical examples From Eq. (30) we knew that the stresses associated with one eigenvalue o contain not only the term of rÿo , but also the rÿo‡1 , rÿo‡2 , . . ., terms. For a joint with given material combination and wedge angles, there are, in general, in®nitely many eigenvalues. Such kinds of in®nite series solutions are inconvenient for practical applications. As we knew well, the singular stress ®eld near the bond edge dominates failure of joint structures. So in this section we develop an asymptotic expression for describing the singular stress ®eld instead of general solutions for the whole joint. For the axisymmetric joints, which have only one non-zero eigenvalue in ÿ1 < o < 1, an asymptotic description can be given as  ÿo ur0 r …36† zij …j, o †: sij …r, j † ˆ s0ij …j † ‡ ssij …r, j † ˆ E1 zij0 …j † ‡ K R R In Eq. (36) there are two unknown parameters K and ur0 : They can be determined by ®tting Eq. (36) with the stress results calculated from a numerical method, such as Boundary Element Method (BEM) or Finite Element Method (FEM). In some cases, there are more than one non-zero eigenvalues in ÿ1 < o < 1 for a given joint. The stress asymptotic expression for such cases can be given as sij …r, j † ˆ E1

 ÿom M X ur0 r zij0 …j † ‡ Km zij …j, om †, R R mˆ1

…37†

where Km …m ˆ 1, 2, . . . ,M † has the same meaning as K. They can be determined by numerical methods. Examples for such cases in the joints under plane strain deformation can be found in [22]. As an examination of the asymptotic expression in Eq. (36), examples having only one non-zero eigenvalue in ÿ1 < o < 1 are taken into account and presented. The ®rst three examples are chosen to posses of same geometry shape and same Dundurs parameters a and b so that same o: By presenting such examples, we want to show that the stress ®elds in the joints are di€erent even when their material combinations have the same a, b and under the same traction boundary condition. Thus, the joint with j1 ˆ j2 ˆ 908 and composite material parameters a ˆ 0:6, b ˆ 0:25298 is supposed. The corresponding singular exponent is o ˆ 0:05: To keep the same a and b, the three material combinations listed in Table 1 are considered. A tensile load p is prescribed to the joint upper and lower ends and L1 ˆ L2 ˆ 2R is taken. In order to obtain the stresses and displacements, the BEM code BEASY [23] is employed to do numerical calculation. Some results are shown in Fig. 2 by the discrete symbol lines. Obviously, the stress ®elds for the three di€erent material combinations are di€erent, even in the small vicinity of the singular point. In the following we will show that the asymptotic expression (36) can describe the stress ®eld. When Eq. (36) is used, the unknown parameters ur0 and K should be determined in advance. The radial displacement ur0 of the singular point can be obtained directly from the numerical results of

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Table 1 Material property constants Combination

1 2 3

Material 1 (E1=100 GPa)

Material 2

n1

E2 (GPa)

n2

a

b

k

0.4877 0.3276 0.03526

30.5454 26.9611 24.7147

0.2623 0.1931 0.1124

0.6 0.6 0.6

0.25298 0.25298 0.25298

0.36 0.30 0.23

Composite parameters

BEM. Then, the regular stress terms s0ij can be calculated analytically by Eq. (35). From Section 4, we knew that the regular stress terms become constants and s0rr ˆ s0rz ˆ 0 when j1 ˆ j2 ˆ 908: For the three examples considered, the corresponding non-zero s0ij are listed in Table 2. The stress intensity factor K can be calculated by using the least square method to ®t the stresses in Eq. (36) with the numerical results from BEM. In the present paper, the axial stress szz along the interface is chosen to determine K, i.e. N h X

Kˆ

nˆ1

i r ÿo ÿ
n 0 … † r sBE 0 , 0 ÿ s n zz zz R ,  N  X rn ÿ2o R nˆ1

…38†

where the de®nition of zzz …0, o† ˆ 1 has been used. In Eq. (38) the values of sBE zz are from the numerical results of BEM and N is the number of the used points. The points with the coordinates …rn , 0† are 0 chosen in the range where the relation between log10 ‰sBE zz …r, 0† ÿ szz …0†Š and log10 …r=R† is shown in a straight line. From Fig. 3, it can be seen that these points are fallen in the range of 0:0001 < r=R < 0:01: By applying Eq. (38) K factors of the three examples are obtained respectively. They are also listed in Table 2. Now with the substitution of the determined s0ij and K, the stresses in the vicinity of the singular point can be calculated analytically by Eq. (36). In order to compare with the results of BEM, we presented the corresponding analytical results in the same ®gure, i.e. Fig. 2, with the solid lines. We can see from the ®gures that the stresses calculated by Eq. (36) are in a good agreement with the results from BEM when r is small. This illustrates that the asymptotic expression of Eq. (36) can describe the stress ®eld in the vicinity of the singular point. Additionally, Fig. 4 demonstrates the contributions of terms s0ij and ssij on sij : It is observed that the contribution of s0yy on syy is much larger than the contribution of s0zz on szz : The regular term s0yy has the same magnitude as ssyy : As the second group of examples, we study the material combination of steel and epoxy. Their material constants are: Steel S45C: E1 ˆ 205:8 GPa, n1 ˆ 0:3; Epoxy Sumitomo 3M1838B/A: E2 ˆ 3:18 GPa, n2 ˆ 0:37 [24]. Since the di€erence between their material properties is larger, larger singularities Table 2 Some parameters in Eq. (36) when j1 ˆ j2 ˆ 908 Combination

s0zz =p

s0yy1 =p

s0yy2 =p

K/p

1 2 3

ÿ0.3687 ÿ0.1388 0.01588

ÿ0.7660 ÿ0.4787 ÿ0.1565

ÿ0.2757 ÿ0.1436 ÿ0.0360

1.1929 0.9958 0.8545

Y.L. Li et al. / Engineering Fracture Mechanics 66 (2000) 153±170

Fig. 2. Stress distributions and comparisons between results from BEM and Eq. (36).

165

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Fig. 3. log±log plots of the axial stress szz along the interface of joints.

Fig. 4. Contributions of s0ij and ssij on sij …sij ˆ s0ij ‡ ssij † in the cylindrical joint with material combination 2.

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167

Table 3 The values of o and zij …j ˆ 308† in the joints of steel and epoxy Case

j1

j2

o

zrr

zyy

zzz

0.2125 0.1837 0.1391

1.0518 1.0939 1.0784

zrz

j ˆ 308 a b c

908 808 808

1008 1008 808

0.3571 0.3491 0.2576

ÿ0.3434 ÿ0.4816 ÿ0.6147

0.1767 0.09162 0.2382

are expected. We present such kind of examples to show that even when o is large, the role of s0ij still cannot be neglected. For the above material combination, the following three geometry shapes are taken into consideration. Case a: j1 ˆ 908, j2 ˆ 1008; Case b: j1 ˆ 808, j2 ˆ 1008; Case c: j1 ˆ 808, j2 ˆ 808: The corresponding singularities o can be obtained by solving Eq. (25). They are given in Table 3. In numerical calculation, the displacement boundary conditions in z-direction are prescribed in the upper and lower ends. They are uz ˆ 0 at z ˆ ÿL2 ˆ ÿ2R and uz ˆ up at z ˆ L1 ˆ 2R: The side of the joint is free of traction. From the BEM results, we can get ur0 directly and K by ®tting szz along the interface with Eq. (38). In Figs. 5±7 we present the stress distributions along the angle j ˆ 308 calculated with Eq. (36) for the three examples, respectively. The associated parameters can be found in Tables 3 and 4. For comparison, the results from BEM are shown as well. Same as the preceding ®gures, the symbol lines are drawn from the BEM results and the solid lines are drawn from the results of Eq. (36). The distance r is displayed in the logarithmic scale for showing the results clearly. It can be seen that both results agree to each other well when r is small. Therefore, Eq. (36) can be used to describe the stress ®eld near the bond edge if there is only one non-zero eigenvalue in ÿ1 < o < 1, no matter whether the value of o is large or small. From the given examples, it seems that the range of dominance of the asymptotic expression is not less than i0 …ˆ r0 =R† ˆ 0:01: In order to show the e€ect of s0yy on syy , the results of the singular term ssyy of syy are also drawn in the same ®gures by the dash lines. From the di€erence between the dash lines and the discrete circle lines, we can conclude that the role of s0yy in the asymptotic expression cannot be neglected even when o is large. However, the e€ects of s0ij …ij ˆ rr, zz, rz† becomes smaller when o is larger. It can be seen from Table 4. Table 4 u The corresponding K and s0ij …j ˆ 308† …p0 ˆ E0 Rp , Case

K=p0

E0 ˆ 1 GPa†

s0rr =p0

s0yy =p0

s0zz =p0

s0rz =p0

ÿ1.7843 ÿ2.4794 ÿ1.1839

3.9429Eÿ3 5.6561Eÿ3 2.9437Eÿ3

ÿ5.4162Eÿ4 ÿ9.9732Eÿ4 1.7856Eÿ4

j ˆ 308 a b c

0.8236 0.8295 0.4120

4.4354Eÿ4 1.7585Eÿ4 ÿ6.8073Eÿ4

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Fig. 5. Stress distributions along j ˆ 308 when j1 ˆ 908 and j2 ˆ 1008:

Fig. 6. Stress distributions along j ˆ 308 when j1 ˆ 808 and j2 ˆ 1008:

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169

Fig. 7. Stress distributions along j ˆ 308 when j1 ˆ 808 and j2 ˆ 808:

6. Conclusions In this work, a general solution in a form of power series is employed to analyze the stress singularities near the bond edge of an axisymmetric deformation joint. Based on this solution, the following results have been obtained: . For each eigenvalue o < 1, the corresponding stresses have the expression of Eq. (30), an in®nite power series in r: . The stress singular exponent o and the angular distribution functions zij …j, o† were veri®ed to be the same as those in the plane strain deformation joint with the same material combination and the same j1 , j2 : P ÿom . Di€erent from plane deformations, only the term Krÿo zij …j, o† or the sum zij …j, om † m Km r cannot correctly describe the stress ®eld near the bond edge of the axisymmetric deformation joint. The e€ect of the regular stress term in Eq. (35) must be taken into account. Eq. (35) has the di€erent meaning to the terms with o ˆ 0: . Eq. (36) or (37) can asymptotically describe the stress ®eld near the bond edge of an axisymmetric joint.

Acknowledgements The author (Y.L. Li) gratefully acknowledges the ®nancial support provided by the Alexander von Humboldt Foundation and by Japan Society for the Promotion of Science. She is also indebted to Prof. D. Munz and Prof. K. Watanabe for valuable discussions.

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