Crack-tip fields in anisotropic shells - CiteSeerX

International Journal of Fracture 113: 309–326, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Crack-tip fields in anisotropic shells F.G. YUAN and S. YANG Department of Mechanical and Aerospace Engineering North Carolina State University, Raleigh, NC 27695, U.S.A. Received 7 April 2000; accepted in revised form 22 October 2001 Abstract. Asymptotic crack-tip fields including the effect of transverse shear deformation in an anisotropic shell are presented. The material anisotropy is defined here as a monoclinic material with a plane symmetry at x3 = 0. In general, the shell geometry near the local crack tip region can be considered as a shallow shell. Based on Reissner shallow shell theory, an asymptotic analysis is conducted in this local area. It can be verified that, up to the second order of the crack tip fields in anisotropic shells, the governing equations for bending, transverse shear and membrane deformation are mutually uncoupled. The forms of the solution for the first two terms are identical to those given by respectively the plane stress deformation and the antiplane deformation of anisotropic elasticity. Thus Stroh formalism can be used to characterize the crack tip fields in shells up to the second term and the energy release rate can be expressed in a very compact form in terms of stress intensity factors and Barnett– Lothe tensor L. The first two order terms of the crack-tip stress and displacement fields are derived. Several methods are proposed to determine the stress intensity factors and ‘T-stresses’. Three numerical examples of two circular cylindrical panels and a circular cylinder under symmetrical loading have demonstrated the validity of the approach. Key words: Asymptotic crack-tip fields, anisotropic shells, energy release rate, Reissner shallow shell theory, stress intensity factor, Stroh formalism, T-stress.

1. Introduction The application of a fail-safe design philosophy can be greatly enhanced by a thorough understanding of the resistance of structures and structural materials to failure in the presence of a discontinuity. Shell-like structural components comprise major load-carrying components in aircraft and space vehicles such as aircraft fuselage, reusable launch vehicles, solid propellant rocket motors, and spacecraft use propellant tanks, etc. Depending on their structural details that a variety of failure modes may occur, the failure behavior and resulting residual strength present formidable challenges to the design of these structures and in service structures beyond their original life goal. Motivated by the role of the curvature on the crack tip behavior of shell-like structures, the stress analysis of a crack in isotropic shells based on ordinary linear elastic fracture mechanics has been attracted considerable attention since 1960s. The presence of the curvature causes the coupling between membrane and bending stresses or a combination of in plane and out of plane displacements. Under the small strain and small deformation assumptions, solutions based on linearized classical shell theory can be found (e.g., Folias, 1965, 1967; Copley and Sanders, 1969; Erdogan and Kibler, 1969; Simmonds et al., 1978). Since Kirchhoff assumption only enables the satisfaction of two boundary conditions along the crack surfaces, the Kirchhoff boundary condition for transverse shear resultant yields the angular distribution of the leading singular stress term near the crack tip depending on the Poisson’s ratio. The

310 F.G. Yuan and S. Yang angular distribution obtained from Kirchhoff theory does not conform to the general threedimensional elasticity solutions, which are independent of the Poisson’s ratio. As a result, the crack tip stress field which incorporated the Kirchhoff conditions may be affected considerably and is inadequate. For instance, the transverse shear resultant obtained from the classical shell theory has a singularity of r −3/2 and the strain energy resulting from the transverse shear resultant is omitted which is responsible for the deficiency of the classical shell theory, etc. These shortcomings similar to those for Kirchhoff flat plate solutions have been recognized (e.g., Yuan and Yang, 2000). The inability of classical shell theory to adequately characterize the crack-tip fields has lead to use other higher-order shell theories for describing the cracktip fields such as the shell theory by Naghdi (1956, 1957). The shallow shell theory by taking the transverse shear deformation into account is free of some of the limitations imposed by the linearized classical shell theory and can accommodate all the physical boundary conditions along the crack surfaces, namely the membrane, in-plane shear, bending, twisting and transverse shear. The crack-tip behavior of shell structures in isotropic material or specially orthotropic materials based on higher-order shell theories has been studied by many researchers. Results of a crack in a spherical shell were provided by Sih and Hagendorf (1974) and Delale and Erdogan (1979). An axial crack in a cylindrical shell was solved by Krenk (1978). A circumferentially cracked cylindrical shell was examined by Delale and Erdogan (1979). A cylindrical shell with an arbitrarily oriented crack was studied by Yahsi and Erdogan (1983). By examining the asymptotic behavior near the crack tip from the infinitely long shell solutions, the order of singularity (r −1/2 ) and the angular distribution of the leading singular term of the stresses are identical to those given by the isotropic flat plate solutions. These solutions have the same forms as those given by respectively the plane stress deformation and the antiplane deformation of isotropic elasticity. Note that the analytical solutions based on the Reissner and shallow shell theories utilized the singular integral equation approach in conjunction with Fourier transform. Although these solutions provide an insight into the effect of curvature on the stress intensity factors by a shell parameter, the techniques only can handle the simple shell geometry and the shells are under simple uniform loading conditions. It is clear that this approach is not suitable for modeling complicated shell configuration under general loading conditions. For example, the complex loading on the shell boundary. In addition, the technique is unlikely to model the shell with material anisotropy. As quoted by Erdogan (1977), for shells made of composites, the treatment of anisotropic or, even the orthotropic shells using the singular integral equation approach is not tractable. The analytical difficulties in determining critical fracture parameters are complicated in shell structures because of the complex shell geometry, loading condition, and material anisotropy involved. Only very few hypothetical cases have been solved in conjunction with numerical techniques. Numerical methods such as finite element methods are considered to be most attractive due to their versatility in handling structural problems with a variety of complexities. Barsoum et al. (1979) was among the first to analyze the axial and circumferential through cracks in cylindrical shells using linear shell elements. The bending stress intensity factor is affected significantly by the transverse shear and depends on radius to thickness (R/ h) ratio even for thin shells. The transverse shear effect is found to be more pronounced for the case of applied bending moment than for internal pressure. In this paper, asymptotic crack-tip fields of anisotropic shells extended from anisotropic plates (Yuan and Yang, 2000) are investigated. Using Reissner transverse shear theory, shallow shell equations and asymptotic analysis, the asymptotic crack-tip fields, up to the second order,

Crack-tip fields in anisotropic shells 311

Figure 1. Force and moment notations of a shallow shell in the vicinity of the crack.

are presented for an anisotropic shell under general boundary conditions. It is shown that the first two terms of the asymptotic crack-tip fields are identical to those of anisotropic plates and two-dimensional plane stress and antiplane deformation solution in anisotropic elasticity, and thus they can be expressed in a compact form based on Stroh formalism. Using numerical calculations, methods of determining stress intensity factors and ‘T-stresses’ are proposed for a crack in an anisotropic shell with general boundary conditions and loadings. Three numerical examples for finite circular cylindrical panels or shells with axial crack or circumferential crack under symmetric loading are presented. 2. Basic equations for shallow shells Since the asymptotic crack-tip fields in shells are analyzed in this paper, a small area near the crack tip may not include the whole crack. In the small area, let R be the smallest radius of curvature,  the characteristic dimension of cut area, and h the shell thickness. If /R  1

and

h/  1,

then the small area in the vicinity of the crack tip can be considered locally as a shallow shell. Therefore, the theory of shallow shells can be utilized in investigating the crack tip fields in an asymptotic sense. Otherwise the three-dimensional elasticity theory may be needed to examine the problem. In the theory of shallow shells, the undeformed middle surface of a shell is determined by the Cartesian coordinates of Figure 1: X3 = f (x1 , x2 ).

(1)

Here x1 and x2 are the coordinates on the middle surface of the shell. A point in the shell is described by the curvilinear coordinates (x1 , x2 , x3 ). where x3 is the distance measured along the normal to the middle surface. A shell of constant thickness, h, is considered. The equilibrium equations (Reissner and Wan, 1969; Sih and Hagendorf, 1974) are Nαβ,β + qα = 0,

Qα,α + (f,α Nαβ ),β + q = 0,

Mαβ,β − Qα = 0,

(2)

where comma denotes differentiation, the subscripts α, β = 1, 2; qα and q are the surface load intensities with respect to x1 , x2 , and x3 respectively; Nαβ , Mαβ and Qα are the membrane forces, moments and transverse shear forces per unit length given by

312 F.G. Yuan and S. Yang
h/2
σαβ dx3 , Mαβ = Nαβ = −h/2




h/2

−h/2

Qα =

σαβ x3 dx3 ,

h/2

−h/2

σα3 dx3 .

(3)

In order to obtain the appropriate strain-displacement relations, we introduce approximations for the displacements in the form uα = uα (x1 , x2 ),

u3 = u3 (x1 , x2 ),

ψα = ψα (x1 , x2 )

(4)

here ui (i = 1, 2, 3) are the components of displacements at the middle surface, and ψα are the changes of the slope of the normal to the middle surface. Then the shell strain-displacement relations are ε11 = u1,1 + u3 /R1 , ε22 = u2,2 + u3 /R2 , 2ε12 = u1,2 + u2,1 , κ11 = ψ1,1 ,

κ22 = ψ2,2 ,

γ13 = u3,1 − u1 /R1 + ψ1 ,

2κ12 = ψ1,2 + ψ2,1 ,

(5)

γ23 = u3,2 − u2 /R2 + ψ2 ,

where εαβ , καβ are the middle surface strains and the changes of curvature of the middle surface, respectively; γα3 the transverse shear strains; and the following curvature measures of the middle surface are used 1 ∂ 2f 1 ∂ 2f 1 ∂ 2f = − , = − , =− . 2 2 R1 R2 ∂x1 ∂x2 R12 ∂x1 ∂x2 It is assumed that the curvatures of the middle surface are constant in the local region of interest. An anisotropic material with a plane symmetry at x3 = 0 (monoclinic material) is considered, that is, s14 = s15 = s24 = s25 = s34 = s35 = s36 = s46 = s56 = 0, where sij (i, j = 1, 2, 3, . . . , 6) are elastic compliances. If one principal material axis of an orthotropic material is parallel to x3 direction and the other two principal axes may not coincide with the coordinate lines x1 and x2 , then the elastic constants of the orthotropic material have the above properties referred to the problem coordinates (x1 , x2 , x3 ). In these cases the relations between strains and resulting stresses for shallow shell become 12 1 1 s 2 Q, (6) ε = s 1 N , κ = 3 s 1 M, γ = h h kh where k = 5/6 and ε = {ε11 , ε22 , 2ε12 }T ,

κ = {κ11, κ22 , 2κ12 }T ,

γ = {γ13 , γ23 }T ,

N = {N11 , N22 , N12 }T , M = {M11 , M22 , M12 }T ,   s11 s12 s16    s55 s45  s s s = , , s s1 =  2  12 22 26  s45 s44 s16 s26 s66

Q = {Q1 , Q2 }T ,

s 1 , s 2 are symmetric matrices. The expressions for the stress components, which are consistent with the approximations for the displacements, can be written as

σαβ =

12x3 Nαβ + 3 Mαβ , h h

σα3

Crack-tip fields in anisotropic shells 313

 2x3 2 3Qα 1− = . (7) 2h h

3. Asymptotic crack-tip fields in shells For crack problems, since the area of interest is the region around the crack tip, the shell can be considered as a shallow shell in this area in general. Hence, applying the shallow shell theory to the area, the asymptotic crack tip fields may be derived. The asymptotic crack-tip fields in a shell containing a through crack are discussed below. Assume the Cartesian coordinate system (x1 , x2 , x3 ) is centered at the crack tip, and the axis x1 is tangential to the crack line. Let (r, θ) be polar coordinates measured from the crack tip. Attention will be focused on the asymptotic crack-tip fields as r → 0. The crack surfaces are assumed to be traction-free. Then the traction-free boundary conditions on the crack surfaces in the shallow shell can be expressed as N22 = N12 = M22 = M12 = Q2 = 0,

at

θ = ±π

(8)

Guided by the governing equations for the crack-tip fields in the shallow shell, as r → 0, the displacement and stress fields near the crack tip under general loading can be expanded in the form (1) ui = u(0) i + ui + · · · ,

i = 1, 2, 3

(9a)

ψα = ψα(0) + ψα(1) + · · · ,

(9b)

(0) (1) + Nαβ + ··· , Nαβ = Nαβ

α, β = 1, 2

(9c)

(0) (1) + Mαβ + ··· , Mαβ = Mαβ

(9d)

(1) Qα = Q(0) α + Qα + · · · ,

(9e)

with δ+1 , u(0) i ∼r

ψα(0) ∼ r δ+1 ,

(0) (0) δ Nαβ , Mαβ , Q(0) α ∼r ,

(10)

δ+2 , u(1) i ∼r

ψα(1) ∼ r δ+2 ,

(1) (1) δ+1 Nαβ , Mαβ , Q(1) , α ∼r

(11)

where δ is eigenvalue to be determined and −1 < Re(δ). Substituting the expansions into the governing equations, and collecting terms of equal powers of r, the following equations can be obtained (0) = 0, Nαβ,β

(0) Mαβ,β = 0,

  

u(0) 1,1

  

 

u(0) 2,2 u(0) + u(0) 1,2 2,1

Q(0) α,α = 0,

 (0)   N11 s 1  (0) = N22  h     (0) N12

      

,

   h 2  

(12a) (0) ψ1,1 (0) ψ2,2 (0) (0) ψ1,2 + ψ2,1

   

 (0)   M11 6  (0) = 2 s 1 M22   h     (0) M12

      

,

(12b)

314 F.G. Yuan and S. Yang  (0)   (0)  u3,1 s 2 Q1 = , kh Q(0) u(0) 3,2

(12c)

2

(0) (0) (0) (0) = N12 = M22 = M12 = Q(0) N22 2 = 0,

at θ = ±π

(12d)

and (1) = 0, Nαβ,β

(1) Mαβ,β − Q(0) α = 0,

 (1) u + u(0)  3 /R1   1,1 (1) u2,2 + u(0) 3 /R2    (1) u1,2 + u(1) 2,1 

(0) Q(1) α,α + f,αβ Nαβ = 0,

 (1)   N11 s 1  (1) = N22  h     (1)  N12

   





   

(0) (0) u(1) 3,1 + ψ1 − u1 /R1 (0) (0) u(1) 3,2 + ψ2 − u2 /R2

s2 = kh

  

,

Q(1) 1

   h 2  

(1) ψ1,1 (1) ψ2,2 (1) ψ1,2

+

(1) ψ2,1

   

 (1)   M11 6  (1) = 2 s 1 M22   h     (1) M12

(13a)       

,

(13b)



Q(1) 2

(1) (1) (1) (1) = N12 = M22 = M12 = Q(1) N22 2 = 0,

(13c)

, at θ = ±π.

(13d)

(0) (0) The first set of Equation (12) governs u(0) α , ψα , u3 , . . . and they are mutually uncou(1) (1) pled; while u(1) α , ψα , u3 , . . . are governed by the second set of Equation (13) in which the (0) (0) resulting uα , ψα , u(0) 3 , . . . appear as inhomogeneous terms. These characteristics can also been seen from the governing equations of Equations (12a) and (13a) and from the analysis of crack-tip fields in elastic-plastic solids (e.g., Yuan and Yang, 1997). (0) (0) (0) (0) (0) , u(0) First, consider Nαβ α , Mαβ , ψα , Qαβ , u3 . Recall the crack-tip field under plane stress deformation in monoclinic materials with a plane symmetry at x3 = 0. The governing equations can be written as (Yuan, 1998)

σαβ,β = 0,     u1,1   σ11   u2,2 = s 1 σ22 ,     u1,2 + u2,1 σ12 σ22 = σ12 = 0,

at θ = ±π.

(14a)

(14b)

(14c)

The crack-tip fields for antiplane deformation in the monoclinic materials are governing by σ3α,α = 0, 

u3,1 u3,2

(15a)



σ32 = 0,

 = s2 at

σ31 σ32

 ,

θ = ±π.

(15b) (15c)

Crack-tip fields in anisotropic shells 315 Comparing the governing equations for shells with those for the plane stress deformation and the antiplane deformation in anisotropic elasticity, it is clear that the equations for (0) / h and u(0) Nαβ α in shells given by Equations (12a)1 , (12b)1 and (12d)1,2 are identical to the equations for σαβ and uα for plane stress deformation governed by Equations (14a–c), so (0) / h2 and ψα(0) h/2 in shells given by Equations (12a)2 , (12b)2 , are the equations for 6 Mαβ (0) and (12d)3,4 . The equations for Q(0) αβ /kh and u3 in shells given by Equations (12a)3 , (12c), and (12d)5 are identical to the equations for σ3α and u3 for antiplane deformation governed (0) (0) , u(0) by Equations (15a–c) (Yuan and Yang, 2000). Therefore, the solutions for Nαβ α , Mαβ , (0) ψα(0) , Q(0) αβ , u3 , can be readily obtained if we directly use the solutions for cases of plane stress and antiplane deformation in anisotropic elasticity. We have δ = −1/2, 0, 1/2, . . . ,

(0) (0) (0) (0) (0) , u(0) for Nαβ α , Mαβ , ψα , Qαβ , u3 .

(16)

(0) (0) (0) (0) (0) , u(0) Each of Nαβ α , Mαβ , ψα , Qαβ , u3 is a series of power of r. Furthermore, with Stroh formalism for two-dimensional deformations of anisotropic elasticity (Stroh, 1958, 1962; (0) (0) (0) (0) (0) Ting, 1996), u(0) α , ψα , u3 and the corresponding resulting forces Nαβ , Mαβ , Qα with δ = −1/2, 0, . . . can be written in a compact form  (0)   u1 √ 1 2 Re[A zB −1 ]k m − Im[A zB −1 ]g m + O(z3/2), (17a) = (0) π h u 2

1 h 1 h h 2



(0) N11



(0) N12

 

(0) N12 (0) N22

ψ1(0)

  p 1 −1 = √ Re[B √ B −1 ]k m + Im[B pB −1 ]g m + O(z1/2), z h 2π

  1 1 = √ Re[B √ B −1 ]k m + O(z1/2), z 2π   √ 6 2 Re[A zB −1 ]k b − 2 Im[A zB −1 ]g b + O(z3/2), = π h



ψ2(0)  (0)    p 6 −1 6 M11 = √ Re[B √ B −1 ]k b + 2 Im[B pB −1 ]g b + O(z1/2), 2 (0) h z h 2π M12 6 h2



(17b)

(0) M12 (0) M22



  1 1 = √ Re[B √ B −1 ]k b + O(z1/2), z 2π

 √ 2k3 2 g3 3/2 Im( z3 ) + Re(z3 ) + O(z3 ), = 3kµ π khµ       1 g3 Im(p3 ) 2 k3 1 Q(0) −p3 1 + O(z1/2), = √ Re{ √ }+ 1 0 kh Q(0) 3k z kh 2π 3 2

u(0) 3

where k m = {km2 , km1 }T , k b = {kb1 , kb2 }T ,

g m = {gm2 , gm1 }T , g b = {gb1 , gb2 }T ,

(17c)

(18a)

(18b)

(18c)

(19a) (19b)

316 F.G. Yuan and S. Yang k3 , and g3 , are real constants to be determined, which are functions of geometry of the cracked shell, loading and material properties, the subscripts or superscripts, m, b, and 3 denote the quantities related to stretching, bending and transverse shear, respectively; F (z, p) = diag[F (z1 , p1 ), F (z2 , p2 )] zα = x1 + pα x2 = r(cos θ + pα sin θ),

Im(pα ) > 0,

pi , A, B are Stroh eigenvalues and matrices which depend on the material constants only. p1 and p2 are the roots of s11 p 4 − 2s16 p 3 + (2s12 + s66 )p 2 − 2s26 p + s22 = 0,

(20)

p3 is root of s55 p 2 − 2s45 p + s44 = 0, A, B are given by  ξ1 ξ2 , A= η1 η2

 B=

ξα = s11 pα2 − s16 pα + s12 ,

(21)

−p1 −p2 1 1

,

B

−1

1 = p1 − p2



−1 −p2 1 p1

,

(22)

ηα = s12 pα − s26 + s22 /pα ,

2 −1/2 ) . µ = (s44 s55 − s45 (1) (1) (1) (1) (1) The solutions of u(1) α , ψα , u3 , Nαβ , Mαβ , Qα can be obtained from the corresponding inhomogeneous governing equations in Equations (13). The resulting expressions are too lengthy to be recorded here. However, without going into details, substituting the solutions (0) (0) (0) (0) (0) , u(0) of Nαβ α , Mαβ , ψα , Qαβ , u3 in Equations (17)–(19) into Equtions (13) shows that (1) (1) 3/2 , u(1) α , ψα , u3 ∼ r

(1) (1) 1/2 Nαβ , Mαβ , Q(1) , α ∼ r

as r → 0

(23)

(1) (1) (1) Comparing the order of r in Equation (23) and in Equations (16), u(1) α , ψα , u3 , Nαβ , (1) , Q(1) Mαβ α enter the third-order terms of the expansions of the displacement and stress fields, uα , ψα , u3 , Nαβ , Mαβ , Qα . Equation (13) shows that the coefficients of the leading order term solution, k m , k b , k3 and curvature parameters Rα and R12 enter the third-order terms explicitly. Based on the above analysis, the first two terms of the asymptotic crack-tip fields in shell can be expressed as    √ 1 2 u1 Re[A zB −1 ]k m − Im[A zB −1 ]g m + O(z3/2), (24a) = u2 π h    c    m p 1 N11 −1 1 N11 σ11 −1 + O(z1/2), (24b) = = √ Re B √ B km + m σ12 0 h N12 z h 2π

1 h



N12 N22



 =

m σ12 m σ22



   1 1 −1 = √ Re B √ B k m + O(z1/2), z 2π

c is constant membrane force, where N11

(24c)

Crack-tip fields in anisotropic shells 317  c N11 = Im[B pB −1 ]g m ; (24d) 0    √ 6 2 h ψ1 (25a) = Re[A zB −1 ]k b − 2 Im[A zB −1 ]g b + O(z3/2), 2 ψ2 π h       b   c  6 M11 p −1 6 M11 x3 −1 σ11 −1 Re B B + O(z1/2), = √ + = k √ b b 2 0 σ12 h2 M12 h/2 z h 2π (25b)



6 h2



M12 M22



 =

b σ12 b σ22



x3 h/2

−1

   p 1 −1 = √ Re B √ B k b + O(z1/2), z 2π

c is constant moment, where M11  c  M11 = Im[B pB −1 ]g b ; 0  2k3 2 g3 √ 3/2 Im( z3 ) + Re(z3 ) + O(z3 ), u3 = 3kµ π khµ       −1  1 Q1 1 2 k3 2 σ13 x32 −p3 = = 1−4 2 + √ Re √ 1 kh Q2 3k σ23 h 3k 2π z3   1 Qc1 + O(z1/2), 0 kh

where Qc1 is a constant transverse shear force,    c Im(p3 ) Q1 = g3 . 0 0

(25c)

(25d)

(26a)

(26b)

(26c)

Hence, for general loading, the singular terms of the crack-tip stress fields in general anisotropic shells are characterized by k m , k b , k3 , while the second-order terms, the constant terms or c c , M11 , Qc11 . Note that the asymptotic crack tip fields of displacements and ‘T-stresses’, by N11 stresses are consistent with all available solutions for shallow shells with simple geometry and simple loading conditions, at least up to the leading order. As far as a cracked shell with tractions specified on the crack surfaces is concerned, the tractions are self-equilibrating loads, that is, equal and opposite loads are applied to the two crack surfaces. For simplicity, uniform surface tractions at crack surfaces are assumed, and they can be a single or a combination of loading of the following five loads, c , N22 = N22

c N12 = N12 ,

c M22 = M22 ,

c M12 = M12 ,

Q2 = Qc2 ,

c c c c , N12 , M22 , M12 , Qc2 are constants. In these cases, the asymptotic crack-tip fields where N22 for traction-free crack surfaces need to be modified. The singular terms of the crack-tip fields have an exact solution as given before, and the second-order terms can be readily obtained by superimposing the additional constant-stress term and the corresponding displacements into the second-order stress field and the deformation field listed above, respectively. The additional deformation can be obtained after simple manipulations. For the sake of brevity, they will not be presented here. With the following definition of the stress intensity factors

318 F.G. Yuan and S. Yang  √  k2 = lim 2π rσ12  θ =0 , r→0

k1 = lim

√

r→0

k3 = lim

r→0

√

  2π rσ22    2π rσ23 

(27a)

x3 =h/2

θ =0 x3 =h/2

θ =0 x3 =h/2

,

(27b)

,

(27c)

we have k2 = km2 + kb2 , k1 = km1 + kb1 . 4. The J-integral and energy release rate For a three-dimensional deformation field, the J -integral (Eshelby, 1956 and Rice, 1968) is defined as

(28) J = (W n1 − ti Ui,1 ) dS = (σij Ui,j n1 /2 − σij nj Ui,1 ) dS, S

S

where Ui represent the displacements. S is a surface surrounding the crack tip, n is the unit outward normal. The integral is independent of the surface of the integration for crack with traction-free surfaces. For the shallow shell with crack subjected to general loading, the J integral can be expressed as

1 (Nαβ εαβ + Mαβ καβ + Qα γα3 )n1 ds − (Nαβ uα,1 + Mαβ ψα,1 + Qβ u3,1 )nβ ds (29) J = 2 C C where C is a contour surrounding the crack tip on the middle surface. The J -integral does not depend on C, thus path-independent. In the absence of body forces, J is equal to the energy release rate G. According to the definition of G, for shallow shells, it becomes  
h/2
7a 1 σ2i (r, 0, ζ )[Ui (7a − r, π, ζ ) − Ui (7a − r, −π, ζ ]drdx3 G = lim 7a→0 27a −h/2 0     
7a    1 7u1 7ψ1    [N12 , N22 ] x1 =r +[M12 , M22 ] x1 =r + Q2 x1 =r 7u3 dr = lim 7u2 7ψ2 7a→0 27a 0 x2 =0 x2 =0 x2 =0  1 2 k32 1 T −1 (30) k m L k m + k Tb L−1 k b + , =h 2 6 9k µ where Uα = uα (x1 , x2 ) + x3 ψα (x1 , x2 ), U3 = u3 (x1 , x2 ) and 7ui = ui (7a − r, π ) − ui (7a − r, −π ),

L−1 = Re(iAB −1 ) = s11 p1 + p2 = a + ib,



b d d e

7ψα = ψα (7a − r, π ) − ψα (7a − r, −π ), (31)

,

p1 p2 = c + id,

e = ad − bc.

Crack-tip fields in anisotropic shells 319 For isotropic shells, L−1 =

2 I E

where I is an identity matrix. 5. Determination of stress intensity factor and ‘T-stress’ Define the following integrals
7a 1 Gm2 = lim N21 (r, 0)7u1 dr, 7a→0 27a 0
7a 1 N22 (r, 0)7u2 dr, Gm1 = lim 7a→0 27a 0
7a 1 M21 (r, 0)7ψ1 dr, Gb2 = lim 7a→0 27a 0
7a 1 M22 (r, 0)7ψ2 dr, Gb1 = lim 7a→0 27a 0
7a 1 Q2 (r, 0)7u3 dr. G3 = lim 7a→0 27a 0 The leading terms of the asymptotic expansions of Nαβ , Mαβ , Qβ , ui , ψα , Equations (24)– (26), are sufficient for evaluating the right sides of the above equations. When all modes of deformation are present, the above integrals yield hs11 km2 (L−1 k m )1 , 2 hs11 km1 (L−1 k m )2 , Gm1 = 2 hs11 kb2 (L−1 k b )1 , Gb2 = 6 hs11 kb1 (L−1 k b )2 , Gb1 = 6 2h 2 k . G3 = 9kµ 3

Gm2 =

(32)

In the above (∗ )i denotes the i-th element of the vector inside the bracket. The integrals of the above equations can be evaluated from numerical results. Therefore they provide five equations for the five unknowns, kmα , kbα , k3 : the first two for k m ; the third and the fourth for k b ; the fifth for k3 . From displacement fields, it can be shown that for a crack with traction-free surfaces we have the following relations on the crack flanks

320 F.G. Yuan and S. Yang  s  11 c r + u01 + O(r 2 ), =u1 = 2 − N11 h  12 c 0 =ψ1 = 2 − 3 s11 M11 r + ψ1 + O(r 2 ), h  s  55 =u3 = 2 − Qc1 r + u03 + O(r 2 ), kh

(33)

where u01 , ψ10 , and u03 are rigid body displacements and =ui = ui (r, π ) + ui (r, −π ),

=ψ1 = ψ1 (r, π ) + ψ1 (r − π )

(34)

To effectively evaluate the ‘T-stresses’ without extracting the rigid body displacements, the other approach is proposed using strain variables. Equations (33) provide the following relations h h d(=u1 ) h c = =− =u1,1 = =ε11 , N11 2s11 dr 2s11 2s11 c =− M11

Qc1 = −

h3 h3 d(=ψ1 ) h3 = =ψ1,1 = =κ11 24s11 dr 24s11 24s11

as r → 0,

(35)

kh kh d(=u3 ) kh = =u3,1 = =γ13 . 2s55 dr 2s55 2s55

The right sides of the above equations can be evaluated from numerical results at the crack flanks. Therefore the ‘T-stresses’ can be calculated accordingly. The errors for evaluating the T-stresses using Equation (35) is O(r). Similarly, from the displacement fields k m , k b and k3 can be expressed as, when r → 0,  1 π Lδu + O(r), (36a) km = 2 2r  h π Lδψ + O(r), (36b) kb = 4 2r  3kµ π δu3 + O(r), (36c) k3 = 4 2r where δui = ui (r, π ) − ui (r, −π ), and 1 L= s11 (be − d 2 )



e −d −d b

δψα = ψα (r, π ) − ψα (r, −π ) (37)

6. Numerical results Based on Riessner shallow shell theory, the theoretical solutions mentioned in the Introduction section are for infinite shells, at least in one dimension. The asymptotic analysis derived for

Crack-tip fields in anisotropic shells 321 crack tip field in this paper is valid for any type of loading conditions and finite geometry of the region near the crack tip satisfies the thin shallow shell conditions, yet the coefficients associated with the singular and higher-order terms cannot be determined by asymptotic analysis alone. The loading, shell curvature, and shell geometry are implicitly reflected in the coefficients. These coefficients, in general, need to be determined from numerical methods. A finite element method is used to three shell cases: (a) a circular cylindrical shell panel with an axial crack under uniform bending moment, (b) a circular cylindrical shell panel with a circumferential crack under uniform membrane load, and (c) a circumferentially cracked circular cylinder subjected to uniform axial tension. The shell geometry and the loading conditions are shown in Figure 2. An orthotropic material with xi as principal material axes is considered. The material properties are given by E1 = 11.21 Msi, G12 = 2.77 Msi,

E2 = 5.09 Msi,

E3 = 1.53 Msi,

G23 = 0.57 Msi,

G13 = 0.64 Msi,

ν12 = 0.44,

ν23 = 0.29,

ν13 = 0.22

In the following examples, the stiffer material principal axis is perpendicular to the axial direction of cylindrical shell. The 8-node thick shell elements (ABAQUS element type S8R, 1997), which take the transverse shear deformation into account, are used to model the example problems and the finite boundary effect on the fracture parameters is studied by changing the crack size relative to the shell length. Each node has six degrees of freedom. The crack tip elements are kept at size of 1% to 2% of the crack length. The three illustrative examples are symmetrical problems with mode-I deformation. The stress intensity factors and ‘T-stresses’ are calculated using the following equations, from Equations (32) and (35),   2Gm1 6Gb1 , kb1 = , (38) km1 = hes11 hes11 c = N11

h =ε11 , 2s11

c M11 =

h3 =κ11 , 24s11

(39)

where s11 is the compliance in the crack line direction, e is the element of the matrix L in Equation (37) which is decided by the crack-tip local coordinates (x1 , x2 , x3 ) where the crack line lies tangentially along the x1 axis. The finite element results are obtained from the finite element code ABAQUS. Using these results and a virtual crack closure method (Rybicki and Kanninen, 1977), the integrals Gm1 and Gb1 can be evaluated from (see Figure 3) Gm1 =

! " ! "# 1 (N22 )m1 (u2 )n3 − (u2 )n4 + (N22 )m2 (u2 )n1 − (u2 )n2 , 27a

Gb1 =

! " ! "# 1 (M22 )m1 (ψ2 )n3 − (ψ2 )n4 + (M22 )m2 (ψ2 )n1 − (ψ2 )n2 , 27a

(N22 )m1 , (N22 )m2 are nodal forces at nodes m1 and m2, (N22 )m1 is the summation of nodal forces evaluated from elements E3 and E4, (N22 )m2 is evaluated from element E4, (u2 )ni , i =

322 F.G. Yuan and S. Yang

Figure 2. Geometry and loading conditions of axially and circumferentially cracked cylindrical shells.

Crack-tip fields in anisotropic shells 323

Figure 3. Shell finite element configuration around the crack tip.

1 to 4, are the displacements for nodes on the crack flank surfaces, and 7a is the size of the element around the crack tip. (M22 )m1 ,. (M22 )m2 , (ψ2 )ni are defined in the similar manner. For the shell panels shown in Figures 2a and 2b, h = 1 in, w = L = 10 in, the span angle θ0 = (2L/R)(180/π ) = 20◦ ; for the circular cylinder shown in Figire 2c, h = 1 in., L = π R = 100 in. The normalized stress intensity factors and direct ‘T-stresses’ are shown in Figures 4–6. Here √ k0 = 6M0 π a/ h2 , for a constant bending moment M0 and N0 = 0, √ k0 = N0 π a/ h, for a constant membrane load N0 and M0 = 0. The membrane T-stress induced by the bending load or vice versa is too small to be plotted in the figures. When R → ∞, it is expected that the results approach those of flat plates. The limit a/R → 0 has been checked for the three cases with fixed values of a/ h. In all cases good agreement is found with flat plate calculations. The exact solutions for a flat √ plate with small c /N0 = s22 /s11 ≈ −1.48. cracks under uniform tension give km1 /k0 = 1, kb1 = 0, and N11 When a/L  1 and a/R  1, the numerical results for cases 2 and 3 for the shells are close to those values for the plate with small cracks. For the shell panel under bending, when a/R → 0, numerical results approach the orthotropic bending results given by Yuan and Yang (2000). For the case 1 where the cylindrical panel with an axial crack is under uniform bending moment as shown in Figure 2a, the results are illustrated in Figure 4. The values of kb1 are c c are negative. The value of km1 /kb1 is relatively small. M11 positive, while both km1 and M11 decreases with a/w for a/w > 0.1. In the case 2, a cylindrical shell panel with a circumferential crack is under uniform membrane load as shown in Figure 2b and the results are c indicated in Figure 5. The values of km1 /k0 and kb1 /k0 both increase with a/w; while N11 decreases monotonically with a/w. As a/w reaches 0.5, the ratio of magnitude of kb1 /km1 is close to 1/3. In case 3 where a circumferentially cracked cylinder subjected to uniform axial c reaches maximum at tension, the sign of values of kb1 alters at around a/ h ≈ 11, and N11 about a/ h ≈ 12.5 shown in Figure 6.

324 F.G. Yuan and S. Yang

c for a cylindrical shell panel with axial crack under Figure 4. Stress intensity factors km1 , kb1 , and ‘T-stress’ N11 uniform bending moment.

c for cylindrical panel with a circumferential crack Figure 5. Stress intensity factors km1 , kb1 , and ‘T-stress’ N11 under uniform membrane load.

7. Conclusions In general, a local area may exist in the vicinity of the crack tip where the shell can be considered as shallow. Using the Reissner shallow shell theory, the crack-tip fields, up to the second order, for a cracked anisotropic shell with general boundary conditions and loadings are presented. It has been proved that the forms of the first two terms in the asymptotic cracktip fields are identical to those given by respectively the plane stress deformation and the antiplane deformation in anisotropic elasticity. Therefore the stress intensity factors and ‘T-

Crack-tip fields in anisotropic shells 325

c for a circular cylinder with a circumferential crack Figure 6. Stress intensity factors km1 , kb1 , and ‘T-stress’ N11 under uniform membrane load.

stresses’ can be used to characterize the crack tip fields in the shells. Methods in determining five stress intensity factors and three ‘T-stresses’ are proposed based on the asymptotic analysis in an anisotropic shell under arbitrary loading. Three numerical examples: cracked shell panels and a cylinder under symmetrical loading for orthotropic materials are illustrated. The numerical results have demonstrated that the proposed methodology is applicable to general shells with material anisotropy. Lastly, for the large displacement and small strain analysis of the entire shell structures, since the relative deformation is small in the local area near the crack tip, the crack tip fields remain valid. For some cracked shells, it may not be possible to find a local area near a crack tip, which satisfies the conditions for shallow shells. Therefore the method presented in this paper is not valid. For example, in a thick cylindrical shell with a crack, we may face a fully threedimensional problem. The asymptotic crack tip fields for the first two terms have a similar structure as those in the shells: leading term has a singularity of r −1/2 and the second-order term is a constant stress. In this case, the singular crack tip field along the crack front can be characterized by a singular field for two-dimensional solutions, except in the region near the two free surfaces. However the stress intensity factors are functions of locations along the crack front. Acknowledgement This research is supported by NASA Grant No. 98-0548 from NASA Langley Research Center under Advanced Composite Technology Program. References ABAQUS, Version 5.7 (1997). Hibbitt, Karlsson & Sorensen, Inc., Pawtuckt, RI. Barsoum R.S., Loomis R.W. and Stewart B.D. (1979). Analysis of through cracks in cylindrical shells by the quarter-point elements. International Journal of Fracture 15, 259–280.

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