International Journal of Fracture 89:, 998. 998 Kluwer Academic Publishers. Printed in the Netherlands. Eigenfunction expansion for penny-shaped and circumferential cracks A.Y.T. LEUNG, and R.K.L. SU Professor, D.Sc., University of Manchester, Manchester M 9PL e-mail: andrew.leung@man.ac.uk PhD, Ove Arup, Hong Kong Received June 997; accepted in revised form March 998 Abstract. Cracks that are developed in pipes, pressure vessels and circular bars due to improper welding are often idealized to be circumferential, penny-shaped or flat toroidal. Based on the three-dimensional Navier s equations in cylindrical coordinates, we derive the general displacement functions for both the penny-shaped and circumferential cracks by eigenfunction expansion method. We find that only those terms of the first order displacement and stress are similar to the case of three-dimensional straight plane cracks and that the higher order terms are coupled through the equilibrium equations. The strength of coupling depends on the ratio r = r/a where r is the radius normal to the crack front and a is the diameter of the circular crack tip. It decreases with r. Key words: Eigenfunction expansion, penny-shaped crack, circumferential crack.. Introduction Cracks in pipes and circular bars due to fatigue or improper welding are often idealized to be circular. To predict the crack growth rate and the critical crack size, we need accurate modeling of the displacement of stress fields around the traction-free crack front. Williams (97; 9 originated the eigenfunction expansion method to tackle two-dimensional in-plane and out-ofplane thin plate crack problems. Based on the traction-free boundary conditions over the crack surfaces, he expanded the stresses and displacements in power series which automatically satisfy the displacement boundary conditions at crack surfaces. Since then, the eigenfunction expansion method had been employed to solve many crack problems. Examples were the three-dimensional straight cracks problems by Hartranft (99, thick plate bending problems by Murthy (98 and Sosa (98 and most recently elliptical cracks problems by Li (988. It is noted that Li applied the Navier s equations to derive the eigenfunction expansion series up to th terms only for elliptical cracks. We find in Leung and Su (998 that more terms are required for accurate computation of stress intensity factors. In this paper, based on Navier s equations, we extend Li s solutions from elliptical crack to both axisymmetric penny-shaped and circumferential cracks. We also derive the displacement solutions up to the 8th term in r, as the higher order terms have significant contributions to the accurate evaluation of the stress intensity factors (Leung and Su, 99; 998. The coupling effects of the coefficients are discussed. The solution obtained is useful for deriving singular crack tip elements and two-level finite element (Leung and Su, 99; 99a,b; 998; Leung and Wong, 989 for axisymmetric crack problems. Corresponding Author: Phone(7-, Fax (7-. INTERPRINT (typeset/copyprep. J.N.B. Frac (frackap:engifam v.. 9.tex; /9/99; 7:8; p.
A.Y.T. Leung and R.K.L. Su Figure. Geometry and coordinate systems of penny-shaped crack.. Governing equations The formulation of the eigenfunction expansion problems can be expressed most conveniently in terms of the local coordinates (r,,ϕ at the front of crack border. Figure shows the geometry of the circular crack and illustrates the orientation of the local and rectangular coordinates, in which x is along the tangent of the circular crack border. The radius of the crack border is a. The transformation from the local curvilinear coordinates to the global cartesian coordinates is found to be x = a cos ϕ + r cos cos ϕ, x = r sin, x = a sin ϕ = r cos sin ϕ. (a (b (c By using relationships (, we express the equations of equilibrium in the local coordinates as σ r r + σ r r + σ r σ + σϕr r + (σ r σ ϕ cos σ r sin =, (a σ r r σ ϕr r + σ r + σ r r + σ ϕ r + σ ϕr r + + σϕ + σ r cos + (σ ϕ σ sin =, (b σϕ + (σ ϕr cos σ ϕ sin =, (c where = a + r cos. The strain-displacement relationships in the local coordinates are ε r = u r r, ε = u r + u r r, ε ϕ = ( uϕ + u r cos u sin, 9.tex; /9/99; 7:8; p.
Eigenfunction expansion for penny-shaped and circumferential cracks 7 ε r = u r r + u r u r, ε ϕ = u + u ϕ r + u ϕ sin, ε ϕr = u r + u ϕ r u ϕ cos. ( For the orthogonal coordinate system, the stress-strain relationships are given by σ r σ σ ϕ σ r = E ( + ν( ν σ ϕ σ ϕr ν ν ν ν ν ν ν ν ν ν ν ν ε r ε ε ϕ ε r ε ϕ ε ϕr. ( Substituting ( and (, one has the stresses, Eν ( ν u r σ r = ( + ν( ν ν r + ( u r + u r + ( u r cos u sin + u ϕ, (a σ = σ ϕ = Eν ur ( + ν( ν r + ( u r + u ( ν + νr ( u r cos u sin + u ϕ Eν ur ( + ν( ν r + ( u r + u r ( + ( ν ν u r cos u sin + u ϕ, (b, (c 9.tex; /9/99; 7:8; p.
8 A.Y.T. Leung and R.K.L. Su E u σ r = ( + ν r + r σ ϕ = σ ϕr = ( ur u, (d E u ( + ν r + ( u + u ϕ sin, (e E uϕ ( + ν r + ( ur u ϕ cos. (f Further substituting ( into the equilibrium equations (, one has three Navier s equations in local coordinates ( u r ( ν r + u r r r u r + ( ν u r r r + u r r ( ν u r + u ϕ r + ( ν u r r cos u sin ( ν r ( ur r u r + ( ν u ϕ cos ( ν(u r cos u sin cos +( ν u r u r r r + ( ν u r r + + sin =, (a ( u + ( ν r ( u ϕ u ( ν r ( ur r u r + ( ν u r + ( ν u r + u r r cos sin + ( ν u ϕ sin + u r r u + ( ν u r r +( ν(u r cos u sin sin =, (b ( u ϕ ( ν r + u r r + + u ϕ r r + u ϕ r u r + u r + ( ν r ( uϕ r cos + u ϕ r sin + ( ( ν u ϕ ur + ( ν cos u sin ( νu ϕ =, (c 9.tex; /9/99; 7:8; p.
Eigenfunction expansion for penny-shaped and circumferential cracks 9. Eigenfunction expansion for penny-shaped crack By the assumption of free surface traction on the crack free surfaces, we have the boundary conditions for the penny-shaped crack, σ = σ r = σ ϕ = for =±π. (7 For simplicity, the displacements, the radius r and the function are non-dimensionalized, such that ū r = u r a, ū = u a, ū ϕ = u ϕ a, r = r a and = a. (8 Further, the functions /, / and / are expanded in ( r cos by using the binomial theorem to = ( r cos k, (9a = k= (k + ( r cos k, (9b k= = k= (k + (k + ( r cos k. (9c It has been shown by Li (988 that the displacement can be written by double summation series ū r = r λm+n U n, (a m= n= ū = r λm+n V n, m= n= ū ϕ = r λm+n W n, m= n= (b (c where U n,v n and W n are some functions of and ϕ, the solution of the eigenvalue problem gives λ m =±m/, m =,,,... ( The negative values of m have been excluded in ( so that the boundedness conditions of the displacements are not violated as r. Therefore ( can be reduced to ū r = r n/ f n (, ϕ, n= (a 9.tex; /9/99; 7:8; p.
A.Y.T. Leung and R.K.L. Su ū = r n/ g n (, ϕ, n= ū ϕ = r n/ h n (, ϕ. n= (b (c Navier s ( become ( n ( νf n ( ν f n + ( n 8ν g n { = ( cos k (n k ( νcos f n k ( νsin f n k k= + ( n + k νsin g n k + (n k h n k +(k + (ν (cos f n k sin g n k cos ( νcos h n k } + ( ν f n k, (a ( n ( νg n ( + n ν f n 8( g n ν = { ( cos k k= ( + (n k ( νcos g n k + cos f n k ( ν sin f n k ( νsin g n k + h n k +(k + (ν (sin g n k cos f n k sin +( νsin h n k } + ( ν g n k, (b ( ( ν n h n + h n { = ( cos k (n k f n k + g n k k= +(n k( νcos h n k 9.tex; /9/99; 7:8; p.
Eigenfunction expansion for penny-shaped and circumferential cracks ( νsin h n k + (k + ( ( ν cos f n k sin g n k } ( ν h n k + ( ν h n k. (c Boundary conditions (7 become ( ( ν + n ( ν g n f n + ν ν = ( cos k cos f n k sin g n k + h n k, (a k= f n + ( + ng n =, h n = ( cos k gn k k= (b + sin h n k, (c for =±π. The solution of ( is composed of two parts, the complementary functions and the particular integrals. The complementary functions can be evaluated by considering the homogeneous parts of ( and making use of the stress free boundary conditions. We have f c n g c n = a( n cos +a n ( sin = a( n h c n = a( n where a (i n sin +a n ( cos ( + n ( + n ( + n ( + n { + ( n ( n 8ν + ( n ( n + n cos ( n ( n 8ν + ( n n sin ( + n 8ν + ( n + n sin ( + n 8ν + ( n n cos cos ( n + ( n ( n ( n sin, (a, (b ( } n, (c = a n (i (ϕ(i =,, are coefficient functions to be determined. Utilizing the recurrence relationship of (, we determine the particular integral up to the th term in r. We give the general solutions here after adding the complementary functions and particular integral. For n = f = a ( cos + a ( sin, (a 9.tex; /9/99; 7:8; p.7
A.Y.T. Leung and R.K.L. Su g = a ( sin + a ( cos, (b h = a (. (c For n = f = a ( cos ( ( 8νcos +a( sin ( 8ν ( sin, (7a g = a ( sin ( (7 8νsin +a( cos (7 8ν ( cos, (7b h = a ( sin. (7c For n = f = a ( (cos + ν ν(a( + a (, (8a g = a ( sin + a (, (8b h = a ( cos a ( sin. (8c For n =. f = a ( cos ( + ( 8νcos +a( { ( 8ν + a ( cos ( ( 9ν + 8ν sin ( 8ν (+ cos cos ( 8ν +a ( sin ( (7 + 9ν 8ν sin 8 a ( a ( ( + ν ( ν sin }, (9a g = a ( sin ( (9 8νsin +a( cos (9 8ν ( cos { ( 8ν + a ( sin (+ ( 9ν + 8ν sin ( 8ν +a ( cos ( (79 9ν + 8ν cos 8 } cos, (9b 9.tex; /9/99; 7:8; p.8
Eigenfunction expansion for penny-shaped and circumferential cracks ( (7 8ν h = a ( sin ( + a ( cos ( + cos +a ( sin + a( sin. (9c For n = f = a ( ( ν cos + cos a ( ( ν( + ν cos a ( + a ( sin + ( νsin ( + ν ν sin + a( cos +a ( ( + ν( + ν cos + a ( g = a ( ( ν sin + a ( a ( ( ν( ν ( + ν( ν sin sin a ( a ( ( + ν ν a ( sin, ( + ν + ν ν sin + a( cos, + a ( cos ( νcos ( + ν ( ν ν cos + a( sin a ( ν cos sin (a (b h = a ( cos (a ( a ( + a( + a( sin + a( sin + a( a(, (c where the primes denote the derivatives with respect to ϕ. The coefficient functions and their derivatives are coupled through the equilibrium. The coupling effects diminish when approaching to the crack front where the singular stress fields are dominant. The determination of the coefficient functions including both the singular and regular terms requires iterative processes.. Axisymmetrical penny-shaped crack In the case of axisymmetric penny-shaped cracks, all the derivatives of ( with respect to ϕ are equal to zero. The coefficient functions become independent on ϕ. Therefore, Navier s equations ( can be simplified to ( n ( νf n ( ν d f n d + ( n 8νdg n d = ( cos {(n k k ( νcos f n k k= 9.tex; /9/99; 7:8; p.9
A.Y.T. Leung and R.K.L. Su + ( n + k νsin g n k ( νsin df n k d } +(k + (ν (cos f n k sin g n k cos, (a ( n ( νg n ( + n ν df n d 8( g n νd d = { ( cos k k= ( n h n + d h n d = { ( cos k k= ( + (n k ( νcos g n k + cos df n k d ( ν sin f n k ( νsin dg n k d } +(k + (ν (sin g n k cos f n k sin, (b (n kcos h n k sin dh } n k (k + h n k. (c d The boundary conditions become ( ( ν + n ( ν dg n f n + ν ν d = ( cos k cos f n k sin g n k, (a k= df n d + ( + ng n =, (b dh n d = ( cos k sin h n k, (c k= for =±π. According to Navier s equations ( and boundary conditions (, we find that mode III is completely independent of mode I and mode II. Therefore the eigenfunction expansion series for mode III can be solved separately. Furthermore, due to the axisymmetric condition, the displacements are no longer depending on ϕ. The solutions of modes I, II and mode III are solved and are expressed as 9.tex; /9/99; 7:8; p.
Eigenfunction expansion for penny-shaped and circumferential cracks For n = f = a ( cos + a ( sin, (a g = a ( sin + a ( cos, (b h = a (. (c For n = f = a ( cos ( ( 8νcos +a( sin ( 8ν ( sin, (a g = a ( sin ( (7 8νsin +a( cos (7 8ν ( cos, (b h = a ( sin. (c For n = f = a ( (cos + ν νa(, (a g = a ( sin + a (, (b h = a ( cos. (c For n =. f = a ( cos ( + ( 8νcos +a( ( 8ν +a ( cos ( ( 9ν + 8ν sin ( 8ν (+ sin cos ( 8ν +a ( sin ( (7 + 9ν 8ν sin 8, (a g = a ( sin ( (9 8νsin +a( cos (9 8ν ( cos ( 8ν a ( sin (+ ( 9ν + 8ν sin ( 8ν +a ( cos ( (79 9ν + 8ν cos 8, (b h = a ( sin (+ a( sin. (c 9.tex; /9/99; 7:8; p.
A.Y.T. Leung and R.K.L. Su For n = f = a ( ( ν cos + cos + a ( sin + ( νsin a ( ( ν( + ν cos + a ( g = a ( ( ν sin sin ( + ν + ν ν sin + a( cos, + a ( cos ( νcos (7a +a ( ( ν( ν h = a ( cos a ( For n = f = a ( +a ( +a ( +a ( +a ( + a( sin a ( cos ( 8ν (7+ cos ( ( + ν ν ν cos a( sin, (7b. (7c + a ( sin ( 8ν (7+ sin 7 ( ( + νcos ( + ν 8ν cos ( 7 ( 8νsin ( + ν 8ν sin ( ( + 8ν cos ( + ν + ν 9ν cos ( ( 8νcos ( ( 8 + 79ν + 89ν sin +(88 7ν 78ν 9ν sin ( (7 νsin (, (8a g = a ( sin ( 8ν (7 + sin ( + a ( cos ( 8ν (7 cos 7 ( +a ( +a ( +a ( ( νsin + (9 9ν + 8ν sin ( 7 ( 8νcos (9 9ν + 8ν cos ( ( + 88ν 8ν sin +(9 ν ν + 9ν sin ( 9.tex; /9/99; 7:8; p.
+a ( h = a ( sin For n = f = a ( Eigenfunction expansion for penny-shaped and circumferential cracks 7 +( 8νsin ( ( + ν + 89ν cos +(99 8ν + ν + 9ν cos (7 νcos, (8b ( a( sin + a( (8sin sin (. (8c (cos ν cos + a( (sin ν sin +a ( +a ( 9 ν ( ν cos a ( ( ν sin ( ν 7 7 + ν + (8 + 9ν + ν cos a ( ( + ν sin +a ( ( + ν 7 + ν + (8 + 9ν + ν cos, (9a g = a ( ( ν sin + sin +a ( +a ( +a ( h = a ( cos a ( For n = 7 f 7 = a ( 7 +a ( +a ( +a ( + a ( cos ( νcos 8 ( ν ν sin + a ( ( ν ( νcos ( + ν ( ν( + ν( + νsin + a( ( νcos ( ν( + ν( + νsin, (9b 88 cos + a( cos a( cos. (9c cos ( + 8ν (9 cos ( + a ( 7 sin ( + 8ν (9 sin 9 ( ( + 8νcos ( ν 8ν cos ( 7 ( + νsin (77 ν 8ν sin ( 8 (79 ν 89ν cos (7 + ν cos ( + (9 ν 978ν 9ν cos ( 9.tex; /9/99; 7:8; p.
8 A.Y.T. Leung and R.K.L. Su +a ( +a ( +a ( 8 (98 87ν 8ν sin + ( + 9ν sin (+ (9 ν ν 9ν sin ( ( + ν 8ν ν cos +(7 ν ν cos ( +(778 + 8ν 7ν 8ν ν cos ( (787 + νcos (7 78 ( 78 + 97ν + 9ν + 88ν sin + (97 9ν 87ν sin ( ( 97ν 7ν 9ν ν sin ( (7 + νsin (7, (a g 7 = a ( 7 sin ( 8ν (9 sin ( +a ( +a ( +a ( +a ( a ( a ( + a ( 7 cos ( 8ν (9 cos 9 ( ( + 8νsin + (99 9ν 8ν sin ( 7 (7 + νcos (79 9ν + 8ν cos ( 8 (98 89ν 89ν sin (787 ν sin ( ( 998ν + ν + 9ν sin ( 8 ( 88 + ν + 8ν cos (7 9ν cos ( + (78 9ν + 88ν + 9ν cos ( (8 ν + 8ν + ν sin +(8 νsin ( +( + 7ν 8ν + 9ν + ν sin ( +(787 νsin (7 78 (77 79888ν + 8ν + 88ν 9.tex; /9/99; 7:8; p.
h 7 = a ( 7 sin For n = 8 +a ( f 8 = a ( 8 +a ( +a ( +a ( +a ( Eigenfunction expansion for penny-shaped and circumferential cracks 9 cos (99 888ν 87ν cos ( +(888 9ν 8ν + 78ν + ν cos ( (7 νcos (7, (b (7 a( sin ( a( ( sin sin ( sin 9 sin (+ sin (. (c ( + ν cos cos + a ( ( + ν 8 sin sin 8 ( + νcos ( ν ν cos ( + νsin ( ν ν sin 7 ( ν ν cos + ( ν ν ν cos ( ν ( + νsin + (7 + ν + ν sin +a ( ( ν 7 (8 + ν + ν cos + (89 + 9ν + 9ν + 8ν cos +a ( ( + ν 88 ( + νsin + (7 + ν + ν sin +a ( ( + ν (8 + ν + ν cos + (89 + 9ν +9ν + 8ν cos, (a g 8 = a ( (7 ν 8 sin sin + a ( (7 ν 8 cos cos +a ( +a ( +a ( a ( 8 ( + νsin + (7 ν + ν sin ( νcos ( ν + ν cos 7 (7 ν ν sin ( 8ν + ν + ν sin ( ν ( + νcos ( ν ν cos 9.tex; /9/99; 7:8; p.
A.Y.T. Leung and R.K.L. Su +a ( ( ν 7 ( 8 + ν + ν sin ( ν 8ν 8ν sin +a ( ( + ν 88 ( νcos + ( ν ν cos a ( ( ν (8 ν ν sin h 8 = a ( 8 cos a ( a ( 8 +( ν 8ν 8ν sin, (b cos + a( ( + cos + a( ( + cos ( + cos. (c 8 By substituting ( into (, we can evaluate all the stresses distribution near the crack tip. The singular stress components are given as follows r / µ σ r ( = a ( (cos ( cos + a( (sin ( sin, (a r / µ σ ( r / = a ( (cos ( + cos µ σ ϕ ( = ν(a ( cos + a( r / µ σ ( r / µ σ ( ϕ r / µ σ ϕr ( r = a ( (sin ( + sin sin a( (sin ( + sin, (b, (c + a( (cos ( + cos, (d = a( cos, (e = a( sin. (f The first order displacement and stress distributions near the crack tip are found to be similar to the three-dimensional case (Hartranft and Sih, 99. The relationships between the stress intensity factors (K I,K II and K III and the coefficient functions are a a ( = π µk I, (a a a ( µk II = π and a ( = a π µk III. (b (c 9.tex; /9/99; 7:8; p.
Eigenfunction expansion for penny-shaped and circumferential cracks Figure. Geometry and coordinate systems of circumferential crack. Table. Sign for coefficients of axisymmetrical circumferential crack. n a (i a (i a (i a (i a (i a (i a (i a (i 7 a (i 8 + + + + + + + + + + 7 + + 8 + + +. Axisymmetrical circumferential crack Considering the local coordinates (r,,ϕat the front of crack border as shown in Figure in which a is the radius of the crack border, one has the coordinate transformation from the local to rectrangular coordinate system as x = a cos ϕ + r cos cos ϕ, x = r sin, x = a sin ϕ r cos sin ϕ. (a (b (c Following the same procedure as described in the previous sections, one can solve the displacement functions. The displacement functions are exactly similar to ( except that the signs of coefficients should be changed according to Table. Although the derivation of series ( for axisymmetric penny-shaped and circumferential cracks is based on the assumption of infinite axisymmetric solid, for finite size components such as pipes and pressure vessels, we can address the problem (Leung and Su, 99; 99a, b; 998; Leung and Wong, 989. The singular crack tip region is modeled by the generalized stiffness element whereas the 9.tex; /9/99; 7:8; p.7
A.Y.T. Leung and R.K.L. Su regular domain and the complex boundary conditions are modeled by conventional finite elements. We extend two-levels finite element method to solve the axisymmetric crack problems in Leung and Su (989.. Conclusion We have extended the eigenfunction expansion method, used previously for analyzing twoand three-dimensional crack problems, to the penny-shaped and circumferential crack problems. The three displacement functions valid for traction-free crack face boundary conditions are derived in close form. Only the first order displacement and stress distribution are similar to the three-dimensional straight plane crack for the purpose of defining the stress intensity factors. The coefficients of the higher order terms are coupled through the equilibrium equations. The coupling effects diminish near the crack tip. The displacement functions derived here are used for formulating the associated singular elements for axisymmetric crack problems. Together with the two-level finite element method, we can find the stress intensity factors for cracks in finite pipes and pressure vessels (Leung and Su, 998. References Hartranft, R.J. and Sih, G.C. (99. The use of eigenfunction expansions in general solution of three-dimensional crack problems. Journal of Mathematics and Mechanics 9(, 8. Leung, A.Y.T. and Su, R.K.L. (99. Mode I crack problems by fractal two-level finite element methods. Engineering Fracture Mechanics 8(, 87 8. Leung, A.Y.T. and Su, R.K.L. (99a. Body-force linear elastic stress intensity factor calculation using fractal two-level finite element method. Engineering Fracture Mechanics (, 879 888. Leung, A.Y.T. and Su, R.K.L. (99b. Mixed mode two-dimensional crack problems by fractal two-level finite element method. Engineering Fracture Mechanics (, 889 89. Leung, A.Y.T. and Su, R.K.L. (998. Two-level finite element study of axisymmetric cracks. International Journal of Fracture (To appear. Leung, A.Y.T. and Wong, S.C. (989. Two-level finite element method for plane cracks. Communications in Applied Numerical Methods, 7. Li, Y. (988. Crack tip stress and strain fields for surface cracks in D body and the calculation of stress intensity factors. Chinese Science A 8, 88 8 (in Chinese Language. Murthy, M.V.V., Raju, K.N. and Viswanath, S. (98. On the bending stress distribution at the tip of a stationary crack from Reissner s theory. International Journal of Fracture 7(, 7. Sosa, H.A. (98. On the Analysis of Bars, Beams and Plates with Defects. Ph.D. Thesis, Stanford University. Williams, M.L. (97. On the stress distribution at the base of a stationary crack. ASME Journal of Applied Mechanics 9. Williams, M.L. (9. The bending stress distribution at the base of a stationary crack. ASME Journal of Applied Mechanics 8, 78 8. 9.tex; /9/99; 7:8; p.8