PSI - Issue 2_A

Larisa Stepanova et al. / Procedia Structural Integrity 2 (2016) 1789–1796 Stepanova L.V., Roslyakov P.S. / Structural Integrity Procedia 00 (2016) 000–000

1791

3

A)

B)

σ

σ

22 4

12 4

σ

σ

σ

σ

α

α

22 4

12 4

22 4

12 4

-a

-a

a

a

-b

-b

b

b

σ

σ

22 4

12 4

Fig. 1. Two collinear cracks of equal lengths in the infinite plane under remote loading

3. Analytical determination of the coe ffi cients of the complete Williams asymptotic series for the stress field in the vicinity of the mode I crack tips Exact stress solutions for the problem investigated are expressed in terms of complex variable z = x 1 + ix 2 in Eq. (4) for mode I loading and Eq. (6) for mode II loading. On the other hand, the Williams series expansion (1) describes the stress field at the vicinity of a crack-tip as a series involving real polar variables r and θ. When a polar coordinate system is considered at the crack tip z = b : z ( r , θ ) = b + re i θ , x 2 = rsin θ one can obtain: ϕ � 1 ( z ) = ( σ ∞ 22 / 2) ∞ n = 0 q n ( z − b ) n − 1 / 2 + ( α − 1)( σ ∞ 22 / 4) , q n = n k = 0 p k d n − k , p n = n k = 0 c k e n − k , e n = n k = 0 a k b n − k , a k = l k ( b − a ) − (2 k + 1) / 2 , b k = l k ( b + a ) − (2 k + 1) / 2 , c k = l k (2 b ) − (2 k + 1) / 2 , l k = ( − 1) k | 2 k − 1 | !! / (2 k k !) , d 0 = b 2 − c , d 1 = 2 b , d 2 = 1 , d k = 0 ∀ k > 2 . (7) Using the Koloso ff – Muskhelishvili formulae (3) and comparing the expansion (7) with the Williams series expansion (1) one can find the expressions for all the coe ffi cients a 1 k of the Williams series expansion a 1 2 k + 1 = σ ∞ 22 q k / f 1 , 22 2 k + 1 ( θ = 0) , a 1 2 = ( α − 1) σ ∞ 22 / 4 , a 1 2 k = 0 , ∀ k > 1 . The first seven coe ffi cients of the Williams series expansion of the stress field in the vicinity of the crack tip z = b in the infinite plane medium with two cracks have the form σ ∞ 22 √ 2 1792 − 5 a 6 c + 137 a 2 b 4 c + 11 a 4 b 2 c + 177 b 6 c + 13 a 6 b 2 + 7 b 8 − 113 a 2 b 6 − 227 a 4 b 4 b 7 / 2 b 2 − a 2 7 / 2 . An analogous approach to the above procedure allows us to find the asymptotic expansion for the complex potential ϕ � 1 ( z ) in the vicinity of the crack tip z = a : z = a + re i θ , x 2 = r sin θ ϕ � 1 ( z ) = σ ∞ 22 / 2 ∞ n = 0 δ n ( z − a ) n − 1 / 2 + ( α − 1) σ ∞ 22 / 4 , δ n = n k = 0 α k χ n − k , χ n = n k = 0 β k ξ n − k , ξ n = n k = 0 ζ k η n − k , η 0 = a 2 − c , η 1 = 2 a , η 2 = 1 , η k = 0 , k > 2 , α k = l k (2 a ) − (2 k + 1) / 2 , β k = l k ( b − a ) − (2 k + 1) / 2 , ζ k = l k ( b + a ) − (2 k + 1) / 2 . (8) The formulae (8) present the dependence of the coe ffi cients of the Williams series expansion on the configuration of the cracked body and the load σ ∞ 22 . Substitution of the complex potential (8) into the Koloso ff – Muskhelishvili presentation (3) and the comparison with the Williams series expansion (1) result in the following expressions for the amplitude coe ffi cients a 1 k : a 1 2 k + 1 = σ ∞ 22 δ k / f 1 , 22 2 k + 1 ( θ = 0) , a 1 2 = ( α − 1) σ ∞ 22 / 4 , a 1 2 k = 0 , ∀ k > 1 . The first seven coe ffi cients of the Williams series expansion a 1 k can be written in the expanded form a 1 1 = σ ∞ 22 √ 2 b b 2 − c √ b 2 − a 2 , a 1 2 = ( α − 1) σ ∞ 22 / 4 , a 1 2 k = 0 , ∀ k > 1 , a 1 3 = σ ∞ 22 √ 2 24 3 b 4 − 7 a 2 b 2 + 5 b 2 c − a 2 c b 3 / 2 b 2 − a 2 3 / 2 , a 1 5 = − σ ∞ 22 √ 2 320 2 a 2 b 2 c − 34 a 2 b 4 − 19 a 4 b 2 + 43 b 4 c + 5 b 6 + 3 a 4 c b 5 / 2 b 2 − a 2 5 / 2 , a 1 7 =

σ ∞ 22 √ 2 24

a 2 − c √ b 2 − a 2

3 a 4 − 7 a 2 b 2 + 5 a 2 c − b 2 c a 3 / 2 b 2 − a 2 3 / 2 ,

σ ∞ 22 √ 2 a

α − 1

a 1

1 2 k = 0 , ∀ k > 1 , a 1

a 1

∞ 22 ,

a

1 = −

3 = −

4 σ

,

2 =