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

1794

6

σ

σ

12 4

/

σ

σ

12 4

12

/

12

1.4

r=r/a=0.05 ^

r=r/a=0.25 ^

1.2

2

N=7,9,13,15,50

1

N=5,6

1.5

0.8

N=3,5,15,35,50,150

N=3,4

0.6

1

N=1,2

0.4

N=1,2

0.5

0.2

θ

θ

-1

2

3

0

2

3

1

0

-1

-2

1

-3

-2

-3

Fig. 6. Angular distributions of the stress components σ 12 near the crack tip z = b at di ff erent distances from the crack tip

Using the asymptotic presentation of the complex potential ϕ � 2 ( z ) at z = a and the Williams series expansion (1) one can derive the coe ffi cients of the series depending on the applied load σ ∞ 12 and the geometrical parameters a and b : a 2 2 k + 1 = σ ∞ 12 δ k / f 2 , 12 2 k + 1 ( θ = 0) , a 1 2 k = 0 , ∀ k > 0 . (11) The first seven coe ffi cients of the Williams series expansion at the crack tip z = a can be presented in the form

σ ∞ 12 √ 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 ∞ 12 √ 2 a σ ∞ 12 √ 2 320

α − 1

a 2

1 2 k = 0 , ∀ k > 2 ,

2 3 =

a 2

∞ 12 ,

a

a

1 = −

4 σ

,

2 =

a 2 2 a 2 b 2 c − 34 b 2 a 4 − 19 b 4 a 2 + 43 a 4 c + 5 a 6 + 3 b 4 c a 5 / 2 b 2 − a 2 5 / 2 , σ ∞ 12 √ 2 1792 − 5 b 6 c + 137 b 2 a 4 c + 11 b 4 a 2 c + 177 a 6 c + 13 b 6 a 2 + 7 a 8 − 113 b 2 a 6 − 227 b 4 a 4 a 7 / 2 b 2 − a 2 7 / 2 . The formulae (12) present the dependence of the coe ffi cients of the Williams series expansion a 2 k on the configuration of the cracked body and the load σ ∞ 12 . Similarly, for pure mode II the coe ffi cients of the Williams series expansion in the vicinity of the crack tip z = b + re i θ can be identified. For skew symmetric problem the complex potential ϕ � 2 ( z ) in the vicinity of the crack tip z = b has the form ϕ � 2 ( z ) = − i ( σ ∞ 12 / 2) ∞ n = 0 q n ( z − b ) n − 1 / 2 + i σ ∞ 12 / 2 , 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 , d 0 = b 2 − c , d 1 = 2 b , d 2 = 1 , η k = 0 , k > 2 , a k = l k ( b − a ) − (2 k + 1) / 2 , b k = l k ( a + b ) − (2 k + 1) / 2 , c k = l k (2 b ) − (2 k + 1) / 2 . (13) Substitution of the complex potential (13) into the Koloso ff – Muskhelishvili presentation (6) and the comparison with the Williams series expansion (1) result in the following expressions for the amplitude coe ffi cients a 2 k in the vicinity of the crack tip z = b a 2 2 k + 1 = σ ∞ 12 q k / f 2 , 12 2 k + 1 ( θ = 0) , a 1 2 k = 0 , ∀ k > 0 . (14) Using formulae (13), (14) one can find the angular distributions of the stress components in the vicinity of the crack tip z = b . The angular distributions of σ 12 at di ff erent distances from the crack tip are shown in Fig. 6 – 8. It can be seen that the domain in which the accuracy of the Williams solution expands with increase of the number of terms in the asymptotic expansion taken into account. a 2 7 = (12) 5 =