PSI - Issue 39

Paolo Livieri et al. / Procedia Structural Integrity 39 (2022) 194–203 Author name / Structural Integrity Procedia 00 (2019) 000–000

197

4

class C 2 with respect to ε, where 0 ≤ ε ≤ 1 is a parameter and 0 ≤ ψ ≤ 2 π is the angle. By means of the Taylor expansion, we have

K (0, ) O( ) 2 I α + ε

2

∂

(4)

K ( , )

ε α =

+ε

I

∂ε

π

(see reference [18] for the non-trivial detailed calculation of K (0, ) I α ∂ε ∂ ). In reference [18] we obtained the following approximation for the Oore-Burns integral (1) as a function of the angle α :

  

  

2 1

∑ +∞ −∞

(5)

in

O( ) 2

K ( , )

c E e n n

α

+ ε

ε α =

+ε

I

π

∂ α → α = S( ) R 0,

( ) α

where c n are the Fourier coefficients of the first order crack front position

in the sense that

∂ε

∑ +∞ −∞

α c e . The E n coefficients are independent of the homotopy R and are reported in Table 1 (for the complete formulae of E n coefficients see reference [18]). In general, by considering that an a -dilatation of Ω under uniform normal tension σ n produces factor a in the expression of K I , from (5) we are able to state the following final equation: α = in n S( )

  

  

a

σ

∑ +∞ −∞

( ) 2 ε

α b E e O in n n +

( , ) 2 α ε =

1

K

ε

+

n

(6)

I

π

a 1

R (0, )

∂

where b n are the Fourier coefficients of ( ) ∂Ω ε . Now let ρ = ρ ( α ) the polar equation of ∂Ω and consider the homotopy R( , ) 1 ( ( ) 1) − = + α ρ ε α ε . If Ω is a slight distortion of a disc of radius a under remote uniform tensile stress σ n , by choosing ε=1, the SIF turns out to be: α ε α ∂ → and R( , ) α ε describes the boundary of

   

   

+ a b 1

σ

∑ ≥ 2 n

in

≈ ( ) 2 α

α

K

n n b E e

+

0

n

2 (7) where, in this case, b n are the Fourier coefficients of ̅ ( ) . Eq. (7) is the first order approximation of the Oore-Burns integral (2). Finally, if we take into account the Fourier series in the form: ∑ ∑ ∞ = ∞ = + = + 1 1 0 sin ( ) cos ( ) ( ) n n n n q n p n b α α α ρ (8) with I π