PSI - Issue 25

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

5

Marco Maurizi et al. / Procedia Structural Integrity 25 (2020) 268–281

272

Φ = r λ 1 + 1 [ A

λ 2 + 1 [ A

s cos (( λ 1 + 1) θ ) + B s cos ((1 − λ 1 ) θ )] + r

a cos (( λ 2 + 1) θ ) + B a cos ((1 − λ 2 ) θ )]

(6) (7)

λ 3 s cos ( λ

λ 3 a sin ( λ

w = D s r

3 s θ ) + D a r

3 a θ ) ,

where λ 1 and λ 2 are the eigenvalues corresponding to Mode I and II, respectively; λ 3 s and λ 3 a are the symmetric and anti-symmetric eigenvalues for the anti-plane shear mode (Mode III), respectively. The constants A s , B s , A a , B a , D s and D a have to be determined by imposing the boundary conditions. Specifically, supposing traction-free conditions on the V-notch flanks ( θ = ± γ in Fig. 1b) and recalling the expressions for the stress components in polar coordinates in relation with Φ and w :

∂ ∂ r −

∂ Φ ∂θ

∂ 2 Φ ∂ r 2

1 r

σ θθ =

τ r θ =

∂ u z ∂θ

z h

1 r

∂ w ∂θ

1 r

= G

τ θ z = G

∂ u z ∂ r

z h

∂ w ∂ r

τ rz = G

= G

,

(8)

two homogeneous systems of equations are obtained (Lazzarin and Zappalorto (2012)), correspondent to the sym metric and skew-symmetric terms. To obtain non-trivial solutions, the determinants are equaled to zero, and the fol lowing equations are deduced:

sin (2 λ 1 γ ) + λ 1 sin (2 γ ) sin ( λ 3 s γ ) = 0 sin (2 λ 2 γ ) + λ 2 sin (2 γ ) cos ( λ 3 a γ ) = 0 .

(9)

(10)

These two equations represent the eigenvalue problem for symmetric (Eq. (9)) and anti-symmetric (Eq. (10)) load ing. It is evident that the terms in square brackets for both equations are the classical in-plane eigenvalue problem (Williams (1952)), whereas the other terms are the out-of-plane eigenvalue equations (Mode III). Moreover, Eq. (9) and (10) match the Kotousov and Lew’s eigenvalue equations (Kotousov and Lew (2006)). Singularities on displace ment fields and strain energy density (SED) averaged in a small volume around the notch tip are not physically possible; hence, the condition Re( λ ) > 0 has to be imposed on the eigenvalues. Besides, singular stress fields are ob tained if and only if Re( λ ) < 1, which together with the previous condition gives rise to the constraint 0 < Re( λ ) < 1. Because λ 3 s = 2 λ 3 a = π / γ > 1 (for 0 < γ < π ), and from Eq. (8), the out-of-plane shear stress singularity occurs only for the skew-symmetric component. Therefore, the anti-plane Mode III shear stresses ahead the notch tip can be written as follows (see Appendix A for details):

K N III √ K N III √

r λ 3 a − 1 sin ( λ

3 a θ )

(11)

τ zr =

2 π

r λ 3 a − 1 cos ( λ

3 a θ ) ,

(12)

τ z θ =

2 π