Issue 49

G. Meneghetti et alii, Frattura ed Integrità Strutturale, 49 (2019) 53-64; DOI: 10.3221/IGF-ESIS.49.06

I NTRODUCTION

D

ealing with cracks under in-plane mixed mode I+II loading conditions, according to Williams [1], the local stress fields expressed in terms of Cartesian stress components as functions of the polar coordinates ( r , θ ), with origin at the crack tip (Fig. 1a), can be written in the following form:

θ 1 2 2 θ 1 2 2

3 2

θ 1 2 2

3 2    

        

 

        

        

           

  - sin θ sin θ    

   

 

cos

-2sin - sin θ cos θ    

3 2      

1/2 1 +T 0 +O(r ) 0              

xx σ     yy       xy τ

K

K

1 2

3 2

  sin θ cos θ    

 

I

II

(1)

σ = cos + sin θ sin θ +  

2πr

2πr

 

     

1 2

3 2

θ 1 2 2

3 2

  sin θ cos θ    

      

  sin θ sin

θ

cos

 

 



With reference to cracks under mode III loading conditions, the asymptotic, singular stress distributions have been determined by Qian and Hasebe [2], following Williams’ procedure [1]. The local stress field in terms of Cartesian stress components as functions of the polar coordinates ( r , θ ), with origin at the crack tip (Fig. 1b), is the following:

θ -sin   

  

2     

 

xz       yz τ τ

III K=

1/2 +O(r )

(2)

θ 2πr cos 2   

 

      

σ nom

(a)

(b)

F

M t

τ nom

σ yy

σ yy

τ yz

y

τ xy

τ xy

y

σ xx

σ xx

σ zz

x

r

x

τ xz

θ

τ nom

z

r

θ

L = W

2α=0°

L = D

2a

a

a

crack bisector

crack bisector

D = 10 ·a

M t

W = 10 ·2a

F

  Figure 1 : Cartesian stress components and polar coordinates with origin at the crack tip for (a) in-plane mixed mode I+II crack problem and (b) out-of-plane mixed mode I+III crack problem. The mode I and mode II Stress Intensity Factors (SIFs) can be defined according to Gross and Mendelson [3] by means of Eqns. (3) and (4), respectively.   0.5 0 0 2 lim I yy r K r             (3)   0.5 0 0 2 lim II xy r K r             (4)

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