Issue 66

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 66 (2023) 233-260; DOI: 10.3221/IGF-ESIS.66.15

Upon differentiation of the expressions of Eqn.(26) and (27) it is obtained that:

 



(1 κ ) σ 

z











δ 1 k (1 k)cos2 β i τ (1 k)sin2 β        





Φ (z)

1



(28)

2 z α     2

2

8 κ

 



(1 κ ) σ 

z











δ 1 k (1 k)cos2 β i τ (1 k)sin2 β        





Ω (z)

1



(29)

2 z α     2

2

8

Substituting in Eqn.(5) from Eqns.(28) and (29), and letting z tend to x on the upper and lower crack lips, the stresses on the crack lips due to the ‘inverse problem’ read as:

2 (1 κ ) σ 

2 (1 κ ) σ 

x









δ 1 k (1 k)cos2 β    



τ (1 k)sin2 β  

σ





(30)

yy,2

8 κ

8 κ

2

2



α

x

2 (1 κ ) σ 

2 (1 κ ) σ 

x









τ (1 k)sin2 β  

δ 1 k (1 k)cos2 β    

τ





(31)

xy,2

8 κ

8 κ

2 α x 

2

The stresses provided by Eqns.(30) and (31) correspond to the inversion of overlapping and could stand for the contact stresses developed on the crack lips, prohibiting overlapping. However, as it will be proven in next section, only specific portions of the above determined stresses may be considered as contact stresses. The contact stresses on the lips of the crack To clarify the last point raised in previous section, and in order to obtain the contact stresses developed on the crack lips, consider the problem of the infinite cracked plate under uniaxial compression normal to the crack (i.e., – σ ∞ , k=0, β =90 ο ), which produces the unacceptable overlapping crack-ellipse L (Fig.6a). In this case, the ‘inverse problem’ of Fig.6b should be superimposed to the ‘initial problem’ of Fig.6. Namely, since there is no relative displacement of the crack lips along the x-direction in the ‘initial problem’, τ should be set, according to the present approach, equal to zero, while δ should be set equal to unity. In this case the boundary displacements on the crack in the ‘inverse problem’ read as: 2 2 1,e u 0, v 1 v       (Fig.6b). Clearly, such boundary displacements on the crack do not permit Poisson’s shrinkage along x-direction, something achieved by the generation of tangential stresses on the crack in the ‘inverse problem’, reading (according to Eqn.(31)) as:   2 o 2 2 xy,p (1 κ ) σ x τ δ 1 k (1 k)cos2 β , (with δ 1, k 0, β 90 ) 8 κ α x             (32)

y

y

y

 

L 

( ) 

( ) 

L

( )  ( ) 

( )  ( ) 









x

x

x

O

O

( ) 

L

 xy,p 

( ) 

2 2 1,e u 0, v 1 v      

 

Figure 6: (a) The unacceptable overlapping, (b) the boundary displacements imposed to the crack lips in the ‘inverse problem’ prevent ing overlapping, (c) the ‘parasitic’ tangential stresses. (a) (b) (c)

242