Issue 66

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

tude varies linearly along the crack, becoming zero at the origin of the reference system (mid-point of the crack). There fore, these terms are responsible for: (i) a rigid body rotation λ , and (ii) a change of the length of the crack, leading to an intermediate deformed position of the crack, denoted as ‘false-crack’ [13, 14] (Fig.2). Then, applying the elliptic terms ( 1,e u  , 1 ,e v  ) of the displacement components to the facing points of the ‘false-crack’, one obtains the final shape of the deformed crack, which (as already mentioned) is always an ellipse, either open or over lapped. The ‘elliptic’ displacement vector 1,e 1,e 1,e υ (u ,v )    forms a constant angle ω with respect to the y-axis of the reference system. Angles λ and ω are given as follows [13, 14]:

c(1 k)sin2 β 

(1 k)sin2 β 





tan λ

, tan ω

(11)

1 c(1 k)cos2 β  

1 k (1 k)cos2 β   

As an example, the above mechanism of crack deformation is shown in Fig.2, for an infinite plate made of Plexiglas (with Young’s modulus E=3.2 GPa and Poisson’s ratio ν =0.36), weakened by a ‘mathematical’ crack of length 2 α =0.10 m and loaded biaxially at infinity by a pair of principal stresses ( σ ∞ , k σ ∞ ). Assigning numerical values equal to σ ∞ =+1 GPa (not realistic but it provides clarity to the figure) and k=0.2, an open crack/ellipse is obtained (see Fig.2a). On the contrary, for the pair σ ∞ = –1 GPa and k=0.1, the crack lips overlap each other (see Fig.2b). It is critical to note at this point that the original crack tips of the undeformed crack (i.e., points ± α ) are no longer the ‘tips’ of the deformed crack/ellipse (either open or overlapped). In other words, points ± α΄ are not the end-points of the major axis of the deformed crack/ellipse.

 

k 0.2     

upper( )  lip open

0.02

40  

Natural deformation

1,e v 



0.01

1,e u 

1,e u 

1, in 

( )  ( ) 

undeformed 



0.00



crack

1, in 



1,e v 

-0.01 y - axis [m]



lip open lower ( ) 

(a)

Ellipse's axes

-0.02

-0.06

-0.03

0.00

0.03

0.06

 

x - axis [m]

k 0.2     

k 0.1     

 

lower( ) 

0.02

lip overlapping



40  

0.01

1,e v 

1, in 

( )  ( ) 



0.00

undeformed 



1,e u 

crack

1, in 

1,e u 

-0.01 y - axis [m]

Unnatural deformation

1,e v 





upper( ) 

(b)

lip overlapping

-0.02

Ellipse's axes

-0.06

-0.03

0.00

0.03

0.06

 

x - axis [m]

k 0.1     

Figure 2: The elliptical shape of the deformed crack for (a) open and (b) overlapping crack lips.

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