Issue 25

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 25 (2013) 94-101; DOI: 10.3221/IGF-ESIS.25.14

T HE KINKED CRACK MODEL Self-balanced microstress field Consider an infinite cracked plate described by the xy coordinate system in Fig. 1, exposed to remote tensile stress ( ) y   along the y-axis and shear stress ( ) xy   . Assume that material microstructural features create a self-balanced (residual) microstress field, which is characterized by a length scale, d , related to a characteristic material length, and amplitudes e.g. governed by material properties’ dispersion. Further let us assume that such a microstress field is a one-dimensional function (of the x - coordinate), defined by the following stress tensor:

   

   

   

   

x    

 





x d      

T 

xy

x,a

xy,a





(1)

(x)

f



y  



 





xy

xy,a

y,a

Without lack of generality, we describe the plane microstress field by taking into account the following two non-zero stress components: y a f(x / d)         and xy a f(x / d)         . An attempt to correlate the above self-balanced microstress to some heterogeneity features of the material microstructure is presented in Ref. [18].

K I

  

k I

K II

y

k II

2

2 

b

k I

k II

  

0

1

x

1 

a

l

K II

K I

Figure 1 : Nomenclature for the kinked crack in an infinite plate (y-axis of symmetry).

Approximate stress intensity factors in the kinked crack According to the present model, the central crack might kink as a result of both remote and microstess fields (see Fig. 1). As will be shown below, the local stress intensity factors (SIFs) at the crack tips ( I k and II k ) can be expressed as a function of those ( I K and II K ) of a straight crack having length equal to the projected length of the kinked crack [2-7]. The total values of SIFs defined with respect to the projected crack are the sum of two contributions (due to remote and microstress fields, respectively), that is: ( ) I I I ( ) II II II K K K K K K         (2) The remote SIFs are defined with respect to the projected crack of semi-length l , aligned with the x - axis (Fig. 1). Hence, under the uniform remote stresses ( ) ( ) y      and ( ) ( ) xy      , we have :

l

1

l



( ) K 2  

( ) 

( ) 









dx

l

I



2

2



0

l

x

(3)

l

1

l



( ) K 2  

( ) 

( ) 









dx

l

II



2

2



0

l

x

95