Issue 75

R. Ince et alii, Fracture and Structural Integrity, 75 (20YY) 435-462; DOI: 10.3221/IGF-ESIS.75.30

a c







x a

  

  

2

a

x

c

c

c



,   



K

G

dx

(8)



Ic

a

d



a

a

c

c

0

Here, x refers to the vertical distance from the crack mouth of the specimen, and G is Green’s function. For instance, Tada et al. [20] derived the following Green’s function for an infinite strip as shown in Fig. 3c:

    

   

1.5

 3.52 1







4.35 5.28 1.3 0.3   











,  

0.83 1.76 1 1     

     







(9)

G



1.5

0.5

0.5













2

2





1









1

1

The results of the experiments on compact quasi-brittle samples by Ince [21] indicated that the fracture quantities ( K un Ic and K ini Ic ) of the double- K model can also be evaluated using the peak-load method.

c Ic K according to the inverse method a) distribution of cohesive stress at peak load b) linear function for  - w

Figure 3: Computing of

c) a body containing crack under wedge forces

D ERIVATION OF FRACTURE MECHANICS FORMULAS OF SNDB AND SNCB SPECIMENS

T

wo prominent energy methods based on the FEM, such as the J-integral and the crack closure integral (CCI) technique[22], which can also be successfully utilized for three-dimensional crack problems, were employed in this study. Although conventional methods commonly used in FEM crack problems, such as extrapolation methods based on stress matching, displacement matching, and the hybrid method, require a large number of fine elements near a crack for accurate results, the aforementioned methods do not have such refined mesh requirements. The J-integral is a contour integral evaluated along a path that encircles the crack tip, beginning and concluding at the surfaces of the crack. Since this integral is path-independent for any of the contours, it has the same value for any similar contour. The use of the CCI technique is not as widespread as the J-integral approach and is not embedded in many commercial FEM programs, but the implementation of the CCI technique is very simple compared to the J-integral method, as will be detailed below. The basic principles of the crack closure integral (CCI) technique can be derived using the fundamental principle of LEFM [1]. To determine the relationship between the energy release rate G and the fracture toughness K , it is necessary to account for the work of the closing forces ( P=  dx ), which corresponds to the initial state of the new crack profile when a crack of length a increases by a small amount (  a ), as shown in Fig. 4a. For a plate with a thickness of b , the energy released as a result of the small extension of the crack ( Gb  a ) must be equal to the work done by the internal forces that closed it. Accordingly, the following equation can be written:

1 0 2 a W vdx Gb a      

(10)

Here, v is the crack opening displacement (COD). The above work expression can easily be applied by running two-step computer numerical approaches, such as the finite element method, where the elastic body model is assumed to be

440