PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1237–1248 Author name / Structural Integrity Procedia 00 (2019) 000–000

1243 7

longitudinal crack of length, a , in the bar. The crack front is a circle of radius 1 R . The bar is clamped in its right hand end. Since the loading consists of an axial force, F , applied at the free end of the internal crack arm, the external crack arm is free of stresses.

Fig. 2. Geometry and loading of a cantilever with a longitudinal crack.

0 * 02  u . It is assumed that the modulus of elasticity is distributed in radial direction according to the

Therefore,

following cotangent law:

     

  

  

R





  1 E E p g cot



,

(24)

0

t

2

4

R

2

where

2 0 R R   .

(25)

In (24), 0 E is the value of modulus of elasticity in the centre of the cross-section, t p is a parameter that regulates the material inhomogeneity in radial direction. The longitudinal fracture behaviour of the cantilever shown in Fig. 2 is studied in terms of the strain energy release rate by applying (1). The axial force in the internal crack arm is written as (Fig. 2) N F  1 . (26) The equations obtained by substituting of (24) and (26) in (10), (11), (12) and (14) should be solved with respect to  , 1 q , 2 q and 3 q by using the MatLab computer program. Then, * 01 u is determined by substituting of  and (24) in (5). Formula (5) is applied also to calculate the complementary strain energy density in the bar cross-section ahead of