PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1226–1236 Author name / Structural Integrity Procedia 00 (2019) 000–000

1229

4

u    

u

,

(6)

0

0

where  is the shear stress,  is the shear strain, 0 u is the strain energy density. The non-linear mechanical behaviour of the material is treated by the following stress-strain relation (Lukash (1998)): n B D      , (7) where B , D and n are material properties. The distribution of B in radial direction is described by the following exponential law:

2 R p R

B B e 

,

(8)

0

where

2 0 R R   .

(9)

In (8), 0 B is the value of B in the centre of the cross-section of the rod, p is a parameter which controls the material inhomogeneity in radial direction. The strain energy density is equal to the area enclosed by the stress-strain curve. Thus, by integrating of (6), one obtains

2 1

1

2   

1

n D 



u B

.

(10)

0

1

n



By substituting of (7) and (10) in (6), one derives

2

n



1

n D 

 u B



 

.

(11)

0

2 1

n



The distribution of  in radial direction is treated by applying the Bernoulli’s hypothesis for plane sections since rods of high length to diameter ratio are under consideration in the present paper. Thus, the distribution of  is expressed as

R et 2    , R

(12)

where

2 1 R R R   .

(13)

In (12), et  is the shear strain at the periphery of the external crack arm. The following equation for equilibrium of the cross-section of the external crack arm is used to determine et  :