PSI - Issue 28

5

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

1230

R

2



2 2  

R dR T 

,

(14)

et

R

1

where et T is the torsion moment in the external crack arm. In order to derive et T , the rod is treated as a structure with one degree of internal static indeterminacy ( et T is taken as a redundant). Since the rod is made of non-linear elastic material, the statically indeterminate problem is solved by using the Castigliano’s theorem for structures which exhibit material non-linearity



et dT dU

0 

.

(15)

The complementary strain energy in the internal crack arm is written as

  0 R in U a u RdR  .   2 1 2

(16)

0

By using (11), the complementary strain energy density, 

in u 0 , in the internal crack arm is expressed as

2

n



1

n in

0  u B in



D

  in



,

(17)

2 1

n



where the distribution of shear strain, in  , is written as



qs

R

in  

,

(18)

R

1

for

1 0 R R   .

(19)

In (18), qs  is the shear strain at the periphery of the internal crack arm. The following equation for equilibrium of the cross-section of the internal crack arm is used to determine qs  :

R

1

 0

2 2  

R dR T 

,

(20)

in

in

where in  is the shear stress, in T is the torsion moment in the internal crack arm. By using the equation for equilibrium of torsion moments, one obtains