PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1212–1225 Author name / Structural Integrity Procedia 00 (2019) 000–000

1219

8

2 1

01  u



.

(31)

By using the Hooke’s law, the stress is expressed as ( ) 0 G t    .

(32)

The strain energy density is written as

0 2 1  u G t  01 ( )

2

.

(33)

The distribution of the shear strains along the radius of the beam cross-section is treated by applying the Bernoulli’s hypothesis for plane sections since beams of high length to diameter ratio are under consideration in the present study. Thus, the distribution along the radius of the cross-section of the internal crack arm is expressed as

R pr    ,

(34)

1 R

3

where

3 0 R R   .

(35)

In (34), 1 pr  is the shear strain at the periphery of the internal crack arm. The strain energy cumulated in the un-cracked beam portion is written as

l R 4   

 2 02

U

u

RdxdR

,

(36)

2

0

a

where the strain energy density, 02 u , is found by formula (33). For this purpose,  is replaced with unc  . Here, unc  is the shear strain in the un-cracked beam portion. The distribution of unc  in radial direction is obtained as

R

unc  



,

(37)

2 R

pr

4

where

4 0 R R   .

(38)

In (37), 2 pr  is the shear strain at the periphery of the un-cracked beam portion. It should be mentioned that the shear strains at the peripheries of the internal crack arm and the un-cracked beam portion are determined by using the equations for equilibrium of the shear stress resultants in cross-sections of the beam. The shear stress, unc  , in the un-cracked beam portion is found by replacing of  with unc  in (32).