PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

433

6

(   

l a u dA )

U

,

(14)

0

U

U

A ( ) U

U A is the beam cross-section area. By using (12), the strain energy

U u 0 is the strain energy density,

where

density is found as

Et

2 1





 U U u E z z e    2 2 2 2 0 2 n



,

(15)

where

h

h z   

.

(16)

2

2

2

In formula (15), U  is the curvature of the beam in the un-cracked portion, 2 z and axis and the coordinate of the neutral axis, respectively. Analogically to (13), the distribution of the time-dependent normal stress,

2 2 n z are the vertical centroidal

U  , along the thickness of the beam

in the un-cracked portion is written as

   Et  n U U E z z e    2 2 2 

.

(17)

In order to determine the curvatures, 2 2 n z , which are involved in the expressions for the strain energy densities, four equations are constituted in the following way. First, one equation is written by using the fact that the bending moment in the lower crack arm is zero D  and U  , and the neutral axes coordinates, 1 1 n z and

1  z dA D A  ( )

0



.

(18)

Further one equation is obtained by using the fact that the axial forces in the lower crack arm and the un-cracked beam portion are equal    ( ) ( ) U D A U A dA dA   . (19) One equation is composed by using the fact that the bending moment in the un-cracked beam portion is equal to the moment of the axial force in the lower crack arm with respect to the centre of the cross-section of the un-cracked beam portion h h z dA z dA 1   . (20)

     2 2

  



 

2

1

U

U ( ) A

D ( ) A

Finally, one additional equation is composed by expressing the displacement, F u , as a function of the curvatures of the lower crack arm and the un-cracked beam portion. For this purpose, the integrals of Maxwell-Mohr are applied.