PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 86–96 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

91

6

L u 0 , is equal to the area enclosed by the stress-strain curve. Thus, by integrating the stress

The strain energy density, strain relation (3), one derives

m

1

+



0 m u K L =



.

(25)

1

+

By combining of (3), (11) and (24), one obtains

m

1

+



* m u K m L 0 =



.

(26)

1

+

It should be mentioned that the strain energy density and the complementary strain energy density are time-dependent since  is function of time. The distribution of strains is analyzed by applying the Bernoulli’s hypothesis for plane sections since the span to height ratio of the beam under consideration is large. Concerning the application of the Berno ulli’s hypothesis in the present paper, it should also be mentioned that since the beam is loaded in pure bending (Fig. 1) the only non-zero strain is the longitudinal strain, ε . Therefore, according to the small strain compatibility equations, ε is distributed linearly in the beam cross-section. Thus, ε in the lower crack arm is written as

y = + +

z    

,

(27)

C

y

z

1

1

1

1

1

1 C  is the strain in the centre of the lower crack arm cross-section,

1 y  and

1 z  are the curvatures of lower

where

crack arm in the 1 1 x y and 1 1 x z planes, respectively. The quantities, 1 C  , 1 y  and

1 z  , are determined from the following equations for equilibrium of the lower

crack arm cross-section:

2 h

2 b

    

    

b   − −

N

dy dz 

=

,

(28)

1

1

1

2 h

2

2 h

2 b

    

         

  − −

M

1 1 z dy dz 

=

,

(29)

y

1

1

2 h

2 b

2 h

2 b

    

  − −

M

1 1 y dy dz 

=

,

(30)

z

1

1

2 h

2 b

1 y M and

1 z M are the axial force and bending moments with respect to 1 y and 1 z axes. Obviously

1 N ,

where

(Fig. 2),