PSI - Issue 26

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

88

3

The fracture behaviour is analyzed in terms of the strain energy release rate, G , with considering the aging. For this purpose, the strain energy release rate is obtained by differentiating the complementary strain energy with respect to the lengthwise crack area

bda G dU * =

,

(1)

where * dU is the change of the complementary strain energy stored in the beam, da is an elementary increase of the crack length.

Fig. 2. The geometry and loading of the free end of the lower crack arm.

The complementary strain energy is obtained by integrating of the complementary strain energy density in the volume of the beam

2 h

2 b

2 h

2 b

2 b

    

    

    

    

    

    

a

a

h

l

  

  

  

  

  

  

  

  

  

*

* dy dz u dx

* u dx U 0

4 4 dy dz u dx R 4 * 0

U

dy dz

+

+

=

, (2)

L

0

1

1

1

2

2

2

2 h

2 b

2 h

2 b

2 b

h − −

a

0

0

− −

− −

where * * 0 R u are the complimentary strain energy densities in the lower crack arm, the upper crack arm ) l a x   3 , respectively. Axes, 1 y and 1 z , are shown in Fig. 2. In formula (2), 2 y and 2 z are the centroidal axes of the cross-section of the upper crack arm ( 2 z is directed downwards), 4 y and 4 z are the centroidal axes of the cross-section of the un-cracked beam portion ( 4 z is directed downwards). The mechanical behaviour of the material is described by applying the following power-law stress-strain relation:    m K = , (3) where σ is the stress, ε is the strain, K and m are material properties,  is a function for describing the effect of the aging. The function,  , is written as (Mihailov (1998)) and the un-cracked beam portion ( 0 L u , * 0 U u and

1

=



,

(4)

f st +

1