PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 75–85 Rizov/ Structural Integrity Procedia 00 (2019) 000 – 000

77

3

in Fig. 1. The left-hand and right-hand crack tips are located in cross-sections of abscissas 3 1 x l = and x l a = + 3 1 , respectively. Here, 3 x is the longitudinal centroidal axis (Fig. 1). The upper and lower crack arms have different heights denoted by 1 h and 2 h , respectively. The beam exhibits continuous material inhomogeneity in the height direction. The material of the beam has non-linear elastic behaviour described by the following stress-strain relation (Kuznetzov and Barzilovitc (1990)):

n m B D H     = − − ,

(1)

where  is the stress,  is the strain, B , D , m and n are material properties. The distribution of material property, B , in the height direction is written as

2 h z

−

  

   +

= + h B B B B K 0

0

,

(2)

3

where

3 h z h −  

.

(3)

2

2

In formula (2), 3 z is the vertical centroidal axis of the cross-section of the beam, 0 B and K B are the values of B in the upper and lower surfaces of the beam, respectively. The longitudinal fracture behaviour of the beam is analyzed in terms of the strain energy release rate, G . For this purpose, by assuming a small increase of the crack at the left-hand crack tip, the strain energy release rate can be written as (Rizov (2018))

h

h

h

1

2

 2

=  − 2 2

 2 h

* u dz ,

* h a G u dz 0 1

* a u dz 0

+

−

(4)

0

3

1

2

2

2 h

1

2

−

−

2

* 0 u are the complementary strain energy densities in the upper and lower crack arms and in

where *

* 0 2 a u and

0 1 a u ,

the un-cracked part of the beam ahead of the left-hand crack tip, respectively. In principle, the complementary strain energy density is equal to the area that supplements the area enclosed by the stress-strain curve to a rectangle. Thus, the complementary strain energy density in the upper crack arm is expressed as

* u = − 

u

,

(5)

a

a

0

0

1

1

1 0 a u , is equal to the area enclosed by the stress-strain curve. Therefore, by integrating

where the strain energy density,

1 0 a u is obtained as

of (1),

m

n

2

1

1

+

+

n H







2 m u B D a 0 1 = −

−

.

(6)

1

1

+

+

By substituting of (1) and (6) in (5), one derives