PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 97–105 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

100

4

It is assumed that the modulus of elasticity is zero in the damage zone. The lengthwise fracture is studied in terms of the strain energy release rate. For this purpose, a time-dependent solution to the strain energy release rate is derived by applying the compliance method. According to this method, the strain energy release rate, G , is expressed as

   ,

   2 2 1 b

da F dC 2

G

=

(9)

where C is the compliance. It should be noted that the right-hand side of (9) is doubled in view of the symmetry. For the beam shown in Fig. 1 the compliance is written as

F C w = ,

(10)

where the vertical displacement, w , of the right-hand end of the beam is obtained by using the integrals of Maxwell-Mohr. The fracture behaviour of the beam is analyzed also by applying the J -integral approach

x p u x  

x p v y  

  

  

  

  

 

J

u

ds

 cos 0

=

−

+

,

(11)

where Γ is a contour of integration going from the lower crack face to the upper crack face in the counter clockwise direction, 0 u is the strain energy density,  is the angle between the outwards normal vector to the contour of integration and the crack direction, x p and y p are the components of stress vector, u and v are the components of displacement vector with respect to the crack tip coordinate system, xy ( x is directed along the crack), ds is a differential element along the contour. The J -integral is solved by using an integration contour,  , that is shown by dashed line in Fig. 1. The integration contour has two segments ( 1 A and B ). The J -integral value is obtained by summation: ) 2( 1 B A J J J = + . (12) It should be mentioned that the term in brackets in (12) is doubled because there are two symmetric cracks in the beam (Fig. 1). First, the J -integral is solved in segment Α 1 of the integration contour (this segment coincides with the lower crack arm cross-section behind the crack tip). The curvature of lower crack arm, 1  , that is needed to perform the integration in (11) is determined by considering the equilibrium of lower crack arm. The equilibrium equations are written as

2 =  − h h 1

N

bdz

( )

0

 

=

,

(13)

1

1

1

2

h

1

 2 h

M

bz dz

( )

=

 

,

(14)

1

1 1

1

−

2

where Ν 1 and Μ 1 are the axial force and the bending moment in lower crack arm, respectively. It is obvious that (Fig. 1)