PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1237–1248 Author name / Structural Integrity Procedia 00 (2019) 000–000

1244 8

the crack front. Equations (10), (11), (12) and (14) are used to obtain g  , 1 g q , 2 g q and

3 g q . For this purpose, after

replacing of 1 R ,  , 1 q , 2 q and 3 q , respectively, with 3 g q , the equations should be solved by using the MatLab computer program. After that the complementary strain energy density is obtained by substituting of g  and (24) in formula (5). Finally, the strain energy release rate is determined by substituting of * 01 u , * 02 u and * 0 u in (1). The MatLab computer program is used to carry-out the integration in (1). The longitudinal fracture behaviour of the cantilever shown in Fig. 2 is analyzed also by considering the energy balance in order to verify the solution of the strain energy release rate. For this purpose, by assuming a small increase, a  , of the crack length, one writes the energy balance as 2 R , g  , 1 g q , 2 g q and



a F u U   

a Gl a C   

,

(27)

where u  is the increases of the longitudinal displacement of the free end of the internal crack arm, U is the strain energy cumulated in the bar, C l is the length of the crack front. From (65), one expresses the strain energy release rate as

1

a l u 

a U

 



l G F C 

.

(28)



C

Since the length of the crack front is 1 2 l R C   ,

(29)

formula (28) is re-written as

R 1 2 1 

  

   .

a U

  a F u



G





(30)



The integrals of Maxwell-Mohr are used to derive the longitudinal displacement of the free end of the internal crack arm. The result is   l a u a g      , (31)

The strain energy cumulated in the bar is expressed as        1 0 01 A A l a u dA U a u dA ,

(32)

where 01 u and 0 u are, respectively, the strain energy densities in the internal crack arm and in the un-cracked bar portion, a x l   , (Fig.2). The strain energy density in the internal crack arm is obtained by the following formula (Rizov (2019)):