PSI - Issue 28

7

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1212–1225 Author name / Structural Integrity Procedia 00 (2019) 000–000

1218

x l   0 .

(23)

In (22), C 2  at the free end of the beam, s is a parameter that controls the distribution of C 2  in the length direction of the beam. The time dependent shear modulus is obtained by substituting of (14) in (12). The result is     2 0 0 0 0 0 0 ( )  G G t g g g G t G G G e      . (24) The time dependent shear moduli obtained by formulae (13) and (24) are used in the fracture analysis in order to investigate the effects of the stress relaxation. The longitudinal fracture behaviour of the inhomogeneous cantilever beam configuration shown in Fig. 1 is studied in terms of the strain energy release rate, G . For this purpose, the strain energy release rate is written as D 2  is the value of

dA G dU  ,

(25)

where U is the strain energy, dA is an elementary increase of the crack area. Since the length of the crack front is

2   , R

l

(26)

3

crf

the elementary increase of the crack area is found as dA R da 3 2   ,

(27)

where da is an elementary increase of the crack length. By combining of (25) and (27), one obtains

G dU 3 2  

.

(28)

R da

The strain energy is written as

2 1 U U U   ,

(29)

where 1 U and 2 U are the strain energies in the internal crack arm and in the un-cracked beam portion, a x l   . The strain energy in the internal crack arm is expressed as

a R 3   

 2 01

U

u

RdxdR

,

(30)

1

0 0

01 u , is found as

where the strain energy density,