PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 63–74 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

65 3

Fig. 1. Geometry and loading of an inhomogeneous beam with a continuously varying radius of cross-section along the beam length.

The longitudinal fracture is studied in terms the strain energy release rate, G . General approach for analyzing the strain energy release rate is developed by considering the energy balance. By assuming a small increase, a  , of the crack length, the energy balance is written as

i n  = = 1 i F



,

(2)

a F u F u U b F i F b i    = +

a Gl a   +

cf

i F u  is the increase of the longitudinal displacement of

where n F is the number of axial forces applied on the beam,

the application point of the i -th force, b F u  is the increase of the longitudinal displacement of the free end of the internal crack arm, U is the strain energy in the beam, c f l is the length of the crack front. Since the crack front is a circle of radius, 3 R , the crack front length is 3 2 l R cf  = (Fig. 1). From (2), the strain energy release rate is derived as

u

u





F =  = = 1 i n i

F

F

a U

1



F

F

.

(3)

G

+

−

i

b

i

b

R

a

R

a

R

2

2

2













3

3

3