PSI - Issue 25

114 Victor Rizov / Procedia Structural Integrity 25 (2020) 112–127 Author name / Structural Integrity Procedia 00 (2019) 000–000 3 A lengthwise crack of length, a , is located arbitrary along the height of the beam. The lengthwise fracture behaviour of the beam is studied in terms of the strain energy release rate. In accordance with linear-elastic fracture mechanics, the strain energy release rate, G , can be expressed as

dA G dU  ,

(1)

where U is the strain energy cumulated in the beam, dA is an elementary increase of the crack area. For a beam configuration of width, pk b , at the level of the lengthwise crack, the elementary increase of the crack area is written as dA b da pk  , (2) where da is an elementary increase of the crack length. By substituting of (2) in (1), the strain energy release rate is obtained as

G dU pk 

.

(3)

b da

The strain energy is found by integrating of the strain energy density, u , in the volume of the beam, V

V ( )  

U

u dV 0

.

(4)

The strain energy density is written as

2 1  u



,

(5)

where  is the normal stress,  is the lengthwise strain. The stress is expressed by using the Hooke’s law   E  , (6) where E is the modulus of elasticity. Since the material is continuously inhomogeneous in height and length directions, E varies continuously along the height and length of the beam. By combining of (5) and (6), the strain energy density is obtained as

2 1  u E 

2

.

(7)

Since beams of high length to height ratio are under consideration in the present paper, the distribution of the strains along the height of the beam cross-section is expressed as   n z z     , (8)