PSI - Issue 26

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

69 7

Fig. 2. Inhomogeneous cantilever beam configuration with a continuously varying radius of cross-section along the beam length.

The mechanical behaviour of the material is described by the following non-linear stress-strain relation (Masterov and Berkovsky (1989)):



,

(18)



=

L Q +



where L and Q are material properties. The beam exhibits continuous (smooth) material inhomogeneity in radial direction. The material property, L , varies in radial direction according to the following exponential law:

R R A

L L e  0 =

,

(19)

where 0 L is the value of L in the centre of the beam cross-section,  is a material property which controls the material inhomogeneity in radial direction, the radius, A R , varies in the interval   R 0; . For the beam configuration in Fig. 2, one writes 1 = F n , 1 1 = CD N , 1 1 = DH N , 1 = b N , 1 = DH N . (20) By substituting of (20) in (11), one derives the following expression for the strain energy release rate for the cantilever beam in Fig. 2: ( ) ( ) − − − + = + 1 int 1 1 3 3 1 2 2 r r p x x b x x R F R G F      

R

R

R

2



2

2





  

  

1

3

 

 

 

0 IN A A u R dR d

0 A A u R dR d EX

UC A A u R dR d

−

+

−







.

(21)

R

2



p

1

1

+

3

R

0

0

0

0

0

3

where the strain, int x  , is found from the equation of equilibrium (2.12). For this purpose, after substituting of (19) and (20) in (12), the equation is solved with respect to strain by using the MatLab computer program. Equation for equilibrium (13) is used to determine the strain, 1 + p x  . For this purpose, EX  is expressed by substituting 1 + p x 