PSI - Issue 47

Victor Rizov / Procedia Structural Integrity 47 (2023) 3–12 Author name / Structural Integrity Procedia 00 (2019) 000–000

5

3

p

   

   



       B 1 1

  

 

,

(1)

H

where  is the stress,  is the strain, B , H and p are material properties,

  is a function that describes the

  is used here (Kishkilov and Apostolov (1994)):

viscoelastic behaviour. The following form of



 1 

 t where t is time,  and  are material properties. The distribution of B in radial direction is 

1  





expressed as

4 R R

q

C B B e 

,

(2)

C B is the value of B in centre of the beam cross-section, q is a material property

4 0 R R   . In (2),

where

that controls the material inhomogeneity in radial direction. The longitudinal fracture behaviour of the beam is analyzed in terms of the strain energy release rate, G , by considering the balance of the energy. For this purpose, first, a small increase, 1 a  , of the length of crack 1 is assumed. The balance of the energy is written as (3) where u and  are, respectively, the longitudinal displacement of the centre of the free end of crack arm 1 and the angle of rotation of the free end of the beam, 1 cf l is the length of the front of crack 1, U is the strain energy in the beam. It is obvious that 1 1 2 l R cf   . (4) By combining of (4) in (3), one obtains By applying the integrals of Maxwell-Mohr, the longitudinal displacement of the centre of the free end of crack arm 1 and the angle of rotation of the free end of the beam are obtained as       3 2 3 1 2 1 3 4 2 3 1 2 l a a a a a a u CUN CD D CD D CDD            , (6)       3 2 3 1 2 1 3 4 2 3 1 2 l a a a a a a UN D D D D DD             , (7) where 1 2 CDD  , 2 3 CD D  , 3 4 CD D  and CUN  are, respectively, the longitudinal strains in the centres of crack arm 1, portion, 2 3 D D , of crack 2 (the boundaries of this portion are 1 x a  , 2 x a  and 2 R R  ), portion, 3 4 DD , of crack 3 (the boundaries of this portion are 2 x a  , 3 x a  and 3 R R  ) and the un-cracked portion of the beam, 1 2 DD  , 2 3 D D  , 3 4 D D  and UN  are, respectively, the curvatures of crack arm 1, beam portions, 2 3 D D and 3 4 DD , and the un-cracked portion of the beam. It is obvious that in beam portion, 1 2 DD , the force, F , loads only the crack arm 1 while the bending moment, M , is distributed on the four crack arms (Fig. 1). Besides, the curvature of the four crack arms is the same. Therefore, the strain, 1 2 CDD  , and the curvature, 1 2 DD  , are obtained from the following equations for equilibrium:   ( ) 1 A dA F  ,     ) ( ( ) 1 RQ A RQ A zdA zdA M   , (8) 1 1 a Gl a cf    1 1 a U F u M       ,    .             1 a U 1 1 1 2 1 R  a M a u F G  (5)