PSI - Issue 28

7

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1226–1236 Author name / Structural Integrity Procedia 00 (2019) 000–000

1232



a T U   

Gl a a cf   

,

(28)

where  is the angle of twist of the end of the rod, U is the strain energy. By substituting of (2) in (28), one derives

 R a R T 2 1     

  

   .

a U



2

G



(29)

2



1

1

It should be mentioned that the right-hand side of (29) is doubled in view of the symmetry (Fig. 1). By applying the integrals of Maxwell-Mohr, one obtains the following expression for the angle of twist:   2 2 R l a R a ft et       . (30)

Fig. 2. The strain energy release rate in non-dimensional form presented as a function of parameter, p (curve 1 – at non-linear behaviour of the material, curve 2 - at linear-elastic behaviour). 3 U are replaced with 1 U , 2 U and 3 U , respectively. Here, 1 U , 2 U and 3 U are the strain energies in the external and internal crack arms, and in the un cracked part of the rod, respectively. The strain energies, 1 U , 2 U and 3 U , are obtained by formulae (5), (16) and (22), respectively. For this purpose,  0 u ,  in u 0 and  uc u 0 are replaced, respectively, with 0 u , in u 0 and uc u 0 where in u 0 and uc u 0 are the strain energy densities in the internal crack arm and the un-cracked portion of the rod. By using (10), the strain energy densities, in u 0 and uc u 0 , are written as The strain energy is obtained by (4). For this purpose,  1 U ,  2 U and 