PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1237–1248 Author name / Structural Integrity Procedia 00 (2019) 000–000

1239

3

n 1

E H        

  

,

(2)

where E is the modulus of elasticity, H and n are material properties,  and  are, respectively, the longitudinal strain and normal longitudinal stress.

Fig. 1. Round bar portion with the crack front (the longitudinal crack is a cylindrical surface with radius, 1 R ). The bar under consideration exhibits continuous (smooth) material inhomogeneity in radial direction. It is assumed that the modulus of elasticity varies gradually in radial direction   E ER  , (3)

where

2 0 R R   .

(4)

The complementary strain energy density in the cross-section of the internal crack arm behind the crack front is determined by using the following formula (Rizov (2019)):

1

n



2

n





n

* 01

u

 

.

(5)

1

2

E



 n

1

n H



In order to calculate * 01 u by (5), first, one has to determine  . However, it is obvious that  can not be determined explicitly from equation (2). Therefore,  is expanded in series of Maclaurin. The first three members of the series are kept. Thus, the series is written as