PSI - Issue 26

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

68 6

1 + p x  by applying the stress-strain

EX  , that that is involved in (13) is expressed as a function of

The normal stress,

1 + p x  by using

EX  in (13), the equation should be solved with respect to

relation. After substituting of (14) and

the MatLab computer program. The strain,

1 + p x  , that is involved in (11) is determined from the following equation for equilibrium of the beam

cross-section in portion, DH , of the un-cracked part of the beam (Fig. 1):

  = R 2



,

(15)

N

R dR d





UC

UC A A

0

0

UC  , is expressed as a function of

1 + p x  by using the stress-strain relation. The axial force,

where the normal stress, UC N , is found as (Fig. 1)

i p i UC N F F = +  = = 1 i

.

(16)

b

UC  and (16) in (15) should be solved with respect to

1 + p x  by using the

The equation obtained by substituting of MatLab computer program. The strain energy densities, and σ UC in (2.8). It should be mentioned that R , 1 0 + p EX u and int x  ,

1 0 UC u , which are involved in (11) are obtained by substituting of EX 

IN u 0 ,

1 0 UC u which are involved in (11) are

1 r x  ,

1 0 + p EX u and

1 + p x  ,

found by (1), (12), (13), (15), and (8) at x a = . The integration in (11) should be performed by using the MatLab computer program for particular beam geometry, loading conditions, stress-strain relation and material inhomogeneity in radial direction. 3. Numerical example The general approach developed in previous section of the paper is applied here for analyzing the strain energy release rate for the longitudinal circular cylindrical crack in the inhomogeneous cantilever beam configuration shown in Fig.2. The beam is clamped in its right-hand end. The beam has a circular cross-section of radius, R , which varies continuously along the beam length according to the following law: where 1 R and 2 R are the radiuses of the cross-section in the free end and in the clamping, respectively, l is the beam length, x is the longitudinal centroidal axis ) (0 x l   . A longitudinal circular cylindrical crack of radius, 3 R , and length, a , is located in the beam as shown in Fig. 2. The internal crack arm has a circular cross-section of radius, 3 R . The external crack arm has a ring-shaped cross section of internal radius, 3 R , and external radius, R , where R is expressed by (17). The external loading of the beam consists of axial force, b F , applied at the free end of the internal crack arm and axial force, 1 F , applied at the external crack arm (Fig. 2). 3 1 3 1 l R R R R − = + 1 2 1 x , (17)