Issue 39

M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17

The Newmark family includes many widely used methods. The average acceleration method is one of them for structural dynamics applications, which is unconditionally stable. In this method, J and ] are equal to 0.5 and 0.25, respectively. We choose the mean acceleration method in this study.

I NTERACTION INTEGRAL AND SIF COMPUTATIONS

T

he general form of domain integral for axisymmetric problems, introduced by Moran and Shih [24] is as follows:

r 1 ¬ : ª ¬

º

T

˜ ¼ q P P q

³ ’ ’ ˜

I

r dA ¬

(4-1)

c

where, c is the energy-momentum tensor and W is summation of strain and kinetic energy. In this work, the interaction integral method is used to compute the mode I stress intensity factor ( K I ). By superimposing the actual and auxiliary fields on the domain integral, in the absence of thermal strains the general axisymmetric form of interaction integral ( MI ) in local Cartesian coordinate systems on crack tip (Fig. 2) for FGMs is obtained as below. r is the crack tip radial coordinate, r q r z e , q , lj lj ij i l P W u , G V

½ ° ¾ ° ¿

aux

§ ¨ ¨ ©

· ¸ ¸ ¹

u

u

q

1

° ® ° ¯

aux

aux

aux

aux

aux

³

r

r

V

V

ij ij V H

V

V

ij ij V H

MI

u

u

q

T

T

ir i r ,

ir

i r ,

r

,

r

r

r

r

c A

*

aux

aux

V

V

u

u q

iz r 1,

iz i r ,

z

,

(4-2)

§ ¨ ¨ ©

· ¸ ¸ ¹

aux

§ ¨ ¨ ©

· ¸ ¸ ¹

u

u

aux

aux aux

aux

aux

r

r

H

H

u B u U

V

V

V

V

C

u

u

T

T

ijkl r kl ,

ij

i

i

i r ,

ij j i r , ,

ir i r ,

r

r

^

`

aux ir

V

u

r q dA

/

i r ,

where q is a weight function varying from unity at the crack tip to zero on boundary of domain A * . For a stationary crack in axisymmetric state, the relation between the M-integral and the SIF is identical to its relation in plane strain state [25].

Figure 2 : Local (r, z) coordinate system.

2

2 1 Q

tip

aux K K K K I I

aux

MI

(4-3)

II II

E

tip

172