PSI - Issue 13

Tianyu Chen et al. / Procedia Structural Integrity 13 (2018) 613–618 T. Chen et al./ Structural Integrity Procedia 00 (2018) 000 – 000

616

4

and i A and i B are determined by initial conditions. In Eq. (9), i  is given in Eq. (10), and i  is a dimensionless parameter determined solely from the boundary conditions.

cos cosh sin sinh i    

i 

i 



.

(10)

i 

i

The solution for the shifting function   1 F x is

1

3

 

.

(11)

3

2

1 F x

x

x

 



3

2

L

L

2

2

and the solution for   2 F x is   2 1 2 sin

cos sinh cosh F x C kx C kx C kx C kx     .

(12)

3

4

4 k A EI     and the coefficients

1 C , 2 C , 3 C and 4 C are determined from

where

1     2                          3 4  0 0 1 0 C C C C     

0 1

1 0

0 1

1 0

     

.

(13)

kL

kL

sinh cosh sinh cosh kL

kL kL

sin

cos

kL

kL kL



sin



cos

2.3. Dynamic energy release rate Substituting Eqs. (8), (11) and (12) into Eq. (6), the deflection of beam under the supposed displacement is derived. Now the bending moment at crack tip B is obtained from     2 1B 0, M EIw t  (14) Finally, by expanding Eq. (14) and combining with Eq. (2), the total dynamic energy release rate at the crack tip B is obtained in Eq. (15), and 1 i K , 2 i K , 3 i K and 4 i K are given in Eq. (16).           2 2 2 2 1 2 2 4 2 2 2 2 1 1 1 sin 2 3 sin i i i j ij i i i A v H C K t EI EI L G b vt k C C H t L                                        . (15)     2 (1) 1 (3) 2 4 4 2 3 2 4 sin sin 1 0 cos cos 0 1 1 sinh sinh 1 0 2 cosh cosh 0 1 2 i i i i i i i i i i K kL kL k L K kL kL L K kL kL k k K kL kL                                                           . (16) and   i x  is