The estimation of the phase velocity of the elastic waves based on the transfer matrix method for...
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The estimation of the phase velocity of the elastic waves
based on the transfer matrix method for binary
systemsNicoleta Popescu-Pogrion-INCDFM BucurestiIonel Mercioniu-INCDFM BucurestiNicolae Cretu –Univ. Transilvania Brasov
Transfer matrix method
1 1
2 22 2 1 1
2 2 1 11 1
2 2
1 1exp 0 exp 01
( )0 exp 0 exp2
1 1
Z Z
Z Zik l ik lT
ik l ik lZ Z
Z Z
1 2
1 2
exp 0'( )
0 exp
eff
eff
ik l lT
ik l l
The equivalent medium
1 2 1 1 2 1
2 2 1 2 2 1
1 2 1 1 2 1
2 2 1 2 2 1
1 2
1 2
(1 )exp ( ) (1 )exp ( )1
2(1 )exp ( ) (1 )exp ( )
exp ( ) 0
0 exp ( )
eff
eff
Z l l Z l li i
Z c c Z c c
Z l l Z l li i
Z c c Z c c
ik l l
ik l l
Equivalent matrices
The matrix T(ω) has the eigenvalues:
The matrix T’(ω) has the eigenvalues:
2
1
21
122122
2
1
21
1221
2
12,1 4cos)1(
2
1cos)1(
2
1
Z
Z
cc
clcl
Z
Z
cc
clcl
Z
Z
1coscos 212
212,1 llkllkq effeff
21
12
21
21
21
1221
2121 )(
ll
lc
ll
lc
cc
lclc
llccceff
Effective velocity
effclc
cl
21
21
Numeric estimation
121
211
22
212
1
21 )()(
cllxllc
llx
c
llxg
The zero’ estimation of g(x)
0 17695 16546 l1 0.147l2 0.0091
c1 2 l1 0
x 2000 6000
c1 5.202103
g x( ) x2 2 l1 l2
c1 x 2 l12 l22 c1 l1 l2( ) 2 l1 l2 c1
g x( )
0
x2000 3000 4000 5000 6000
5 104
0
5 104
1 105
x 4000
root g x( ) x( ) 3.638103
Error estimation
The behaviour of c2 around fexp. The vertical bars show the range of values of c2 around c2exp obtained for deviations of f around fexp. The maximum relative deviations of f are shown above the vertical bars.
References
[1] N.Cretu, G.Nita, Pulse propagation in finite elastic inhomogeneous media, Computational Materials Science 31(2004) 329-336 [2] Z. Wesolowski, Wave speed in periodic elastic layers, Arch. Mech. 43 (1991) 271-282.[3] Y. A. Godin, Waves in random and complex media, Vol. 16, No. 4, November 2006, pp 409–416[4] S.A.Molchanov, Ideas in the theory of random media, Acta Applicandae Mathematicae, 22 (1991) 139–282.[5] N. Cretu, Acoustic measurements and computational results on material specimens with harmonic variation of the cross section, Ultrasonics, 43 (2005) 547-550. [6] P.P. Delsanto, R. S. Schechter, H. H. Chaskelis, R. B. Mignogna, and R. Kline, CM Simulation of the Ultrasonic Wave Propagation in Materials, Wave Motion 16 (1992) 65-80[7] P. P. Delsanto, R. S. Schechter, H. H. Chaskelis, R. B. Mignogna, and R. Kline, CM Simulation of the Ultrasonic Wave in Materials 2, Wave Motion 20 (1994) 295-307[8] S. Guo, Y. Kagowa, T. Nishimura, H. Tanaka, Elastic properties of spark plasma sintered ZrB2-ZrC-SiC composites, Ceramics International, 34 (2008) 1811-1817