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