Reflection and Transmission Characteristics of Coupled Wave through Micropolar Elastic Solid Interla

International Journal of Automation and Power Engineering (IJAPE) Volume 2 Issue 6, September 2013 www.ijape.org

Reflection and Transmission Characteristics of Coupled Wave through Micropolar Elastic Solid Interlayer in Micropolar Fluid Xu Hongyu*1, Dang Songyang2, Sun Qingyong 3, Liang Bin4 Schematizing and Architectural Engineering School, Henan University of Science and Technology, No.263, Kaiyuan Road, Luoyang, China *1

xuhongyu@haust.edu.cn; 2790952752@qq.com; 3sun‐qing‐yong‐wo@163.com; 4liangbin4231@163.com

Abstract

to the translation assumed in classical theory of elasticity. This theory is known as ‘Cosserat theory’ after their names. Mindlin (1964) presented a linear theory of a three‐dimensional continuum having the properties of a crystal lattice, including the idea of unit cell. Later, Eringen incorporated micro‐inertia and renamed the ‘Cosserat elasticity’ as the ‘Micropolar elasticity’.The linear theory of micropolar elasticity developed by Eringen(1966) is basically an extension of the classical theory of elasticity.With rapid advance of the science and technology, the research in the space of the sandwich structure with the spread of the wave based on micropolar fluid theory has got more and more attention by scholars. The research of wave‐ absorbing materials is the most significant. Because the stealth and electromagnetic compatibility (EMC) technology become more and more important, the effection of electromagnetic wave absorption material is very outstanding. It becomes a modern military in the magic weapon of the electronic counter and a ʺsecret weaponʺ.

Based on micropolar fluid theory and micropolar solid elasticity theory, reflection and transmission characteristics of three kinds of micropolar elastic waves involving longitudinal displacement wave and two coupled waves, are studied when incident coupled wave propagates through micropolar elastic solid plate in micropolar fluid. Theoretical and numerical analytical results reveal that in general the amplitude ratios of various reflected and transmitted waves are functions of angle of incidence, frequency of the incident wave and of the material properties of the medium through which they travel. At normal incidence, the reflection and transmission of only coupled wave takes place and no other wave is found to reflect or transmit. At grazing incidence, no reflection or transmission phenomena take place and the same wave propagates along the interface. The rules of the reflection waves and refraction waves amplitudes varied with incident angle are also discussed.

Keywords Micropolar Theory; Reflection; Transmission; Amplitude Ratios; Micropolar Elastic Wave

Introduction

In recent decades, many problems of reflection and refraction of micropolar elastic waves at a plane interface have been studied by several researchers[1‐10].

Under continuum hypothesis of an elastic body, the classical theory of elasticity is based on linear stress– strain law (Hooke’s law). In this theory, the transmission of load across a surface element of an elastic body is described by a force stress and the motion is characterized by translational degrees of freedom only. For materials possessing granular structure, it is found that the classical theory of elasticity is inadequate to represent the complete deformation. Certain discrepancies are observed between the results obtained experimentally and theoretically, particularly, in dynamical problems of waves and vibrations involving high frequencies.

Cosserat and Cosserat (1909) was the first who gave importance to the microstructure of a granular body and incorporated a local rotation of points, in addition

In this paper, based on micropolar fluid theory and micropolar solid elasticity theory, reflection and transmission characteristics of three kinds of micropolar elastic waves involving longitudinal displacement wave and two coupled waves are studied when incident coupled wave propagates through micropolar elastic solid plate in micropolar fluid. The change rules of the reflection waves and refraction waves amplitudes varied with incident angle are also discussed.

Equations of Motion and Constitutive Relations In the absence of body force density and body couple

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r r B and D are the vector potentials. Plugging Eqs.(5)

density, for micropolar fluid medium and micropolar solid medium, the equations of motion are given as Eqs.(1) and (2), respectively.

c

2 1f

 c3f2      u f    c2f2  c3f2       u f   f   c5f2      Φ  f  c5f2      Φ

2 4f

 f Φ  f  c6f2    u f  2Φ

c

2 1s

c

2 3s

    u   c

c

2 3s

     u  s

 c3s2   Φ s  u  s

c

2 4s



c

2 5s

    Φ   c

r



2 5s

s

where c1r2   r  2 

c4r2   r  

2 2s

s

r

    Φ

s



 s  c6s2    u s  2Φ s   Φ



1 Af  0,  2 C  0 (6) f

    (1)        (2)   

 f u  c3f2   Φ  f

c

into Eqs.(1), we can obtain

c

f B  f   c3f2   2B f  c3f2   D  (7) f 2 2 2 2 f f f     c5f  D  c6f   B  2c6f D  D  2 2f

where

  1   c1f2  c3f2   2   t  t 

It can be seen that equations in Eqs.(6) are un‐coupled in scalar potentials A r and C r while Eqs. (7) are coupled in vector potentials B r and D r .

r r r r  , c2r2    , c3r2  K r  ,

r r r r r r r  j , c5r2    j , c6r2  c3r2 j ,  is

Plane Wave of Micropolar Fluid and Solid

r

the density of the medium, j is the micro‐inertia and

Take the form of a plane wave propagating in the positive direction of a unit vector n as

r r u and Φ are, respectively, the displacement and

microrotation vectors for the micropolar elastic half‐ spaces. Here, the quantity having superscript r corresponds to the fluid and solid medium when r = f and r = s, respectively.  f ,  f and K f are the fluid

 A , C , B , D   a , c , b , d  exp ik  n  r - Vt  (8) r

   u  kl    u  u

K

f

u

f l ,k

f



s

s

s

s

s

s

s

s s m kl    r , r kl    k ,l    l , k s

s

s

s

s









(ii) two sets of coupled waves, each consists of a transverse displacement wave and a transverse microrotational wave perpendicular to it, propagating with phase velocities V s 2 and V s3 given by

s

 (4) 



2 Vs2,s3  b  b 2  4ac

where  is the force stress tensor, m is the couple stress tensor, the ‘comma’ in the subscript denotes the spatial derivative,  kl and  klp are Kronecker delta and r kl

r

propagating with phase velocity Vs1  c1s2  c3s2

 kl   u r , r kl    u k ,l  u l , k   K  u l , k   klp p   s

r

(i) an independent longitudinal displacement wave

  klp p   (3) f f f f f f f  m kl    r , r kl    k ,l   l , k f l ,k

r

Parfitt and Eringen (1969) have already shown that there exist four waves in an infinite micropolar elastic solid medium propagating with distinct phase velocities. The four phase velocities are given as follows.

For micropolar fluid medium and micropolar solid medium, the constitutive relations are given by Eqs.(3) and (4), respectively. f k ,l

r

angular frequency.

elastic constants for the micropolar elastic solid.

f

r

is the position vector, V is the phase velocity in the direction of n, k (=  V ) is the wave number,  is the

constant and K s ,  s ,  s and  s are the micropolar

f r ,r

r



viscosity coefficients responsible for gyrational dissipation of the micropolar fluid,  s ,  s are Lame’s

f

r

where a r , c r , b r and d r are constants, r  xi  yj  zk

viscosity coefficients and  f ,  f and  f are the fluid

f kl





and  2   c4f2  c5f2   2  2c6f2   . t  t 

r kl



2a , where

02  c6s2 , a  1  202  2 , b  c2s2  c3s2  c5s2   2c2s2  c3s2  02  2 , c  c5s2  c2s2  c3s2 

the alternating tensors respectively. Other symbols have their usual meanings.

(iii) an independent longitudinal microrotational wave propagating with phase velocity

Using Helmholtz theorem, we can write

Vs42   c4s2  c5s2 1  202  2  .

 ur   Ar  Br  B r       ,    r  r  r  r   0 (5) C  D  D  Φ  r r where A and C are the scalar potentials, while

Similarly, Singh and Tomar have geiven that there exist four waves in an infinite micropolar fluid medium propagating with distinct phase velocities.

348

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The four phase velocities are given by

 c  c V   2 f1

2 1f

Vf42

 ,V  c   2 3f

2 f2,f3

2

2 4f



 b  b  4ac 2



 0  ik1  sin  0 x  cos 0 z   i1t ,

2a ,

 i  iki  sin i x  cosi z   ii t ,

 c5f2   2c6f2  ,

 i  iki  sin ix  cosiz   iit , (i  1, 2,3) .

When d  z , in micropolar fluid medium M 1 , for

1

where

a    2c6f2 ,

 c   c  c b    2 5f

2 2f



2 3f

longitudinal displacement wave:

  2c   c 2 6f



2 3f

  A1exp   1   B1exp   10  ,   3  As , B2s , s2   3  0 0  Bi exp     , i Bi exp       2   2  (12)   3  B exp   , B exp   ,       i i 1  1   2  Af , B2f , f2   3  iBi exp   i    2 For couple wave I,  j2,3

c  2 6f

and c   3 c5f2 c2f2  c3f2 .

Reflection and Transmission of Coupled Wave Introducing the Cartesian coordinates x, y and z such that xy plane( z  0 ) is interface. Thickness of d micropolar solid placed between micropolar fluid , the z‐axis is taken perpendicular to the interface ( z  0 ) and pointing downward into the medium M 2 .We shall consider a two‐dimensional problem in xz plane, so that the followings are the displacement and microrotational vectors in micropolar elastic solid and in micropolar fluid:

  0  B4 exp   4     6 T 0  Bi exp       As , B2s ,2s    A2 exp   2     5   6   B exp  0   j i i     5  (13)    B4 exp   4   6 T  Bi exp   i   Af , B2f ,2f     5   6   B exp     j i  i     5 For couple wave II,  j2,3

r r r r r u   u1  x, z  ,0, u3  x, z   , Φ   0,  2 ,0   r  f,s  (9)

The plan couple wave incident to interface z  0 with phase velocity Vf1 and incident angle  0 within micropolar fluid propagation and then entrance to micropolar solid medium M 2 and incident into interface z  d go through medium M 2 ,then entrance to micropolar fluid medium M 1 .

We take the following form of potentials: Interface z  0 :when z  0 , in micropolar fluid medium M 1 ,

  0   B7 exp   7    9  s s s T 0 A , B ,  B exp    2 2    i  i  8   9  A exp    B exp  0   3   i  j i  8  3  (14)    B7 exp   7   9 T  Bi exp   i   Af , B2f ,2f     8   9   B exp     i  j i   8  where Bi and Bi are the amplitude of reflected and

   A0 exp   0   A1exp  1   3   f f f T   A , B ,  A exp  ,     2 2    (10) i i  2   3  A exp      i i i   2 And when 0  z  d , in micropolar solid medium M 2

A , B s

s 2

, 2

s T

   A1 exp   1   3     Ai exp      (11)  2   3  A exp     i i     2 

refracted longitudinal displacement wave or coupled wave, respectively.

where A0 , A1 and A1 are the amplitude of the incident,

 i0  iki  sin i x  cos i z   ii0 t ,

reflected and refracted longitudinal displacement wave, respectively. A2 , A3 and A2 , A3 are the

 i  iki  sin ix  cos iz   iit ,  i  1, 2, ,9  . The  are given by: coupling parameters  2,3 and  2,3

amplitude of the reflected and refracted couple wave .

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 s  2.09730  1010 dyne cm 2 ,

1

 2  c2  2 6s2  c5s2  ,  2,3  c Vs2,s3 k2,3   2 6s

 s  0.91822  1010 dyne cm 2 , K s  0.22965  1010 dyne cm 2 ,

1

V  ic 2   ic  f2,f3  2 6f2  c5f2  .  2,3   k2,3  k2,3  2 6f

 s  0.0000291  1010dyne ,  s  0.000045  1010dyne ,

Boundary conditions to be satisfied at the interface z  0 and z  d , are the continuity of force stress, couple stress, displacement and microrotation.

Mathematically, these boundary conditions can be written as:

 s  0.0000423  1010dyne , j s  0.037cm 2 ,

 s  0.0034 g cm 3 .

   ,   , m  m , u  u , u  u ,    (15) s zz

f zz

s zy

f zy

s zy

f zy

s 1

f 1

s 3

f 3

s 2

f 2

And   0  100 and thickness of the interlayer: d=0.01m. The system of equations given by Eqs.(16) are solved by Gauss elimination method. The values of the amplitude ratios are computed at different angles of incidence.

Employing the Snell’s law given by:

sin  0 sin  i sin  i sin i sin i  3 sin i  6      Vf1 Vsi Vfi Vsi Vsi Vsi

sin i sin i 3 sin i 6 , (i  1, 2,3)    Vfi Vfi Vfi

From FIG.1 to FIG.6,we get the curve of three kinds of wave reflection and transmission coefficient with the angle of incidence changed after coupled wave through micropolar elastic solid plate in micropolar fluid.

and assuming that all frequencies are equal at the interface, 24 homogeneous equations can be obtained and can be written in a matrix form as: PZ  Q (16) where

P   aij  24 24 , Z   zk 

T

FIG.1 shows that the variation of the amplitude ratios of reflected longitudinal displacement waves with the incidence angle  0 , when a plane coupled wave

 k  1, 2, , 24  ,

Q  11 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .

T

propagates from the micropolar fluid half‐space with phase velocity Vf2 . The amplitude ratio of longitudinal

The entries of the matrix P and the elements of the matrix Z are given in Appendix.

displacement wave has two peak value when incidence angle  0 is 27.7 or 68.6 , respectively. And

Numerical Results and Discussions

it is zero when  0  45 .

The paper presents numerical example to explore the transmission characteristics of incident longitudinal displacement wave when throughing the micropolar fluids of micropolar elastic plate. For numerical computations, we take the values of the relevant parameters from Hsia(2006)[9].

Reflection coefficidents

2.5

For micropolar viscous fluid M 1 :

  1.5  10 dyne s cm , f

10

2

1.5

1.0

0.5

 f  0.3  1010dyne s cm 2 , K f  0.00223  1010dyne s cm 2 ,

0.0 0

 f  0.00111  1010dyne s ,

10

20

30

40

50

60

70

80

90

Angle of incidence(in degrees)

 f  0.0022  1010dyne s ,

FIG. 1 REFLCTION COEFFICIENT OF LONGITUDINAL DISPLACEMENT WAVE

 f  0.000222  1010dyne s ,

FIG.2 shows that the variation of the amplitude ratios of reflected coupled wave I with the incidence angle  0 .The amplitude ratio of coupled wave I has

j f  0.0400cm 2 ,

 f  0.8 g cm 3 .

the minimal value when  0 is 19.1 . It decreases

For micropolar elastic solid M 2 :

350

2.0

monotonically from the value 1.0 to the value 0.7784 at

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 0  19.1 and then it increases rapidly. The amplitude

incidence angle  0 is 28.3 or 73 ,respectively. The

ratio of coupled wave II gets to the maximum value when  0 is about 23 . And then, the change of

curve fluctuates at 23 . The amplitude ratio of of transmission longitudinal displacement wave is zero when  0 is about 45 .

incidence angle has no effect on the amplitude ratio of coupled waves.

-5

6.0x10

1.00

-5

Transmission coefficidents

Reflection coefficidents

5.0x10

0.95

0.90

0.85

-5

4.0x10

-5

3.0x10

-5

2.0x10

-5

1.0x10

0.80

0.0 0

0.75 0

10

20

30

40

50

60

70

80

10

90

20

30

40

50

60

70

80

90

Angle of incidence(in degrees)

Angle of incidence(in degrees)

FIG. 2 REFLCTION COEFFICIENT OF COUPLE WAVE I

FIG. 4 TRANSMISSION COEFFICIENT OF LONGITUDINAL DISPLACEMENT WAVE

FIG.3 shows that the variation of the amplitude ratios of reflected coupled wave II with the incidence angle  0 . The variation of the amplitude ratio of coupled

FIG.5 shows that the variation of the amplitude ratio of transmission couple wave I with the incidence angle  0 . The variation of the amplitude ratio of coupled

wave II decreases slowly at first. It reaches the

wave I changed slowly at first. Then it reaches to the

minimal value when  0 is about 20 . It suddenly rears

peak value when  0 is about 30 . It declines sharply in

up at about 23 in the range of 5 . It declines sharply

the range of 10 . And then, the curve changes gently

to zero when  0 is about 45 . And then it gradually

at about 67 . At last it decreases gradually to zero

increases to the peak value at about 73 . At last, it decreases to zero gradually.

when  0 is 90 .



-5

-6

8.0x10

3.0x10

Transmission coefficidents

-5

-6

Reflection coefficidents

2.5x10

-6

2.0x10

-6

1.5x10

-6

1.0x10

7.0x10

-5

6.0x10

-5

5.0x10

-5

4.0x10

-5

3.0x10

-5

2.0x10

-5

-7

1.0x10

5.0x10

0.0 0.0

0 0

10

20

30

40

50

60

70

80

90

Angle of incidence(in degrees)

10

20

30

40

50

60

70

80

90

Angle of incidence(in degrees)

FIG. 3 REFLCTION COEFFICIENT OF COUPLE WAVE II

FIG. 5 TRANSMISSION COEFFICIENT OF COUPLE WAVE I

FIG.4 shows that the variation of the amplitude ratio of transmission longitudinal displacement wave with the incidence angle  0 , when a plane coupled wave

FIG.6 shows that the variation of the amplitude ratio of transmission couple wave II with the incidence angle  0 . The shape of the curve likes the transmission

propagates from the micropolar fluid half‐space with phase velocity Vf2 . The amplitude ratio of longitudinal

couple wave I, but the amplitude ratio of couple wave II is small than that of couple wave I about four orders of magnitude.

displacement wave has two peak value when

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-8

1.0x10

transmission coefficient for coupled wave. There exist peak values of reflection and transmission coefficient for longitudinal displacement wave.

Transmission coefficients

-9

8.0x10

(3) There exist zero values of reflection and transmission coefficient for longitudinal displacement wave when incidence angle is 45 , at the same time, transmission coefficient presents wave phenomenon when incidence angle is 23 .

-9

6.0x10

-9

4.0x10

-9

2.0x10

ACKNOWLEDGMENT

0.0 0

10

20

30

40

50

60

70

80

90

Angle of incidence(in degrees)

The authors acknowledge the financial support of National Natural Science Foundation of China (51075124).

FIG. 6 TRANSMISSION COEFFICIENT OF COUPLE WAVE II

From FIG.1 and FIG.6, it can be seen that at normal incidence the reflection and transmission of only reflected couple wave I take place and no other wave is found to reflect or transmit. At grazing incidence no reflection or transmission phenomena take place and the same wave propagates along the interface.

Appendix The component aij of the matrix P in non‐dimensional form are given by a11  L1 L0 , a12   L2 L0 , a12   L3 L0 , a15  L4 L0 ,

a16  L5 L0 , a21  M 1 M 0 , a22  M 4 M 0 , a23  M 5 M 0 , a25   M 6 M 0 , a26   M 7 M 0 , a32   N 2 , a33   N 3 , a35   N 4 , a36   N 5 , a41  P1 P0 , a42   P2 P0 , a43   P3 P0 , a45  P4 P0 , a46  P5 P0 , a51  Q1 Q0 , a52  Q2 Q0 , a53  Q3 Q0 , a55   Q4 Q0 , a56   Q5 Q0 , a62   2 , a63   3 , a65   2 , a66  3 , a71  L111 , a77  L11 ,a78  L2 2 ,a79  L33 ,a710   L01 ,a711   L4 2 ,

Conclusions Based on micropolar fluid theory and micropolar solid elasticity theory, the reflection and transmission phenomena of an coupled wave propagating through the micropolar elastic solid infinite plate in micropolar pluid was discussed. Theoretical and analytical results reveal that, in general, the amplitude ratios of various reflected and transmitted waves are functions of angle of incidence, of frequency of the incident wave and of the material properties of the medium through which they travel. Get three kinds of micropolar elastic waveʹs curves of reflection coefficient or transmission coefficient with the change of incident angle through the numerical results, and get the varying pattern of reflection and transmission coefficient: there is a maximum value in the curve of reflection coefficient or transmission coefficient of coupled wave,and there are two extremum in the curve of reflection coefficient or transmission coefficient of longitudinal displacement wave.

a712   L53 , a81  M 111 , a87   M 11 , a88  M 4 2 , a89  M 53 , a810   M 01 , a811   M 6 2 , a812   M 73 , a98  N 2 2 , a99  N 33 , a911   N 4 2 , a912   N 5 3 , a101  P111 , a107  P11 , a108  P2 2 , a109  P33 , a1010   P01 , a1011   P4 2 , a1012   P53 , a111  Q111 , a117  Q11 , a118  Q2 2 , a119  Q33 , a1110  Q01 , a1111  Q4 2 , a1112  Q5 3 , a128   2 2 , a129   3 3 , a1211  2 2 , a1212  33 , a1312   L2 2 , a1313  L11 , a1314  L2 2 , a1315  L33 , a1316   L01 , a1317   L4 2 , a1318   L53 , a142  M 2 2 1 , a1413   M 11 , a1414  M 4 2 , a1415  M 53 , a1416   M 01 , a1417   M 6 2 , a1418   M 73 , a1514  N 2 2 , a1515  N 33 , a1517   N 4 2 , a1518   N 5 3 ,

Based on the specific medium material parameters given by the paper ,we conclude that:

a162   P2 2 1 , a1613  P11 , a1614  P2 2 , a1615  P33 , a1616   P01 , a1617   P4 2 , a1618   P53 , a172  Q2 2 1 ,

(1) At normal incidence, the reflection and transmission of only coupled waves take place and no longitudinal displacement wave is found to reflect or transmit. At grazing incidence, no reflection or transmission phenomena take place and the same wave propagates along the interface.

a173  3 , a1713  Q11 , a1714  Q2 2 , a1715  Q33 , a1716  Q01 , a1717  Q4 2 , a1718  Q5 3 , a1814   2 2 , a1815  3 3 , a1817  2 2 , a1818  3 3 , a1919  L11 , a1920  L2 2 , a1921  L33 , a1922   L01 , a1923   L4 2 , a1924   L53 , a203  M 33 1 , a2019   M 11 , a2020  M 4 2 ,

(2) There exist maximum values of reflection and

352

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a2021  M 53 ,a2022   M 01 ,a2023   M 6 2 , a2024   M 73 , a213  N13 1 , a2120  N 2 2 , a2121  N 33 , a2123   N 4 2 , a2124   N 53 , a2219  P11 , a2220  P2 2 , a2221  P33 , a2222   P01 , a2223   P4 2 , a2224   P53 , a2319  Q11 , a2320  Q2 2 , a2321  Q33 , a2322  Q01 , a2323  Q4 2 ,

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