**Appendix**

The non-zero components of the matrix [ ] **h** appearing in equation (59) are followed as:

$$\begin{aligned} h\_{1,1} &= A\_{11} \left( -j\_{\varphi\_z}^{\varepsilon} \right)^2 + 2A\_{16} \frac{1}{R\_l} \{-j\_{\varphi\_z}^{\varepsilon} \} (-n) + A\_{66} \frac{1}{R\_l^2} (-n^2) - I\_i o^2 \\\\ h\_{1,2} &= A\_{12} \frac{1}{R\_l} \{-j\tilde{\varphi}\_z\} (n) + A\_{16} \left( -j\tilde{\varphi}\_z \right)^2 + B\_{12} \frac{1}{R\_l^2} \{-j\tilde{\varphi}\_z\} (n) + B\_{16} \frac{1}{R\_l} \{-j\tilde{\varphi}\_z \}^2 + A\_{62} \frac{1}{R\_l^2} (-n^2) + B\_{16} \frac{1}{R\_l^2} \{-j\tilde{\varphi}\_z \} (n) + A\_{66} \frac{1}{R\_l} \{-j\tilde{\varphi}\_z \} (n) + B\_{16} \frac{1}{R\_l} \{-j\tilde{\varphi}\_z \} (n) \end{aligned}$$

$$\begin{aligned} A\_{66} \frac{1}{R\_l} \{-j\tilde{\varphi}\_z\} (n) + B\_{62} \frac{1}{R\_l^3} (-n^2) + B\_{66} \frac{1}{R\_l^2} \{-j\tilde{\varphi}\_z \} (n) \end{aligned}$$

$$\begin{split} \mathcal{H}\_{1,3} &= A\_{12} \frac{1}{R\_{\!\!\!i}} \big( -j\xi\_{z} \big) - B\_{11} \big( -j\xi\_{z} \big)^{3} - B\_{12} \frac{1}{R\_{\!\!\!i}^{2}} \big( -j\xi\_{z} \big) \big( -n^{2} \big) - \mathfrak{D}\_{16} \frac{1}{R\_{\!\!\!i}} \big( -j\xi\_{z} \big)^{2} \big( -n \big) + \mathfrak{D}\_{6} \\\ A\_{62} \frac{1}{R\_{\!\!\!i}^{2}} \big( -n \big) - B\_{62} \frac{1}{R\_{\!\!\!i}^{3}} \big( n^{3} \big) - 2B\_{66} \frac{1}{R\_{\!\!\!i}^{2}} \big( -n^{2} \big) \big( -j\xi\_{z} \big) \end{split}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 57

 2 2 2 2,1 12 16 12 2 2 16 62 2 66 62 3 2 66 <sup>1</sup> 1 11 ( ) ( ) ( ) 1 11 () ( ) ( ) *zz z z i i i i z z i i i A j nA j B j nB j A n R R R R A j nB n B j n <sup>R</sup> R R* 2 2 2 2,2 22 2 26 22 3 2 26 66 2 2 2 2 66 22 43 2 23 33 11 11 ( ) 2 () 2 ( ) 4 () 1 11 1 <sup>2</sup> ( )2 ( ) *z z z i i i i z z z i i ii i A n A j n B n B j nA j <sup>R</sup> <sup>R</sup> R R B j D n D j nD j I <sup>R</sup> RR R* 2 3 2 2,3 22 2 12 22 3 2 23 62 3 2 2 3 61 66 22 32 4 21 22 3 2 2 23 3 2 32 61 33 2 11 1 1 1 () () ()3 ( ) 1 11 1 2 () ( ) () () 1 11 1 3 () 2 ( *z z z i i i i i z z z i ii i z zz z i i i i A nB n j B n B j n A j <sup>R</sup> R R R R B j B n j B nD j nD n <sup>R</sup> RR R D j nB j D j D j n R R <sup>R</sup> <sup>R</sup>* ) 2 3 2 3,2 22 2 12 22 3 2 23 3 2 62 61 66 22 3 2 3 2 21 2 43 22 23 3 2 32 2 2 61 33 11 1 1 () () ()3 ( ) 1 11 2 () ( ) 1 11 ( ) ()3 ( ) 11 1 2 ( *z z i i i i z z z i i i z z i ii zz z i i i A nB n j B n B j n <sup>R</sup> <sup>R</sup> R R A j B j B nj B n R R <sup>R</sup> D j nD n D j n R RR B jD j D j n R R <sup>R</sup>* ) 2 4 2 3,3 22 2 21 22 3 2 23 11 2 3 2 3 12 2 3 16 62 2 2 42 33 2 4 22 11 1 1 <sup>2</sup> 2 ( )4 ( ) 1 11 2 () 4 ( )4 ( ) 1 1 4 () *z z z i i i i zz z i i i z i i i A B j B n B j nDj R RR <sup>R</sup> D n j D j nD jn R R <sup>R</sup> D j nD nI R R* 1 3,8 2 ( ), *H R n ri* , <sup>2</sup> 3,9 2 ( ), *H R n ri* <sup>1</sup> 3,10 3 ( ) *H R n ri <sup>z</sup> i i i z <sup>i</sup> <sup>z</sup> i z z i n j <sup>R</sup> <sup>n</sup> <sup>B</sup> <sup>R</sup> <sup>n</sup> <sup>B</sup> R A <sup>j</sup> <sup>n</sup> <sup>R</sup> <sup>j</sup> <sup>n</sup> <sup>B</sup> R j B j B R A* ( ) <sup>1</sup> ( ) <sup>2</sup> <sup>1</sup> ( ) <sup>1</sup> ( ) <sup>1</sup> ( ) <sup>3</sup> <sup>1</sup> <sup>1</sup> 2 2 66 3 3 62 2 62 2 16 2 2 12 3 3,1 <sup>12</sup> <sup>11</sup>

56 Wave Processes in Classical and New Solids

Angle of incidence

Angular frequency

ˆ Bulk Poisson's ratio

Ratio of specific heats

62 23 2 62 66

*R*

*ii i*

11 1 ( ) ()2 ( )

( ) <sup>1</sup> ( ) <sup>1</sup> ( ) <sup>1</sup>

*j n A j B*

*z z*

2

*i i*

*A nB n B n j RR R*

Re , Real part and the complex conjugate

ˆ Loss factor

Maximum incident angle

Material properties

Rotational strain in the solid phase

Wave number in external and cavity media

one shear waves

2D dimension

2D dimension

The non-zero components of the matrix [ ] **h** appearing in equation (59) are followed as:

*R*

*i*

1,1 11 16 66 2

1,3 12 11 12 2 16

2 66

*<sup>j</sup> <sup>n</sup> <sup>R</sup> <sup>n</sup> <sup>B</sup>*

2

3 2

<sup>2</sup> 2 2

( ) <sup>1</sup> <sup>1</sup> ( ) <sup>1</sup> ( ) <sup>1</sup>

16 2 12

*<sup>A</sup> j A j nA n I <sup>R</sup> <sup>R</sup>* 

*<sup>A</sup> j Bj B j n B j n R R <sup>R</sup>*

*z*

*i i i*

1 11 ( )3 ( )

*zz z z*

 

*z*

3 2 2

*<sup>R</sup> <sup>j</sup> <sup>n</sup> <sup>B</sup>*

*<sup>i</sup> <sup>z</sup>*

1 1 2 () ( ) *zz i i i*

 

2

*j A*

*z*

2

2 62

*<sup>n</sup> <sup>R</sup>*

*i*

 

<sup>ˆ</sup>*<sup>x</sup> <sup>u</sup>* , ˆ*<sup>y</sup> <sup>u</sup>* , ˆ*Ux* , ˆ*Uy* Displacement components in the solid and fluid phases,

*i* Potential of the incident wave, 2D dimension

Complex wave numbers of the two compression and

Stresses in the solid and fluid phases, 2D dimension

*<sup>l</sup> v* Poisson's ratios in the directions 1 and 2 of the *l* th ply,

*m* 

> , ,

1 , <sup>3</sup> 

 , , *<sup>t</sup>* 

ˆ*s y* , ˆ *xy* , ˆ *<sup>f</sup>* 

( ) 12 *<sup>l</sup> v* , ( ) 21

**Appendix** 

*R A*

*i*

3 66 62

*z*

*R A*

*i* 

*j n B*

1,2 12 16

$$\mathcal{H}\_{4,4} = A\_{11} \left( -j\xi\_z \right)^2 + 2A\_{16} \frac{1}{R\_\epsilon} \{-j\xi\_z\} (-n) + A\_{66} \frac{1}{R\_\epsilon^2} (-n^2) - I\_\epsilon \alpha^2$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 59

> 

 

> 

 <sup>2</sup> 7,6 1 *s*

, 8,9 2 2 2 ( ) *<sup>R</sup> n n re r pH R* 

10,3 3 *s*

*z z*

)

2 3 2

62 61 66 22 3

*A j B j B nj B n R R <sup>R</sup>*

*z z z e e e*

1 11 ( ) ()3 ( )

*z z*

2 3 2

*Dj D nj D j n <sup>R</sup> <sup>R</sup>*

*z zz e e*

62 32 4 33 22

6,8 2 ( ), *H R n re* 

9,8 2 2 ( ), *H R n ri r* 

*ee e*

*D jn D j nD nI RR R*

11 1 4 ()4 ( )

6,6 22 2 21 22 3 2 23

, <sup>1</sup>

1 10,10 3 3 ( ) *H R n ri r* 

[1] Bolton J S, Shiau N M, Kang Y J (1996) Sound Transmission through Multi-Panel Structures Lined with Elastic Porous Materials. Journal of Sound and Vibration. 191:

[2] Lee J H, Kim J (2001) Simplified method to solve sound transmission through structures lined with elastic porous material. J. of the Acoust. Soc of Am. 110(5):

[3] Daneshjou K, Nouri A, Talebitooti R (2006) Sound transmission through laminated composite cylindrical shells using analytical model. Arch. Appl. Mech. 77: 363-379. [4] Daneshjou K, Talebitooti R, Nouri A (2007) Analytical Model of Sound Transmission through Orthotropic Double Walled Cylindrical Shells. CSME Transaction. 32(1).

<sup>2</sup>

<sup>123</sup> , *zzzz*

 

4 23 2

*e e e e*

1 11 2 () ( )

6,5 22 2 12 22 3 2 23

3 2

*A nB n j B n B j n <sup>R</sup> <sup>R</sup> R R*

*e e e e*

11 1 1 () () ()3 ( )

 

> 

*z z*

6,9 <sup>2</sup> ( ), *H R n re* 

9,9 2 2 ( ), *H R n ri r* 

 2 2 2 2 . *r z*

 .

<sup>2</sup>

> <sup>2</sup>

 

2 3 2 42

8,8 2 2 ( ), *H R n re r* 

3 2

2 2

1 1 2 () 4 ( )

*<sup>A</sup> B j B nB jn <sup>R</sup> <sup>R</sup> R R*

*zz e*

<sup>2</sup>

11 1 1 <sup>2</sup> 2 ( )4 ( )

*zz z*

*B jD j D j n R R <sup>R</sup>*

11 12 2 16

<sup>1</sup>

8,6 2 *s*

11 1 2 (

*D j nD n D j n R RR*

21 2 43 22 23

*e e e*

*e ee*

32 2 2 61 33

2 6,7 1 ( ), *H R n re* 

<sup>2</sup>

, <sup>1</sup>

 , *<sup>d</sup> dr*

2

2 7,7 1 1 ( ), *H R n re r* 

9,3 2 *s*

Here:

**12. References** 

317-347.

2282-2294.

$$\begin{split} \mathcal{H}\_{4,5} &= A\_{12} \frac{1}{R\_{\varepsilon}} (-j\check{\varphi}\_{z})(n) + A\_{16} \left(-j\check{\varphi}\_{z}\right)^{2} + B\_{12} \frac{1}{R\_{\varepsilon}^{2}} (-j\check{\varphi}\_{z})(n) + B\_{16} \frac{1}{R\_{\varepsilon}} (-j\check{\varphi}\_{z})^{2} + A\_{62} \frac{1}{R\_{\varepsilon}^{2}} (-n^{2}) + \\ &A\_{66} \frac{1}{R\_{\varepsilon}} (-j\check{\varphi}\_{z})(n) + B\_{62} \frac{1}{R\_{\varepsilon}^{3}} (-n^{2}) + B\_{66} \frac{1}{R\_{\varepsilon}^{2}} (-j\check{\varphi}\_{z})(n) \end{split}$$

$$\begin{split} \mathcal{H}\_{4,6} &= A\_{12} \frac{1}{R\_{\epsilon}} \big( -j\check{\varphi}\_{z} \big) - B\_{11} \big( -j\check{\varphi}\_{z} \big)^{3} - B\_{12} \frac{1}{R\_{\epsilon}^{2}} \big( -j\check{\varphi}\_{z} \big) \big( -n^{2} \big) - 3B\_{16} \frac{1}{R\_{\epsilon}} \big( -j\check{\varphi}\_{z} \big)^{2} \big( -n \big) + 1 \\\ A\_{62} \frac{1}{R\_{\epsilon}^{2}} \big( -n \big) + -B\_{62} \frac{1}{R\_{\epsilon}^{3}} \big( n^{3} \big) - 2B\_{66} \frac{1}{R\_{\epsilon}^{2}} \big( -n^{2} \big) \big( -j\check{\varphi}\_{z} \big) \end{split}$$

$$\begin{split} \mathcal{H}\_{5,4} &= A\_{12} \frac{1}{R\_{\varepsilon}} (-j\check{\varphi}\_{z})(n) + A\_{16} \left(-j\check{\varphi}\_{z}\right)^{2} + B\_{12} \frac{1}{R\_{\varepsilon}^{2}} (-j\check{\varphi}\_{z})(n) + B\_{16} \frac{1}{R\_{\varepsilon}} (-j\check{\varphi}\_{z})^{2} + A\_{62} \frac{1}{R\_{\varepsilon}^{2}} (-n^{2}) + B\_{22} \frac{1}{R\_{\varepsilon}^{2}} (-n^{2}) \\ &A\_{66} \frac{1}{R\_{\varepsilon}} (-j\check{\varphi}\_{z})(n) + B\_{62} \frac{1}{R\_{\varepsilon}^{3}} (-n^{2}) + B\_{66} \frac{1}{R\_{\varepsilon}^{2}} (-j\check{\varphi}\_{z})(n) \end{split}$$

$$\begin{split} \mathcal{H}\_{5,5} &= A\_{22} \frac{1}{R\_{\varepsilon}^{2}} (-n^{2}) + 2A\_{26} \frac{1}{R\_{\varepsilon}} (-j\xi\_{z}) \{n\} + 2B\_{22} \frac{1}{R\_{\varepsilon}^{3}} (-n^{2}) + 4B\_{26} \frac{1}{R\_{\varepsilon}^{2}} (-j\xi\_{z}) \{n\} + A\_{66} \left(-j\xi\_{z}\right)^{2} + 4B\_{26} \frac{1}{R\_{\varepsilon}^{3}} (-n^{2}) \\ &+ 2B\_{66} \frac{1}{R\_{\varepsilon}} (-j\xi\_{z})^{2} + D\_{22} \frac{1}{R\_{\varepsilon}^{4}} (-n^{2}) + 2D\_{23} \frac{1}{R\_{\varepsilon}^{3}} (-j\xi\_{z}) \{n\} + D\_{33} \frac{1}{R\_{\varepsilon}^{2}} (-j\xi\_{z})^{2} - I\_{\varepsilon}o^{2} \end{split}$$

$$\begin{split} \mathcal{H}\_{5,6} &= A\_{22} \frac{1}{R\_{\epsilon}^{2}} (-n) - B\_{12} \frac{1}{R\_{\epsilon}} (-n) \left( -j\dot{\boldsymbol{\varepsilon}}\_{z} \right)^{2} - B\_{22} \frac{1}{R\_{\epsilon}^{3}} (n^{3}) - 3B\_{23} \frac{1}{R\_{\epsilon}^{2}} (-j\dot{\boldsymbol{\varepsilon}}\_{z}) (-n^{2}) + \cdots \\ & A\_{62} \frac{1}{R\_{\epsilon}} \Big( -j\dot{\boldsymbol{\varepsilon}}\_{z} \Big) + -B\_{61} \Big( -j\dot{\boldsymbol{\varepsilon}}\_{z} \Big)^{3} - 2B\_{66} \frac{1}{R\_{\epsilon}} (-n) \Big( -j\dot{\boldsymbol{\varepsilon}}\_{z} \Big)^{2} + B\_{22} \frac{1}{R\_{\epsilon}^{3}} (-n) - \\ & D\_{21} \frac{1}{R\_{\epsilon}^{2}} \Big( -j\dot{\boldsymbol{\varepsilon}}\_{z} \Big)^{2} (-n) - D\_{22} \frac{1}{R\_{\epsilon}^{4}} (n^{3}) - 3D\_{23} \frac{1}{R\_{\epsilon}^{3}} (-j\dot{\boldsymbol{\varepsilon}}\_{z}) (-n^{2}) + B\_{32} \frac{1}{R\_{\epsilon}^{2}} (-j\dot{\boldsymbol{\varepsilon}}\_{z}) - \\ & D\_{61} \frac{1}{R\_{\epsilon}} \Big( -j\dot{\boldsymbol{\varepsilon}}\_{z} \Big)^{3} - 2D\_{33} \frac{1}{R\_{\epsilon}^{2}} (-j\dot{\boldsymbol{\varepsilon}}\_{z})^{2} (-n) \end{split}$$

$$\begin{split} \mathcal{H}\_{6,4} &= A\_{12} \frac{1}{R\_{\varepsilon}} \big( -j\underline{\varepsilon}\_{z} \big) - B\_{11} \big( -j\underline{\varepsilon}\_{z} \big)^{3} - B\_{12} \frac{1}{R\_{\varepsilon}^{2}} \big( -j\underline{\varepsilon}\_{z} \big) \big( -n^{2} \big) - 3B\_{16} \frac{1}{R\_{\varepsilon}} \big( -j\underline{\varepsilon}\_{z} \big)^{2} \big( -n \big) + 1 \\\ A\_{62} \frac{1}{R\_{\varepsilon}^{2}} \big( -n \big) + -B\_{62} \frac{1}{R\_{\varepsilon}^{3}} \big( n^{3} \big) - 2B\_{66} \frac{1}{R\_{\varepsilon}^{2}} \big( -n^{2} \big) \big( -j\underline{\varepsilon}\_{z} \big) \end{split}$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 59

$$\begin{split} \mathcal{H}\_{6,5} &= A\_{22} \frac{1}{R\_{\epsilon}^{2}} (-n) - B\_{12} \frac{1}{R\_{\epsilon}} (-n) \left( -j\underline{\varepsilon}\_{z} \right)^{2} - B\_{22} \frac{1}{R\_{\epsilon}^{3}} (n^{3}) - 3B\_{23} \frac{1}{R\_{\epsilon}^{2}} (-j\underline{\varepsilon}\_{z}) (-n^{2}) + 1 \\ &A\_{62} \frac{1}{R\_{\epsilon}} \left( -j\underline{\varepsilon}\_{z} \right) + -B\_{61} \left( -j\underline{\varepsilon}\_{z} \right)^{3} - 2B\_{66} \frac{1}{R\_{\epsilon}} (-n) \left( -j\underline{\varepsilon}\_{z} \right)^{2} + B\_{22} \frac{1}{R\_{\epsilon}^{3}} (-n) - \\ &D\_{21} \frac{1}{R\_{\epsilon}^{2}} (-j\underline{\varepsilon}\_{z})^{2} \left( -n \right) - D\_{22} \frac{1}{R\_{\epsilon}^{4}} (n^{3}) - 3D\_{23} \frac{1}{R\_{\epsilon}^{3}} (-j\underline{\varepsilon}\_{z}) (-n^{2}) + \\ &B\_{32} \frac{1}{R\_{\epsilon}^{2}} (-j\underline{\varepsilon}\_{z}) - D\_{61} \frac{1}{R\_{\epsilon}} (-j\underline{\varepsilon}\_{z})^{3} - 2D\_{33} \frac{1}{R\_{\epsilon}^{2}} (-j\underline{\varepsilon}\_{z})^{2} (-n) \end{split}$$

$$\begin{split} h\_{6,6} &= A\_{22} \frac{1}{R\_{\varepsilon}^{2}} - 2B\_{21} \frac{1}{R\_{\varepsilon}} \big( -j\xi\_{z} \big)^{2} - 2B\_{22} \frac{1}{R\_{\varepsilon}^{3}} (-n^{2}) - 4B\_{23} \frac{1}{R\_{\varepsilon}^{2}} \big( -j\xi\_{z} \big) (-n) + 1 \\ &- D\_{11} \big( -j\xi\_{z} \big)^{4} - 2D\_{12} \frac{1}{R\_{\varepsilon}^{2}} (-n^{2}) \big( -j\xi\_{z} \big)^{2} + 4D\_{16} \frac{1}{R\_{\varepsilon}} \big( -j\xi\_{z} \big)^{3} (-n) + \\ &4D\_{62} \frac{1}{R\_{\varepsilon}^{3}} \big( -j\xi\_{z} \big) (n^{3}) + 4D\_{33} \frac{1}{R\_{\varepsilon}^{2}} \big( -j\xi\_{z} \big)^{2} (-n^{2}) + D\_{22} \frac{1}{R\_{\varepsilon}^{4}} \big( n^{4} \big) - I\_{\varepsilon} o^{2} \end{split}$$

$$\boldsymbol{\hbar}\_{6,7} = \boldsymbol{H}\_n^2(\boldsymbol{\xi}\_{1r}\boldsymbol{R}\_e), \ \boldsymbol{\hbar}\_{6,8} = \boldsymbol{H}\_n^1(\boldsymbol{\xi}\_{2r}\boldsymbol{R}\_e), \ \boldsymbol{\hbar}\_{6,9} = -\boldsymbol{H}\_n^2(\boldsymbol{\xi}\_{2r}\boldsymbol{R}\_e), \ \boldsymbol{\hbar}\_{7,6} = -\boldsymbol{s}\_1\boldsymbol{\alpha}^2$$

$$\boldsymbol{\hbar}\_{7,7} = \boldsymbol{H}\_{\boldsymbol{n}}^{\mathcal{I}} \left( \boldsymbol{\xi}\_{1r} \, \boldsymbol{R}\_{\boldsymbol{\varepsilon}} \right) \boldsymbol{\xi}\_{1r'} \quad \boldsymbol{\hbar}\_{8,6} = -\boldsymbol{s}\_2 \, \boldsymbol{\alpha}^2, \quad \boldsymbol{\hbar}\_{8,8} = \boldsymbol{H}\_{\boldsymbol{n}}^{\mathcal{I}} \left( \boldsymbol{\xi}\_{2r} \, \boldsymbol{R}\_{\boldsymbol{\varepsilon}} \right) \boldsymbol{\xi}\_{2r'} \quad \boldsymbol{\hbar}\_{8,9} = \boldsymbol{p}\_{\boldsymbol{n}2}^R \, \boldsymbol{H}\_{\boldsymbol{n}}^{\mathcal{I}} \left( \boldsymbol{\xi}\_{2r} \, \boldsymbol{R}\_{\boldsymbol{\varepsilon}} \right) \boldsymbol{\xi}\_{2r'} \quad \boldsymbol{\hbar}\_{9,1} = \boldsymbol{p}\_{\boldsymbol{n},1}^R \, \boldsymbol{H}\_{\boldsymbol{n}}^{\mathcal{I}} \left( \boldsymbol{\xi}\_{2r} \, \boldsymbol{R}\_{\boldsymbol{\varepsilon}} \right) \boldsymbol{\xi}\_{2r'} \quad \boldsymbol{\hbar}\_{1,2} = \boldsymbol{p}\_{\boldsymbol{n},1}^R \, \boldsymbol{H}\_{\boldsymbol{n}}^{\mathcal{I}} \left( \boldsymbol{R}\_{\boldsymbol{\varepsilon}} \, \boldsymbol{R}\_{\boldsymbol{\varepsilon}} \right) \boldsymbol{\$$

$$\begin{aligned} \hbar\_{9,3} = -\mathbf{s}\_2 \,\mathrm{o}\boldsymbol{\theta}^2, \quad \hbar\_{9,8} = \mathrm{H}\_{\mathrm{n}}^{\mathrm{I}}\left(\boldsymbol{\xi}\_{2r}\,\mathrm{R}\_{\mathrm{i}}\right)\boldsymbol{\xi}\_{2r}, \quad \hbar\_{9,9} = \mathrm{H}\_{\mathrm{n}}^{\mathrm{2}'}\left(\boldsymbol{\xi}\_{2r}\,\mathrm{R}\_{\mathrm{i}}\right)\boldsymbol{\xi}\_{2r}, \quad \hbar\_{10,3} = -\mathbf{s}\_3 \,\mathrm{o}\boldsymbol{\alpha}^2, \end{aligned}$$

$$\boldsymbol{\hbar}\_{10,10} = \boldsymbol{\hbar}\_{\mathrm{n}}^{\mathrm{I}'} (\boldsymbol{\xi}\_{3r}\,\mathrm{R}\_{\mathrm{i}})\boldsymbol{\xi}\_{3r}$$

Here:

58 Wave Processes in Classical and New Solids

<sup>2</sup> 2 2

<sup>1</sup> 1 11 ( ) ( ) ( )

 

*A j A j nA n I <sup>R</sup> <sup>R</sup>* 

1 1 2 () ( ) *zz e e e*

2 2 2

   

> 

4,4 11 16 66 2

*zz z z e e e e*

> 

*e e e*

*zz z z e e e e*

> 

2 2 2 2

*z z z*

*<sup>A</sup> j Bj B j n B j n R R <sup>R</sup>*

1 11 ( )3 ( )

*zz z z*

 

*z*

3 2

*A nB n j B n B j n <sup>R</sup> <sup>R</sup> R R*

*e e e e*

11 1 1 () () ()3 ( )

 

3 2 2

*A n A j n B n B j nA j <sup>R</sup> <sup>R</sup> R R*

*A j nA j B j nB j A n R R R R*

5,4 12 16 12 2 2 16 62

5,5 22 2 26 22 3 2 26 66

*z z z e*

11 11 ( ) 2 () 2 ( ) 4 ()

*A j nA j B j nB j A n R R R R*

*<sup>A</sup> j Bj B j n B j n R R <sup>R</sup>*

1 11 ( )3 ( )

*zz z z*

<sup>1</sup> 1 11 ( ) ( ) ( )

 

 

*z*

3 2 2

2 2 2

*z z*

*z z z*

 

2 3 2

2 2 2

 

 

> 

 

 

4,5 12 16 12 2 2 16 62

4,6 12 11 12 2 16

*B j D n D j nD j I <sup>R</sup> RR R*

 

6,4 12 11 12 2 16

3 2

62 61 66 22 3

*A j B j B nj B n R R <sup>R</sup>*

*z z z e e e*

2 3 2 21 2 43 2 22 23 32

*D j nD n D j n B j R RR R*

1 11 1 ( ) ()3 ( )

*e ee e*

)

*e e e*

1 11 2 () ( )

*e e e e*

5,6 22 2 12 22 3 2 23

3 2

1 11 () ( ) ( )

*z z*

1 11 ( ) ()2 ( )

*A nB n B nj R RR*

66 62 3 2 66

*e e e*

*A j nB n B j n <sup>R</sup> R R*

62 2 32 62 66

*e ee*

1 11 () ( ) ( )

*z z*

66 62 3 2 66

*e e e*

*A j nB n B j n <sup>R</sup> R R*

2

66 22 43 2 23 33

*e ee e*

1 11 1 <sup>2</sup> ( )2 ( )

*D j D jn <sup>R</sup> <sup>R</sup>*

62 2 32 62 66

*e ee*

*z z*

1 1 2 (

61 33 2

*e e*

3 2

1 11 ( ) ()2 ( )

*A nB n B nj R RR*

2

$$\left(\begin{array}{c} \\ \end{array}\right)' = \frac{d}{dr}\prime\prime\_z = \xi\_{1z} = \xi\_{2z} = \xi\_{3z} \,, \quad \xi\_{2r} = \sqrt{\xi\_2^2 - \xi\_z^2} \,, \dots$$

#### **12. References**

	- [5] Daneshjou K, Talebitooti R, Nouri A (2009) Investigation model of sound transmission through orthotropic cylindrical shells with subsonic external flow. Aerospace Science and Technology. 13: 18-26.

**Chapter 0**

**Chapter 3**

**Linear Wave Motions in Continua with Nano-Pores**

The first attempts to describe the behaviour of porous materials with the use of an additional degree of kinematical freedom, in order to refine the Cauchy's theory, are due principally to Nunziato and Cowin [1, 2] and co-workers. Nevertheless, their voids theory can be considered as a particular case of a general theory of continua with microstructure [3] and so, when we have to consider more complex media with nano-pores, we need to use suggestions of this last theory [4]. In fact, a nano-pore in a thermoelastic solid is roughly ellipsoidal, unlike small lacunae finely dispersed in the solid matrix that can be supposed all spherical and for which the volume fraction suffices to describe the microdeformation (see, also, [5, 6]). Cowin itself remarked the importance of the shape of the holes in the description of lacunae containing osteocytes or of bone canaliculi [7, 8]: in the human bone, *e.g.*, the lacunae are almost ellipsoidal with mean values along the axes of about 4 *μ*m, 9 *μ*m and 22 *μ*m. And, as a matter of fact, the voids theory does not predict size effects in torsion of bars in an isotropic material, while they occur both in torsion and in bending, as observed for bones and polymer foam materials in [9]. Even if some problem of physical concreteness or of mathematical hardness could arise [10, 11], a better improvement of the voids theory, within a microstructured scheme, is necessary in order to characterize the more complex structure. A direct way to proceed is to consider the thermoelastic solid with nano-pores as a continuum with an ellipsoidal microstructure (see [4, 12]) which describes media whose each material element contains a large cavity, that does not diffuse through the skeleton, filled by an elastic inclusion, or an inviscid fluid, both of negligible mass (*e.g.*, composite materials reinforced with chopped elastic fibers, porous media with elastic granular inclusions, real ceramics, *etc.*): this cavity is able to have a microstretch different from, and independent of, the local affine deformation deriving from the macromotion and so can allow distinct microstrains along the principal axes of microdeformation, in absence of microrotations . The "tortuosity" matrix, a macroscopic geometrical symmetric tensor that expresses the effects of the geometry of the microscopic pores' surface, was previously presented in [13], but in [14] the model of a microstretched medium has been firstly used to study materials with distributions of aligned ellipsoidal vacuous pores and explicit computations have been carried

> ©2012 Giovine, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

Pasquale Giovine

**1. Introduction**

http://dx.doi.org/10.5772/50768

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cited.

