**1. Introduction**

34 Wave Processes in Classical and New Solids

[49] Pride S R, Berryman J G, Harris J M (2004) Seisimic attenuation due to wave-induced

[50] Mavko G A, Jizba D (1994) The relation between seismic P- and S-wave velocity

[51] Zaitsev V, Sas P (2004) Effect of high-compiant porosity on variations of P- and S- wave velocities in dry and saturated rocks: comparison between theory and experiment.

[52] Ba J, Carcione J M, Nie J X (2011) Biot-Rayleigh theory of wave propagation in doubleporosity media. J. Geophys. Res. solid earth, 116: B06202, doi: 10.1029/2010JB008185.. [53] Ba J, Carcione J M, Cao H, Du Q Z, Yuan Z Y, Lu M H (2012) Velocity dispersion and attenuation of P waves in partially-saturated rocks: Wave propagation equations in

double-porosity medium. Chinese J. Geophys. (in Chinese), 55(1): 219-231.

flow. J. Geophys. Res., 109, B01201, doi:10.1029/2003JB002639.

dispersion in saturated rocks. Geophysics, 59(1): 87-92.

Physical Mesomechanics, 7(1-2): 37-46.

Although the researchers have done many efforts to perform the numerical model such as FEM (Finite Elements Method) to investigate the wave prorogation through the shells, the analytical vibro-acoustic modeling of the composite shells is unavoidable because of the accuracy of the model in a broadband frequency. Bolton *et. al.* [1] investigated sound transmission through sandwich structures lined with porous materials and following Lee *et. al.* [2] proposed a simplified method to analyze curved sandwich structures. Daneshjou *et. al.* [3-5] studied an exact solution to estimate the transmission loss of orthotropic and laminated composite cylindrical shells with considering all three displacements of the shell. Recently the authors [6] have presented an exact solution of free harmonic wave propagation in a double-walled laminated composite cylindrical shell whose walls sandwich a layer of porous material using an approximate method. This investigation is focused on sound transmission through the sandwich structure, which includes the porous material core between the two laminated composite cylindrical shells to predict the reliable results for all structures used foam as an acoustic treatment.

Wave propagation through a composite cylindrical shell lined with porous materials is investigated, based on classical laminated theory. The porous material is completely modeled using elastic frame. The vibro-acoustic equations of the shell are derived considering both the shell vibration equations and boundary conditions on interfaces. These coupled equations are solved simultaneously to calculate the Transmission Loss (TL). Moreover, the results are verified with a special case where the porosity approaches zero. Finally, the numerical results are illustrated to properly study the geometrical and physical properties of composite and porous material. In addition, the effects of the stacking sequence of composite shells and fiber directions are properly studied.

© 2012 Daneshjou et al., 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 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, and reproduction in any medium, provided the original work is properly cited.

#### **2. Propagation of sound in porous media**

If the porous material is assumed a homogeneous aggregate of the elastic frame and the fluid trapped in pores, its acoustic behavior can be considered by the following two wave equations (See Eq. (22) and Eq. (25) of [1]):

$$
\nabla^4 e\_s + A\_1 \nabla^2 e\_s + A\_2 e\_s = 0 \tag{1}
$$

$$
\nabla^2 \varpi + \xi\_t^2 \varpi = 0 \tag{2}
$$

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 37

> 

and

 

(7)

(8)

(9)

(11)

1 *f f*

*E <sup>E</sup>* , <sup>1</sup> 1 *s f E <sup>E</sup>* , <sup>3</sup> 1 *f f*

*E <sup>E</sup>* , <sup>3</sup> 1 *s f E <sup>E</sup>* , <sup>5</sup> 1 *s f E E*

<sup>0</sup>( 1) *<sup>a</sup>*

<sup>2</sup> 22 11 12 ( 2) 2( )

<sup>2</sup> 22 11 12 ( 2) 2( )

 

2 2 2 22 11 12

> 

22 ( ) *<sup>t</sup>*

22 11 12 22 11 12 (

 

As the full method is too complicated to model the porous layer in the curved sandwich structures, thus a simplified method is expanded for this category of structures [2]. The foundation of this approximate method considers the strongest wave between those ones. It includes two steps. At the first step a flat double laminated composite with infinite extents with the same cross sectional construction is considered using the full method. Then, only the strongest wave number is chosen from the results and the material is modeled using the wave number and its corresponding equivalent density. Thus, the material is modeled as an

The strain energy which is related to the displacement in the solid and fluid phases can be defined for each wave component. The energy terms can be represented as follows; 1*<sup>s</sup> E* and <sup>1</sup> *<sup>f</sup> E* for the airborne wave, 3*<sup>s</sup> E* and 3 *<sup>f</sup> E* for the frame wave and 5*<sup>s</sup> E* for the shear wave, which the subscripts *s* and *f* represent the solid and fluid phase, respectively. For each new problem, comparing the ratios of the energy carried by the frame wave and the shear

 

> 

wave to the airborne wave in the fluid and solid phases: i.e., <sup>1</sup>

show the strongest wave component in the entire frequency range.

 

 

 

 

22 2

2 ) 4( )( ) (10)

 

The complex wave numbers of the two compression (longitudinal) waves,

2

2

 

 

 

2

2

and the wave number of the shear (rotational) wave is:

is the shear modulus of the porous material.

**3. Simplified method** 

equivalent fluid.

are:

where

Eqs. (1) and (2) determine 2 elastic longitudinal waves and 1 rotational wave, respectively. In Eqs. (1) and (2) is the vector differential operator, . *<sup>s</sup> e* **u** is the solid volumetric strain, **u** is the displacement vector of the solid, **u** is the rotational strain in the solid phase, 2 2 <sup>1</sup> 22 11 12 *A* ( 2 )/( ) , 4 22 2 22 11 12 *A* ( ( ) )/( ) , and *t* is the wave number of the shear wave (See Eq. 10). 11 , 12 and 22 are equivalent masses given by:

$$
\hat{\rho}\_{11} = \rho\_1 + \rho\_a - j\sigma\_r \phi^2(\frac{1}{o\nu} + \frac{4j\alpha\_\alpha^2 \kappa\_v \rho\_0}{\sigma\_r^2 \Lambda^2 \phi^2}) \tag{3}
$$

$$
\hat{\rho}\_{12} = -\rho\_a + j\sigma\_r \phi^2 \left(\frac{1}{a\nu} + \frac{4j\alpha\_\alpha^2 \kappa\_v \rho\_0}{\sigma\_r^2 \Lambda^2 \phi^2}\right) \tag{4}
$$

$$
\hat{\rho}\_{22} = \phi \rho\_0 + \rho\_a - j\sigma\_r \phi^2 (\frac{1}{a\nu} + \frac{4j\alpha\_\alpha^2 \kappa\_\upsilon \rho\_0}{\sigma\_r^2 \Lambda^2 \phi^2}) \tag{5}
$$

where *j* is the imaginary unit <sup>2</sup> *j* 1 . 1 and 0 are the densities of the solid and fluid parts of the porous material. Moreover, parameters , *<sup>v</sup>* , , *<sup>r</sup>* , and are tortuosity, air viscosity, viscous characteristic length, flow resistivity, porosity, and angular frequency, respectively. , and represent material properties: (1 ) *G* , *G* , *A* 2 , *E* / 2(1 ) and *A E* / (1 )(1 2 ) . *E* and *v* are the in vacuo Young's modulus and Poisson's ratio of the bulk solid phase, respectively. Assuming that pores are shaped in cylindrical form, an expression for *G* is:

$$G = \rho\_0 c\_2^2 \left[ 1 + \left[ (\boldsymbol{\zeta} - 1) \sqrt{\boldsymbol{\phi} \sigma\_r} \;/\, \text{N}\_{\text{Pr}}^{0.5} \sqrt{-2 \dot{\boldsymbol{\rho}} \boldsymbol{\alpha} \rho\_0 \boldsymbol{\alpha}\_w} \right] \left[ \frac{J\_1 \left( 2 \text{N}\_{\text{Pr}}^{0.5} \sqrt{-2 \boldsymbol{\alpha} \rho\_0 \boldsymbol{\alpha}\_w \mathbf{j} / \, \boldsymbol{\phi} \sigma\_r} \right)}{J\_0 \left( 2 \text{N}\_{\text{Pr}}^{0.5} \sqrt{-2 \boldsymbol{\alpha} \rho\_0 \boldsymbol{\alpha}\_w \mathbf{j} / \, \boldsymbol{\phi} \sigma\_r} \right)} \right] \right]^{-1} \tag{6}$$

where is the ratio of specific heats, 2 *c* is the speed of sound in the fluid phase of porous materials, *N*Pr represents the Prandtl number, and 0*J* and 1*J* are Bessel functions of the first kind, zero and first order, respectively. *<sup>a</sup>* is the inertial coupling term:

Acoustical Modeling of Laminated Composite Cylindrical Double-Walled Shell Lined with Porous Materials 37

$$
\rho\_a = \phi \rho\_0 (\alpha\_\alpha - 1) \tag{7}
$$

The complex wave numbers of the two compression (longitudinal) waves, and are:

$$\xi\_{\alpha}^{\varepsilon^2} = \frac{\alpha^2}{2(\varrho\mu - \chi^2)} (\varrho\hat{\rho}\_{22} + \mu\hat{\rho}\_{11} - 2\chi\hat{\rho}\_{12} + \sqrt{\wp}) \tag{8}$$

$$\left|\xi\right|^2\_{\beta} = \frac{\alpha^2}{2(\varrho\mu - \chi^2)} (\varrho\hat{\rho}\_{22} + \mu\hat{\rho}\_{11} - 2\chi\hat{\rho}\_{12} - \sqrt{\wp})\tag{9}$$

where

36 Wave Processes in Classical and New Solids

**2. Propagation of sound in porous media** 

strain, **u** is the displacement vector of the solid,

solid phase, 2 2

 (

where *j* is the imaginary unit <sup>2</sup>

 and *A E* 

cylindrical form, an expression for *G* is:

 , and 

*t* 

masses given by:

respectively.

*E* / 2(1 )

where  <sup>1</sup> 22 11 12 *A*

 

 

parts of the porous material. Moreover, parameters

is the wave number of the shear wave (See Eq. 10). 11

 

 

equations (See Eq. (22) and Eq. (25) of [1]):

If the porous material is assumed a homogeneous aggregate of the elastic frame and the fluid trapped in pores, its acoustic behavior can be considered by the following two wave

> 4 2 <sup>0</sup> 1 2 *e A e Ae s ss*

> > 2 2 <sup>0</sup> *<sup>t</sup>*

Eqs. (1) and (2) determine 2 elastic longitudinal waves and 1 rotational wave, respectively. In Eqs. (1) and (2) is the vector differential operator, . *<sup>s</sup> e* **u** is the solid volumetric

> 

11 1 2 22

12 2 22

22 0 2 22

 and 0 

air viscosity, viscous characteristic length, flow resistivity, porosity, and angular frequency,

Poisson's ratio of the bulk solid phase, respectively. Assuming that pores are shaped in

 

materials, *N*Pr represents the Prandtl number, and 0*J* and 1*J* are Bessel functions of the

is the ratio of specific heats, 2 *c* is the speed of sound in the fluid phase of porous

represent material properties:

1 Pr 0 2 0.5

 

0 2 Pr 0 0.5

*a r*

*<sup>j</sup> <sup>j</sup>*

*<sup>j</sup> <sup>j</sup>*

*a r*

*a r*

 

*j* 1 . 1 

 / (1 )(1 2 ) 

1 1/ 2

 

*G c N j*

 

first kind, zero and first order, respectively. *<sup>a</sup>*

*r*

1 4 ( ) *<sup>v</sup>*

*<sup>j</sup> <sup>j</sup>*

2 )/( ) , 4 22

2 2 0

 (3)

*r*

2 2 0

2 2 0

 (5)

> (1 ) *G* , *G* , *A* 2

0.5

0 Pr 0

is the inertial coupling term:

. *E* and *v* are the in vacuo Young's modulus and

 

 

22 /

*JN j*

*JN j*

22 /

*r*

 

 

<sup>1</sup> <sup>4</sup> ( ) *<sup>v</sup>*

 

*r*

 

1 4 ( ) *<sup>v</sup>*

> , *<sup>v</sup>* , , *<sup>r</sup>* , and

 

 

 

 , 12 

(4)

(1)

(2)

2 22 11 12 *A*

**u** is the rotational strain in the

( ( ) )/( )

are the densities of the solid and fluid

 

are equivalent

are tortuosity,

1

*r*

*r*

,

(6)

, and

 

 and 22 

$$\hat{\rho} = (\hat{\rho}\hat{\rho}\_{22} + \mu\hat{\rho}\_{11} - 2\,\hat{\chi}\hat{\rho}\_{12})^2 - 4(\hat{\rho}\mu - \chi^2)(\hat{\rho}\_{22}\hat{\rho}\_{11} - \hat{\rho}\_{12}^2) \tag{10}$$

and the wave number of the shear (rotational) wave is:

$$
\xi\_t^2 = \frac{\alpha^2}{\delta} (\frac{\hat{\rho}\_{22}\hat{\rho}\_{11} - \hat{\rho}\_{12}^2}{\hat{\rho}\_{22}}) \tag{11}
$$

is the shear modulus of the porous material.

## **3. Simplified method**

As the full method is too complicated to model the porous layer in the curved sandwich structures, thus a simplified method is expanded for this category of structures [2]. The foundation of this approximate method considers the strongest wave between those ones. It includes two steps. At the first step a flat double laminated composite with infinite extents with the same cross sectional construction is considered using the full method. Then, only the strongest wave number is chosen from the results and the material is modeled using the wave number and its corresponding equivalent density. Thus, the material is modeled as an equivalent fluid.

The strain energy which is related to the displacement in the solid and fluid phases can be defined for each wave component. The energy terms can be represented as follows; 1*<sup>s</sup> E* and <sup>1</sup> *<sup>f</sup> E* for the airborne wave, 3*<sup>s</sup> E* and 3 *<sup>f</sup> E* for the frame wave and 5*<sup>s</sup> E* for the shear wave, which the subscripts *s* and *f* represent the solid and fluid phase, respectively. For each new problem, comparing the ratios of the energy carried by the frame wave and the shear wave to the airborne wave in the fluid and solid phases: i.e., <sup>1</sup> 1 *f f E <sup>E</sup>* , <sup>1</sup> 1 *s f E <sup>E</sup>* , <sup>3</sup> 1 *f f E <sup>E</sup>* , <sup>3</sup> 1 *s f E <sup>E</sup>* , <sup>5</sup> 1 *s f E E* show the strongest wave component in the entire frequency range.
