Abstract
This paper applies acoustic analysis of Sound Transmission Loss (STL) through infinite Functionally Graded (FG) thick plate employing Hyperbolic Shear Deformation Theory (HSDT). The procedure for applying a FG plate is followed by considering the material properties are changed continually based on powerlaw distribution of the materials in terms of volume fraction. The main benefit of HSDT can be justified knowing the fact that, it uses parabolic transverse shear strain across thickness direction. Therefore, no need to enter the extra effect of shear correction coefficient factor. Besides, the displacement field is extended as a combination of polynomial as well as hyperbolic tangent function by neglecting the effect of thickness stretching. Furthermore, the equations of motion are obtained employing Hamilton’s Principle. To provide an analytical solution based on HSDT, equations of motion are combined with acoustic wave equations. Moreover, some comparisons are made with the known theoretical and experimental results available in literature to verify the accuracy and efficiency of the current formulation. These comparisons reveal an excellent agreement. Consequently, some configurations are presented to demonstrate which parameters appear to be effective to improve the behavior of STL including the effects of modulus of elasticity and density in the thickness direction with respect to various powerlaw distributions.
Keywords
Functionally Graded Material; power transmission; Hyperbolic Shear Deformation Theory; orthotropic thick plate; acoustic
1 INTRODUCTION
The application of the Functionally Graded Material (FGM) as new type of inhomogeneous composite materials in many various typical aspects including space rockets, launchers, space shuttles, nuclear reactors and chemical plants is extensively well defined in literature. Besides, another application of these materials can be addressed in high temperature environments. As it is obvious, the importance of using these materials was remarked for the first time by Japanese scientists in 1984 so that many years later the first conference around them was held in Japan based on Yamanouchi et al. (1991Yamanouchi, M., Koizumi, M., (1991). Functionally gradient materials. Proceeding of the first international symposium on functionally graded materials.). The main benefit of this material is devoted to the smoothness as well as continuousness variation of the material properties in various directions. In addition, when the constituent volume fraction is altered, the various properties of these materials are changed continuously. Accordingly, the properties of these materials are gradually changed based on specific application and environmental loading. As another aspect, the other application of FGM can be found at the interface due to preventing the interface disbanding as well as cracking.
Noise transmission, measured by Sound Transmission Loss (STL) through flat or curved wall has been proposed by many authors in the past and it is still continuing. Accordingly, in an analytical model presented by London (1950London, A., (1950). Transmission of reverberant sound through double walls. The journal of the acoustical society of America 22 : 270279.), acoustic transmission through two identical parallel plates was determined in both theoretical as well as experimental methods. Next, Maestrello (1995Maestrello, L., (1995). Responses of finite baffled plate to turbulent flow excitations. AIAA Journal 33: 1319.) presented a formulation for acoustic and dynamic response of the finite baffled plate using both experimental and analytical approaches. In the following, Galerkin’s method was applied by Clark et al. (1996 aClark, R. L., Frampton, K. D., (1996a). Sound transmission through an aeroelastic plate into an acoustic cavity. The Journal of the Acoustical Society of America 99: 25862603., ^{1996 b}Clark, R. L., (1996b). Transmission of stochastic pressures through an aeroelastic plate into a cavity. In 37th Structure, Structural Dynamics and Materials Conference 1445.) across STL of the convected fluid loaded plate employing singular value decomposition method. In another work, sound transmission of the elastic plate was achieved to show the transmission of turbulent boundary layer noise in the presence of external full potential flow considering acoustic energy in the cavity. Afterwards, the wave propagation across finite plate backed with or without cavity was done by Bhattachary et al. (1971)Bhattacharya, M. C., Guy, R. W., Crocker, M. J., (1971). Coincidence effect with sound waves in a finite plate. Journal of sound and vibration 18: 157169.. Moreover, Koval (1976Koval, L. R., (1976). Effect of air flow, panel curvature, and internal pressurization on field‐incidence transmission loss. The Journal of the Acoustical Society of America 59: 13791385.) suggested an analytical model across STL of the singlewalled panel in the diffuse sound field considering the influences of panel curvature, external air flow and internal fuselage pressurization. Then, Roussos (1984Roussos, L. A., (1984). Noise transmission loss of a rectangular plate in an infinite baffle. The Journal of the Acoustical Society of America 75: S2S3.) calculated noise transmission through composite finite plate considering classicalthin plate theory under simply supported boundary condition on all four edges. The procedure was followed based on considering an oblique plane sound wave with an arbitrary angle. Renji et al. (1997Renji, K., Nair, P. S., Narayanan, S., (1997). Critical and coincidence frequencies of flat panels. Journal of Sound and Vibration 205: 1932.) presented coincidence and critical frequencies of the isotropic and composite thick panel under acoustic excitation. In another work presented by Tang et al. (2006aTang, H., Zhao, X. P., Luo, C. R., (2006a). Sonic responses of an electrorheological layer with one side of grating electrodes. Journal of Physics D: Applied Physics 39(3):552, ^{2006b}Tang, H., Luo, C. R., Zhao, X. P., (2006b). Sound transmission behavior through a sandwiched electrorheological layer. Fuhe Cailiao Xuebao/Acta Mater Compos Sinica 23: 128132.), acoustic transmission of the triple layered panel composed of two plastic plates was offered. Next, last research was developed by Xin et al. (2009Xin, F. X., Lu, T. J., Chen, C. Q., (2009). External mean flow influence on noise transmission through doubleleaf aeroelastic plates. AIAA J 47: 19391951., ^{2010}Xin, F. X., Lu, T. J., (2010). Analytical modeling of sound transmission across finite aeroelastic panels in convected fluids. The Journal of the Acoustical Society of America 128: 10971107. and ^{2011}Xin, F. X., Lu, T. J., (2011). Analytical modeling of sound transmission through clamped triplepanel partition separated by enclosed air cavities. European Journal of MechanicsA/Solids 30: 770782.) through doubleleaf plate in the presence of external flow. In addition, another work on simply supported rectangular aero elastic panel was presented. Besides, the wave propagation on clamped triplepanel was done to show STL of the structure as a function of excitation frequency. Chandra et al. (2015Chandra, N., Gopal, K.N., Raja, S. (2015). Vibroacoustic response of sandwich plates with functionally graded core. Acta Mechanica: 228(8):277589. ) obtained STL of the sandwich plate with functionally core. Recently, Talebitooti et al. (2018aTalebitooti, R., Johari, V., Zarastvand, M. R. (2018a). Wave transmission across laminated composite plate in the subsonic flow Investigating Twovariable Refined Plate Theory. Latin American Journal of Solids and Structures, vol. 15 (5).) considered twovariable refined theory to obtain acoustic transmission of a plate in the external flow.
In the following, it is attempted to present some literature review through acoustic transmission of the cylindrical shell due to various technical applications including aircraft and launcher as a fuselage skin. Liu et al. (2016Liu, Y., He, C. (2016). Analytical modelling of acoustic transmission across doublewall sandwich shells: Effect of an air gap flow. Composite Structures 136: 149161.) determined STL through doublewalled sandwich shell with further effects of air gap. Next, Talebitooti et al. (2016aDaneshjou, K., Talebitooti, R., Tarkashvand, A. (2016a). Investigation on sound transmission through thickwall cylindrical shells using 3Dtheory of elasticity in the presence of external and mean airgap flow. Journal of Vibration and Control, 24(5):9751000. and ^{2016b}Talebitooti, R., Zarastvand, M. R., Gheibi, M. R. (2016b). Acoustic transmission through laminated composite cylindrical shell employing Third order Shear Deformation Theory in the presence of subsonic flow. Composite Structures 157 : 95110.) investigated 3D elasticity theory to designate STL of the orthotropic cylindrical shell with arbitrary thickness. In another work, third order shear deformation theory as a one of the derivate of higher order theories was considered to obtain power transmission through laminated composite cylindrical shell in the presence of external flow. Following the last works, Talebitooti et al. (2017aTalebitooti, R., Gohari, H. D., Zarastvand, M. R. (2017a). Multi objective optimization of sound transmission across laminated composite cylindrical shell lined with porous core investigating Nondominated Sorting Genetic Algorithm. Aerospace Science and Technology 69: 269280. and ^{2018b}Talebitooti, R., Choudari Khameneh, A.M. Zarastvand, M.R., Kornokar, M. (2018b). Investigation of threedimensional theory on sound transmission through compressed poroelastic sandwich cylindrical shell in various boundary configurations, Journal of Sandwich Structure and Matererial, 1099636217751562.) determined STL of the multilayered cylindrical shell so that in the first model 3D elasticity theory was employed for the special case of doublewalled composite shell subjected to porous material. On the other hand, Nondominated sorting Genetic algorithm was applied in another work for optimization of STL through cylinder interlayered with porous material.
The inspection of the last literature shows that although the wave propagation across various types of plate including isotropic, orthotropic and laminated composite based on various kinds of theories such as Classical Laminated Plate Theory(CLPT) , Firstorder Shear Deformation Theory (FSDT) and Higher order Theory Reddy (1984Reddy, J. N. (1984). A simple higherorder theory for laminated composite plates. Journal of applied mechanics 51: 745752.) has been done, the shortage of presenting the work that considers Hyperbolic Shear Deformation Theory (HSDT) through acoustic transmission of the thick plate is considerably concerned. Accordingly, this paper applies HSDT with the assumption of Mahi et al. (2015Mahi, A., Tounsi, A. (2015). A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling 39: 24892508.) in which inplane displacements are extracted as a combination of polynomial as well as hyperbolic tangent functions by neglecting the effect of thickness stretching. It is also gives parabolic distribution of transverse shear strains and shear stresses across each layer. Therefore, it is not essential to consider the shear correction factor. Finally, the results are validated and the effective parameters on STL are discussed in numerical results.
2 System description
Consider a Functionally Graded (FG) infinite plate in both sides with thickness h and mass density of , as shown in Fig.1. The construction is excited by an oblique plane sound wave with two angles of incidence as
The configuration of a FG plate under excitation of a plane wave with incident angle
3 Fundamental formulation:
3.1 Displacement field:
Since Hyperbolic Shear Deformation Theory (HSDT) is employed to acoustic analyze of a FG plate. Therefore, the following displacement fields are considered as (Mahi et al., 2015Mahi, A., Tounsi, A. (2015). A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling 39: 24892508.):
In Eqs. (1) (3),
3.2 Strains relationship
According to HSDT, the strains are considered as fallow (Zhou et al., 2013Zhou, J., Bhaskar, A., Zhang, X. (2013). Sound transmission through a doublepanel construction lined with poroelastic material in the presence of mean flow. Journal of Sound and Vibration 332: 37243734.):
3 Functionally Graded plate
In this section, it is noteworthy that the material properties of a FG plate are formulated through thickness coordinate as below (see Fig.2):
In Eq. (11),
In above equation,
For the orthotropic shells the stressstrain components can be written as below (Mahi et al., 2015Mahi, A., Tounsi, A. (2015). A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling 39: 24892508.):
In Eq. (13)
It should be considered that these stiffness constants in Eqs. (14) and (15) are devoted to the FG and orthotropic plates, respectively. As clearly defined in Eq. (15),
3.4 Moments and forces resultants
By substituting Eqs. (14) and (15) into Eq. (13), the forces and moments related to FG plate are achieved as a result of integrating the stresses over the shell thicknesses as below:
As another consequence, the following equations are considered:
In Eqs. (20)  (22),
The moment’s inertia terms are presented in the following form:
In Eq. (26),
3.5 Equations of motion
Since Hyperbolic Shear Deformation Theory (HSDT) is employed to achieve the results, therefore the following equations are considered based on Hamilton’s principle as:
Since the structure is excited acoustically, therefore
3.6 Boundary condition
In this section, it is essential to create a coupling between fluid particle and shell surface in the internal and external spaces, therefore the following equations should be taken into account (Talebitooti et al., 2018cTalebitooti, R., Zarastvand, M.R., Gohari, H.D. (2018c). The influence of boundaries on sound insulation of the multilayered aerospace poroelastic composite structure. Aerospace Science and Technology 80: 452471.):
It is also essential to note that in Eqs. (33) and (34),
Note that
These equations in the transmitted side are presented as below:
3.7 Solution procedure
In this section the displacements and rotation terms are considered as:
Now, it is well defined to insert Eqs. (35)  (37) and (40) into Eqs. (28)  (32) along with Eqs. (33)  (34). Afterwards, these seven equations are arranged in a matrix form, as below:
Where
Note that in above equation, F demonstrates the acoustic forces. However, L describes the coefficient matrix by considering that the detailed descriptions of these variables are presented in Appendix 1. Eventually, by solving Eq. (41), the unknown constants in Eq. (42) can be achieved.
4 Sound Transmission Loss:
The power transmission coefficient
Finally, STL of the construction in the logarithmic scale can be prepared as below:
5 Discussion
5.1 Validation
At the beginning of this section, it is nominated to bring forward one major frequency, known as coincidence frequency, due to equating the speed of the forced bending wave with the speed of the free bending wave based on (Talebitooti et al., 2017bTalebitooti, R., Zarastvand, M. R., Gohari, H. D. (2017b). Investigation of power transmission across laminated composite doubly curved shell in the presence of external flow considering shear deformation shallow shell theory. Journal of Vibration and Control, 24(19):4492504.) as below:
In Eq. (45),
In order to provide accuracy of the present model (HSDT), the obtained results are verified with those available in literature. Therefore, the results are compared with those of (Roussos, 1984Roussos, L. A., (1984). Noise transmission loss of a rectangular plate in an infinite baffle. The Journal of the Acoustical Society of America 75: S2S3.), (Abid et al., 2012Abid, M., Abbes, M. S., Chazot, J. D., Hammemi, L., Hamdi, M. A., Haddar, M. (2012). Acoustic response of a multilayer panel with viscoelastic material. International Journal of Acoustics and Vibration 17(2):82.) and (Howard et al., 2006Howard, C. Q., Kidner, M. R. (2006). Experimental validation of a model for the transmission loss of a plate with an array of lumped masses. In Proceedings of Acoustics :169177.). Hence, some configurations are brought up based on isotropic plate considering the same specifications at their papers.
In Fig.3, the obtained STL from present formulation (HSDT) for the special case of isotropic plate made of Aluminum according to Table 1 is compared with those of (Roussos, 1984Roussos, L. A., (1984). Noise transmission loss of a rectangular plate in an infinite baffle. The Journal of the Acoustical Society of America 75: S2S3.). As it is seen, the achieved results from two theories are corroborated to each other in entire range of frequency as a consequence of presenting the result for thin shell. In fact, since isotropic thin plate is employed, therefore no discrepancies between HSDT and applied classical shell theory with Roussos can be observed. Moreover, the effects of the shear and rotation that offered by HSDT due to thin shell are not highlighted in this comparison. In addition, both theories predict the coincidence frequency at the same location.
STL comparison between present formulation (HSDT) and obtained results by (Roussos, 1984Roussos, L. A., (1984). Noise transmission loss of a rectangular plate in an infinite baffle. The Journal of the Acoustical Society of America 75: S2S3.).
As illustrated in Fig.4, another comparison is made between present work (HSDT) and that of (Abid et al., 2012Abid, M., Abbes, M. S., Chazot, J. D., Hammemi, L., Hamdi, M. A., Haddar, M. (2012). Acoustic response of a multilayer panel with viscoelastic material. International Journal of Acoustics and Vibration 17(2):82.) for an isotropic thick plate made of Steel with the listed specifications in Table 1. Although, the obtained results demonstrate a good validity in entire range of frequency, a little discrepancy is observed below the coincidence frequency as a result of some numerical deviations occurred in Transfer matrix approach applied by Abid et al. Since in the present result the rotary inertia terms are extended up to higher order terms, therefore the accuracy of the current formulation would be assured.
The comparison of present study (HSDT) and (Abid et al., 2012Abid, M., Abbes, M. S., Chazot, J. D., Hammemi, L., Hamdi, M. A., Haddar, M. (2012). Acoustic response of a multilayer panel with viscoelastic material. International Journal of Acoustics and Vibration 17(2):82.).
In Fig.5, the obtained STL from the present formulation is compared with Experimental results offered by (Howard et al., 2006Howard, C. Q., Kidner, M. R. (2006). Experimental validation of a model for the transmission loss of a plate with an array of lumped masses. In Proceedings of Acoustics :169177.) for the especial case of isotropic plate made of Aluminum with the thickness of
STL comparison between present study (HSDT) and those of Experimental results obtained by (Howard et al., 2006Howard, C. Q., Kidner, M. R. (2006). Experimental validation of a model for the transmission loss of a plate with an array of lumped masses. In Proceedings of Acoustics :169177.).
Additionally, the accuracy of the current results (HSDT) is also provided with comparing the obtained dimensionless natural frequencies from HSDT with those available in literature (Chandra et al., 2014Chandra, N., Raja, S., Gopal, K. N. (2014). Vibroacoustic response and sound transmission loss analysis of functionally graded plates. Journal of Sound and Vibration 333: 57865802.) and (Vel et al., 2004Vel, S. S., Batra, R. C. (2004). Threedimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration 272: 703730.) for the especial case of FG square plate made of
5.2 Numerical results
In this section, some configurations are presented based on HSDT to analyze acoustic transmission of a FG plate constituted of AlAlumina with the powerlaw index of
In Fig.6, the effects of various powerlaw distributions on STL are presented and discussed. The achieved results demonstrate that increasing the powerlaw exponents will enhance STL in frequency region above coincidence frequency. Besides, the coincidence frequency shifts downward in this case. Furthermore, it is essential to note that in high frequency region the trend is also similar to low frequency zone. However, when the powerlaw exponent goes to zero, the whole of the structure constituted of pure Aluminum, presents the minimum level of STL. On the other hand, when the powerlaw exponent is set to be infinite, the structure composed of pure Alumina, reveals the maximum level of STL in entire range of frequency. Moreover, the obtained results for
Another configuration is plotted in Fig.7 to illustrate the influence of various thicknesses on STL of the FG plate. The obtained result from the figure demonstrates that the thickness of the structure can significantly influence on STL so that by increasing this parameter, STL is enhanced in entire range of frequency. In fact, by thickening the plate, the incident wave cannot penetrate the structure which results in enhancing STL. As another consequence, with increasing the thickness of the FG plate, the coincidence frequency shifts downward.
As depicted in Fig.8, the effects of various isotropic and FG materials with
Fig.9 illustrates the effects of various graded elasticity modulus on STL curves for different powerlow distribution. Herewith, Aluminum is selected as a basis so that Young’s modulus is continually changed from 72 (GPa) at the bottom of the plate to 380 (GPa) at the top of the plate based on powerlaw distribution of volume fraction. As depicted in this figure, by varying this parameter, no considerable improvement on STL can be observed in frequency region below
The effects of various graded elasticity modulus on STL curves for different powerlaw distributions.
As indicated in Fig.10, the influences of graded density from 2760 (top of the plate) to 3800 (bottom of the plate) with respect to various powerlaw distributions on STL of the FG plate are presented and discussed. The results show that with increasing the powerlaw index, the value of STL is enhanced in low frequency domain. Although, coincidence frequency
In Fig.11, the effect of graded Poisson’s ratio on acoustic transmission of the FG plate is shown based on two various quantities of
Fig.12 is representative of variations in STL resulted from various
As shown in Fig.13, STL comparison between FG (AlAlumina) and various orthotropic material plates is presented and discussed. It is easily seen that structure made of ALAlumina presents the maximum level of STL in entire range of frequency. However, when structure with loss weight becomes important in practical application, Glass/Epoxy in low frequency zone is employed. Meanwhile, FG material is considerably used where the resistance temperature becomes important. Since the higher modulus of elasticity leads to improving the behavior of STL in high frequency zone, therefore the lowest value of STL from Glass/Epoxy can be observed.
STL comparison between achieved results from FG (AlAlumina) plate in contrast to orthotropic plate.
6 Concluding remarks
In this paper, Hyperbolic Shear Deformation Theory was considered to obtain acoustic transmission of the FG plate. Accordingly, in the first step, the equations of motion were determined. Consequently, an analytical solution was provided to solve the obtained equation besides acoustic wave equation. Moreover, the accuracy of the current formulation was prepared by bringing up some validations with previously published data. Finally, following results can be remarked:
Since Hyperbolic Shear Deformation Theory is considered, therefore the displacements are developed up to cubic order of thickness coordinate so that the effects of the main terms including coupling, bending and inertia are extended up to higher order components. Accordingly, in obtaining STL of the thick plate, the more precise results can be achieved.
The results illustrate the direct effect between powerlaw distribution
For the structure made of FG material, graded Young modulus is known as parameter which appears to be effective to produce noticeable improvement on STL in comparison with graded poison's ratio and graded density.
References
 Yamanouchi, M., Koizumi, M., (1991). Functionally gradient materials. Proceeding of the first international symposium on functionally graded materials.
 London, A., (1950). Transmission of reverberant sound through double walls. The journal of the acoustical society of America 22 : 270279.
 Maestrello, L., (1995). Responses of finite baffled plate to turbulent flow excitations. AIAA Journal 33: 1319.
 Clark, R. L., Frampton, K. D., (1996a). Sound transmission through an aeroelastic plate into an acoustic cavity. The Journal of the Acoustical Society of America 99: 25862603.
 Clark, R. L., (1996b). Transmission of stochastic pressures through an aeroelastic plate into a cavity. In 37th Structure, Structural Dynamics and Materials Conference 1445.
 Bhattacharya, M. C., Guy, R. W., Crocker, M. J., (1971). Coincidence effect with sound waves in a finite plate. Journal of sound and vibration 18: 157169.
 Koval, L. R., (1976). Effect of air flow, panel curvature, and internal pressurization on field‐incidence transmission loss. The Journal of the Acoustical Society of America 59: 13791385.
 Roussos, L. A., (1984). Noise transmission loss of a rectangular plate in an infinite baffle. The Journal of the Acoustical Society of America 75: S2S3.
 Renji, K., Nair, P. S., Narayanan, S., (1997). Critical and coincidence frequencies of flat panels. Journal of Sound and Vibration 205: 1932.
 Tang, H., Zhao, X. P., Luo, C. R., (2006a). Sonic responses of an electrorheological layer with one side of grating electrodes. Journal of Physics D: Applied Physics 39(3):552
 Tang, H., Luo, C. R., Zhao, X. P., (2006b). Sound transmission behavior through a sandwiched electrorheological layer. Fuhe Cailiao Xuebao/Acta Mater Compos Sinica 23: 128132.
 Xin, F. X., Lu, T. J., Chen, C. Q., (2009). External mean flow influence on noise transmission through doubleleaf aeroelastic plates. AIAA J 47: 19391951.
 Xin, F. X., Lu, T. J., (2010). Analytical modeling of sound transmission across finite aeroelastic panels in convected fluids. The Journal of the Acoustical Society of America 128: 10971107.
 Xin, F. X., Lu, T. J., (2011). Analytical modeling of sound transmission through clamped triplepanel partition separated by enclosed air cavities. European Journal of MechanicsA/Solids 30: 770782.
 Chandra, N., Gopal, K.N., Raja, S. (2015). Vibroacoustic response of sandwich plates with functionally graded core. Acta Mechanica: 228(8):277589.
 Talebitooti, R., Johari, V., Zarastvand, M. R. (2018a). Wave transmission across laminated composite plate in the subsonic flow Investigating Twovariable Refined Plate Theory. Latin American Journal of Solids and Structures, vol. 15 (5).
 Liu, Y., He, C. (2016). Analytical modelling of acoustic transmission across doublewall sandwich shells: Effect of an air gap flow. Composite Structures 136: 149161.
 Daneshjou, K., Talebitooti, R., Tarkashvand, A. (2016a). Investigation on sound transmission through thickwall cylindrical shells using 3Dtheory of elasticity in the presence of external and mean airgap flow. Journal of Vibration and Control, 24(5):9751000.
 Talebitooti, R., Zarastvand, M. R., Gheibi, M. R. (2016b). Acoustic transmission through laminated composite cylindrical shell employing Third order Shear Deformation Theory in the presence of subsonic flow. Composite Structures 157 : 95110.
 Talebitooti, R., Gohari, H. D., Zarastvand, M. R. (2017a). Multi objective optimization of sound transmission across laminated composite cylindrical shell lined with porous core investigating Nondominated Sorting Genetic Algorithm. Aerospace Science and Technology 69: 269280.
 Talebitooti, R., Choudari Khameneh, A.M. Zarastvand, M.R., Kornokar, M. (2018b). Investigation of threedimensional theory on sound transmission through compressed poroelastic sandwich cylindrical shell in various boundary configurations, Journal of Sandwich Structure and Matererial, 1099636217751562.
 Reddy, J. N. (1984). A simple higherorder theory for laminated composite plates. Journal of applied mechanics 51: 745752.
 Mahi, A., Tounsi, A. (2015). A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling 39: 24892508.
 Talebitooti, R., Zarastvand, M. R., Gohari, H. D. (2017b). Investigation of power transmission across laminated composite doubly curved shell in the presence of external flow considering shear deformation shallow shell theory. Journal of Vibration and Control, 24(19):4492504.
 Talebitooti, R., Zarastvand, M.R., Gohari, H.D. (2018c). The influence of boundaries on sound insulation of the multilayered aerospace poroelastic composite structure. Aerospace Science and Technology 80: 452471.
 Talebitooti, R., Zarastvand, M.R., (2018d). The effect of nature of porous material on diffuse field acoustic transmission of the sandwich aerospace composite doubly curved shell. Aerospace Science and Technology 78: 157170.
 Zhou, J., Bhaskar, A., Zhang, X. (2013). Sound transmission through a doublepanel construction lined with poroelastic material in the presence of mean flow. Journal of Sound and Vibration 332: 37243734.
 Talebitooti, R., Zarastvand, M.R. (2018e). Vibroacoustic behavior of orthotropic aerospace composite structure in the subsonic flow considering the Third order Shear Deformation Theory. Aerospace Science and Technology 75: 227236.
 Abid, M., Abbes, M. S., Chazot, J. D., Hammemi, L., Hamdi, M. A., Haddar, M. (2012). Acoustic response of a multilayer panel with viscoelastic material. International Journal of Acoustics and Vibration 17(2):82.
 Howard, C. Q., Kidner, M. R. (2006). Experimental validation of a model for the transmission loss of a plate with an array of lumped masses. In Proceedings of Acoustics :169177.
 Chandra, N., Raja, S., Gopal, K. N. (2014). Vibroacoustic response and sound transmission loss analysis of functionally graded plates. Journal of Sound and Vibration 333: 57865802.
 Vel, S. S., Batra, R. C. (2004). Threedimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration 272: 703730.

Available online: November 05, 2018
Appendix 1
Publication Dates

Publication in this collection
2019
History

Received
31 Jan 2018 
Reviewed
05 Sept 2018 
Accepted
31 Oct 2018