Investigating Hyperbolic Shear Deformation Theory on vibroacoustic behavior of the infinite Functionally Graded thick plate

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 power-law 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 power-law distributions.


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. (1991). 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.

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 and , in which denotes the angle between wave and horizontal plane and is known as azimuthal angle. In addition, and denote to the speed of sound and density in which the subscript presents the incident side ( = 1) and transmitted side ( = 2) of the structure, respectively.
(1)-(3), and present the rotations of the transverse normal in and directions, respectively. Moreover, , and describe the displacements on the middle surface of the structure. However, and show the shape functions as a combination of polynomial and hyperbolic tangent functions as below:

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), ( ) denotes to the volume fraction in the following form: ( ) = ( /ℎ + 1/2) , −ℎ/2 ≤ ≤ ℎ/2 In above equation, describes the power-law exponent in the interval of [0, ∞]. Moreover, ( ) is one of the specifications of a FG plate including Young modulus , Poisson's ratio and density in which the subscript can be ( , ). Note that the parameters and are the material properties at the bottom and top of the structure, respectively. It is noteworthy that these material properties are continuously changed through thickness coordinate with respect to power-law component. As another consequence, and denote the material properties at top and bottom of the construction.
For the orthotropic shells the stress-strain components can be written as below (Mahi et al., 2015): In Eq. (13) and illustrate stress and strain relation in the orthotropic layer and present reduced stiffness constants in terms of coordinate , which can be obtained by following equations:  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), in which = (1, 2), denotes to the modulus of elasticity, in which = (1, 2) and = (2,3), present the modulus of rigidity. Moreover, shows the Poisson's ratio.

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 = ( 1 + 1 − 1 ) presents the external force.

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., 2018c): It is also essential to note that in Eqs. (33) and (34), 1 devotes to the incident wave, 2 presents the transmitted wave and 1 denotes to the reflected wave as below (Talebitooti et al., 2018d): Latin American Journal of Solids and Structures, 2019, 16(1), e139 7/17 1 ( , , , ) = 0 Note that in Eqs. (35)-(37) demonstrates the angular frequency, 0 presents the amplitude of the incident wave. As another consequence, 1 = 1 ( 1 1+ 1 ( ) ) illustrates the wave number which propagates in the , and directions as follow: These equations in the transmitted side are presented as below:

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: 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.

Sound Transmission Loss:
The power transmission coefficient can be obtained as a ratio of transmitted power and incident power per unit area of the structure as follow (Talebitooti et al., 2018e): Finally, STL of the construction in the logarithmic scale can be prepared as below: Latin American Journal of Solids and Structures, 2019, 16(1), e139 8/17

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., 2017b) as below: In Eq. (45), represents the modulus of elasticity, denotes the density and devotes the Poisson's ratio, respectively. Moreover, the symbol ℎ is related to the wall thickness.
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, 1984), (Abid et al., 2012) and (Howard et al., 2006). 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, 1984). 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.  (Roussos, 1984). As illustrated in Fig.4, another comparison is made between present work (HSDT) and that of (Abid et al., 2012) 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.

Figure 4: The comparison of present study (HSDT) and (Abid et al., 2012).
In Fig.5, the obtained STL from the present formulation is compared with Experimental results offered by (Howard et al., 2006) for the especial case of isotropic plate made of Aluminum with the thickness of 1.5 in which the excitation of the structure is performed by an oblique plane sound wave with the angle of 30° .As it is obvious, some discrepancies can be observed in entire range of frequency. The inspection of the results indicates that although both of structures are excited acoustically, the type of the incident wave is different which result in these variations. In fact, in the present study the excitation of the structure is done by an oblique plane sound wave whereas the field incidence acoustic condition is considered by Howard et al. (Howard et al., 2006).

Figure 5: STL comparison between present study (HSDT) and those of Experimental results obtained by
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., 2014) and (Vel et al., 2004) for the especial case of FG square plate made of Al/ZrO 2 , as shown in Table.2. The inspection of this table shows that not only a good agreement is observed in comparing these natural frequencies but also the obtained results are appeared to be much closer to 3D method. On the other hand, some deviations exist in the natural frequencies of First order Shear Deformation Theory as a result of applying the shear correction factor by (Chandra et al., 2014) whereas the present work is not employed this correction factor in its equations.

Numerical results
In this section, some configurations are presented based on HSDT to analyze acoustic transmission of a FG plate constituted of Al-Alumina with the power-law index of = 1 according to Table 1. The thickness of the plate is considered to be 1 and it is excited by an oblique plane sound wave with the angle of 30°. In Fig.6, the effects of various power-law distributions on STL are presented and discussed. The achieved results demonstrate that increasing the power-law 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 power-law 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 power-law 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 = 10 (physical power) are nearly corroborated with those of = ∞ (theoretical concept) in frequency region above coincidence frequency. 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 Latin American Journal of Solids and Structures, 2019, 16(1), e139 11/17 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.

Figure 7: The influences of various FG plate thicknesses on STL curves.
As depicted in Fig.8, the effects of various isotropic and FG materials with = 1 on power transmission of the structure are discussed. The obtained results show that in low frequency domain, Steel as a result of having the maximum density, present the highest level of STL. Since the isotropic material includes high weight (which is addressed as a negative point for these structures) therefore employing the combination of Isotropic-FG materials such as Steel-Alumina which has the acceptable STL and less weight has the high proficiency. Besides, the location of the coincidence frequency is changed by varying this material so that AL-Alumina includes the lowest value and Al contains the highest value of coincidence frequency, respectively. It is also essential to note that at high frequency domain, Steel-Alumina, as a result of taking the upper stiffness, presents the maximum level of STL. below 1600 . However, over this frequency, the trend is quite different so that STL will enhance with increasing the power-law exponent as a result of shifting the coincidence frequency downward. This event can be fulfilled due to enhancing stiffness of the FG plate. 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 power-law distributions on STL of the FG plate are presented and discussed. The results show that with increasing the power-law index, the value of STL is enhanced in low frequency domain. Although, coincidence frequency ( ) shifts forward in this case, no considerable improvement on STL can be observed in low frequency zone. 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 0.3 and 0.28 for the bottom and top of the plate, respectively. It is easily seen that by changing this parameter, no significant deviations are appeared in entire range of frequency. By increasing awareness of this issue, it is noteworthy that in interpreting acoustic transmission of the structure, the graded Poisson's ratio in corresponding equations can be ignored to decrease the volume of calculations.   The results show that the direct effect exists between increasing this angle and STL in frequency region below coincidence frequency which results in improving the behavior of STL. However, at high frequency zone, STL behaves in opposite way due to decreasing the wave number in , directions. As another aspect, by varying this parameter, coincidence frequency shifts upward. As shown in Fig.13, STL comparison between FG (Al-Alumina) and various orthotropic material plates is presented and discussed. It is easily seen that structure made of AL-Alumina 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.

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 power-law distribution ( ) and STL of the structure. Besides, by increasing , coincidence frequency is reduced.
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.