Thermo-acoustic analysis of higher-order shear deformable laminated composite sandwich flat panel

Abstract

The acoustic radiation responses of laminated sandwich baffled flat panels subjected to harmonic loading in an elevated thermal environment are investigated via a novel coupled finite and boundary elements formulation based on the higher-order shear deformation shell theory. The structural stiffness and mass tensors are obtained using competent finite element steps engaging the Hamilton’s principle followed by computation of acoustic responses by resolving the Helmholtz partial differential equation. An in-house MATLAB code is developed based on the present formulation for the computation of all the desired responses. The accuracy and robustness of the present scheme are recognized by the close conformance of the critical buckling temperature, natural frequencies and the sound power level values with the available benchmark solutions alongside the values obtained via a simulation model implemented using commercially available finite element (ANSYS) and boundary element (LMS Virtual.Lab) packages. Subsequently, the present model is employed to solve wide variety of numerical illustrations and the useful inferences related to the influence of elevated temperature, core-to-face thickness ratio, core-to face modular ratio and lay-up scheme on the sound emission characteristics of sandwich composite flat panels are deliberated in detail.

Keywords

Laminated composite sandwich flat panel higher-order mid-plane kinematics coupled finite element method–boundary element method scheme thermo-acoustic radiation MATLAB computer code

Introduction

Laminated composite structures in the form of flat and curved shell panels are widely used in several areas such as automotive, aircrafts, submarine, trains and buildings to name a few. A large number of structures and/or structural components in the present day are being made from sandwich materials those are formed by sandwiching (mechanical bonding by adhesives or welding) a thick and lightly-dense core between two relatively thin face sheets having high strength and stiffness. The choice of the material for the face and the core is largely dependent on their functionalities. It is well known that the core should be adequately stiff to endure compression to maintain a constant gap between the face sheets and at the same time inhibit the slipping of face sheets during flexure. On the other hand, the face sheets should be able to withstand the in-plane tensile, compressive and shear stresses. The laminated composite materials as face sheet along with the conventional isotropic materials such as aluminum as the core are commonly being used in weigh sensitive and high-performance engineering applications. This layered pattern not only leads to a structure having high strength-to-weight ratio but also renders the structures with accentuated radiation and transmission of acoustic noise. However, the exposure of the structures to environmental hostilities (such as elevated temperature and/or moisture) while in service may potentially influence their dynamic characteristics and thereby aggravate the problem for certain applications where acoustic comfort/stealth is of primary importance. Therefore, the vibroacoustic behaviour of sandwich shell panel structure has become a matter of intense scrutiny in the recent years [1]. The development lead time, lead cost and trial products associated with the experimental study calls for the accurate numerical modeling for computing their acoustic responses more precisely. However, it is also apparent that the exactness of computing the acoustic responses of any vibrating structure is greatly dependent on the accuracy in estimating its free vibration responses (natural frequencies) which in turn dependent on the displacement field (mid-plane kinematics) used for structural modeling [2]. To address this issue, several equivalent single layer theories including the classical theories, lower-order as well as higher-order shear deformation theories (HSDT) [3 –7] and various refined theories [8 –10] are being used to study the bending [11], free vibration and buckling responses [12 –14] of laminated composite and sandwich structure subjected to dynamic loading in an ambient and/or thermal environment. Various studies incorporating isogeometric approach [15,16] and mesh-free approach [17] have also been reported. The vibroacoustic behaviour of laminated composite flat and curved panels subjected to ambient/thermal/hygroscopic environment in the framework of the aforementioned single-layer theories [18 –23] has been a topic of great interest in the recent past. However, it has been recognized that the HSDT leads to a more realistic estimation of the flexure of the structure so far as the shear deformable laminated composite structures are concerned and is relatively easier to implement at a sensible computational cost [18,19]. We also note that the Rayleigh integral formulation and the coupled finite element–boundary element (FE-BE) technique has been utilized through commercial software packages to study the acoustic radiation caused by laminated composite shell panels [24 –27] in unlike environment.

Numerous studies investigating the sound radiation responses of the isotropic/composite sandwich panels have also been reported in open literature. The vibroacoustic behaviour of sandwich flat panel with isotropic core and faces under the influence of thermal load have been studied using the equivalent classical theory [28]. The sound emission traits of sandwich flat panels having faces made up of orthotropic materials subjected to increased temperature environment have also been analysed in the framework of piecewise low-order shear deformation theory in conjunction with Rayleigh integral formulation [29]. Further, coupled FEM–BEM scheme implemented via commercial software have been utilized to study the acoustic radiation emanating from sandwich flat [30] and sandwich composite cylindrical shell panels with viscoelastic core [31] in thermal environment. Hwang et al. [32] proposed a novel approach with empirical feedback to the conventional model to predict the sound transmission loss (STL) of double sandwich flat panels. Zhou and Crocker [33] used the statistical energy analysis (SEA) to predict the STL of foam-filled honeycomb sandwich panels and compared the theoretical results with the experimental values. Guillaumie [34] provided analytical solutions for the vibroacoustic responses of honeycomb sandwich panels and compared results with numerical values obtained in the framework of the Classical Laminated Plate Theory (CLPT). Yang et al. [35] performed an experimental investigation on the sound absorption and transmission loss properties of the glass fibre-filled honeycomb sandwich structures. Further, Petrone et al. [36] analysed the sound power radiated by aluminium foam sandwich panels experimentally and compared the results with the numerical solution obtain using commercial software NASTRAN in conjunction with Rayleigh’s integral. Sadri and Younesian [37] studied the vibroacoustic characteristics of a sandwich panel coupled with an enclosure cavity analytically in Laplace domain. The active control of sound power through soft-cored sandwich panels through multiple piezoelectric actuators [38] and also by using volume velocity cancellation approach [39,40] have also been reported.

Literature review affirms that the analytical and experimental studies related to the sound transmission and absorption characteristics of the sandwich panels are more in number in comparison to the numerical analysis of their vibroacoustic responses. However, studies dealing with the acoustic emission responses of the laminated composite sandwich panels are scarce. Further, studies investigating the influence of unlike environment, such as elevated temperature and/or moisture on the acoustic radiation responses sandwich composite structure are far from existence. We also note that the HSDT offers reasonable accuracy, is capable of modeling the in-plane, shear as well as the bending stresses alongside eliminating the need for a shear correction factor. Therefore, in this paper, the vibroacoustic radiation responses of laminated composite sandwich flat panel acted upon by harmonic point excitation under an elevated temperature environment are investigated using a novel HSDT-based coupled FE–BE numerical scheme. The core of the panels is taken to be soft and isotropic, whereas the faces are considered to be made of laminated composite material. A nine-noded isoparametric Lagrangian element having nine degrees of freedom per node is utilized for the discretizing the structural model. The influence of temperature, core-to-face thickness ratio, core-to-face modular ratio and the lamination scheme on the sound emission characteristics of sandwich panels is brought out and deliberated in detail through appropriate numerical experimentation.

Mathematical formulation

The sandwich composite flat panels considered in this work consist of the core which is relatively thick and isotropic and the faces made up of laminated composite material. The geometry (length a, width b and thickness h) and the stacking sequence of core and faces of the composite sandwich panel is shown in Figure 1. The thickness of the core is $h_{c}$ and the thickness of each face is $h_{f}$ such that $h = h_{c} + 2 h_{f}$ .

In the present analysis, HSDT is utilized to model the displacement field of the sandwich composite flat panels as it not only accounts for the parabolic variation of stress and strain through the thickness but also does away with the need of shear correction factor. Accordingly, the displacement field $(x_{1}, x_{2}, x_{3})$ representing the overall directional shift in the position of any point on the panel mid-plane can be stated as [4]:

\begin{array}{l} x_{1} (x, y, z, t) = x_{0} (x, y, t) + z u_{1} (x, y, t) + z^{2} v_{1} (x, y, t) + z^{3} w_{1} (x, y, t) \\ x_{2} (x, y, z, t) = y_{0} (x, y, t) + z u_{2} (x, y, t) + z^{2} v_{2} (x, y, t) + z^{3} w_{2} (x, y, t) \\ x_{3} (x, y, t) = z_{0} (x, y, t) \end{array}

(1)

where

(x_{0}, y_{0}, z_{0})

are displacements of any point on the mid-plane; the shear rotations about the x and y-axes are represented as

u_{2}

and

u_{1}

, respectively and

v_{1}, v_{2}, w_{1}, w_{2}

are the higher-order terms of the Taylor series expansion defined in the mid-plane to attain a parabolic distribution of the shear stress. The expression for strain displacement in terms of the global displacement fields (given by equation (1)) can be stated as

{ε} = {\begin{matrix} ε_{x_{1} x_{1}} \\ ε_{x_{2} x_{2}} \\ ε_{x_{1} x_{2}} \\ ε_{x_{1} x_{3}} \\ ε_{x_{2} x_{3}} \end{matrix}} = {\begin{matrix} {(x_{1})}_{, x_{1}} \\ {(x_{2})}_{, x_{2}} \\ {(x_{1})}_{, x_{2}} + {(x_{2})}_{, x_{1}} \\ {(x_{1})}_{, x_{3}} + {(x_{3})}_{, x_{1}} \\ {(x_{2})}_{, x_{3}} + {(x_{3})}_{, x_{2}} \end{matrix}}

(2)

where

{ε}

is the strain tensor and the notation

{(ℓ)}_{, x}

represents the partial derivative of

ℓ

with respect to x. On substituting

(x_{1}, x_{2}, x_{3})

from equation (1) in equation (2), the strain tensor can further be written as

{ε} = {\begin{matrix} ε_{x_{1} x_{1}} \\ ε_{x_{2} x_{2}} \\ ε_{x_{1} x_{2}} \\ ε_{x_{1} x_{3}} \\ ε_{x_{2} x_{3}} \end{matrix}} = {\begin{matrix} [{\tilde{ε}}_{1} + z {\hat{κ}}_{11} + z^{2} {\hat{κ}}_{12} + z^{3} {\hat{κ}}_{13}] \\ [{\tilde{ε}}_{2} + z {\hat{κ}}_{21} + z^{2} {\hat{κ}}_{22} + z^{3} {\hat{κ}}_{23}] \\ [{\tilde{ε}}_{6} + z {\hat{κ}}_{61} + z^{2} {\hat{κ}}_{62} + z^{3} {\hat{κ}}_{63}] \\ [{\tilde{ε}}_{5} + z {\hat{κ}}_{51} + z^{2} {\hat{κ}}_{52} + z^{3} {\hat{κ}}_{53}] \\ [{\tilde{ε}}_{4} + z {\hat{κ}}_{41} + z^{2} {\hat{κ}}_{42} + z^{3} {\hat{κ}}_{43}] \end{matrix}}

(3)

The strain tensor given by equation (3) can be expressed in the product form as

{ε} = [Z] {\tilde{ε}}

(4)

where

{\tilde{ε}} = {\begin{matrix} {\tilde{ε}}_{1} & {\tilde{ε}}_{2} & {\tilde{ε}}_{6} & {\tilde{ε}}_{5} & {\tilde{ε}}_{4} & {\hat{κ}}_{11} & {\hat{κ}}_{21} & {\hat{κ}}_{61} & {\hat{κ}}_{51} & {\hat{κ}}_{41} \\ {\hat{κ}}_{12} & {\hat{κ}}_{22} & {\hat{κ}}_{62} & {\hat{κ}}_{52} & {\hat{κ}}_{42} & {\hat{κ}}_{13} & {\hat{κ}}_{23} & {\hat{κ}}_{63} & {\hat{κ}}_{53} & {\hat{κ}}_{43} \end{matrix}}^{T}

and

[Z]

is the thickness coordinate matrix [4]. The subscript i in the term

{\hat{κ}}_{(…) i}

belonging to

{\tilde{ε}}

corresponds to bending (i = 1), curvature (i = 2) and the higher-order terms (i = 3).

The generalized stress tensor reflecting the effect of in-plane stresses generated due to an applied constant temperature load can be obtained from the material constitutive relationship for the kth composite lamina with an arbitrary fibre orientation ϕ and conceded as

{{\hat{σ}}_{i j}}^{k} = {[{\hat{Q}}_{i j}]}^{k} {ε_{i j} - α_{i j} Δ T}^{k}

(5)

where

{{\hat{σ}}_{i}_{j}}^{k} = {{\begin{matrix} {\hat{σ}}_{1} & {\hat{σ}}_{2} & {\hat{σ}}_{6} & {\hat{σ}}_{5} & \hat{σ} \end{matrix}}_{4}}^{T}

is the stress vector,

{ε_{i j}}^{k} = {\begin{matrix} ε_{1} & ε_{2} & ε_{6} & ε_{5} & ε_{4} \end{matrix}}^{T}

is the corresponding strain vector,

{[{\hat{Q}}_{i j}]}^{k}

is the matrix of the transferred reduced elastic constant for the kth layer,

{α_{i j}}^{k} = {\begin{matrix} α_{1} & α_{2} & 2 α_{12} \end{matrix}}^{T}

is the thermal expansion coefficient vector and

Δ T

the uniform change in temperature. The stress vector can further be unambiguously represented by expanding the shapes of the matrices forming the product in equation (5) and given as

{{\hat{σ}}_{i j}}^{k} = {[\begin{matrix} {\hat{Q}}_{11} & {\hat{Q}}_{12} & {\hat{Q}}_{16} & 0 & 0 \\ {\hat{Q}}_{21} & {\hat{Q}}_{22} & {\hat{Q}}_{26} & 0 & 0 \\ {\hat{Q}}_{16} & {\hat{Q}}_{26} & {\hat{Q}}_{66} & 0 & 0 \\ 0 & 0 & 0 & {\hat{Q}}_{55} & {\hat{Q}}_{54} \\ 0 & 0 & 0 & {\hat{Q}}_{45} & {\hat{Q}}_{44} \end{matrix}]}^{k} [{\begin{matrix} ε_{1} \\ ε_{2} \\ ε_{6} \\ ε_{5} \\ ε_{4} \end{matrix}}^{k} - {\begin{matrix} α_{1} \\ α_{2} \\ 2 α_{12} \\ 0 \\ 0 \end{matrix}}^{k} Δ T]

(6)

In order to obtain the laminar in-plane forces generated because of the applied temperature load, the stress can be integrated over the thickness of the sandwich panel and expressed as

[\begin{matrix} {Γ_{Δ T}^{1}} & {Γ_{Δ T}^{2}} & {Γ_{Δ T}^{3}} \end{matrix}] = [\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} {{[\hat{Q}]}^{k} {{α}^{k} (1, x_{3}, {(x_{3})}^{3}) Δ T}} d x_{3}]

(7)

where the vectors

{Γ_{Δ T}^{1}}

{Γ_{Δ T}^{2}}

{Γ_{Δ T}^{3}}

represent the compressive in-plane thermal force, the moment and the higher order terms, respectively.

The work done by the in-plane thermal force ${Γ_{Δ T}^{1}} = {\begin{matrix} {(Γ_{Δ T}^{1})}_{1} & {(Γ_{Δ T}^{1})}_{2} & {(Γ_{Δ T}^{1})}_{12} \end{matrix}}$ on the shell panel is given by

{W_{T}} = \int_{V} \frac{1}{2} {\begin{cases} [{{(x_{0})}_{, x_{1}}}^{2} + {{(y_{0})}_{, x_{1}}}^{2} + {{(z_{0})}_{, x_{1}}}^{2}] {Γ_{Δ T}^{1}}_{1} \\ + [{{(x_{0})}_{, x_{2}}}^{2} + {{(y_{0})}_{, x_{2}}}^{2} + {{(z_{0})}_{, x_{2}}}^{2}] {Γ_{Δ T}^{1}}_{2} \\ + 2 [{{(x_{0})}_{, x_{1}}} {{(x_{0})}_{, x_{2}}} + {{(y_{0})}_{, x_{1}}} {{(y_{0})}_{, x_{2}}} + {{(z_{0})}_{, x_{1}}} {{(z_{0})}_{, x_{2}}}] {Γ_{Δ T}^{1}}_{12} \end{cases}} d V

(8)

The material property matrix $[Π]$ and the geometric strain vector ${ε_{G}}$ can be introduced in equation (8) (for detailed steps, refer Cook et al. [41]) to represent the work {W_T} given by equation (8)

{W_{T}} = \frac{1}{2} \int_{A} {ε_{G}}^{T} [Π] {ε_{G}} d A

(9)

The global displacement field represented by equation (1) can be expressed in vector form as

{d} = {x_{1} x_{2} x_{3}}^{T} = [f] {δ}

(10)

where

{δ}

is the displacement vector at any point in the mid-plane. The kinetic energy (P₁) of the sandwich composite flat panel is expressed as

P_{1} = \frac{1}{2} \iint {\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} ρ^{k} {\dot{d}}^{T} {\dot{d}} d x_{3}} d x_{1} d x_{2}

(11)

where N is the number of layers,

ρ^{k}

is the mass density of the kth layer,

\dot{d} = \frac{\partial (d)}{\partial t}

x_{3, k - 1}

and

x_{3, k}

coordinates represent the altitude of the lower and upper surface of a laminate along

x_{3}

direction. Subsequently, on substituting the global displacement vector from equation (10) in the expression for

P_{1}

, equation (11) could be written as

\begin{array}{l} P_{1} = \frac{1}{2} \int_{A} (\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} ρ^{k} {\dot{δ}}^{T} {[f]}^{T} [f] {\dot{δ}} d x_{3}) d A \\ = \frac{1}{2} \int_{A} {\overset{\cdot}{δ}}^{T} [m] {\overset{\cdot}{δ}} d A \end{array}}

(12)

where

[m] = \sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} ρ^{k} {[f]}^{T} [f] d x_{3}

is the inertia matrix.

The strain energy (P₂) of the shell panel is written as

P_{2} = \frac{1}{2} \iint {\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} {ε}_{i}^{T} {\hat{σ}} d x_{3}} d x_{1} d x_{2}

(13)

By using equation (5), the strain energy given by equation (13) can further be stated as

\begin{matrix} P_{2} = \frac{1}{2} \iint {\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} {ε}_{i}^{T} [\hat{Q}] {ε_{i} - α_{i} Δ T} d x_{3}} d x_{1} d x_{2} \\ = \frac{1}{2} \iint \sum_{k = 1}^{N} \int {ε}_{i}^{T} [\hat{Q}] {ε}_{i} d x_{1} d x_{2} d x_{3} - \frac{1}{2} \iint {\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} {ε}_{i}^{T} [\hat{Q}] {α_{i} Δ T} d x_{3}} d x_{1} d x_{2} \end{matrix}}

(14)

Subsequently, using equation (4), equation (14) can further be modified to have the following form

P_{2} = \frac{1}{2} \int_{A} ({\tilde{ε}}_{i}^{T} [Λ] {\tilde{ε}}_{i}) d A - \frac{1}{2} \iint {\sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} {ε}_{i}^{T} [\hat{Q}] {α_{i} Δ T} d x_{3}} d x_{1} d x_{2}

(15)

where

[Λ] = \sum_{k = 1}^{N} \int_{x_{3, k - 1}}^{x_{3, k}} {[Z]}^{T} [\hat{Q}] [Z] d x_{3}

The FEM is now employed for the discretization of the structural model. A nine-noded isoparametric element having a total of 81 degrees of freedom $(9 per node, given as x_{0}, y_{0}, z_{0}, u_{1}, u_{2}, v_{1}, v_{2}, w_{1}, w_{2})$ is considered for this purpose. Following the FEM steps, the elemental matrix representing the nodal displacement field at each node is derived as

{δ^{*}} = \sum_{i = 1}^{9} [N_{i}^{s}] {δ i}

(16)

where

N_{i}^{s}

is the interpolation/shape function corresponding to the ith node,

{δ i} = {[\begin{matrix} x_{0 i} & y_{0 i} & z_{0 i} & u_{1 i} & u_{2 i} & v_{1 i} & v_{2 i} & w_{1 i} & w_{2 i} \end{matrix}]}^{T}

is the nodal displacement vector and

{δ^{*}} = {[\begin{matrix} x_{0} & y_{0} & z_{0} & u_{1} & u_{2} & v_{1} & v_{2} & w_{1} & w_{2} \end{matrix}]}^{T}

([41]).

The strain field represented as in equation (4) could be stated as

{ε}_{i} = [ϒ] {δ^{*}}_{i}

(17)

where

[ϒ]

is the matrix of strain displacement relation.

The work (W_T) done by the thermal force is reworked by replacing equation (17) into equation (9) and expressed as

{W_{T}} = \frac{1}{2} \int_{A} ({δ^{*}}_{i}^{T} {[ϒ]}_{i}^{T} [Λ_{G}] {[ϒ]}_{i} {δ^{*}}_{i}) d A

(18)

The expression for strain energy (P₂) as specified in equation (15) is reformulated by using equation (17) and takes the following form

P_{2} = \frac{1}{2} \int_{A} ({δ^{*}}_{i}^{T} {[ϒ]}_{i}^{T} [Λ] {[ϒ]}_{i} {δ^{*}}_{i}) d A - {F_{Δ T}}_{i}

(19)

where the thermal load is defined as

{F_{Δ T}}_{i} = \frac{1}{2} \int_{A} {δ^{*}}_{i}^{T} {[ϒ]}_{i}^{T} {Γ_{Δ T}^{1}} d A

. The Hamilton’s principle is now utilized to derive the equation of motion for free vibration

\int_{0}^{t} T_{E} d t = 0

(20)

where

T_{E} = P_{1} - [P_{2} + W_{Δ T}]

The equation of motion as given by equation (20) is reformulated by substituting the expressions for P₁, W_T and P₂ from equations (12), (14) and (18), respectively, to take the following form

[Ψ] {\overset{\cdot \cdot}{δ}} + [Φ_{L}] {δ} = {F_{Δ T}}

(21)

where

[Ψ]

[Φ_{L}]

and

{F_{Δ T}}

represent the global mass matrix, linear stiffness matrix and thermal load vector, respectively.

The eigenvalue form of equation (21) is obtained by incorporating the effects of temperature loading in terms of geometry stiffness matrix $[Φ_{G}]$ and represented as

{([Φ_{L}] - [Φ_{G}]) - ω^{2} [Ψ]} {δ} = 0

(22)

The buckling analysis is first performed (before solving equation (22)) by solving the subsequent eigenvalue problem given as

([Φ_{L}] - ℓ_{i} [Φ_{G}]) {ζ_{i}) = 0

(23)

where

ζ_{i}

is the ith eigenvector and

ℓ_{i}

is the corresponding eigenvalue. The critical buckling temperature T_cr =

ℓ_{\min} Δ T

, where

Δ T

is the temperature rise.

The equation (22) is now rewritten as

{[Φ_{e q}] - ω^{2} [Ψ]} {δ} = 0

(24)

where

[Φ_{e q}] = [Φ_{L}] - [Φ_{G}]

After obtaining the modal parameters, the forced vibration response of the flat shell panels is obtained. The general form of governing equation for the vibrating flat panel is given as

[Φ_{e q}] {δ} + [X] {\overset{\cdot}{δ}} + [Ψ] {\overset{\cdot \cdot}{δ}} = M (t)

(25)

where

[X]

is the damping matrix which is a function of frequency and stiffness of the shell panel, and

M

is the excitation load vector. The mass matrix

[Ψ]

and the stiffness matrix

[Φ_{e q}]

are further used in performing the coupled vibroacoustic analysis to compute the sound radiation characteristics of the panels using a coupled FE–BE approach. The structure is modeled using FEs, whereas, the boundary element method (BEM) is utilized to model the fluid in contact with the surrounding structure. The BEM enables the reduction of the spatial dimensions of the problem and offers high accuracy at reasonable computation cost in comparison to other available techniques. The dynamics of the sound radiation of vibrating panels is governed by the Helmholtz wave equation

\nabla^{2} p + k^{2} p = 0

(26)

where

p

is the acoustic pressure. The discreet form of the governing Helmholtz wave equation in terms of BEs is given as

S p = i ρ_{s} ω Z v_{p}

(27)

where the matrix p represents pressure in the surroundings,

S

and

Z

are the boundary influence matrices dependent on the excitation frequency,

v_{p}

represents the normal component of the panel velocity and

ρ_{s}

is the density of the fluid medium contiguous to the vibrating shell panel. The panel velocity

(v_{p})

can be expressed in the form of displacement vector as

v_{p} = i ω δ \bar{\forall}

(28)

where

\bar{\forall}

is coupling matrix that transforms the panel displacement to the panel normal velocity. The coupled structural–acoustic equation of the system is conceded as [42,43]

(\begin{array}{l} Φ_{e q} + i ω X - ω^{2} Ψ L \\ ρ_{s} ω^{2} Z \bar{\forall} & S \end{array}) {\begin{matrix} δ \\ p \end{matrix}} = {\begin{matrix} M \\ 0 \end{matrix}}

(29)

where L is the FE–BE coupling matrix. The structural damping matrix [X] is defined as:

[X] = \frac{2 ξ}{ω} [Φ_{e q}]

where

ξ

is the constant damping ratio. The panel displacement

δ

and the sound pressure p in the surroundings can be obtained by solving the equation (29) (refer Mariem and Hamdi [42] and Atalla and Sgard [43] for detailed solution procedure). The other acoustic response indicators dependent on the values of p and

δ

are obtained as follows:

The sound power emitted by vibrating panel can be acquired by evaluating

W_{r a d} = \frac{1}{2} real [\iint_{s} p v_{p} d S]

(30)

where S represents the surface area of the vibrating panel.

The sound power level radiated by the panel structure is given by

Sound power level = 10 \times \log (\frac{W_{r a d}}{W_{r e f}})

(31)

where the reference power

W_{r e f} = 10^{- 12} W

The surface normal root mean square velocity ( $〈 v_{n} 〉$ ) is defined as

〈 v_{n} 〉 (ω) = \sqrt{\frac{1}{2 S} \int_{S} | {v_{p}}^{2} | d S}

(32)

The radiation efficiency ( $Ω$ ) is defined as

Ω = \frac{W_{r a d}}{ρ_{s} c S 〈 v_{n}^{2} 〉}

(33)

The sound pressure level (SPL) $P (dB)$ at a point in the surrounding fluid (field point) is defined as

P (dB) = 20 \times \log (\frac{p}{p_{r e f}})

(34)

where

p

and

p_{r e f}

are the field point pressure and the reference pressure (

= 20

µPa), respectively.

Results and discussion

In this section, the vibroacoustic responses of the laminated composite sandwich flat panels with isotropic core and orthotropic faces subjected to harmonic point excitation in a thermal environment are investigated and discussed in detail. The responses are computed using a domestic MATLAB computer code based on the proposed higher-order FEM–BEM scheme. The schematic of the steps followed to obtain the coupled vibroacoustic responses is illustrated in Figure 2. The numerical results presented in this work have been obtained using a full (3 × 3) integration rule. The critical buckling temperature (T_cr) corresponding to each configuration is first obtained by finding the solution to equation (23) and the uniform thermal load applied to the panels is designed below the T_cr and the glass transition temperature of the considered composite material. Sandwich panels with the following dimension, material properties and lay-up scheme is considered throughout the present analysis unless stated otherwise: a = 0.4 m, b = 0.3 m, h = 0.01 m, t_c/t_f = 15, core (isotropic) [28]: Ec = 7 GPa, ν _c = 0.3, ρ _c = 1000 kg/m³ and α _c = 1.8 × 10⁻⁵/°C, face (laminated composite) [20]: E₁_, _f = 132 GPa, E₂_, _f = 10.3 GPa, G₁₂_, _f = G₁₃_, _f =6.5 GPa, G₂₃_, _f = 3.91 GPa, ν₁₂_, _f = ν₁₃_, _f = ν₂₃_, _f = 0.25, ρ _f = 1570 kg/m³, α₁_, _f = 1.2 ×10⁻⁶/°C, α₂_, _f = 2.4 × 10⁻⁵/°C, and stacking sequence [0°/90°/Core/90°/0°]. A constant structural damping ratio of 1% is used throughout the analysis. The panels are excited at (0.1 m, 0.1 m, 0 m) by a harmonic point load of 1 N acing along the x₃-direction and the SPL is obtained at a field point lying at a distance of 2 m in x₃-direction directly above the point of excitation. Similarly, the SPL directivity pattern is observed for the points lying on a circle of radius 1 m, lying in the plane x₂ = b/2 and centred at the central node of the panels. The various support conditions to which the panel edges are subjected are classified as:

Clamped (C):

$x_{0} = y_{0} = z_{0} = u_{1} = u_{2} = v_{1} = v_{2} = w_{1} = w_{2} = 0$ at x₁ = 0 and a; at x₂ = 0 and b.

b. Simply supported (S):

$y_{0} = z_{0} = u_{2} = v_{2} = w_{2} = 0$ at x₁ = 0 and a; $u_{0} = w_{0} = u_{1} = v_{1} = w_{1} = 0$ at x₂ = 0 and b.

On the basis of these definitions, the all sides clamped (CCCC) and clamped-simply supported [CSCS] support conditions are utilized in the present analysis.

Convergence and validation study

Firstly, the natural frequencies of sandwich panels in ambient environment are obtained using the present scheme and compared with the values reported by Kant and Swaminathan [44]. The panel dimensions, material properties and boundary conditions are taken to be similar to as in the reference. The results are computed by varying the core-to-face thickness ratio (t_c/t_f) as: t_c/t_f = 4, 10, 50 and 100 and presented in Table 1. It can be observed that the natural frequency values computed using the present model is converging well with the mesh refinement. Also, the present values are in close conformance with the reference results, thus establishing the validity of the current approach. Further, the natural frequency of a simply supported sandwich panel as reported by Liu and Li [28] is reproduced using the present scheme and the results are shown in Table 2 for comparison. The panel is considered to be having an isotropic core and isotropic face (properties same as that of Liu and Li [28]). It is evident that the present values agree very well with the reference results. Based on the convergence studies (12 × 12) mesh is utilized for the computation purpose throughout the analysis.

Table 1.

Convergence and validation of non-dimensional fundamental frequency of a square sandwich composite (isotropic face and core) plate in ambient environment (ΔT = 0°C).

	Mesh size						Kant and Swaminathan [44]
t_c/t_f	4 × 4	6 × 6	8 × 8	10 × 10	12 × 12	14 × 14	Kant and Swaminathan [44]
4	9.1084	9.1043	9.1036	9.1034	9.1033	9.1033	8.9948
10	4.9523	4.9503	4.9499	4.9497	4.9497	4.9496	4.8594
50	2.8680	2.8672	2.8671	2.8670	2.8670	2.8670	2.8625
100	3.0295	3.0287	3.0286	3.0286	3.0286	3.0286	3.0290

Table 2.

Convergence and validation of fundamental frequency (Hz) of a square sandwich (isotropic face and core) flat panel in thermal environment (ΔT = 50°C).

	Mesh size						Liu and Li [28]
Mode	4 × 4	6 × 6	8 × 8	10 × 10	12 × 12	14 × 14	Liu and Li [28]
1	259.64	256.46	255.70	255.47	255.34	255.37	255.44
2	696.51	670.96	664.32	662.05	661.14	660.72	659.75
3	1025.87	981.32	970.70	967.31	965.99	965.41	963.44
4	1412.76	1366.22	1332.45	1318.57	1312.90	1310.28	1304.5
5	1515.47	1372.62	1355.34	1351.87	1350.53	1349.95	1346.7
6	2163.49	2029.46	1996.19	1985.19	1980.82	1978.83	1970.6

Additionally, another example from Liu and Li [28] is solved to compute the critical buckling temperature of a simply supported sandwich panel using the present scheme as well as a simulation model developed using commercial FE software ANSYS. From the results shown in Table 3 it is evident that the present scheme yields valid results for T_cr indicating the competence of present scheme to capture the influence of thermal loads on the structural characteristics of the sandwich flat panels.

Table 3.

Comparison of critical buckling temperature (T_cr) of simply supported sandwich (isotropic face and core) flat panel.

	Liu and Li [28]	Present numerical (HSDT)	Present numerical (simulation model)
Critical buckling temperature (T_cr)	94°C	93.76°C	94.21°C

HSDT: higher-order shear deformation theory.

Finally, in order to build more confidence in the present coupled higher-order FEM–BEM scheme for computing the vibroacoustic responses of laminated composite sandwich panels in thermal environment, the numerical examples related to laminated composite and sandwich panels available in the published literature are solved for sound power level and compared with the reference results. Additionally, the radiated sound power values are also computed using commercially available FE (ANSYS) and BE (LMS Virtual.Lab) packages. The structural model is developed in ANSYS environment using ANSYS parametric design language (APDL) code and discretized using the Shell 281 (an eight-noded serendipity element with six degrees freedom per node) element available in ANSYS element library. The modal data obtained using the simulation model is then imported into LMS Virtual. Lab environment for computing the radiated sound power by employing indirect BEM technique. For the computation purpose clamped laminated composite [0°/90°/0°/90°/0°] flat panels (without an isotropic core, t_c = 0) under the influence of a uniform thermal load of ΔT = 40°C as taken by Li et al. [20] are considered having the similar geometry, material and point harmonic excitation. Figure 3 depicts the comparison of present sound power level values together with the reference results alongside the results computed using the simulation model. It can be observed that the present values have a good agreement with the reference as well as the present simulation results. The marginally greater values computed via the present scheme as compared to the reference values may be attributed to the different mid-plane kinematics employed for the structural modeling. It is vital to note that the reference utilized the First-order shear deformation theory (FSDT) based kinematic model in contrast to a more accurate HSDT-based mid-plane kinematics as used in the current approach. On the other hand, as expected, the simulation model that is also based on the FSDT follows the reference curve closely. The differences may also be attributed to the fact that the present HSDT-based FEM–BEM formulation uses a nine noded isoparametric element having a 9 degrees of freedom per node. On the other hand Li et al. [20] utilized an analytical approach using the Rayleigh integral to compute the radiated sound power, whereas, the simulation model utilized an 8 noded serendipity element.

Further, a simply supported isotropic core and isotropic face sandwich flat panel (as composite sandwich panel examples are not available in open literature) subjected to a uniform temperature load ΔT = 50°C as considered by Liu and Li [28] with analogous properties and dimensions is taken for the validation of radiated sound power. The sound power values obtained using the present higher-order FEM–BEM scheme is compared with the reference values alongside the values obtained from the simulation model and depicted in Figure 4. In consistence with the earlier comparison results for laminated composite panels, the present sound power values also agrees well with the reference results for sandwich panel case. The slight higher values obtained using the present scheme reveals the necessity of the implementation of the HSDT in the numerical model for the computation of acoustic radiation responses of sandwich composite structure in comparison to the FSDT as used by the considered reference and the commercial FE packages.

Numerical illustrations

The necessity and correctness of the proposed higher-order FEM-BEM scheme to predict the acoustic responses of vibrating layered composite as well as sandwich structures in thermal environment have been confirmed in the convergence and comparison study. Now, specific numerical examples have been solved using the present scheme to unveil the influence of temperature, core-to-face thickness ratio (t_c/t_f), core-to-face modular ratio (E_c/E_1, _f ) and fibre orientation on the vibroacoustic radiation responses of composite sandwich panels in elevated thermal environment.

Firstly, the influence temperature loads (ΔT = 0°C, 25°C, 50°C, 75°C, 100°C and 120°C) on the vibration and acoustic radiation behaviour are investigated. In order to do so, the natural frequency, average root mean square velocity (RMS) velocity, radiation efficiency and the radiated sound power for all sides clamped sandwich composite flat panel (T_cr = 472.76°C) corresponding to each temperature load is computed and illustrated in Figures 5(a), (b), (c) and (d), respectively. It is inferred that the increasing temperature has a softening influence on the panels leading to reduced stiffness. Consequently, the natural frequency of all the modes decreases with increasing temperature. With reference to the no thermal load case (ΔT = 0°C), the first natural frequency of the panel decreases by 2.59%, 5.27%, 8.03%, 10.88%, and 13.24% for 25°C, 50°C, 75°C, 100°C and 120°C loads, respectively. Accordingly, the resonance peaks in average RMS velocity plot and radiated sound power level plot tend to shift to lower frequencies and the same can be observed from Figure 5(b) and (d), respectively. The average RMS velocity follows a generally increasing trend with the increasing temperature. The amplitude of the first resonance peak increases by 14.85% for a temperature rise of ΔT = 120°C. This increase is substantial and is attributed to the presence of soft isotropic core whose stiffness variation with temperature is expectedly larger compared to relatively stiffer and thinner composite faces. The radiation efficiency varies marginally unto the first resonance peaks. Post first resonance, the radiation efficiency follows a decreasing trend with increasing temperature loads as shown in Figure 5(c). Further, the upward jump in the resonance peaks in the radiated sound power curves is marginal as compared to the leftward frequency shift for increasing temperature loads. The overall radiated sound power for the power spectrums shown in Figure 5(d) is 111.89 dB, 111.93 dB, 112.02 dB, 112.24 dB, 112.48 dB and 112.57 dB, respectively. To have deeper insight on the sound radiation pattern, the overall sound power is plotted for constant frequency bands of width 300 Hz and depicted in Figure 6. The overall sound power strictly increases with increasing temperatures in (0–300) Hz and (0–600) Hz frequency bands. This is because the modal participation factors for the first few modes only contribute towards the acoustic radiation in the low frequency range. Also, the difference between the highest and the lowest sound power is higher (3.75%) in (0–600) Hz as compared to that (12.5%) in (0–300) Hz. On the contrary, the overall sound power decreases with increasing temperature in (600–900) Hz. After 900 Hz, the variation of sound power with temperature is minimal as a larger number of modes contribute at higher frequencies than at the lower frequencies.

Subsequently, the influence of core-to-face thickness ratio (t_c/t_f) on the vibroacoustic responses of clamped composite sandwich flat panel is investigated. The displacement of the point of excitation, radiation efficiency, SPL at the field point and the radiated sound power are obtained for ΔT = 0°C and 120°C conditions and shown in Figures 7 and 8, respectively. The core-to-face thickness ratio is varied (keeping the overall thickness h as constant) to have the following values: t_c/t_f = 0, 3, 6, 12 and 18 and the corresponding T_cr values are 631.63°C, 781.84°C, 676.03°C, 521.50°C and 435.45°C, respectively. This indicates that the panels with a thicker core buckle at a lower temperature as compared to the panels with a thinner core. The t_c/t_f = 0 case representing a purely laminated composite panel with no core is solved to have a comparative analysis for core and no-core configurations. In general, the increasing thickness of the soft core would lead to a panel with reduced stiffness. Softer panels are expected to have lesser natural frequencies. Consequently, the peaks in response curves cascade to lower frequencies with increasing t_c/t_f values and the same is evident from Figures 7 and 8. For ΔT = 0°C case, the displacement of the point of excitation is higher for the panels with higher t_c/t_f. The response curves of the t_c/t_f = 0 case float in-between t_c/t_f = 6 and 12 case. The radiation efficiency of the panels decreases with increasing t_c/t_f as shown in Figure 7(b) which is expected. Consequently, the radiated sound power generally increases as the core gets thicker. Further, as observed from Figure 7(c), the SPL at the field point follows a trend similar to that followed by the radiated sound power.

Additionally, the temperature greatly influences the sound radiation characteristics. At a uniform temperature load of ΔT = 120°C, the resonance peaks move towards the lower frequencies (in comparison to ΔT = 0°C case) and the same can be observed from Figure 8. The overall pattern of variation of displacement, radiation efficiency, SPL and radiated sound power with t_c/t_f is similar to that for ΔT = 0°C case. However, thermal stresses cause more modes to be excited within the considered frequency range and therefore more number of resonance peaks and valleys are captured. To have a quantitative evaluation of the influence of temperature, the overall SPL in constant frequency bands is plotted for ΔT = 75°C and shown in Figure 9. It can clearly be seen that the t_c/t_f = 18 contributes to maximum SPL throughout all of the frequency bands considered. In low frequency bands (0–300) Hz, (0–600) Hz the t_c/t_f = 3 case radiates the least, whereas in higher frequency range the t_c/t_f = 0 case causes the least sound radiation.

Also, the overall radiated sound power for the considered t_c/t_f values are plotted for ΔT = 0°C, 75°C, 120°C and shown in Figure 10. It is observed that the overall response is higher for higher temperatures irrespective of t_c/t_f values. Interestingly, the difference in the overall responses for different temperatures is accentuated for t_c/t_f = 0 case as compared to the other cases.

The influence of core-to-face modular ratio (E_c/E_1, _f ) on the vibroacoustic responses in thermal environment is now studied in this example. Clamped flat panels are considered and the E_c/E_1, _f is varied to take the values 0.05, 0.1, 0.5 and 1 while keeping E_1, _f value as constant. The corresponding T_cr values are 489.62°C, 328.60°C, 173.88°C and 152.55°C, respectively. The variation of average RMS velocity, radiation efficiency and radiated sound power with E_c/E_1, _f for ΔT = 0°C and 120°C is observed and illustrated in Figures 11, 12 and 13, respectively. Clearly, as E_c/E_1, _f approaches unity, the core becomes increasingly stiffer. However, as observed from the previous examples, the increasing temperature leads to reduction in the stiffness of the panels. Indeed, these two phenomena are coupled and implicit in the present results. The average RMS velocity decreases with increasing E_c/E_1, _f for both ΔT = 0°C and 120°C. However, due to elevated temperature load, larger number of modes are excited for ΔT = 120°C case and therefore the number of resonance peaks are more compared to ΔT = 0°C case. The radiation efficiency is the highest for E_c/E_1, _f =1 case and decreases with decreasing core-to-face modular ratio. Also, the radiation efficiency for every E_c/E_1, _f value is lesser for ΔT = 120°C case as compared to ΔT = 0°C case and the same can be observed from Figure 12.

Now, Figure 13 shows that the radiated sound power decreases with increasing core-to-face modular ratio (E_c/E_1, _f ). The overall sound power of the panels for ΔT = 0°C, 120°C is listed in Table 4. It is evident that the sound power emitted by the panels increases with increasing temperature. However, the difference is larger for higher values of E_c/E_1, _f .

Table 4.

Variation of overall radiated sound power (dB) with E_c/E_1, _f .

	Overall sound power (dB)
E_c/E_1, _f	0.05	0.1	0.5	1
ΔT = 0°C	111.93	111.66	105.05	97.90
ΔT = 120°C	112.62	111.64	110.86	104.91

Finally, the influence of fibre orientation on the sound radiation characteristics of sandwich composite panels under elevated thermal environment and subjected to CSCS support condition is investigated. The lay-up of the sandwich panel is chosen as $[0 ° / θ / - θ / Core / - θ / θ / 0 °]$ , where $θ$ is the fibre angle which is chosen to have the values 0°, 15°, 30°, 45°, 60° and 90°. The corresponding T_cr values are found to be 188.97°C, 205.08°C, 251.91°C, 322.85°C, 401.36°C and 449.63°C, respectively. The radiation efficiency and radiated sound power are plotted for the uniform temperature loads of ΔT = 0°C, ΔT = 50°C, and ΔT = 120°C, as shown in Figures 14 and 15, respectively. It is obvious from Figure 14 that the radiation efficiency increases for increasing value of fibre orientation angle (θ) irrespective of the temperature loads indicating that panels become stiffer with increasing fibre angle. It must be noted that the radiation efficiency decreases with increasing temperature load. However, the resonance peaks in the sound power radiation patterns move to higher frequencies with increasing value of θ for each temperature load case considered. Also, the radiated sound power for each θ case increases with increasing temperature load. Additionally, the SPL directivity patterns are observed for ΔT = 120°C case for frequencies 400 Hz and 1200 Hz and shown in Figure 16(a) and (b), respectively. For 400 Hz, the patterns are like monopoles because of influence of first radiation mode at low frequency. Interestingly, θ = 45° case radiates the most while θ = 90° case causes the least sound pressure in the surroundings. At 1200 Hz, the patterns are complex because of the contribution of multiple modes at higher frequencies.

Conclusion

In this work, the vibroacoustic responses of sandwich composite flat panels having isotropic core and orthotropic faces subjected to harmonic point excitation in an elevated temperature environment have been investigated in the frame work of a novel HSDT-based coupled FE–BE approach. Firstly, the natural frequency, critical buckling temperature and the sound power radiated by the panels are validated with the available benchmark results alongside the results obtained via a simulation model using commercial software ANSYS and LMS Virtual.Lab. The validation study shows the necessity of the implementation of the HSDT mid-plane kinematics during the numerical analysis of laminated composite and sandwich structure. It is observed that the elevated temperature influences the stiffness of the panels substantially. The natural frequencies of the panels decrease with increasing temperature and the first natural frequency of the panels decreases by 13.5% for a temperature increase of 120°C. Consequently, the peaks in the response curves shift to lower frequency regions. The radiation efficiency is observed to decrease and the overall radiated sound power is observed to increase with increasing temperature loads. Further, the radiated sound power generally increases as the core gets thicker. The t_c/t_f = 3 case cause minimum sound pressure in low frequency range of (0–600) Hz, whereas post 900 Hz t_c/t_f = 0 causes the least sound pressure in the surroundings. The radiation efficiency is the highest for E_c/E_1, _f = 1 case and decreases with decreasing core-to-face modular ratio. The overall sound power decreases with increasing E_c/E_1, _f . The SPL directivity patterns are monopoles at 400 Hz with θ = 45° case causing the most while θ = 90° case causing the least sound pressure in the surroundings.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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