Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support

Abstract

This study presents dynamic responses of a composite thick beam with a functionally graded porous layer under dynamic sine pulse load. The boundary conditions of the composite beam are considered as viscoelastic supports. Three layers are considered, and face sheet layers have porous functionally graded materials in which the distribution of material gradation through the graded layer is described by the power law function, and the porosity is depicted by three different distributions (i.e., symmetric distribution, X distribution, and ◊ distribution). The layered composite thick beam is modeled as a two-dimensional plane stress problem. The equation of motion is obtained by Lagrange’s equations. In formation of the problem, the finite element method is used with a 12-node 2D plane element. In the solution process of the dynamic problem, a numerical time integration method of the Newmark method is used. In numerical analyses, influences of stiffness and damping coefficients of viscoelastic supports, material gradation index, porosity parameter, and porosity models on the dynamic response of thick functionally graded porous beam are investigated under the pulse load.

Keywords

Viscoelastic support porosity models functionally graded deep beam layered structure finite element method

1. Introduction

Beams and plates are used in various applications, such as aerospace, marine, automotive, and defense sectors, where continuous elements of these types are required. Necessity for concurrent thermal and mechanical strengths has led to the invention of plate and beam structures with functionally graded material (FGM) properties (Ghayesh, 2019). The FGM is described by a continuous and smooth variation in both composition profile and material properties along one or more than one direction to accomplish desired structural performance (Zhao et al., 2020a). Structures with functionally graded (FG) porous constituents have been deemed by research communities and industries because of their outstanding energy absorption capability, high specific strength, low thermal conductivity, and other special characteristics (Kannan et al., 2008; Miyamoto et al., 2013). Recently, there has been increasing interest in the development of porous bone implant FG materials that exhibit an architecture mimicking that of human bone because the porous structure can provide space for new bone tissue ingrowth and body fluid circulation (Yuan et al., 2019).

For one-dimensional beam structure, Şimşek and Kocatürk (2009) investigated the free and forced vibrations of an FG simply supported beam subjected to concentrated moving harmonic load. Ergut and Altintas (2013) presented the effect of application area distributions of mass, force, and support on free and forced vibration behaviors of viscoelastically supported plates. Demir and Oz (2014) exploited the finite element method (FEM) to investigate resonance frequencies of the FG beam under viscoelastic boundary conditions. Hamed et al. (2016) presented vibration characteristics of both nonlinear symmetric power and sigmoid FG nonlocal nanobeams by using the nonlocal differential Eringen’s elasticity model. Frikha et al. (2016) presented a 2-node, 4 degree of freedom/node beam element based on a higher-order shear deformation theory for axial–flexural–shear FGM. Akbas (2016a) studied wave propagation of edge-cracked damped cantilever FG beam excited by a transverse triangular force impulse by the FEM. Akbaş (2016b, 2017) exploited the modified couple stress theory and the Kelvin–Voigt viscoelastic model to investigate the vibration response of full and cracked viscoelastic nanobeams rested on Winkler–Pasternak elastic foundation under forced excitation. Ding et al. (2017) presented effects of viscosity on the natural frequency of transverse vibration of an axially moving viscoelastic beam modeled by the Euler–Bernoulli beam and Kelvin model. Şimşek and Al-Shujairi (2017) examined static, free, and forced vibrations of FG sandwich beams under the action of double moving harmonic loads traveling with constant velocities using the Timoshenko beam theory. Fan et al. (2018) studied low-velocity impact response of FG graphene-reinforced composite laminated beam rested on visco-Pasternak foundations and subjected to transverse impact load in thermal environments. Eltaher et al. (2018) presented a modified porosity model that represents porosity and Young’s modulus in an implicit form to study static and dynamic behaviors of FG porous nonlocal nanobeams. Li et al. (2018) exploited the even and uneven dispersion patterns of porosity to present size-dependent the nonlinear free vibration behavior of porous beams with geometric imperfections. Emam et al. (2018) developed an analytical model to study postbuckling and free vibration of multilayer imperfect composite nanobeams under a prestress load. Yang et al. (2018, 2019) investigated the nonlinear instability of FG multilayer graphene reinforced composite shallow arches under a central point load and mechanical and thermal loading.

Ebrahimi et al. (2019) investigated frequency response of curved magneto–electro–viscoelastic FG nanobeams resting on viscoelastic foundation by incorporating the Euler–Bernoulli beam model and Eringen’s nonlocal theory. Hamed et al. (2019) studied effects of four porosity models on the static behavior of size-dependent nonlocal FG nanobeams. Jrad et al. (2020) developed analytical and finite element solutions of free and forced vibrations of unrestrained and braced thin-walled beams. Ghayesh (2019) investigated resonant vibrations of viscoelastic FG imperfect Timoshenko beams by using the Kelvin–Voigt scheme. Silva and Bavastri (2019) developed an optimal design of physical parameters, positions, and viscoelastic materials of simple dynamic absorbers for passive vibration control. Hamed et al. (2020a) and Eltaher and Mohamed (2020) studied the buckling stability of sandwich composite beam structure under varying axial loads and resting on elastic foundation by using the unified beam theory. Hamed et al. (2020b) presented the influence of axial load function and optimization on static stability of sandwich FG beams with porous core. Melaibari et al. (2020a, 2020b) studied the buckling stability of FG beams under variable axial load with different gradation (i.e., power, symmetric, and sigmoid) functions. Yang et al. (2020a, 2020b) investigated in-plane and out-of-plane free vibrations of FG composite arches with graphene reinforcements under a step central point load. Faroughi et al. (2020) studied wave propagation in 2D FG porous rotating nanobeams using a general nonlocal higher-order beam model. Abu-Alshaikh and Almbaidin (2020) used the Caputo and Caputo–Fabrizio fractional derivative models to investigate the response of the FG Euler beam under moving mass. Abida et al. (2020) investigated a viscoelastic–viscoplastic model with hygromechanical coupling for flax fiber–reinforced polymer composites. Zghal and Dammak (2020) investigated the vibrational behavior of beams made of FGMs using double field of displacements and stresses, which offers the possibility of the development of low-order linear elements with enhanced accuracy. Zghal et al. (2020) studied static bending analysis of beams made of functionally graded porous materials using a refined mixed finite element beam model.

Recently, lots of researchers modeled thick beam structures by two-dimensional constitutive equations. Eltaher et al. (2018) and Soliman et al. (2018a, 2018b) presented nonlinear transient analysis and cracked FG pipe subjected to internal pressure and unsteady temperature by using the 2D elasticity model and FEM. Trabelsi et al. (2018) studied thermal postbuckling analysis of FGM structures by using a modified first-order shear deformation theory (FSDT) with a parabolic shape function. Akbaş (2018) exploited the plane solid continua model to investigate forced vibration response of FG porous deep beams by using even and uneven porosity models. Akbaş (2019a) modified the previous model to investigate the forced response of the sandwich FG thick beam by using the FEM. Akbaş (2019b) analyzed the nonlinear behavior of a FG nonuniform hygrothermal cantilever thick beam using the plane solid continua model. Trabelsi et al. (2019, 2020) investigated thermal buckling analysis of functionally graded plates and cylindrical shells by using the Kirchhoff–Love model and modified FSDT-based four-node finite shell element. Foroutan et al. (2020) presented analytical and semianalytical approaches to study the nonlinear dynamic and static hygrothermal buckling analysis of imperfect FG porous cylindrical shells under hygrothermal loading including effect of the von Kármán strain–displacement kinematic nonlinearity. Moleiro et al. (2020) addressed the modeling and analysis of multilayered plates with embedded FGM layer(s) under hygro–thermo–mechanical loadings by using the layerwise mixed model. Zhao et al. (2020b) exploited closed-cell cellular solids under the Gaussian random field scheme to describe the relationship between the elastic modulus and mass density of the material and then to study dynamic instability of FG porous arch reinforced with uniformly distributed graphene platelets. Akbaş et al. (2020) developed a comprehensive model to investigate the vibration response of FG porous thick beams under the dynamic sine pulse load including the damping effect. Mohseni and Shakouri (2020) studied free vibration, damping, and forced responses of sandwich plates with viscoelastic core and graphene nanoplatelet–reinforced face sheets.

As per authors’ knowledge and literature, the transient dynamic behaviors of FG deep 2D porous beams under pulse load and supported with viscoelastic end conditions have not been analyzed and investigated elsewhere. So, this article tries to fill this gap and study influences of viscoelastic supports, gradation index, porosity parameter, and porosity type on the response of FG deep beam by using the numerical FEM. The rest of this article is organized as follows: mathematical formulations, constitutive equations, and numerical formulation are introduced through Section 2. The validation of model and parametric investigation of gradation parameter, porosity model, porosity index, and viscoelastic supports on dynamic response of thick multilayer porous FG beams are examined in Section 3. Conclusion points and main remarks are summarized through Section 4.

2. Problem formulation

2.1. Mathematical formulation

A viscoelastic supported thick beam with three layers under a pulse load $P (t)$ at midpoint is shown in Figure 1. The beam has a length of L through the axial direction x-axis and thickness h through the transverse z-axis. The stacking sequence of the beam is two face layers with porous functionally graded properties and a homogeneous core layer. The stiffness and damping coefficients of the two viscoelastic supports are indicated as K_s and C_s, respectively.

Assuming that, the gradation of material through the layer thickness is assumed to be symmetrical to the midplane axis, and each layer has the same thickness. Based on this assumption, the properties of material constituent are varied through the beam thickness according to the following power law, (Eltaher et al., 2014)

P (z) = (P_{T} - P_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + P_{B}

(1)

in which

P

is the material properties (i.e., Young’s modulus

E

, Poisson’s ratio

ν

, or density

ρ

), n is the positive power gradation index of gradation, and subscripts T and B are the top and bottom properties of the layers. By imposing the porosity profile distribution on the gradation power law, three different porosity models, displayed in Figure 2, can be depicted as follows (Akbaş, 2018)

Uniform distribution E (z) = (E_{T} - E_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + E_{B} - (E_{T} - E_{B}) a

(2a)

Uniform distribution ν (z) = (ν_{T} - ν_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + ν_{B} - (ν_{T} - ν_{B}) a

(2b)

Uniform distribution ρ (z) = (ρ_{T} - ρ_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + ρ_{B} - (ρ_{T} - ρ_{B}) a

(2c)

⋄ - Distribution E (z) = (E_{T} - E_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + E_{B} - (E_{T} - E_{B}) \frac{a}{2} (1 - \frac{2 | z |}{h})

(3a)

⋄ - Distribution ν (z) = (ν_{T} - ν_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + ν_{B} - (ν_{T} - ν_{B}) \frac{a}{2} (1 - \frac{2 | z |}{h})

(3b)

⋄ - Distribution ρ (z) = (ρ_{T} - ρ_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + ρ_{B} - (ρ_{T} - ρ_{B}) \frac{a}{2} (1 - \frac{2 | z |}{h})

(3c)

X - Distribution E (z) = (E_{T} - E_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + E_{B} - (E_{T} - E_{B}) \frac{a}{2} (\frac{2 | z |}{h})

(4a)

X - Distribution ν (z) = (ν_{T} - ν_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + ν_{B} - (ν_{T} - ν_{B}) \frac{a}{2} (\frac{2 | z |}{h})

(4b)

X - Distribution ρ (z) = (ρ_{T} - ρ_{B}) {[\frac{z}{h} + \frac{1}{2}]}^{n} + ρ_{B} - (ρ_{T} - ρ_{B}) \frac{a}{2} (\frac{2 | z |}{h})

(4c)

in which a is the volume fraction of porosities. The FG layer becomes fully FGM without any porosity when a = 0 and fully empty if a = 1, which is unlogic. So, in most cases, the value of porosity is less than a << 1.

The current analysis focuses on the analysis of FG deep beam structures. Thus, to describe the mechanical behavior of this system accurately, 2D plane stress constitutive relation is exploited. Therefore, the kinematic strain–displacement relations are described by (Akbaş, 2018)

{\begin{matrix} ε_{x x} \\ ε_{z z} \\ γ_{x z} \end{matrix}} = [\begin{matrix} \frac{\partial}{\partial x} & 0 \\ 0 & \frac{\partial}{\partial z} \\ \frac{\partial}{\partial z} & \frac{\partial}{\partial x} \end{matrix}] {\begin{matrix} u \\ w \end{matrix}}

(5a)

In the condensed matrix form, equation (5a) becomes

{ε} = [B] {d}

(5b)

in which

ε_{x x}

and

ε_{z z}

are the normal strains, and

γ_{x z}

is the shear strain. Displacements in x and z directions can be described by

u

and

w

, respectively. Thus, the stress–strain governing equation can be depicted by (Akbaş, 2018)

{\begin{matrix} σ_{x x} \\ σ_{z z} \\ σ_{x z} \end{matrix}} = [\begin{matrix} Q_{11} (z, a) & Q_{12} (z, a) & 0 \\ Q_{12} (z, a) & Q_{22} (z, a) & 0 \\ 0 & 0 & Q_{33} (z, a) \end{matrix}] {\begin{matrix} ε_{x x} \\ ε_{z z} \\ γ_{x z} \end{matrix}}

(6a)

In the abbreviated matrix form, equation (6a) turns into

{σ} = [Q] {ε}

(6b)

in which the stiffness coefficients can be portrayed as follows

Q_{11} (z, a) = Q_{22} (z, a) = \frac{E (z, a)}{1 - {[v (z, a)]}^{2}}, Q_{33} (z, a) = \frac{E (z, a)}{2 [1 + v (z, a)]}, Q_{12} (z) = \frac{v (z, a) E (z, a)}{1 - {[v (z, a)]}^{2}}

(7)

2.2. Finite element formulation

Through this section, the Lagrangian equation is exploited to derive the governing equation of motion of FG porous thick beam structure. The Lagrangian equation can be described by

I = T - (U_{i} + U_{e})

(8)

where strain energy

(U_{i})

, kinetic energy

(T)

, and potential energy of the external loads

(U_{e})

are described by the following expressions

U_{i} = \frac{1}{2} {d}^{T} (\int_{V} {[B]}^{T} [C] [B]) {d} d V

(9)

T = \frac{1}{2} {\dot{d}}^{T} (\int_{V} ρ (z)) {\dot{d}} d V

(10)

U_{e} = - P (t) v (x_{p}, t)

(11)

where

v

is the velocity vector and

x_{p}

is the coordinate of the applied load.

The potential energy $(U_{s})$ and dissipation function $(R_{s})$ of the two viscoelastic supports at any time are given as follows

U_{s} = \frac{1}{2} k_{s} q_{1}^{2} + \frac{1}{2} k_{s} q_{2}^{2}

(12a)

R_{s} = \frac{1}{2} c_{s} {\dot{q}}_{1}^{2} + \frac{1}{2} c_{s} {\dot{q}}_{2}^{2}

(12b)

where q₁ and q₂ are the vertical displacements of left and right viscoelastic supports, respectively. To get more realistic and sensitive results for the thick beam problem, the 12-node 2D plane element is used. By using the 12-node 2D plane element model shown in Figure 3, it is aimed to minimize the unknowns and discontinuities of displacements and stresses in interface of finite elements.

The displacement vector for the 12-node plane element can be represented by

{d} = [\begin{matrix} Ø_{1} & Ø_{2} & \dots & Ø_{12} \end{matrix}] {\begin{matrix} d_{1} \\ d_{2} \\ ⋮ \\ d_{n} \end{matrix}}

(13a)

{d_{n}} = {\begin{matrix} u_{1} \\ u_{2} \\ \cdot \\ \cdot \\ u_{12} \\ w_{1} \\ w_{2} \\ \cdot \\ \cdot \\ w_{12} \end{matrix}}

(13b)

where

{d_{n}}

indicates the nodal displacement vector, and its components u_i and w_i are the displacement components for the i node. The displacement of any generic element can be represented by its nodal values and corresponding functions as

u = (u_{1} \emptyset_{1} + u_{2} \emptyset_{2} + u_{3} \emptyset_{3} + u_{4} \emptyset_{4} + u_{5} \emptyset_{5} + u_{6} \emptyset_{6} + u_{7} \emptyset_{7} + u_{8} \emptyset_{8} + u_{9} \emptyset_{9} + u_{10} \emptyset_{10} + u_{11} \emptyset_{11} + u_{12} \emptyset_{12})

(14a)

w = (w_{1} \emptyset_{1} + w_{2} \emptyset_{2} + w_{3} \emptyset_{3} + w_{4} \emptyset_{4} + w_{5} \emptyset_{5} + w_{6} \emptyset_{6} + w_{7} \emptyset_{7} + w_{8} \emptyset_{8} + w_{9} \emptyset_{9} + w_{10} \emptyset_{10} + w_{11} \emptyset_{11} + w_{12} \emptyset_{12})

(14b)

where

\emptyset_{i}

is the nonlinear interpolation shape function that can be described by (Akbaş et al., 2020)

\emptyset_{1} = \frac{1}{32} (1 - \frac{2 X}{L_{x}}) (1 - \frac{2 Z}{L_{z}}) (- 10 + 9 (\frac{4 X^{2}}{L_{x}^{2}} + \frac{4 Z^{2}}{L_{z}^{2}})), \emptyset_{2} = \frac{9}{32} (1 - \frac{2 X}{L_{x}}) (1 - \frac{4 Z^{2}}{L_{z}^{2}}) (1 - \frac{6 Z}{L_{z}})

\emptyset_{3} = \frac{9}{32} (1 - \frac{2 X}{L_{x}}) (1 - \frac{4 Z^{2}}{L_{z}^{2}}) (1 + \frac{6 Z}{L_{z}}), \emptyset_{4} = \frac{1}{32} (1 - \frac{2 X}{L_{x}}) (1 + \frac{2 Z}{L_{z}}) (- 10 + 9 (\frac{4 X^{2}}{L_{x}^{2}} + \frac{4 Z^{2}}{L_{z}^{2}}))

\emptyset_{5} = \frac{9}{32} (1 - \frac{2 Z}{L_{z}}) (1 - \frac{4 X^{2}}{L_{x}^{2}}) (1 - \frac{6 X}{L_{x}}), \emptyset_{6} = \frac{9}{32} (1 + \frac{2 Z}{L_{z}}) (1 - \frac{4 X^{2}}{L_{x}^{2}}) (1 - \frac{6 X}{L_{x}})

\emptyset_{7} = \frac{9}{32} (1 - \frac{2 Z}{L_{z}}) (1 - \frac{4 X^{2}}{L_{x}^{2}}) (1 + \frac{6 X}{L_{x}}), \emptyset_{8} = \frac{9}{32} (1 + \frac{2 Z}{L_{z}}) (1 - \frac{4 X^{2}}{L_{x}^{2}}) (1 + \frac{6 X}{L_{x}})

\emptyset_{9} = \frac{1}{32} (1 + \frac{2 X}{L_{x}}) (1 - \frac{2 Z}{L_{z}}) (- 10 + 9 (\frac{4 X^{2}}{L_{x}^{2}} + \frac{4 Z^{2}}{L_{z}^{2}})), \emptyset_{10} = \frac{9}{32} (1 + \frac{2 X}{L_{x}}) (1 - \frac{4 Z^{2}}{L_{z}^{2}}) (1 - \frac{6 Z}{L_{z}})

\emptyset_{11} = \frac{9}{32} (1 + \frac{2 X}{L_{x}}) (1 - \frac{4 Z^{2}}{L_{z}^{2}}) (1 + \frac{6 Z}{L_{z}}), \emptyset_{12} = \frac{1}{32} (1 + \frac{2 X}{L_{x}}) (1 + \frac{2 Z}{L_{z}}) (- 10 + 9 (\frac{4 X^{2}}{L_{x}^{2}} + \frac{4 Z^{2}}{L_{z}^{2}}))

(15)

Substituting equation (13) into energy equations (9)–(2) and Lagrange’s equations (8), the dynamic equation of motion can be presented as

[K + K_{s}] {d_{n}} + [C_{s}] {{\dot{d}}_{n}} + [M] {{\ddot{d}}_{n}} = {F}

(16)

where

[K]

[M]

{F}

{d_{n}}

, and

{{\ddot{d}}_{n}}

are the stiffness matrix, mass matrix, load vector, displacement vector, and acceleration vector, respectively. Also,

K_{s}

and

C_{s}

indicate stiffness and damping matrices of viscoelastic supports. The expansions of finite element matrices are represented as

[K] = b \int_{A} {[H]}^{T} [Q] [H] d A

(17a)

[M] = b \int_{A} ρ (z) {[Ø]}^{T} [Ø] d A

(17b)

{F} = \int_{Γ} {δ d_{n}}^{T} {[Ø]}^{T} P (t) d Γ

(17c)

where

[H] = [\begin{matrix} \frac{\partial}{\partial X} & 0 \\ 0 & \frac{\partial}{\partial Y} \\ \frac{\partial}{\partial Y} & \frac{\partial}{\partial X} \end{matrix}] [Ø]

(18)

The load point harmonic pulse function $P (t)$ can be described in time domain as follows

P (t) = P_{0} sin (Ω t) 0 \leq t ≪ t_{0} sin pulse

(19)

in which the amplitude of load is

P_{0}

, and the time duration and frequency of the load are

t_{0}

and Ω, respectively. Numerically, the implicit Newmark average acceleration

(α = 0.5, β = 0.25)

method is exploited to solve the equation of motions represented in equation (16). By this procedure, the dynamic problem is solved incrementally as a static problem in each step as

[\bar{K} (t, X)] {d_{n}}_{j + 1} = {\bar{F} (t)}

(20)

in which

[\bar{K} (t, X)] = [K] + \frac{[M]}{β Δ t^{2}} + \frac{[C] α}{β Δ t}

(21a)

{\bar{F} (t)} = {F (t)}_{j + 1} + B_{1} {d_{n}}_{j} + B_{2} {{\dot{d}}_{n}}_{j} + B_{3} {{\ddot{d}}_{n}}_{j}

(21b)

where

B_{1} = \frac{[M]}{β Δ t^{2}} + \frac{[C] α}{β Δ t}, B_{2} = \frac{[M]}{β Δ t} + [C] (\frac{α}{β} - 1), B_{3} = [M] (\frac{1}{2 β} - 1) + [C] (\frac{α}{2 β} - 1)

(22)

Once calculating ${d_{n}}_{j + 1}$ at time $t_{j + 1} = t_{j} + Δ t$ , the acceleration and velocity vectors can be determined by

{{\ddot{d}}_{n}}_{j + 1} = \frac{1}{β Δ t^{2}} ({d_{n}}_{j + 1} - {d_{n}}_{j}) - \frac{[M]}{β Δ t} {{\dot{d}}_{n}}_{j} - (\frac{α}{2 β} - 1) {{\ddot{d}}_{n}}_{j}

(23a)

{{\dot{d}}_{n}}_{j + 1} = {{\dot{d}}_{n}}_{j} + Δ t (1 - α) {{\ddot{d}}_{n}}_{j} + Δ t α {{\ddot{d}}_{n}}_{j + 1}

(23b)

The dimensionless stiffness and damping coefficients of viscoelastic supports are given as follows

\bar{K_{s}} = \frac{K L^{3}}{E_{c} I}, \bar{c_{s}} = \frac{C_{s} E_{c} I A}{L^{2}}

(24)

where

E_{c}

is Young’s modulus of the core material, I is the inertia of moment, and A is the area.

3. Numerical results

3.1. Validation

To validate the proposed model, a comparison study with ANSYS software is presented. In the comparison study, the vertical displacements at the free end of a homogeneous zirconia/homogeneous aluminum/homogeneous zirconia beam are obtained and compared with ANSYS Workbench program under pulse load without damping and porosity effects for L/h = 7, P₀ = 1000 kN, Ω = 5 rad/s, and $t_{0} = 0.01 s$ , as shown in Figure 4. It is seen from Figure 4 that the result of the proposed model is very close to the ANSYS Workbench result.

3.2. Parametric studies

In the numerical study, effects of stiffness and damping coefficients of viscoelastic supports, material gradation index, porosity parameter, and porosity models on the time response of thick multilayer FGM beams under pulse load with viscoelastic support will be investigated. The materials of FG layers are considered as a combination of aluminum metal (Al; E = 70 GPa, ν = 0.3, and ρ = 2702 kg/m³) and zirconia ceramic (E = 151 GPa, ν = 0.3, and ρ = 3000 kg/m³). The material of the core layer is considered aluminum, and the material of the two face layers are considered the gradation of aluminum and zirconia (1. layer: zirconia–Al, 2. layer: fully Al, and 3. layer: Al–zirconia). The dimensions of the FG thick beam are considered as follows: b = 0.15 m, h = 0.15 m, and the length of beam varied according to the aspect ratio L/h = 7 in the numerical process. The height of each layer is equal.. The five-point Gaussian integration is used for calculation of element matrices. In the numerical process, the finite element number is taken as 30 in both X and Z directions.

In Figure 5, the vertical displacements (v_m) at the midpoint of thick layered FGM beam with different porosity factor (a) and gradation parameter (n) are presented for porosity models 1, 2, and 3, respectively. For these figures, the parameters of sinusoidal pulse load are taken in frequency Ω = 2 rad/s and P₀ = 10,000 kN. The viscoelastic support parameters are taken as $\bar{K_{s}} = 1$ and $\bar{C_{s}} = 0.1$ . The displacements are obtained for t = 0.2 s. As shown for a specific distribution of the porosity model, the displacement of the FG beam decreased dramatically as the gradation index n increased from 0 to 0.5. As the gradation index increased from 0.5 to 3, the displacement increased with a small rate. It is noted that by increasing the porosity percentage, the displacement is decreased because of the softening of the stiffness of beam structure. As shown in Figure 4, it is noted the displacement for uniform porosity model 1 is higher than that for the other two porosity models, ◊ and X distribution. Thus, ◊ and X distribution of porosity models are more significant on the dynamic response rather than the uniform porosity distribution.

In Figures 6–8, effects of stiffness of viscoelastic support on the time history of the thick beam are presented for different porosity factors for porosity model 1, porosity model 2, and porosity model 3, respectively. In these figures, the damping of the support is neglected, and the following parameters are used: n = 1, $\bar{C_{s}} = 0$ , Ω = 2 rad/s, and P₀ = 10,000 kN. As shown, the vertical displacements at the midpoint of the beam have a high amplitude of oscillation at $\bar{K_{s}} = 1$ and lowest oscillation rather at $\bar{K_{s}} = 100$ . Thus, by increasing the stiffness of elastic boundary condition, the amplitude of vibration and time of oscillation are affected significantly. It is also noted that the porosity has a little bit effect on the amplitude and oscillation of vibration in comparison with the effect of elastic support.

In Figures 9–11, the effects of damping of viscoelastic support on the time history of the thick beam are presented for different porosity factors for porosity model 1, porosity model 2, and porosity model 3, respectively. In these figures, the following parameters are used: n = 1, $\bar{K_{s}} = 1$ , Ω = 2 rad/s, and P₀ = 10,000 kN. It is observed that the undamped phenomena and higher oscillation amplitude are predicted as the damping coefficient of support $\bar{C_{s}} = 0$ . By adding the damping to the elastic boundary condition, the amplitude of vibration is reduced significantly, and the oscillation period is increased. By comparing the effect of stiffness and the effect of damping on the dynamic behavior of FGM porous beam at the same conditions, it is noted that the damping effect is more pronounced than the stiffness effect on the amplitude of vibration and oscillation period.

4. Conclusions

A dynamic response of FG porous multilayer thick beams under pulse load and rested on the viscoelastic supports is studied in the framework of the 2D plane stress problem. The numerical finite element formulation and Newmark integration method are exploited to solve the equation of motion incrementally. Material properties are graded through the beam thickness and modeled by the generalized power law. Uniform distribution, ◊ distribution, and X distribution of porosity are the three models of porosity assumed through the analysis. The most findings of this article can be summarized as follows:

The gradation parameter (n) has significant effects on dynamic response and amplitude of oscillation. The displacement of FG beam decreased dramatically as the gradation index n increased from 0 to 0.5. As the gradation index increased from 0.5 to 3, the displacement increased with a small rate.

The displacement for uniform porosity model 1 is higher than that for the other two porosity models, ◊ and X distributions. Thus, O and X distributions of porosity models are more significant on the dynamic response rather than the uniform porosity distribution.

The dynamic response at $\bar{K_{s}} = 1$ is higher than that at $\bar{K_{s}} = 100$ . Thus, by increasing the stiffness of elastic boundary condition, the amplitude of vibration and time of oscillation are affected significantly.

By adding the damping to the elastic boundary condition, the amplitude of vibration is reduced significantly, and the oscillation period is increased.

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.

ORCID iD

MA Eltaher

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