Abstract
Based on Mindlin plate theory and finite element method (FEM), dynamic response analysis of sandwich composite plates with auxetic honeycomb core resting on the elastic foundation (EF) under moving oscillator load is investigated in this work. Moving oscillator load includes spring-elastic k and damper c. The EF with two coefficients was modelled by Winkler and Pasternak. The system of equations of motion of the sandwich composite plate can be solved by Newmark’s direct integration method. The reliability of the present method is verified through comparison with the results other methods available in the literature. In addition, the effects of structural parameters, material properties, and moving oscillator loads to the dynamic response of the auxetic honeycomb plate are studied.
1. Introduction
The composite auxetic material with negative Poisson’s ratio is an advanced material with many outstanding benefits due to the excellent performance of ultra-light, higher stiffness, strength-to-weight ratio, and well energy absorption capacity from loads, especially the types of loads that make the structures vibrate as large as blast loads, moving loads, and impact loads, etc. Therefore, it has been used in many areas including civil and defence. With the advantageous features in many practical applications, this material has been extensively interested in the word by scientists with many different methods, including the field of mechanical calculation.
Considering material with negative Poisson’s ratio, Almgren [1] presented an isotropic three-dimensional structure with equal Poisson’s ratio. Whitty et al. [2] designed sandwich composite plates with increased mechanical and thermal properties by the effect of Poisson ratios in in-plane. In [3], Ruzzene and coworkers investigated the wave propagation in sandwich structures with the cyclical auxetic core layer. Jensen and his colleagues [4] survey the effects of Poisson’s ratio on the membrane deformation of the structures under indentation. Qian-Tian and Zhi-Chun [5] studied the wave propagation in sandwich composite plates with the auxetic core. Hadjigeorgiou and Stavroulakis [6] proposed the use of auxetic materials in advanced structures. In [7–9], Teik-Cheng considered the shear deformation, auxetic foundation in plates, and thermal stresses in both plates and shell structures. Prawoto [10] studied and reviewed the structure with negative Poisson’s ratio. Strek and coworkers [11] analyzed the response of dynamic sandwich plates with auxetic cores. Scarpa and Tomlinson [12] considered the theoretical characteristics of the vibration of sandwich composite plates with in-plane negative Poisson’s ratio values. Yang and his colleagues [13] used the FEM to study the geometric effects on the micropolar elastic honeycomb structure. Imbalzano et al. [14] studied the three-dimensional modelling of auxetic sandwich composite plates for localized impact resistance and considered sandwich composite plates with auxetic core layer under blast loadings in Ref [15]. Huang and coworkers [16] investigated the in-plane mechanics of a novel zero Poisson’s ratio honeycomb core. Jin et al. [17] examined the effects of size on the mechanical behavior of hexagonal honeycombs in in-plane. Arbaoui and coworkers [18] analyzed the static bending of multilayer sandwich structures by numerical simulation and experimental methods. Yongcun and his colleagues [19] presented the buckling of a rectangular plate with auxetic material under uniaxial compression. Xu and Deng [20] considered the wave propagation characteristics in thick conventional and auxetic cellular plates. Recently, Duc and his colleagues have a relatively complete research on the mechanical behavior of the sandwich composite structures with negative Poisson’s ratio in the auxetic honeycomb core layer such as investigating dynamic nonlinear response and vibration of sandwich composite plates with auxetic honeycomb core layer by the analytical method [21]; static bending analysis of plate using FEM [22]; nonlinear forced and free vibration response of sandwich composite cylindrical panels [23]; the geometrically nonlinear dynamic response of eccentrically stiffened circular cylindrical shells [24]; double curved shallow shells using TSDT [25]; and evaluation of mechanical properties of composites reinforced by randomly distributed spherical particles with negative Poisson’s ratios [26].
In the case of structures on EF, Zenkour and Radwan [27] have recently proposed an exact analytical approach to free vibration analysis of laminated composite and sandwich plates resting on EF using a plate theory with four unknown variables. Duc et al. [28] presented nonlinear thermal dynamic response of shear deformable functionally graded (FG) plates resting on EF. In [29], he and coworkers analyzed the dynamic response of composite double curved shallow shells with auxetic honeycomb core layer resting on EF under blast and damping loads. In his book [30], the results of the nonlinear dynamic analysis of FGM structures on EF are presented. Mahmoudi et al. [31] developed a refined quasi-three-dimensional shear deformation theory to analyze for FG sandwich plates resting on the two-parameter EF under thermomechanical loading.
Moving load is one of the most common types of load that affects the structures in practice. Besides, the influence of this load on the dynamic response of the system is very significant. Thus, the structures under the action of the moving load are studied by many scientists around the world. Other typical works such as Song et al. [32] presented the oscillation of FG plates with a moving volume, using Kirchhoff's plate theory and analytical method based on Rayleigh–Ritz’s solution and penalty function. Malekzadeh and coworkers [33] analyzed the dynamic of laminated composite plates subjected to moving load based on the analytical method. Ouyang [34] briefly reviewed a variety of moving-load problems and several analytical solution methods. Nguyen and coworkers [35] programed triple-layer composite plates with layers connected by shear connectors subjected to moving load by FEM. Song et al. [36] investigated the dynamic of rectangular thin plates of arbitrary boundary conditions under moving loads using the analytical method. Fryba [37] has summarized a variety of engineering problems that analyzed the dynamics of structures under moving loads. The above literature reviews indicate mainly the analysis of free vibration and forced vibration when moving load or moving mass is on a straight line parallel to an edge of the plate. The research results on plate structure are influenced by the moving load with oscillation system which takes into account the interaction between structure and load as well as the investigation of any motion orbit of the load is not much.
According to the best of authors’ knowledge, the researchers only focus on investigating static bending, free vibration, and the dynamics of the sandwich structures with Poisson’s ratio in auxetic honeycomb layer. Nevertheless, the dynamic response of the sandwich composite plate with honeycomb core layer subject to moving oscillator load has not been published yet. In this article, the dynamic response of plates with the auxetic honeycomb core layer subjected to moving oscillator load resting on EF is investigated numerically. The results of the present work may be applied for computation in many civil engineering problems such as vehicles running on roads, the rails resting on the EF subject to moving oscillator loads, and aeroplanes moving on runways. The sandwich composite plates used in the present study have three layers in which the top and bottom layers are isotropic materials; the core layer is auxetic honeycomb using the same material. In the present formulation, the first-order shear deformation theory (FSDT) combining with FEM is used to build the equation of motion for the sandwich auxetic honeycomb plate and to validate the results of the proposed equations with published works to verify accuracy and reliability. Finally, the influence of geometric parameters, material, load, etc., on the dynamic response of the referred plate under the moving oscillator load is investigated.
2. Computational Model and Assumptions
Consider plates have the length a, width b, and thickness h. The plate is made from materials with negative Poisson’s ratio in auxetic honeycomb subjected to the moving oscillator load consisting of two masses m1 and m2, and they are connected with spring-elastic k and viscous bonds c as shown in Figure 1.
The moving load moves with any orbit depending on (x (t), y (t)) which is the position of mass m1.
The dynamic response of the plate consists of displacement, velocity, acceleration, and stress at any point in the plate. The sandwich composite plate resting on EF is shown in Figure 1. The reaction-deflection relation of Winkler–Pasternak foundation is given by [21, 27]where and is the displacement of the sandwich composite plate with honeycomb core layer; K1 and K2 are, respectively, Winkler foundation stiffness and shear layer stiffness of the Pasternak foundation.
3. Material Properties of the Auxetic Honeycomb Core Layer
This paper researches the sandwich composite plate with the auxetic honeycomb core layer with negative Poisson’s ratio. Unit cells of auxetic honeycomb core layer have geometric parameters as [5, 21] shown in Figure 2, in which l is the length of the inclined cell rib, h is the length of the vertical cell rib, is the inclined angle, and and define the relative cell wall length and the wall’s slenderness ratio, respectively, which are important parameters in honeycomb property. The material property of the honeycomb core layer structures is given by [5, 21]in which “c” represents core material, and E, G, and are Young’s moduli, shear moduli, and mass density of the material.
From Figure 3, it can be observed that when the angle turns to be negative values gradually, Poisson’s ratio also comes with negative values. In the range from to , when , value is equal to zero. However, in the case of and h/l = 0.5, value is undefined, and therefore, this line is discontinuous.
4. Finite Element Model for Dynamic Analysis of Sandwich Composite Plates with Negative Poisson’s Ratio in the Auxetic Honeycomb Core Layer
4.1. The Equation of Motion for Plate Element
The displacement field of Reddy’s FSDT is of the following form [38]:where are the displacements at midplane (z = 0) and are the transverse normal rotations of the y- and x-axes.
The deformed components due to bending at the point of coordinates are given bywhere
The deformed components due to shear at points with coordinates are given by
Stress due to bending deformation at the point of coordinates is as follows:in which
The components in equation (9) are determined by the following equation:
Stress due to shear strain at a point is given by
The components in equation (12) are determined as follows:
In the present study, an eight-node isoparametric element with five degrees of freedom (DOFs) for each node is employed. The DOFs of the ith node and plate element are defined as follows:where Ni (i = 1,…, 8) is the shape function and specified in Appendix A.
The equation of motion for a plate element can be derived from the virtual work principle as follows:in which and are the area and volume of the element, respectively.
By substituting equations (5)–(8) and (15) into equation (17), the equation of motion according to the finite element formulation for the plate element is obtained:where the stiffness matrix is given bywhere is the shear correction factor. The gradient strain matrices in equation (19) are determined by the following formulas:Note that , , and are shown in Appendix B.
Material stiffness matrices are determined by the following formulas:
Stiffness matrices of elastic foundation:(i)For Galerkin foundation:(i)For Pasternak foundation:where is shown in Appendix C.
Mass matrix of the plate element:in whichwith is the unit matrix rank 5 × 5, and
The load vector in the case of distributed uniform load p (x, y) on the surface of the element is given byin which
The stiffness matrices, mass matrices, and nodal load vectors of the element are calculated by using the Gaussian integral method.
4.2. Formulation of the Moving Oscillator Load
Consider the plate element subjected to the moving oscillator load as shown in Figure 4. At time t, the moving oscillator load is located on an element at the coordinate .
The interaction force caused by the moving load on the plate element is written as follows:where represents the displacement along the z-axis of mass and
Rewrite the equation in brief form:in which the matrices are defined as follows:
Using delta Dirac function, force is distributed by the following formula:
4.3. The Equation of Motion of the Plate under Moving Oscillator Load
The differential equations of motion of the plate element on EF can be written as follows:
In order to find the moving oscillator load , substituting equation (31) into equation (29) and then into equation (34), the following equation is obtained:in which receives the value at the coordinate which is the position of the moving load.
By substituting equation (36) into equation (35), we obtained the following equation:in which and are Rayleigh drag coefficients defined in [37].
The differential equations of mass m1 on moving oscillator load are as follows:with
By substituting equation (39) into equation (38), we obtain
By combining equation (37) and equation (40), we have the system of differential equations of motion of the plate element and the mass m2 of the moving load is as follows:where
The system of differential equations of motion of the whole system is obtained after gathering matrices and element load vectors into the matrix and the vector global according to the general algorithm of FEM. Note that when there is a moving load with oscillation system on the plate, the DOFs of the system increase by one, and the extension of element matrices when there are moving loads in equation (41) is added. Rows and columns are extended in the global matrix according to the corresponding total DOFs. They are linear differential equations, which have the coefficient depending on time. To solve these equations, we use the Newmark-beta method [37]:Step 1:discrete the mesh and determine the element matrixStep 2:determine the global matrix when there is no moving load by assembling element matricesStep 3:solve a separate oscillation problem and determine natural frequenciesStep 4:in each integral time step, determine the element matrix depending on the moving load, assembled into the global matrix (similar to Step 2)Step 5:solve the system of equations in each time step and determineStep 6:in each time step, determine stresses and internal forcesStep 7:export the dynamic response results of the system
Flowchart of the Newmark-beta method is shown in [39, 40].
5. Numerical Results
5.1. Accuracy Study
Example 1. Consider a rectangular plate with the simply supported-free-simply supported-free (SFSF—two short edges are simply supported and two long edges are free) boundary condition (BC). The plate has length a = 1 m, width b = a/2, thickness h = a/100, Young’s modulus E = 206.8 GPa, Poisson’s ratio , mass density ρ = 7820 kg/m3 under moving load with mass m = 2.3 kg along y = b/2 and = 10 m/s as in [36]. The authors use the established program with the different cases of meshes: , , , , , to examine the displacement of the central point of the plate versus time. The numerical results with different meshes are compared with analytical solutions as shown in Table 1. From this, it is seen that using the mesh , the present results are in good agreement with the previous literature [36] in both the shape and magnitude as presented in Figure 5. This proved the reliability and the convergence of our finite element formulation.
Example 2. To be more reliable, we consider a fully simple-support (SSSS) FGM plate with geometry parameters: the length a = 1 m, the width b = a, and the thickness h = a/20. The FGM plate is made of ceramic (ZrO2) and metal (Al) with material characteristics as shown in Table 2 and volume factor ratio k = 1. The nondimensional free vibration is computed as the following formula . We use uniform mesh . The numerical results of the present method are compared with the analytical solutions [41] in Table 3.
From Table 3, it can be observed that the results of this work are in good agreement with the results of [41], and the error of the first four frequencies is less than 2%.
5.2. Free Vibration of Sandwich Composite Plates with Negative Poisson’s Ratio in Auxetic Honeycomb Core Layer
In this section, an SSSS sandwich composite plate with three layers, the top layer, the bottom layer, and the auxetic honeycomb core layer, are considered. The top layer (thickness h1) and bottom layer (thickness h3) are isotropic aluminium material with geometrical parameters and materials properties: h1 = h3, E = 69 GPa, G = 26 GPa, , and kg/m3. The core layer is the same material of other layers; geometrical parameters of the core layer are h2, , and ratio h/l and t/l. The plate rests on EF with Winkler foundation stiffness K1 (GPa/m) and shear stiffness of Pasternak K2 (GPa·m). Geometrical parameters of the sandwich composite plate are thickness h = h1 + h2 + h3, length a, and width b = a.
The first natural frequencies with different values of the EF parameter and geometrical parameters of the core layer are presented in Table 4. It can be seen that the influence of geometrical parameters of the core layer on free vibration of the sandwich composite plates with the auxetic honeycomb core layer does not follow any rule. Note that when the EF parameters increase, natural frequencies of plate increase. Furthermore, numerical results show the EF parameters make the plate become stiffener and Pasternak’s EF (K2) is more effective than Winkler’s foundation (K1). For example, with , h/l = 0.5 when K1 = 0.1 GPa/m and K2 = 0, natural frequency f1 = 224.5746 Hz, and if K1 = 0 and K2 = 0.05 GPa·m, natural frequency f1 = 284.8829 Hz. The first six mode shapes of the sandwich composite plate with honeycomb core layer are shown in Figure 6. In the case of the SSSS plate, the second and the third mode shapes are similar to each other. This phenomenon is suitable for the symmetrical plate structures under the same BC in the practice.
5.3. Numerical Results of the Dynamic Analysis of Sandwich Composite Plates with Negative Poisson’s Ratio in Auxetic Honeycomb Core Layer
In this section, the authors consider an SFSF sandwich composite plate in which the core layer is auxetic honeycomb as Section 5.2. All layers have the same material E = 69 GPa, G = 26 GPa, , and kg/m3, the geometrical parameters of the core layer are h2, , and ratio h/l and t/l. Geometrical parameters of the sandwich composite plate are thickness h = h1 + h2 + h3 in which h1 and h3 are the thickness of the top and bottom layers, respectively, length a, and width b = a/2. The plate rests on EF with two parameters K1 and K2. The moving oscillator load (see Figure 1) with parameters m1, m2, k, and c travels on the plate with any orbit with velocity and acceleration a.
5.3.1. Influence of the Ratio h2/h1
Firstly, let us consider a rectangular plate with h = 0.1 m, a = 150h, , h2/l = 2, t/l = 0.1385, and K1 = K2 = 0. The moving oscillator load with m1 = m2 = 100 kg, k = 1.6 MN/m, and c = 2500 Ns/m moves along y = b/2 with velocity . The ratio h2/h1 = 0, 1, 1.5, 2, and 2.5 (h2/h1 = 0 is case of isotropic plate). The deflection, velocity, and stress of the plate central point are shown in Figure 7, and maximum deflections, velocities, and stress are illustrated in Table 5. Besides, the weight reduction of the sandwich composite plate with honeycomb core layer compared to the isotropic plate is also shown in this table.
(a)
(b)
(c)
From Figure 7 and Table 5, it can be observed that when the ratio hc/ht is increased from 0 to 2.5 (i.e., increasing the thickness of the middle layer since the plate thickness h is fixed), the deflection and the velocity of the central point of the plate are almost not changed. It is clear that the sandwich composite plates with honeycomb core layer help significantly absorb the vibration of structures and thus make the central deflection and velocity of plate reduce. In the case of the ratio h2/h1, it can be found that the mass of plate decreases significantly. Specifically, when h2/h1 = 2.5, the mass of the plate has decreased by 45% compared to the uniform plate. This is an impressive result. Therefore, to reduce the weight of the structure, we suggest using sandwich composite plates with honeycomb core layer instead of the anisotropic plates.
5.3.2. Influence of the Velocity
Secondly, we consider an SFSF rectangular plate with h = 0.1 m, a = 150h, h/l = 2, t/l = 0.1385, K1 = K2 = 0, and h2/h1 = 1.5. The moving oscillator load with m1 = m2 = 100 kg, k = 1.6 MN/m, and c = 2500 Ns/m travels along y = b/2 with velocity = 5, 10, 15, 20, and 25 m/s and acceleration a = 0. The deflection, velocity, and stress of the plate central point are shown in Figure 8, and maximum deflections, velocities, and stress are illustrated in Table 6.
(a)
(b)
(c)
With material properties and geometrical parameters as in Section 5.3.1, the change in velocities is from 5 to 25 m/s. The response of the displacement, velocity, and stress of the plate has very little change. Once again, it can be seen that the advantage of honeycomb sandwich composite plates is that it has a very good absorption capacity.
5.3.3. Influence of the Acceleration a
Next, let us consider a SFSF rectangular plate with h = 0.1 m, a = 150h, h/l = 2, t/l = 0.1385, K1 = K2 = 0, and h2/h1 = 1.5. The moving oscillator load with m1 = m2 = 100 kg, k = 1.6 MN/m, and c = 2500 Ns/m travels along y = b/2 with velocity = 15 m/s and various accelerations a = 0, 5, 10, and 12.5 m/s2. The deflection, velocity, and stress of the plate central point are shown in Figure 9, and maximum deflections, velocities, and stress are illustrated in Table 7.
(a)
(b)
(c)
From Figure 9 and Table 7, it can be observed that when the acceleration of the moving load is increased from 0 to 12.5 m/s2, deflection and velocity of the plate central point slightly increase. This proves that the acceleration of the moving load does not affect much the dynamic response of the sandwich composite plate with a honeycomb core layer.
5.3.4. Influence of K1
Now, we consider a SFSF rectangular plate with h = 0.1 m, a = 150h, h/l = 2, t/l = 0.1385, and h2/h1 = 1.5, Winkler foundation stiffness K1 = 0, 0.1, 0.2, 0.3, 0.4 MPa/m, and K2 = 0. The moving oscillator load with m1 = m2 = 100 kg, k = 1.6 MN/m, and c = 2500 Ns/m moves along y = b/2 with velocity = 15 m/s and acceleration a = 0. The deflection, velocity, and stress of the plate central point are shown in Figure 10, and maximum deflections, velocities, and stress are illustrated in Table 8.
(a)
(b)
(c)
Figure 10 and Table 8 show the effects of Winkler foundation stiffness on the dynamic response of the sandwich plates with negative Poisson’s ratio. From these results, it can be seen that the EF makes the vibration amplitude of sandwich composite plates with negative Poisson’s ratio reduce strongly. Furthermore, note that the EF according to Winkler's model significantly influences the dynamic response of the sandwich composite plate.
5.3.5. Influence of K2
Finally, we investigate a SFSF rectangular plate with h = 0.1 m, a = 150h, h/l = 2, t/l = 0.1385, K1 = 0, and shear layer stiffness of Pasternak foundation K2 = 0, 0.05, 0.10, 0.15, 0.20 MPa·m. The moving oscillator load with m1 = m2 = 100 kg, k = 1.6 N/m, and c = 2500 Ns/m moves along y = b/2 with velocity = 15 m/s and acceleration a = 0. The ratio h2/h1 = 1.5. The deflection, velocity, and stress of the plate central point are shown in Figure 11, and maximum deflections, velocities, and stress are shown in Table 9.
(a)
(b)
(c)
Figure 11 and Table 9 show the effects of the Pasternak foundation on the dynamic response of the sandwich composite plates with the honeycomb core layer. When K2 increases, the response of deflections, velocities, and stress of the plate central point reduces and also it has a significant effect on the dynamic response of structures.
6. Conclusion
In the paper, the dynamic response of the sandwich composite plates with negative Poisson’s ratio under moving oscillator load on EF is mainly studied. Reddy’s first-order shear deformation plate theory is used to form the basic equations. By using FEM, the equation system of motion of the sandwich composite plates subjected to moving load is solved. From the numerical results, we can withdraw some of the following points:(i)Authors built a numerical method to analyze the dynamical response of sandwich composite plates with honeycomb core layer under moving oscillator load on EF.(ii)This investigation further clarifies the superiority of sandwich composite plates with negative Poisson’s ratio in auxetic honeycomb core layer under moving oscillator load. It can be noted that using sandwich auxetic honeycomb plates significantly reduces the mass and increases the vibration absorbability of the structures.(iii)This study also analyzes and discusses the effects of material and geometrical properties and EF on the dynamic response of sandwich composite plates with negative Poisson’s ratio in auxetic honeycomb core layer.(iv)The proposed method can analyze the static bending, dynamic response, and nonlinear problems of structures with the different honeycomb core layers, which can be not easily solved by analytical methods.
Appendix
A. Shapes Function in Equation (16)
Shape function in equation (16) is given bywhere are natural coordinates.
B. The Matrices in Equation (20)
The matrices in equation (20) are given bywith i = 1,…, 8 (number nodes per element).
C. The Matric in Equation (23)
The matrix in equation (23) is shown as follows:
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 107.02-2019.330.