Dynamic Response of FGM Plates Under Blast Load
Subject Areas : composite materialsreza azarafza 1 , puya pirali 2 , Ali Davar 3 , majid ghadimi 4
1 - Faculty of Materials and Manufacturing Technologies, Malek Ashtar University of Technology, Tehran, Iran
2 - Faculty of Materials and Manufacturing Technologies,Malek Ashtar University of Technology, Tehran, Iran
3 - Faculty of Materials and Manufacturing Technologies,Malek Ashtar University of Technology, Tehran, Iran
4 - Faculty of Materials and Manufacturing Technologies,Malek Ashtar University of Technology, Tehran, Iran
Keywords: Dynamic Response, Explosive Loading, Functional Graded Materials, Rectangular FGM Plates ,
Abstract :
The present study investigates the deformation of FGM plates under blast load. Hamilton's principle is used to obtain the dynamic Equations. The two constituent phases, ceramic and metal, vary across the wall thickness according to a prescribed power law. Boundary conditions are assumed to be Simply Supported (SS). The type of explosive loading considered is a free in-air spherical air burst and creates a spherical shock wave that travels radially outward in all directions. For the pressure time of the explosion loading, Friedlander’s exponential relation has been used. In order to determine the response analytically, the stress potential field function is considered. Using the Galerkin method, the final Equations are obtained as nonlinear and nonhomogeneous second-order differential Equations. The effect of temperature including thermal stress resultants and different parameters on the dynamic response have been investigated. Results have been compared with references and validated. Results showed that the amplitude of the center point deflection of the FGM plate is less than the pure metal plates when exposed to blast load, by increasing the volumetric index percentage of FGM, center point deflection is increased and in the FGM plates, deformation of symmetrical plates is smaller than the asymmetric plates. Also by applying the damping coefficient of the FGM plates, the amplitude of center point deflection is reduced, and by increasing the aspect ratio of the FGM plate, its center point deflection against explosion waves is reduced and by considering the effects of thermal resultant forces and moments, center point deflection is increased.
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Int. J. Advanced Design and Manufacturing Technology, 2023, Vol. 16, No. 4, pp. 37-48
DOI: 10.30486/ADMT.2021.1920150.1239 ISSN: 2252-0406 https://admt.isfahan.iau.ir
Dynamic Response of FGM Plates Under Blast Load
Reza Azarafza *, Puya Pirali, Ali Davar, Majid Ghadimi
Faculty of Materials and Manufacturing Technologies,
Malek Ashtar University of Technology, Tehran, Iran
E-mail: azarmut@mut.ac.ir, ppirali@mut.ac.ir, davar78@gmail.com, ghadimi_m2@yahoo.com
*Corresponding author
Received: 9 January 2021, Revised: 2 April 2021, Accepted: 9 April 2021
Keywords: Dynamic Response, Explosive Loading, Functional Graded Materials, Rectangular FGM Plates
Biographical notes: Reza Azarafza is an Associate Professor of Mechanical Engineering at Malek Ashtar University of Technology, Tehran, Iran. His current research focuses on composite structures, plates and shell analysis, and vibrations. Puya Pirali is an Assistant Professor at the Department of Mechanical Engineering, at Malek Ashtar University of Technology, Tehran, Iran. His current research focuses on plates and shell analysis. Ali Davar is currently an Assistant Professor at the Department of Mechanical Engineering, at Malek Ashtar University, Tehran, Iran. His current research interest includes composite structures. Majid Ghadimi is currently a PhD student at Malek Ashtar University and his main research interests are composite structures and Modal Analysis.
1 Introduction
Increasing the knowledge of scientific and industrial communities always has been a factor in improving technology and achieving superior technology. In this regard, the use of new materials to achieve specific functional properties always have been a consideration of engineering researchers. Functionally graded Material (FGM) is a kind of modern material that has been widely used in recent years. Extensive research has been conducted to produce the FGM in order to resist high temperature and thermal shock for use in the body of spacecraft and nuclear power plants. FGMs are a kind of composite materials that are heterogeneous in terms of infrastructure, and the volume fraction of its constituent material is a function of the spatial position in each body, so that, in accordance with volume fraction, the other mechanical properties are also exhibiting continually gradual changes along the thickness from one plane to another. This feature of these materials not only increases their resistance to mechanical loadings but also makes them tolerable in extreme temperature gradient environments. The common type of this material is the combination of ceramic and metal, in which case it has a metal face and the other face is ceramic, and mechanical properties change continually from metal to ceramic through the thickness [1].
Recently, more attention has been paid to the design and development of explosion-resistant structures. As in most countries, extensive studies have been done on the reaction of structures, buildings, facilities, and equipment against explosions caused by explosive materials. Therefore, considering the remarkable characteristics of FGM materials of ceramic-metal based combination, in carrying loads such as explosion and penetration, investigating and understanding the behavior and response of these materials under various dynamic loads such as the explosion wave has been attractive to the researchers.
So far, extensive research has been done to study the mechanical effects of explosion on plates made of FGM materials, which can be mentioned below. Turkmen and Mecitoglu [2] compared the results of experimental tests and numerical solutions with the finite element method for a composite plate with reinforced layers under explosive load, and the effect of reinforcements and applied load on the dynamic response of the plate has been studied. Chi and Chung [3] have studied the mechanical behaviour of an elastic rectangular plate supported on the FGM bed exposed to transversal loading with SS boundary conditions. Alibeigloo [4] has studied the three-dimensional thermoelastic analysis of FGM rectangular plates with small deformation on SS boundary conditions. In this study, the thermoelastic properties of the plate change with exponential function in the thickness direction. Tung and Duc [5] studied the nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads. Hause [6] has investigated the deflection of the functionally graded plates under the influence of the explosion theoretically. The theory of classic plates (CPT) has been used and plates have been exposed to a Friedlander exponential explosive loading. Aksoylar et al. [7] investigated the nonlinear transient analysis of FGM and FML composite plates under non-destructive explosive loads using experimental and FE methods.
Sreenivas et al. [8] studied and investigated the transient dynamic response of functionally graded materials. Goudarzi and Zamani [9] have investigated the maximum deflection of circular plates under the effect of uniform and nonuniform shock waves due to explosion by experimental and numerical analysis. Duong and Duc [10] considered the evaluation of the elastic properties and thermal expansion coefficient of composites reinforced by randomly distributed spherical particles with negative Poisson ratios. Duc et al. [11] presented the nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads. Duc et al. [12] presented the nonlinear dynamic and vibration of the S-FGM shallow spherical shells resting on elastic foundations including temperature effects. Cong et al. [13] investigated the nonlinear vibration and dynamic response of ES-FGM plates using third-order shear deformation theory (TSDT).
Hajlaoui et al. [14] presented the nonlinear dynamics analysis of FGM shell structures with a higher order shear strain enhanced solid-shell element. Tong et al. [15] studied the thermo-mechanical buckling and post-buckling of cylindrical shells with functionally graded coatings reinforced by stringers. Duc et al. [16] presented the nonlinear dynamic response of functionally graded porous plates on elastic foundations subjected to thermal and mechanical loads. Cong and Duc [17] studied analytical solutions for the nonlinear dynamic response of ES-FGM plates under blast load. Kim et al. [18] studied the nonlinear vibration and dynamic buckling of eccentrically oblique stiffened FGM plates resting on elastic foundations in a thermal environment. Duc et al. [19] presented the free vibration and nonlinear dynamic response of imperfect nanocomposite FG-CNTRC double-curved shallow shells in a thermal environment.
The main goal of this paper is to calculate the dynamic response of FGM plates under explosive load by considering the thermal forces and moments. Studying the effect of thermal forces and moments on the response of FGM plate is one of the innovations and advantages of this research that less has been studied in the previous research and the literature. Also, the effect of the damping coefficient, power law index of FGM plate, aspect ratio (L1/L2) of FGM plate, and symmetric and asymmetric FGM plate on dynamic response have been investigated.
2 GOVERNING EQUATIONs
2.1. Mechanical Properties of FGM Material
As shown in “Fig. 1ˮ, a rectangular FGM plate with dimensions L1 and, L2 and thickness h is in Cartesian coordinates, so that the origin of the coordinate system (x, y, z) is located at the middle surface of the plate.
|
Fig. 1 Geometric characteristic of FG plate. |
If Vm and Vc are the volume fractions of ceramic and metal in the FGM, respectively, then the relation of each mechanical property related to the volume fractions will be as follows [20]:
| (1) |
As Pm and Pc are, respectively, metal and ceramic properties in the FGM, therefore, with respect to the above relations, the properties of a FGM material such as modulus of elasticity, linear coefficient of expansion, shear modulus, or density can be obtained as:
| (2) |
Two cases for grading and gradual changes of ceramic and metal phases along the thickness of FG plates can be considered:
Symmetrical case: The ceramic and metal phase elements are changed symmetrically in the thickness direction of plate, so that both outer surfaces of the plate are completely ceramic and the middle surface of the plate is full metal. In this case, the volume fractions of ceramic VC(z) is expressed in the form of Equation (3) [6]:
| (3) |
In the above Equation, N is the volumetric percentage index of the material FGM, h is the thickness of the plate and z is the coordinates perpendicular to the middle surface along the thickness direction.
Asymmetrical case: Ceramic and metal phase elements change asymmetrically in the thickness direction of the plate, the upper surface of the plate is full ceramic and the bottom surface of the plate is full metal. In this case, as in “Fig. 1ˮ, the bottom of the plate is completely metallic and its upper surface is completely ceramic and between these two surfaces will be a combination of ceramic and metal. In this case, the volume fraction of ceramic VC(z) is expressed in the form of Equation (4) [6]:
| (4) |
Regarding the function of FGM materials in high-temperature environments, according to Reddy [1], the mechanical properties of its constituents have significant changes with temperature. Therefore, according to Equation 6, properties such as the Ef modulus of elasticity, the Poisson ratio , the thermal expansion coefficient , and the thermal conductivity coefficient Kf could be related to the temperature [1].
| (5) |
2.2.The Fundamental Equation of FGM Plate
2.2.1. Displacement components
According to the classical theory of plates, the displacement field is as follows [21]:
| (6) |
In the above relations u0 and v0, respectively, represent the displacements of the middle surface in thex and y directions, and w0 is the transverse displacement along the z direction. Also, the functions ψx and ψy are rotations of the middle surface around x and y axes, respectively, and are as follows:
| (7) |
2.2.2. Nonlinear strain- displacement relations
The nonlinear relations of Von-Karman between strain and displacement at any point in the thickness of the plate at distance z from the middle surface according to the strains and curvatures of the middle surface are as follows [22]:
| (8) |
In the above relations , and , are the strain of the middle surface and κx, κy and κxy are the curvatures of the middle surface, which are related to the displacement components u, v, w as follows:
| (9) |
According to Hooke's law, the stress-strain relations are defined by the following Equation:
| (10) |
In the above relations, which include mechanical and thermal strains, α is the thermal expansion coefficient and Qij are the elements of the stiffness matrix, which are functions of the plate thickness and temperature; their relations are as follows:
| (11) |
2.2.3. The Force and moment resultants
The vector of forces N and moments M caused by stresses in unit length are expressed in terms of strain components as follows [21]:
| (12) |
By replacing stress relations from the above Equations, we can find the forces and moment resultants in terms of the following matrix strain:
| (13) |
In the above relations NT and MT, are the thermal forces and moments resultants, matrices A, B, and D are the extensional, coupling, and bending stiffness matrices, respectively, and they are as follows:
| (14) |
In the above relations, the values of E1, E2, and E3 are based on Young's modulus Em, Ecm, the thickness of plate (h), and volumetric percentage index (N) for two cases of FGM plate (symmetric and asymmetric) as follows [5-6]:
Asymmetric
| (15) |
Symmetric
| (16) |
Therefore, by replacing the stresses (Equation. (10)) and the strains (Equation (8)) and the effective Young's modulus in Equations (12), the resultant forces and moments are obtained as follows:
| (17) |
| (18) |
Also, the thermal force and moment resultants (NT and MT) are defined as follows:
| (19) |
Where, DT is the increase of temperature relative to the reference temperature T0 (without thermal strain) is as follows:
| (20)
|
2.3. Equations of Motion of the Plate
The dynamic Equations of motion of the plate and its boundary conditions are extracted from Hamilton's principle, which are defined as four boundary conditions at the edges of the plate and three equilibrium Equations as follows [1], [6]:
| (21) |
Where, P (x, y, t) is the distribution of external force on the upper surface of the plate (z = + h / 2) and μ is the coefficient of viscous damping per unit area of the plate. Also, I0 is the inertial of the plate in the z-direction, and is defined as follows:
| (22) |
Where is the density of the plate.
2.4. Boundary Condition
The boundary condition of the plate is defined as follows:
At x=0 and x= L1:
| (23) |
At y=0 and y= L2:
| (24) |
2.5. Extraction of The Equation of FGM Plate Deflection
In order to establish the first two Equations of motion (Equations (21)), the function of the potential stress field f is considered as follows [6]:
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