LUẬN văn THẠC sĩ effects of porosity on free vibration and nonlinear dynamic response of multi layered functionally graded materials subjected to blast loads

INTRODUCTION

Overview

1.1.1 Composite material – Functionally Graded Materials

Composite material is a material composed of two or more different types of component materials in order to achieve superior properties such as light weight, high stiffness and strength, ability of heat resistance and chemical corrosion resistance, good soundproofing, thus it plays a crucial role in advanced industries in the world that are extensively applied across wide range of fields such as: aviation, aerospace, mechanics, construction, automotive [1] [2] However, this material has a defect as a sudden change of material properties at the junction between the layers is likely to generate large contact stresses at this surface One of the solutions to overcome this disadvantage of layered composite material is to use Functionally Graded Material (FGM) which is a material made up of two main component materials as ceramic and metal, in which the volume ratio of each component varies smoothly and continuously from one side to the other according to the thickness of the structure so the functional materials avoid the common disadvantages in composite types such as the detachment between layers material, fibers breakage and high stress in the surface, which can cause material destruction and reduce the efficiency of the structure, especially in heat-resistant structures Due to the high modulus of elasticity E , the thermal conduction coefficient K and the very low coefficient of thermal expansion

 , the ceramic composition makes the material highly variable with high hardness and very good heat resistance While the metal components make the modified materials more flexible, more durable and overcome the cracks that may occur due to the brittleness of ceramic materials when subjected to high temperature (Table 1.1)

Table 1.1 Properties of component materials of FGM material [3]

Depending on the power law of the volume ratio of component materials, we can classify different types of FGM Each of these FGM materials is characterized by different mechanical and physical properties by a function that determines the material properties (effective properties), and the value of the function varies with thickness Mathematical functions of material properties used to classify materials

[4] Specifically, there are three main types of FGM

A power-law distribution P-FGM: is a type of material having a volume fractions of ceramic and metal components which is assumed to vary according to thickness of structure and conforming to the power-law function [5, 6]:

V V : the volume fractions of metal and ceramic, respectively

The effective properties P eff of the P- FGMs are established using the modified mixed rules as follows [7]: e ff ( ) Pr c c ( ) Pr m m ( ),

P z  V z  V z (1.2) in which Pr denotes a specific property of the material such as elastic modulus E , thermal expansion coefficient  or density , thermal conduction K

Sigmoid-law distribution S-FGM: is a type of material having a volume fractions of ceramic and metal components which is assumed to vary according to thickness of structure and conforming to the Sigmoid-law function as:

The effective properties P eff of the S- FGMs are established using the modified mixed rules as follows:

An exponential-law distribution E-FGM: is a type of material having a volume fractions of ceramic and metal components which is assumed to vary according to thickness of structure and conforming to the exponential-law function as:

E t is the elastic modulus of structure on the top  z  h / 2 

E b is the elastic modulus of structure on the bottom  z   h / 2 

FGM sandwich material: The multi-layered sandwich structure is a particularly important type of structure in the aerospace industry as well as in a number of other industries such as ships, automobiles, construction Sandwich structure consists of 3 main layers: core layer and two face-sheets The core layer is made of lightweight material, low hardness between two face-sheets made of very high hardness material The great advantage of sandwich structure is that it increases the stiffness and bending resistance of the structure while ensuring a small volume, because the core layer is made of light material that can be made with a large thickness that will have an effect to transfer the two face-sheets away from the neutral axis

To avoid the phenomenon of flaking between the layers as well as the phenomenon of stress caused as with conventional multi-layer structures, it was thought that FGM sandwich material with ceramic or metal core layer and two face - sheets made of FGM material a Sandwich FGM- Metal-FGM b Sandwich FGM- Metal-FGM

Fig.1.1 The distribution types of FGM sandwich material

The effective properties of this materials vary according to the extended Sigmoid distribution law as follows:

(1.7) with the case of type 1-1a: ic j, m,and the case of type 1-1b: im j, c.

Blast load: In recent years, the safety of important buildings and infrastructure around the globe has become more fragile by extreme dynamic loads due to the increase in terrorist activities, explosions The damage from such events cannot be determined, not just economically because many of these ones are symbolic and important heritage, significant architectures and the spirit of the times Nowadays, considerable efforts in architecture and structural engineering in recent years are often focused towards optimal design and economic efficiency in construction It is essential to guarantee the safe and secure protection of important infrastructure for the present and future

Explosion loads usually act in a very short time (usually in milliseconds) but transmit very high pressure pulses (10 1  10 3 kPa) As a result, damage to structural systems can take many forms, such as damage to the outer surface and structural frame of a building; collapse of walls and bearing columns; blow debris of concrete, glass windows and furniture; and damaging safety systems Most existing buildings are not designed to withstand such extreme dynamic loads, so a comprehensive understanding of the explosion phenomena and the dynamic response of structures is required to be essential for the scientific basis improving the design and material improvement in a feasible manner, in order to improve explosion resistance and ensure the safety of structures Therefore, the investigation of the effects of explosive loading on structures should be focused.

Research objectives

The research objective of this thesis is to investigate the effects of porosity on vibration and nonlinear dynamic response of multi-layered FGM subjected to blast load

 Studies the effects of porosity to FGM sandwich plates and comparison with different cases: FGM plates without porosity, porous -I FGM plates and porous-II FGM plates

 Investigations on nonlinear dynamic analysis on the structure in FGM plates on elastic foundation subjected to blast load In numerical results, the effects of the material properties, geometrical parameters, blast load… on the nonlinear dynamic response will be analyzed.

The layout of the thesis

The thesis includes an introduction, five chapters, conclusions, references and appendices The main contents of the chapters involve:

The thesis presents an overview of FGM materials Porosities are also mentioned in this chapter

Chapter 2 presents some studies which have been reported to this thesis’s field In those publications, I also pointed out their main outstanding results obtained from their research as well as those research’s limitation

Chapter 3 introduces the analytical method by using high-order shear deformation to approach and solve problems…

 Chapter 4: Numerical results and discussion

The numerical results are presented in this chapter for a FGM sandwich plate on elastic foundation in terms of natural frequencies, effects of geometrical parameters, materials properties on nonlinear dynamic response

Chapter 5 summarize the main results obtained from this thesis.

LITERATURE REVIEW

Structures

Because of their remarkable properties, in recent years, sandwich FGM structure has been attracted a lot of attention of scientists Among those, Zhaobo Chen et.al [8] presented the free vibration of the functionally graded material sandwich doubly-curved shallow shells under simply supported conditions according to a new shear deformation theory with stretching effects The wave propagation of FGM sandwich plates with porosities putting on viscoelastic foundation was studied by Chen Liang and Yan Qing Wang [9] based on a quasi-3D trigonometric shear deformation theory Singh and his co-authors [10] used a semi-analytical approach to analyze thermo-mechanical of porous sandwich S-FGM plate for different boundary conditions using Galerkin Vlasov's method Furthermore, the nonlinear vibration of imperfect sandwich plates with FGM face sheets also investigated by Kitipornchaii et al [11] basing on a semi-analytical approach Hoang Van Tung [12] are analyzed nonlinear bending and post buckling behavior of FGM sandwich plates under thermomechanical loading by using the first order shear deformation theory The effect of time constant, temperature, mid radius to thickness ratio and time on transient thermo-elastic behavior of sandwich plate with the core as FGM are taken into consideration by Alibeigloo [13] In his analysis, the sandwich plate’s time dependent response is built from generalized coupled thermo-elasticity when applying the Lord-Shulman expression Moreover, Xia and Shen [14] introduced an analytical using higher-order shear deformation and a general von Kármán-type function to obtain small- and large-amplitude vibration of compressive and thermal post-buckling sandwich plates with FGM face sheets under uniform and non-uniform temperature fields Behzad Mohammadzadeh [15] combined higher-order shear deformation with Hamilton’s principle to analyze nonlinear dynamic responses of sandwich plates with FGM faces on elastic foundation subjected to blast loads

Basing on a new four-variable shear deformation plate theory, Mohammed Sobhy

[16] evaluated the hydrothermal vibration and buckling of various types of FGM sandwich plates resting on elastic foundations exposed moisture condition, rising temperature, Winkler–Pasternak foundation coefficients and power- law distribution index Chien et al [17] used isogeometric approach to investigate static, free vibration and buckling analysis of FGM isotropic and sandwich plates Tao Fu et al [18] adopted the space harmonic approach and virtual work principle to describe analytically sound loss when transmitting through two types of porous FGM sandwich structures.

Porosity

The effects of porosities generated during actual manufacturing process to the vibration characteristics of FGM structures have been studied by several author s

However, the number of researches in terms of the mechanical behaviors of porous materials is still limited The most recent investigations on structures with porosity are listed in the following

Ashraf M.Zenkour [19] used a quasi-3D shear deformation theory to investigate the bending responses of porous functionally graded single-layered and multi-layered thick rectangular plates By taking Galerkin Vlasov's method into account in thermo-mechanical analysis of sandwich S-FGM plate with three different types of porosity for diverse boundary conditions, Singha and co-author [20] obtained the approach for bending and stress under the thermal environment They deduced that the deflection and stress escalate significantly for even porosity distribution (P-1) to bring into comparison with uneven symmetric (P-2) or uneven non-symmetric (P-3) porosity distribution; and the effects of temperature on transverse shear stresses of the multi-layered plates Mojahedin [21] employed higher order shear deformation theory to investigate the buckling of functionally graded porous circular plates Polat et al [22] utilized an atmospheric plasma spray system to obtain functionally gradient coatings from five layers which were prepared on Ni substrates from Y2O3 stabilized ZrO2 (YSZ) and NiCoCrAlY powders In their research, they found that an escalation in porosity ratio of layers lead to the decrease of residual stresses Chen et al [23] employed Chebyshev-Ritz method to analyze buckling and bending loads of a novel functionally graded porous plates Wang et al

[24] focused on effects of parameters on the vibrations of functionally graded material rectangular plates with two types of porosity, namely, even and uneven distributed porosity, and transferring in thermal environment Based on a sinusoidal shear deformation theory in combination with the Rayleigh–Ritz method, Yuewe Wang et al [25] depicted the effects of porosity, boundary conditions, and geometrical parameters on free vibration of the functionally graded porous cylindrical shell An isogeometric finite element model and the nonlocal elasticity were introduced by Phung-Van et al [26] to investigate the transient responses of functionally graded nanoplates with porosity Small size effects, nonlocal parameters, and porosity distributions, volume index, the characteristics of dynamic load have considerably influenced on the plate nonlinear transient deflections Cong et al [27] acquired closed-form expression in regard to critical bucking loads and post-buckling paths of a porous functionally graded plates on elastic foundations subjected to the coupling of mechanical and thermal loads by applying Reddy's higher-order shear deformation plate theory in conjunction Galerkin method Analytical solutions and numerical results revealed that porosity I (evenly distribution) behaves better than porosity II (unevenly distribution) according to the static buckling investigations Chien et al

[28] adopted the first-order shear deformation theory taking the out-of-plane shear deformation into account to calculate the fundamental frequencies and nonlinear dynamic responses of porous functionally graded sandwich shells with double curvature under the influence of thermomechanical loads This study proved that porosities help the shell structures stiffen to some extent.

Blast load

In recent years, explosive loads and their impacts on the safety and efficiency of building and structures have received considerable attention Tuan et al [29] presented the results of an empirical investigation conducted in Woomera, Southern

Australia, in May 2004 on the explosion-resistance of concrete-panel created by ultrahigh-strength concrete material A finite-element method was used to analyze concrete structures under blast and impact loading In the study conducted by Tin and co-authors [30] , they proposed using the explicit finite element software LS-DYNA to induce stress wave propagation and the impacts on structural responses of precast concrete segmental columns subjected to simulated blast loads Balkan et al [31] examined the effects of sandwich stiffeners on the dynamic response of laminated composite plates under the non-uniform blast loading Moreover, the dynamic behavior of stiffened plates exposed to confined blast loads are carried out by Zhao et al [32] through experimental and numerical studies Geretto et al [33] analyzed a series of experiments of square monolithic steel plates to assess the effects of the degrees of confinement of the deformation to blast loads Asoylar et al [34] studied the transient stability analysis metal-fiber laminated composite plates under no-ideal explosion load by experiment and finite element methods In addition, Uybeyli and colleagues [35] used SiC reinforced functionally gradient material via powder metallurgy to investigate the impact of armor piercing projectile Bodaghi et al [36] studied non-linear active control of dynamic response of functionally graded beams with rectangular cross-section in thermal environments under blast loadings

Based on meticulous investigations in the available literature, it can be concluded that there are few free vibration and nonlinear dynamic behaviors of porous functionally graded sandwich plates resting on elastic foundations regardless of the high demand for understanding In particular, literature review indicates lack of investigations on effects of porosity on this structure exposed to blast loads This study has been implemented to meet the demand.

METHODOLOGY

Configurations of analyzed models

The geometry configuration of the rectangular FG sandwich plate with two FGM face-sheets and the core as ceramic resting on elastic foundations under blast load are as follows (Figs 3.1 and 3.2) The plate is referred to a Cartesian coordinate system

, , x y z , where xy is the mid-plane of the plate and z is the thickness coordinator,

, a b : the length and width of the plate

, c , f : h h h thickness of the total plate, the core and the face-sheets a b h z y x shear layer

Fig 3.1 FGM sandwich plate resting on elastic foundation

Fig 3.2 FGM-ceramic- FGM model

The sandwich plate is composed of three elastic layers, namely: “Layer 1”,

“Layer 2”, “Layer 3” corresponding with FGM face-sheet, the core and FGM face- sheet The FGM face-sheets are made from a mixture of metal and ceramic The face- sheets and the core satisfy power-law distribution and the constituent volume fraction varying continuously along thickness direction The assumption of the metal and ceramic volume is written as:

The ceramic volume of each layer are expressed as follows:

(3.2) in which the volume ratio of the metal in the plate is denoted as V z m ( ) The volume ratio of the ceramic is denoted as V z c ( ) The volume fraction index N defines the distribution of component materials in the structure and N   [0, )

In the case of N = 0, the face-sheet is made entirely of ceramic In the case of

N  , the face-sheet is made entirely of metal

The reaction–deflection relation of Pasternak foundation is defined as follows:

  , w is the deflection of the sandwich plate, k k 1 , 2 are respectively Winkler foundation modulus and the shear layer foundation stiffness of Pasternak model

The FG sandwich plate contains porosities in its structure, which can be dispersed evenly or unevenly along the plate thickness Two types of porosity are considered, namely evenly distribution (Porosity I) and unevenly distribution (Porosity II) as shown in Fig.3.3

Fig 3.3 Porosity – I: evenly distributed, Porosity – II: unevenly distributed

The effective properties P z ( , , ) T such as the elastic moduli E z ( , , ) T , the mass density  ( , , ) z T and the thermal expansion coefficient  ( , , ) z T are defined as [28]:

(3.5) where  denotes the coefficient of porosity which is defined by calculating the ratio the void volume and the total volume (0   1)

The material properties of each face-sheet and the core with two types of pore distribution such as elastic modulus E , thermal expansion coefficient  and mass density  are determined for the details by the formula:

(3.5c) where E cm E c E m , cm    c  m , cm   c  m ,K cm K c K m , and the Poisson ratio    z is assumed to be constant ( )z v

The blast load q t ( ) is a short–term load and is a rapid release of stored energy from an explosion, a shock-wave disturbance or a supersonic projectile… Part of the energy is released as thermal radiation (flash); and part is coupled into the air as air blast and into the soil (ground) as ground shock, both as radially expanding shock waves It can be considered as the models in [37] and shown in Fig 3.4

Fig 3.4 Blast pressure function where the "1.8" factor accounts for the effects of a hemispherical blast, Ps max is the maximum (or peak) static over-pressure, b is the parameter controlling the rate of wave amplitude decay and T s is the parameter characterizing the duration of the blast pulse.

Methodology

Natural frequencies and nonlinear dynamic responses of functionally graded sandwich plates with porosities under the influence of blast loadings are studied in this thesis The investigated FGM sandwich plate consists of two face-sheets and a core layer as FGM-ceramic-FGM which satisfy the continuity requirement of material properties The strain-displacement relations taking into account the Von Karman nonlinear terms and the higher order shear deformation theory deduce motion and geometric compatibility equations of the sandwich plate After introducing the Airy stress function, the number of primary variables diminishes from five to three This thesis utilizes the Bubnob-Galerkin procedure to solve the governing equation of the dynamic system The natural frequencies of the sandwich plate are analytically determined directly by taking the smallest value as solving eigenvalue problems Applying the fourth-order Runge-Kutta method obtains numerical results to illustrate the nonlinear dynamic behaviors of the porous multi- layered FGM plates exposed to the effects of diverse geometry configurations, the types of porosity, the volume fraction index.

Theoretical formulation

Suppose that the FGM plate is subjected to blast loads The Reddy’s higher order shear deformation theory (HSDT) in conjunction with the stress function are used to establish the governing motion, compatibility equations and determine the nonlinear dynamic response and vibration of the multi-layered FGM plate

The strain components at the distance from the mid – plane taking into consideration von Karman nonlinear terms is the following [38] :

; , x x x x xz xz xz y y y y yz yz yz xy xy xy xy k k z k z k z k k k k

(3.7) where c 1 4 / 3 ,h 2   x , y are normal strains,  xy is the in-plane shear strain, and   xz , yz are the transverse shear deformations Also u v , , w are the displacement components parallel to the coordinates ( , , ) x y z , respectively, and   x , y are correspondingly the slope rotations in the   x z , and   y z , planes

The compatibility equation for a multi-layered functionally graded are given by:

The thermal stress-strain relation of the functionally graded sandwich plate are presented by Hooke's law as follows:

2(1 ) x y x y y x xy xz yz xy xz yz

(3.9) in which  T is temperature rise from stress free initial state or temperature difference between two surfaces of the FGM plate

The force and moment resultants of the plate can be expressed in terms of stress components across the plate thickness as:

The constitutive relations are established by introducing Eqs (3.4), (3.5) and (3.9) into Eq (3.10) as follows:

2(1 ) xy xy xy x x xz xz y y yz yz

From Eq (3.11), we obtain    x 0 , y 0 , xy 0 as follows:

The equations of motion basing on higher order shear deformation theory are defined as [38] :

(3.15) and the detail of coefficients I i i (  0 4, 6) may be found in Appendix A, and k 1 is Winkler foundation modulus, k 2 is the shear layer foundation stiffness of Pasternak model, q is blast load distributed on the surface of the sandwich plate,  is the viscous damping coefficient

The Airy’s stress function f x y t  , ,  is introduced as :

Substituting Eq (3.16) into the two first Eqs (3.14) yields:

The three last Eqs (3.14) can be rewritten by replacing Eqs (3.17a) and (3.17b) into the three last Eqs (3.14):

Introduction Eqs (3.13) and (3.16) into the deformation compatibility equation (3.8), we have:

By inserting Eq (3.7) into Eq (3.11) and then into Eqs (3.17), the system of motion Eqs (3.18) is rewritten as follows:

          (3.20c) and the linear operatorsH i ij   1 3, j 1 3 and the nonlinear operator P are give n in Appendix B

The system of fours Eqs (3.19, 3.20) in conjunction with boundary conditions and initial conditions are used to analyze the nonlinear dynamical analysis of FGM sandwich plates in the next section.

Solution procedure

Four edges of the sandwich plate are assumed to be simply supported and immovable (IM) In this case, boundary conditions are:

      (3.21) whereN x 0 ,N y 0 are the forces are the jets when the edges are immovable in the plane of the sandwich plate

The possible solutions of the system of Eqs (3.19) and (3.20) satisfying the boundary condition (3.21) can be chosen as:

    and m n, 1, 2, are the natural numbers of half waves in the corresponding direction ,x y, and W,  x , y - the amplitudes of the deflection and rotation angles which are expression depending on time

The stress function f x y t ( , , ) are defined as follows by substituting displacement functions (3.22) into compatibility equation (3.19):

By replacing Eqs (3.22) and (3.23) into the equations of motion (3.20) and then applying Galerkin method we obtain the resulting equation yields:

  (3.24c) where m n , are odd numbers and

      , detail expressions of coefficients h i ij   1 3,j 1 3 , , n n 1 2 are given in Appendix C

The equations (3.24) are used to investigate the nonlinear dynamic response of FGM sandwich plates on the elastic foundations applying the higher order shear deformation theory.

Vibration analysis

The condition expressing the immovability on the edges, u 0(on x0,a) and v0 (ony0,b), is satisfied in an average sense as follows:

From Eqs (3.7) and (3.13) of which mention relations (3.16 ) we obtain the following expressions:

2 a yy xx x x x x xx a xx yy y y y y yy u E c E w f vf w x E E E x E v E c E w f vf w y E E E y E

Putting Eqs (3.22) and (3.23) into Eq (3.26) then into Eq (24) leads to:

Introducing Eqs (3.27a) and (3.27b) into the equations of motion (3.24), the system Eqs (3.24) can be expressed as follows:

  (3.28c) and specific expressions of coefficients h i 1 i   4 7 are given in Appendix D

Taking linear parts of the set of Eqs (3.28) and putting q0 , the natural frequencies    of the sandwich plate can be determined directly by solving determinant :

Solving Eqs (3.29) yields three angular frequencies of the FGM plate in the axial, circumferential and radial directions, and the smallest one is being considered.

NUMERICAL RESULTS AND DISCUSSION

Validation of the present results

In order to validate the exactitude of present study, the natural frequency of homogenous plate without porosity are determined and compared in Table 4.1 with numerical results of Duc et al [39] using higher order shear deformation theory Two values of the power law index are considered N  0 and N   In these cases, the FGM sandwich plate becomes a homogenous plate which is totally made of ceramic and metal, respectively The geometrical and material parameter are chosen as

/ 1, / 20, 0, 0 a b a h k  k  and   T 0 Table 4.1 shows that our results are in excellent agreement with existing ones It can be concluded that the present method is reliable

Table 4.1 Comparison the natural frequencies   s  1 of homogenous plates with a b/ 1, a h/ 20, k 1 0, k 2 0 and   T 0.

Table 4.2 illustrates comparison of the value of non-dimensional natural frequency parameter

  of a FGM sandwich plate with two face-sheets and homogenous core with numerical results of Li et al [40] based on various theories including classical plate theory (CPT), the first-order shear deformation plate theory (FSDT), sinusoidal shear deformation plate theory (SSDT) third-order shear deformation plate theory (TSDT) and three-dimensional linear theory (TDLT) The Young’s modulus and mass density of components are E c 380GPa and

  (alumina) and E m 70GPa ,  m  2707 kg m / 3 (aluminum) Poisson’s ratio is chosen as 0.3 throughout the analyses The thesis’s results are in excellent consonances with other previous results There are small differences due to different theory and solution

Table 4.2 Comparison of natural fundamental frequency parameters  of simply square FGM plates with other theories ( h b /  0.1 )

N Theories The ratio thickness of each layer

Natural frequency

The influence of two types of porosity on natural frequencies of multi-layered sandwich with the material mode as metal-ceramic-ceramic-metal exposed to blast load is depicted in Table 4.3 The FGM sandwich plate is characterized as the following: a b /  1, h a /  0.05, h c  h f , v  0.3, k 1  0, k 2  0 It is observed from Table 4.3 that the frequency parameters decline with the increase of volume fraction index

With the same geometry configuration and escalating of the volume fraction index

N , it is interesting that the porosity fraction  does not affect the natural frequency of the porous-I FGM sandwich plate By contrast, the rising of porosity II ratio leads to a decrease of the natural frequency In other words, the unevenly distributed porosity have negative effects on natural frequency of FGM sandwich plate From observations obtained by analyzing data in Table 4.3, it can be concluded that since design as well as during manufacturing, we should restrict the appearance of porosity as much as possible

Table 4.4 demonstrates the influences of temperature increment  T , the volume fraction index N and elastic foundations with two coefficient

1( / ), 2( / ) k GPa m k GPa m corresponding to the Winker and Pasternak foundations on the linear frequency of the porous FGM sandwich plates The properties of the sandwich plate are chosen as a b /  1, b h /  20, 0.1 The results indicate that an increase of temperature increment leads to a decrease of natural frequencies

Moreover, the higher stiffness coefficients of Winkler and Pasternak foundations results in higher natural frequencies In other words, the elastic foundations enhance the whole stiffness of porous sandwich FGM plates With these observations, it could be concluded that the shear layer foundation stiffness of Pasternak model has a remarkable influence to frequency higher than the Winkler foundation modulus

Table 4.3 The effects of porosity ratio on natural frequency of FGM sandwich plates a/b N

Table 4.4 Influences of temperature increment, elastic foundations and the volume fraction index on natural frequencies of the FGM sandwich plate with porosity I

Dynamic response

Fig 4.1 Influences of power law index N on the nonlinear dynamic response of the FGM sandwich plates with porosity I

Fig 4.2 Influences of power law index N on nonlinear dynamic response of the FGM sandwich plates with porosity II

Figs 4.1 and 4.2 illustrate the effects of the power law index on nonlinear dynamic response of porous multi-layered FGM plates and without elastic foundation acted by blast load The amplitude of the nonlinear dynamic response of FGM sandwich plate increases when increasing the power law index It is explained that if the value of volume fraction increases, the volume of ceramic component of sandwich increases In addition, the elastic modulus of ceramic is higher than metal of this ones, which means that power law index increases leading to increase elastic modulus of structure These reason result in the deflection amplitude of porous FGM plate decreasing

Fig 4.3 Influences of porous ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I

The influences of porous ratio on the deflection amplitude- time curves of the FGM sandwich plates with porosity I is examined in Fig 4.3 The sandwich plate with the lower porosity volume fraction behave better when being subjetcd to blast load.As can be detected, the higher porous ratio is, the higher deflection amplitude of the FGM sandwich plate is The dynamic deflection of the FGM sandwich plate in case of without porosity   0 is smallest Thus, the existence of porosity deteriorates the dynamic response of the FGM sandwich exposed to blast load

Fig 4.4 Influences of type of porosity on nonlinear dynamic response of the FGM sandwich plate

The effects of Porosity I, Porosity II and without porosity on the nonlinear dynamic response of the FGM sandwich plate under blast load are depicted in Fig

4.4 The existence of porosity reduces the dynamic performance of FGM sandwich plate In addition, the dynamic deflection of porosity I is larger than this ones of

The influences of the plate length to width a b / ratio and length to thickness

/ a h ratio on the deflection amplitude – time curves of the multi-layered FGM plates subjected to blast loading are shown in Fig 4.5 and Fig 4.6, respectively It can be seen that the a b / ratio or a h / ratio amplitudes leads to increase of the deflection amplitude of plate From these scrutiny, it could be deduced that the stiffness of the sandwich plate becomes weaker when a h/ ratio or a b/ ratio is increased

Fig 4.5 Influences of a b / ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I

Fig 4.6 Influences of a h / ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I

Fig 4.7 Influences of Pasternak foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I

Fig 4.8 Influences of Winkler foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I

The beneficial effects of Winkler foundation with stiffness coefficient k 1 and Pasternak foundation with the modulus k 2 on the deflection amplitude – time curves of the FGM sandwich plate are described in Fig 4.7 and Fig 4.8 Obviously, an increase of coefficients k 1 or k 2 results in a reduction of the deflection amplitude It can be seen that the stiffness of sandwich plate becomes stronger with the support of elastic foundations

Fig 4.9 Effect of parameter characterizing the duration of the blast pulse T s on nonlinear response of the FGM sandwich plate with porosity I under blast load

Fig 4.9 depicts the effect of parameter characterizing the duration of the blast pulse on nonlinear response of the multi-layered FGM plate with porosity for three cases T s 0.005,0.01,0.02 From this figure, the value of the parameter characterizing the duration of the blast pulse increase results in the amplitude of vibration increases and vice versa.

CONCLUSIONS

Based on higher order shear deformation, thesis investigates the effects of porosity on nonlinear dynamic response and free vibration of the multi-layered FGM plate subjected to blast load Thesis obtains main results as follows:

- The differential motion equations analyze nonlinear dynamic response of porous FGM sandwich plates on elastic foundations subjected to blast load and natural frequency

- Generally, the existence of porosity reduces the stiffness of FGM sandwich plate

Consequently, the deflection amplitude increases and the natural frequency decreases as the porosity fraction rises However, the natural frequency is not affected by the porosity I fraction of porous FGM sandwich plate Besides, the dynamic deflection of porosity I sandwich plate is larger than this ones of porosity II as considering the same porosity value

- The elastic foundations have positive impacts on dynamic behavior of the sandwich plate, and the Pasternak foundation have more beneficial effect than the Winkler one

- Temperature increment significantly effects on the nonlinear vibration of the porous sandwich plate Temperature increment is considered as external impact which is disadvantage to natural frequency and deflection amplitude of the sandwich plate

- The effects of geometrical parameters (a b a h/ , / ratios), the volume ratio N and parameter characterizing the duration of the blast pulse T s on the nonlinear dynamic responses of the FGM sandwich plates with porosity are remarkable

448 cm cm m m cm cm m cm m cm h Nh h

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