Influence of porosity distribution on nonlinear free vibration and transient responses of porous functionally graded skew plates

Nveen Kumr H S , Subhschndr Kttimni ,*, T. Nguyen-Thoi

a Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, 575025, India

b Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

c Faculty of Civil Engineering Ton Duc Thang University, Ho Chi Minh City, Viet Nam

Keywords:Porous functionally graded skew plates Porosity distributions Nonlinear transient response Nonlinear frequency ratio Nonlinear frequency parameter

ABSTRACT This article deals with the investigation of the effects of porosity distributions on nonlinear free vibration and transient analysis of porous functionally graded skew(PFGS)plates.The effective material properties of the PFGS plates are obtained from the modified power-law equations in which gradation varies through the thickness of the PFGS plate.A nonlinear finite element(FE)formulation for the overall PFGS plate is derived by adopting first-order shear deformation theory (FSDT) in conjunction with von Karman’s nonlinear strain displacement relations. The governing equations of the PFGS plate are derived using the principle of virtual work.The direct iterative method and Newmark’s integration technique are espoused to solve nonlinear mathematical relations. The influences of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the PFGS plate for different skew angles are studied in various parameters.The effects of volume fraction grading index and skew angle on the plate’s nonlinear dynamic responses for various porosity distributions are illustrated in detail.

1. Introduction

In recent years,great attention has been focused on functionally graded materials(FGM)because of their excellent performance and widespread applications in the field of thermal,structural,defence,military, armor applications, biomedical, electronics, and optoelectronics [1]. The FGM’s are first proposed by material scientists from Japan in 1984 [2]. These are made up of two constituent materials:metal and ceramic,produced by the powder metallurgy technique. The material properties of the FGM vary smoothly and continuously in the preferred direction.As a result, the generation of residual stresses, stress concentration, delamination, and the occurrence of cracks can be avoided. Hence, FGM’s are most suitable and preferred for many structural applications that may be in the form of FG plates/shells to enhance the mechanical properties.For a plate, if the gradation is through the thickness, then the top and bottom surfaces have isotropic constituent materials of ceramic and metal, respectively. The portion between the top and bottom surfaces has a varying composition of two constituent materials.Thus,in a single FG material, one can obtain the properties of two different materials with a variable composition of two materials in a chosen direction. During the manufacturing of FGM plates, a generation of porosities cannot be avoided due to the limitations of fabricating techniques. It is challenging to infiltrate the first material entirely into the middle portion of the FG plate along with the second constituent material without porosities.But for the top and bottom portion of the plate, the materials can be easily infiltrated with the less amount of presence of porosities. Besides, porosities may generate due to the difference in the solidification temperature of the two materials[3,4].Consequently,the existence of porosities can weaken the strength of FGMs. Therefore, it is essential to consider the effect of porosity on the dynamic characteristics of FGM plates.

An enormous amount of work is presented on the linear vibration and dynamic characterizations of FGM plates. Moreover,the recent works on the static and dynamic analyses of the FG structures with various solution methods, including temperature effects,are presented by several authors[5-10].However,research work is in progress for non-linear vibration and dynamic analysis of FGM plates and has become the most significant problem in recent years. Many research works focused on the dynamic responses of the functionally graded (FG) plates due to the geometric nonlinearity presented in the literature. For example, Praveen and Reddy[11] examined the non-linear transient thermo-elastic analysis of FG plates subjected to steady-state temperature fields and dynamic loads by adopting the von Karman’s geometric nonlinearity. The non-linear transient thermo-elastic analysis of FG Plates was carried out in the framework of third-order shear deformation theory considering von Karman’s type of geometric nonlinearity [12,13].Huang and Shen [14,15] studied non-linear free vibration and dynamic response of the FG plates with and without piezoelectric actuators under the thermal environment. A semi-analytical approach was developed along with transverse and in-plane loads in the framework of classical laminate plate theory (CLPT)to investigate the large deflection and post-buckling response of FG rectangular plates by Yanga and Shen [16]. Further, the non-linear transient response of geometrically imperfect FG plates subjected to uniform thermal loading on surfaces of the plate are investigated by Yang and Huang [17] using Higher-order shear deformation theory (HSDT). Yang et al. [18] presented a work on non-linear vibration and non-linear transient analysis of FG plate having a surface crack exited by transverse force.Wang and Zu [19]developed an analytical formulation by considering the effects of both material and geometrical nonlinearities for the large amplitude motions to investigate the dynamic behavior of FG rectangular plates moving with longitudinal velocity under thermoelastic loadings.In addition, Chang-Cheng Du [20] proposed the effect of in-plane inertial disturbance, which occurs from the environment on the dynamic response of the FG plates. Jrad et al. [21] presented the geometrically non-linear analysis of FG shells by using first-order shear deformation theory (FSDT). Furthermore, Mallek et al. [22]continued investigating the non-linear dynamic behavior of the piezo-laminated FG carbon-nanotube reinforced composite shell based on the improved FSDT.

Owing to the flexibility of FG structures, vibrations induced in the FG structures may lead to the development of large amplitudes.Thus,it is crucial to study the vibration behavior of FG structures by considering the geometrical nonlinearity. Yang and Shen [23]investigate the free and forced vibration of initially stressed FG plates in the thermal environment by developing the formulations using Reddy’s HSDT and von Karman’s geometric nonlinearity.Chen et al. [24,25] derived the formulation for the non-linear equations of motion for initially stressed and non-uniform initial stresses on FG plates based on CLPT to investigate the linear and non-linear frequencies. Woo et al. [26] presented an analytical solution to obtain non-linear free vibration of FG plates. The governing equations are obtained using von Karman’s assumptions for large transverse deflections and mixed Fourier series analysis.Talha et al.[27]investigated the non-linear free vibration of FG plates by using non-linear FE equations based on HSDT associated with Green-Lagrange’s non-linear relations that are considered for the large deflections of the plate.Alijani et al. [28] presented the nonlinear free vibration of FG rectangular plates coupled with thermal effects by adopting a multi-modal energy method,pseudo-arclength, and collocation methods in their study. Malekzadeh et al.[29] proposed in-plane material gradation effects and found the non-linear vibration of FG plates for different boundary conditions by taking the differential quadrature method into account. Shen et al. [30] investigated the non-linear vibration of FG graphene reinforced with composite laminated plates under the thermal environment. Parida et al. [31] studied the non-linear vibration behavior of the FG plate,which rests on an elastic foundation in the thermal environment.

Recently, FG porous material structures, such as plates, beams,and shells,are widely used in structural design problems.The most recent works on the linear vibration and dynamic analyses to investigate the influence of porosity distributions of porous FG structures with several solution methods are presented by various authors[32-36].Further,many authors researched the non-linear vibration analysis and dynamic responses of FG porous structures.For example,Wattanasakulpong et al.[37]presented a problem on linear and non-linear vibration behaviors of porous FG beams with elastic boundary conditions by adopting a differential transformation method.The non-linear vibration behavior of porous FG Timoshenko beams was investigated by Ebrahimi et al. [38], and they found that non-linear free vibration characteristics are mainly affected by material gradation, porosity parameters, and aspect ratios. Wang et al. [39,40] presented the non-linear vibration of porous FG plates, and the solution was obtained by adopting a harmonic balance method. Also, Wang et al. [41] continued investigating non-linear forced vibration of porous FG piezoelectric plates. Gupta and Talha [42,43] investigated the non-linear vibration and stability of FG plates with geometric imperfections and porosity in the framework of HSDT. Huang et al. [44] focused on finding out the non-linear natural frequency and dynamic response of sigmoidal FG porous plates with non-linear elastic boundary conditions. They found that the effect of non-linear foundation on porous FG plates has less impact. Phung-Van et al. [45-47] investigated with and without introducing the geometrically sizedependent non-linear transient responses of porous FG nanoplates in the framework of the iso-geometric FE method. Further,the authors carried out the same work under hygro-thermal and mechanical loads. Xie et al. [48] investigated the non-linear free vibration of rectangular porous FG plates based on the energy balance approach in conjunction with von Karman’s assumptions and Lagrange’s equations to obtain the governing equations of the FG plates.Thanh et al.[49]investigated the porous FG micro-plate’s static bending analysis based on the iso-geometric approach coupled with von Karman’s geometric nonlinearity.

The literature reveals that most of the non-linear free vibration and non-linear transient response analysis of the porous FGM structures focused on beams and rectangular plates. However,Functionally Graded Skew plates are found to be acquired minimal space even though it facilitates an engineer to have a variety of alignment opportunities in case of obstacles. The skewness in the geometry of the plate decreases the plate area. As a result, the stiffness of the plate increases.This exhibits a considerable effect on the non-linear frequency and transient responses of the plates.The skew plates find their structural applications in the fields of aerospace, automotive, machinery and equipment, civil, defence, military,energy,optoelectronics and biomedical,etc.[1].Sundararajan et al. [50] developed non-linear governing equations by adopting von Karman’s equations of motion to examine the non-linear free vibration characteristics of FG rectangular and skew plates under the thermal environment. These are solved by the FE method conjugated with a direct iteration technique.Upadhyay and Shukla[51] presented non-linear static analysis and non-linear dynamic responses of FG skew plates. The analytical solutions are formulated using the HSDT in connection with von Karman’s nonlinearity.Taj et al. [52,53] presented the static and dynamic analysis of FG skew plates with mechanical load and carried out the FG skew sandwich plates’ bending responses. Zhang et al. [54] investigated the large deflection analyses of the FG skew plates reinforced with carbon nanotubes using an element-free IMLS-Ritz method.Further,Lei et al.[55]adopted the same approach for the buckling analysis of FG skew plates reinforced with carbon nanotube composites rests on Pasternak foundation. Adineh et al. [56] investigated the multi-directional FG skew plates based on an elastic foundation by adopting a three-dimensional thermoelastic method in conjunction with thermal and mechanical loadings.Parida et al.[57] presented the non-linear free vibration behavior of FG skew plates under thermal environments using the HSDT along with Green Lagrange’s geometric nonlinearity. Tomar and Talha [58]developed the non-linear vibration analysis of the FG laminated skew plates under a thermal environment.The practical aspects of the present study such as axles housings, armour plates and bulletproof vests,and latches are the emerging applications in the field of defence [59-61] whereas rocket nozzle, wings, rotary launchers, landing gear doors, structures blades, spacecraft truss structure,cutting tools,turbine blades,bones and dental implants,solar panels, electronic components are some of the applications made of FGMs [62].

To the best of authors’knowledge,no work has been reported on demonstrating the effect of porosity distributions and skewness on the porous functionally graded skew (PFGS) plates. In this regard,the present work makes a first attempt at understanding the nonlinear free vibration behavior and transient analysis of PFGS plates.The necessity of this work seems to more evident for the structures used in defence, energy sector, aerospace, and civil fields. Hence,the present study aims to develop a non-linear FE model for the PFGS plates based on the FSDT concatenated with von Karman’s assumptions. The governing equations are derived using the principle of virtual work, and the non-linear natural frequency of the PFGS plate is computed by using a direct iterative technique. The Newmark’s integration technique is employed to extract the dimensionless non-linear transient results. The PFGS plate’s effective material properties are derived from the modified power-law distribution in which gradation varies through the thickness of the plate in terms of volume fraction of the material constituents.The predominant effect of porosity distributions and skewness on PFGS plate on the large amplitude free vibration and non-linear transient responses are thoroughly investigated.

2. Materials and methods

Fig.1. The geometry of the PFGS plate.

The dimensional parameters of the PFGS plate are length a,width b,and thickness t,as shown in Fig.1.The plate’s skewness is considered by rotating the y-axis at an angle φ called the skew angle. The PFGS plate is made up of two isotropic and homogeneous materials, metal and ceramic. The volume fraction of the material constituents of the PFGS plate changes continuously in the thickness direction. i.e.,z-axis.The top surface of the PFGS plate is solely ceramic （z=t/2） and the bottom surface is solely metal-（z=-t/2）as depicted in Fig.1.The effective material properties of the PFGS plate （Pfg（z）） are Young’s modulus （Efg（z））, the density（ρfg（z））,and the Poisson’s ratio（?fg（z））at any point in the plate are determined by employing the modified power-law distribution method [63] by using Eq. (1).

a) Evenly distributed porosity(Pe):The porosities are distributed evenly throughout the PFGS plate area, as shown in Fig. 2(a).Further, the effective material properties for evenly distributed porosities can be obtained by putting the valuePd=1 in Eq.(4)and the modified equation becomes as follows:

b) Centrally distributed porosity (Pc): The high density of porosities is distributed at the middle portion of the PFGS plate,as shown in Fig. 2(b). The effective material properties for the centrally distributed porosities can be obtained as follows:

Fig. 2. Porosity distribution in PFGS plate (a) Evenly distributed porosity (Pe), (b)Centrally distributed porosity (Pc), and (c) High porosity distribution near the top and bottom surface while narrow at the middle (Ptb).

c) High porosity distribution near the top and bottom surface while narrow at the middle(Ptb):In this case,the high density of porosities are distributed nearer to the top and bottom surface while the low density of porosity distribution at the middle span of the PFGS plate as shown in Fig. 2(c). The effective material properties for this type of porosity distribution, i.e., high porosity distribution closer to the top and the bottom surface while shallow at the middle span of the PFGS plate. The modified equation can be obtained as follows:

The constituent material properties of PFGS plates used in the present study are given in Table 1.The variation of Young’s modulus as a function of porosity distributions of Si3N4/SUS304 porous functionally graded plate (FGP) is shown in Fig. 3. For the evenly distributed porosity (Pe) case, the variation profiles of Young’s modulus of Peare the same as ideal FGP, but a decrease in amplitude can be observed. For the centrally distributed porosity (Pc)case, the amplitude matches with the ideal FGP at the top and bottom surface; meanwhile, it matches with the Peat the middle surface.This is due to the distribution of porosities are concentratedin the middle portion and tends to zero at the top and bottom surface for Pccase.In contrast,for Ptbcase,the amplitude matches with the ideal FGP at the middle portion of the plate;meanwhile,it matches with the Peat the top and middle surface of the FGP.This is because of the distribution of porosities is concentrated at the top and bottom portion of the FGP and tends to zero at the middle surface.

Table 1 The Properties of isotropic materials [14].

3. Kinematics of deformations

The kinematics of the deformation of the PFGS plate is considered to be grounded on the FSDT [11,65,66]. Accordingly, the displacements u0,v0and w0at any point in the overall plate along x,y,and z-coordinate directions, respectively, are given by

where (u1, v1,and w1) are the generalized displacements at a reference point(x,y)lying on the midplane(z=0)of the PFGS plate in the direction of x, y, and z, respectively. θxandθyare the corresponding generalized rotations of a normal to the reference point at the reference plane about the y-axis and x-axis,respectively.For the simplicity of analysis,the generalized displacements are separated into variables of translational｛dt｝and variables of rotational｛dr｝as follows.

Eq. (11) can be written as

Fig. 3. Variation of Young’s modulus of porous FGP made of Si3N4/SUS304.

where, the generalized strain vectors ｛εtb｝, ｛εrb｝, ｛εtbNL｝, ｛εts｝and｛εrs｝appeared in Eq. (14) are given by

The state of stress and the corresponding state of strain at any point in the overall PFGS plate is described by

The terms appearing in Eq. (17) σxxand σyyare the normal stresses in the direction of the x-axis and y-axis,respectively.τxyis the in-plane shear stress, τxzand τyzare the transverse shear stresses. T indicates transpose, [Qb（Z）] and [Qs（Z）] are the elastic coefficient matrices and are the function of Z- coordinate.

where, Efg（Z） and ?fg（Z） are the effective young’s modulus and Poisson’s ratio of PFGS plate, respectively.

The in-plane stress resultants, moment resultants and the transverse force resultants acting on the plate can be expressed as

where｛F｝ = ｛0 0 p｝Tis externally applied surface traction acting over a surface area,p is transverse load,ρ is the mass density of the PFGS plate,and δ is the symbol that describes the first variation.

4. Finite element modeling of PFGS plate

The overall PFGS plate is discretized into eight-node isoparametric quadrilateral elements with five degrees of freedom(three translational and two rotational) for each node. Making use of Eq. (9), the generalized displacement vectors ｛dtn｝ and ｛drn｝associated with the nth(n=1,2,3,…,8)node of an element can be written as

where nnis the shape function in which the natural coordinates are associated with node n.Itand Irare the identity matrices.Now,from Eqs.(14)and(26),the generalized strain vectors at any point within an element are obtained as follows:

In which,[?tb], [?1],[?2],[?rb],[?ts] and[?rs] are the generalized nodal strain displacement matrices and are presented in the Appendix-A.

Substituting Eqs. (14), (16) and (27) into Eq. (23) and applying the principle of minimum total potential energy, i.e., δTep= 0, and simplifying, we obtain elemental equations of motion for PFGS plate as follows

The elemental stiffness matrices evolved in Eq. (30) along with the various rigidity matrices and vectors appearing in the elemental matrices are exclusively presented in Appendix-B. The elemental governing equations are assembled into the global space in a straight forward manner to obtain the global equations of equilibrium as follows:

In which, [M]is the global mass matrix, [Ktt], [Ktr], [Krt]and [Krr]are the global stiffness matrices, it may be noted that the stiffness matrices [Ktt], [Ktr]and [Krt] include both global linear and nonlinear components. ｛F｝ is the global mechanical load vector, ｛Xt｝and ｛Xr｝ are the global generalized translational and rotational displacement vectors, respectively. After enforcing the displacement boundary conditions, the global rotational degrees of freedom are reduced to obtain the global equilibrium equations concerning the global nodal translational degrees of freedom as follows:

Eq.(34)presents the non-linear FE model for the PFGS plate.To obtain the eigenvalues and eigenvectors of the PFGS plate,Eq.(32)can be rearranged in the form as follows

where, ω and Xare the natural frequency and corresponding eigenvector of the PFGS plate,respectively. The transient response of the plate can be obtained by incorporating Newmark’s integration technique. Further, the non-linear frequency-amplitude relationship is determined from Eq.(34)using a direct iterative method associated with FE formulations.

5. Skew boundary transformation

It can be observed from the PFGS plate shown in Fig.1 that the boundary elements are supporting two adjacent edges.These edges may or may not be parallel to the global coordinate axes (x, y, z).Therefore,the nodal displacements are defined in terms of axes of local coordinates. Then, the axes of local coordinates are transformed into the global coordinate axes. Further, the boundary conditions are defined in terms of the axes of the global coordinate.The transformation of coordinates between the global degrees of freedom and local degrees of freedom of each node can be written as [68,69].

where,c = cos（φ）and s = sin（φ）.

It can be noted from the above transformation matrices that the transformation from the global coordinate system to the local coordinate system is not necessary for the nodes which are not resting on the skew edges of the FG plate. In such a case, the transformation matrix has diagonal matrices and the values assigned to the principal diagonal elements are equal to one. The transformation matrices for the complete element is:

6. Solution methodology

6.1. Non-linear frequency parameter analysis

The solutions to the vibration problems are obtained using the eigenvalue formulation, while an iterative method is adopted to solve the non-linear eigenvalue problems of the PFGS plate[50,70].Firstly, utilizing a standard eigenvalue extraction algorithm, Eq.(34)is used to extract the linear eigenvalue and eigenvectors from the global mass and stiffness matrices by neglecting non-linear stiffness matrices. Next, by normalizing the mode shape vector and then the normalized eigenvector is scaled up to the amplitude ratio. Thus, the desired value should be equal to the maximum displacement values.This provides theinitial vector.The initial nonlinear vector is then obtained by computing the non-linear stiffness matrices by numerical integration technique and then solved for the non-linear eigenvalues and corresponding eigenvectors. These steps are repeated until the values of non-linear frequency and its mode shape converge [71] at a specified rate as given by

where the suffix itr indicates the iteration number.

6.2. Non-linear transient analysis

In the non-linear dynamic analysis, the non-linear governing equation(Eq.(32))depends on both linear and non-linear stiffness matrices. The non-linear stiffness matrix (KNL) depends on both unknown displacement(Xt)and time-domain(ti).Therefore,for the complete discretization of the non-linear governing equation, the time derivatives appearing in Eq. (32) are approximated using Newmark’s direct integration technique and Picard’s methods.Here,the initial values of displacements,velocity,and acceleration are set to zero at time ti= 0.Now,to determine the velocities and accelerations at ti= （′n + 1）Δ tican be written as

It may be noted that all the parameters can be determined at the initial time ti= ′n Δtifrom Eqs. (43) and (44) except for the nonlinear stiffness matrix (KNL), which depends on the unknown displacementX′n+1. Therefore, the non-linear equation from Eq. (43)is re-approximated by adopting the Picard type iteration method.After re-approximation, the equation appears as follows

7. Results and discussions

The PFGS plate is considered to investigate the effects of porosity distributions and skewness on the large amplitude free vibration and non-linear transient analysis.The PFGS plate consists of a ceramic surface at the top of the plate and a metallic surface at the bottom of the plate,in between there is a gradation of these two constituent materials in conjunction with porosities. Validation studies have been executed to evaluate the accuracy of the proposed FE model.The properties of the various constituent materials used in the present study are given in Table 1. Parametric studies are conducted for various parameters to evaluate the non-linear frequency and dynamic response of the PFGS plate.

Table 2 Comparing the fundamental natural frequency parameter for Si3N4/SUS304 FGM square plate (a = 0.2; b = 0.2; t = 0.025, SSSS).

7.1. Validation

To examine the accuracy and effectiveness of the present FE model.The fundamental natural frequency parameter for a simply supported Si3N4/SUS304 FG square plate for different volume fraction index(α)are compared with the results reported by Huang and Shen[14]for the similar geometrical parameters and material properties of the plate. It can be observed from Table 2 that the results obtained with the present model resemble very well with the reference literature [14]. The mesh size of 8 × 8 is found adequate based on the progressive mesh refinement to model the PFGS plate. An additional validation study is carried out for the dimensionless natural frequency of the isotropic skew plate with skew angles of 0°,15°, 30°, and 45°with SSSS boundary condition for the aspect ratio a/b=1 and thickness ratio a/t=5 and 10.It may be seen that the results reported in Table 3 display very good agreement with the reference literature Liew et al.[72].To validate further, the dimensionless frequency parameter for porous FG plates are determined and compared with the results reported in the literature by Rezaei et al. [73] for the Al2O3/Al FG porous plate with simply supported boundary condition for various volume fraction index (α) and porosity parameter index (ep). It can be witnessed from Table 4 that the results hold a firm agreement for both evenly distributed porosity (Pe) and centrally distributed porosity(Pc)with reference[73].The non-linear to linear frequency ratio for different amplitude ratio Wmax/t is validated with the results reported by Sundarajan [50], and good agreement with the present model for the simply supported Si3N4/SUS304 FG plate are observed as shown in Table 5.

Furthermore, to check the reliability and accuracy of the approach adopted for the analysis of nonlinear transient response,the present method is also validated with the Chen et al.[74]for an orthotropic plate with a = b = 0.250, t = 0.005, E1= 525 GPa,E2=21 GPa,G12=G23=G13=10.50 GPa,?=0.25,and ρ=800 kg/m3under the uniform step load of q0= 1 N/mm2. It may be observed from Fig. 4 that in this case also, the dimensionless nonlinear transient deflection （w︶c=w/t） exhibit a very close agreement in comparison with reference [74].

Table 3 Comparison of fundamental frequency for an isotropic skew plate (a/b = 1, SSSS).

Table 4 Comparison of fundamental natural frequency for different porosity parameter index for simply supported square Al2O3/Al porous FGM plate.

Table 5 Comparison of nonlinear frequency ratio(ωnl/ω)with different values of amplitude ratios for Si3N4/SUS304 FGM square plate(a/b = 1, a/t = 10,α = 1, and SSSS).

Fig. 4. Comparison of dimensionless non-linear transient deflection of the simply supported orthotropic plate.

8. Parametric studies

Based on the comparison studies presented in section 7.1, the proposed approach extracts the accurate results for carrying out the parametric studies on the PFGS plate.Influence of various boundary conditions like SSSS(simply supported),CCCC(Clamped clamped),SCSC(Simply supported and clamped),SSSF(Simply supported and free edge),and CCCF(Clamped and Free edge)are considered in the present analysis. The mesh size (8 × 8) used for the validation is adopted to extract the results for all parametric studies.The porous FG plate consists of two sets of combinations of materials: Si3N4/SUS304 and ZrO2/Ti-6Al-4V. The properties such as Young’s modulus(E),density(ρ),and Poisson’s ratio(?)of these constituent materials are provided in Table 1.

Table 6 Effect of volume fraction grading index for different porosity distributions on the fundamental NFR (ωnl/ω) of Si3N4/SUS304 PFGS plate (a/b = 1, a/t = 50, ep = 0.10,φ = 0° and SSSS).

8.1. Non-linear frequency parameter analysis

In this section, the non-linear vibration characteristics of the PFGS plate is analyzed considering the parameters which influence the non-linear frequency ratio (NFR) such as skew angle, porosity distributions, porosity parameter index, volume fraction grading index, and geometrical parameters of the plate. The results presented in Table 6 is focused on investigating the influence of porosity distributions for the skew angle φ = 0°, volume fraction grading index (α = 0, 0.5,1, 2, 5, and 10), and porosity parameter index ep=0.10 on the fundamental NFR(ωnl/ω).It can be seen that for a given porosity distribution, increasing the value of volume fraction grading index (α) leads to decrease in the NFR till α = 2.0 whereas from α = 5, it enhances. It indicates that the NFR has the lowest values in the gradation region α = 1 to 5 for all porosity distributions. The same trend was observed by the researchers[27,50]. This is due to the decrease of ceramic composition as the volume fraction grading index increases and thus reduces the stiffness of the plate. As the stiffness of the plate decreases for higher values of volume fraction index, the linear frequency also decreases rapidly,and the difference between non-linear frequency to linear frequency increases. Hence, the frequency ratio increases for higher values of the volume fraction index. Analogously, for agiven value of α, the effect of porosity distributions on the NFR follows the trend as centrally distributed porosity (Pc) < evenly distributed porosity (Pe)

Table 7 Influence of Skew angle for different porosity distributions on the fundamental NFR(ωnl/ω) of Si3N4/SUS304 PFGS plate.(a/b = 1, a/t = 50, ep = 0.10,α = 2, and SSSS).

Table 7 shows the influence of porosity distributions for different skew angles on the fundamental NFR(ωnl/ω).The effect of porosity distributions (Pe, Pc, and Ptb) with porosity parameter index ep= 0.10 for different skew angles (φ = 0°,10°, 20°, 30°, and 40°) are considered. Further, the porous plate made up of Si3N4/SUS304 with the boundary conditions SSSS is considered for the analysis. From Table 7, it can be observed that in the presence of porosities, the NFR increases as the amplitude ratio Wmax/t increases for all the types of porosity distributions and skew angles.This indicates that the amplitude ratio exhibits the hardening type behavior in the presence of porosity and the amplitude ratio determines the extent of nonlinearity. Where Wmaxindicates the maximum flexural amplitude of the plate and t is the thickness of the plate. If the amplitude ratio is set to zero, then the porous FGplate undergoes linear vibration and gives a linear frequency.Similar observations were made by researchers for FGM [50].Further,it can also be observed from Table 7 that with the increase in skew angle, the NFR decreases for all porosity distributions,porosity parameter, and volume fraction index. This is due to the increase in linear frequency with an increase in stiffness of the plate as the skew angle enhances.Hence,it reduces the NFR.Besides,for a given skew angle, the effect of porosity distributions on the NFR follows the same trend followed in volume fraction grading index such as centrally distributed porosity (Pc) < evenly distributed porosity (Pe)

Table 8 Effect of thickness ratio for various skew angles on the fundamental NFR(ωnl/ω)for different porosity distributions of Si3N4/SUS304 PFGS plate(a/b=1,α=2,ep=0.10,Wmax/t = 1.0, and SSSS).

Table 8 depicts the influence of side to thickness ratio (a/t) for different skew angles and porosity distributions on the NFR for simply supported Si3N4/SUS304 PFGS plate for α=2,ep=0.10,and Wmax/t = 1.0. It can be witnessed that the NFR decreases as the thickness ratio increases.The physical significance of this behavior is the decrease of the plate’s stiffness as the thickness of the plate decreases and due to the presence of porosities. Similar kind of observations is made by the researchers [27]. Besides, porosity distribution Pchas the lowest NFR while Ptbexhibits the highest for all the cases of thickness ratios and skew angles(Pc

The influence of variation of aspect ratio (b/a) for various skew angles and porosity distributions on the non-linear behavior of simply supported PFGS plate has been studied considering the volume fraction α=2,ep=0.10,and Wmax/t=1.0.It can be noticed from Table 9 that for an increase in aspect ratio,the NFR decreases and then increases for lower skew angles.However,for higher skew angles (φ = 30°, and 40°), the NFR increases gradually with an increase in aspect ratio. This is due to the decrease in linear frequencies as the aspect ratio increases with the presence of porosities; however, it is not presented here for the sake of brevity.Furthermore, the lowest NFR is observed for the porosity distribution Pcwhile Ptbexhibits the highest for all the instances of aspect ratio(Pc

Table 9 Effect of aspect ratio for various skew angles on the fundamental NFR (ωnl/ω) for different porosity distributions of Si3N4/SUS304 PFGS plate. (a/t = 50, α = 2,ep = 0.10, Wmax/t = 1.0, and SSSS).

Table 10 Effect of porosity parameter index for various skew angles on the fundamental NFR(ωnl/ω)for different porosity distributions of simply supported Si3N4/SUS304 PFGS plate.(a/t = 20, a/b = 1,α = 1, and Wmax/t = 1.0).

Table 10 presents the effect of various porosity parameter indices for different skew angles on the NFR of simply supported Si3N4/SUS304 PFGS plate for α = 1 at an amplitude ratio of Wmax/t=1.0.It can be observable that,irrespective of skew angles for the porosity distributions Peand Pc,the NFR decreases with an increase in porosity parameter index while increases for Ptbtype of porosity distribution.This is due to the degradation of the properties of the PFGS plate in the presence of distributions of porosities and geometric nonlinearity in the gradation region.Comparatively,Ptbtype of porosity distribution has a higher value of NFR for all the porosity parameter indices, and Pcexhibits the lowest values of NFR(Pc

Fig. 5. Influence of porosity distributions on the NFR for various boundary conditions for ZrO2/Ti-6Al-4V PFGS plate.

Fig.5 displays the plots of the influence of porosity distributions on the NFR for various boundary conditions. The geometrical parameters used for the PFGS plate are a/b = 1, a/t = 10, α = 1,ep=0.10,and φ=15°and the porous material is ZrO2/Ti-6Al-4V.It can be seen from Fig.5 that the NFR values are minimum for the CCCF boundary condition and maximum for the simply supported condition.For the cases of both fully clamped and simply supported edges of the plate, the simply supported plates exhibit maximum values of NFR compared to a fully clamped plate. The effect of boundary conditions on the NFR follows the trend as SSSS > SSSF > SCSC > CCCC > CCCF. It may be attributed to the presence of porosity and an increase of linear frequencies for clamped boundary conditions. Also, the NFR increases as the amplitude ratio increases for all the edge constraining cases of the PFGS plate. Besides, irrespective of boundary constraints, porosity distribution Ptbdominates the other two (Peand Pc) by exhibiting the highest NFR and Pcdisplays the lowest NFR(Pc

Fig. 6. Effect of various volume fraction indices on NFP of simply supported and clamped boundary conditions for ZrO2/Ti-6Al-4V PFGS plate for evenly distributed porosity (Pe).

To investigate further, the non-linear frequency parameter(NFP) of ZrO2/Ti-6Al-4V PFGS plates for various geometrical parameters, the following non-dimensional NFP (ωnl) equation is adopted.

where,ρcand Ecare the density and Young’s modulus,respectively,for the ceramic material.

Fig. 7. Effect of various volume fraction indices on NFP of simply supported and clamped boundary conditions for ZrO2/Ti-6Al-4V PFGS plate for centrally distributed porosity (Pc).

Fig. 8. Effect of various volume fraction indices on NFP of simply supported and clamped boundary conditions for ZrO2/Ti-6Al-4V PFGS plate for Ptb type of porosity distribution.

The influence of various volume fraction indices on the NFP for clamped and simply supported boundary conditions of the PFGS plate for all the types of porosity distributions are depicted in Figs. 6-8. The geometrical parameters a/b = 1, a/t = 40, φ = 30°,and ep=0.2 are considered for the investigation.It can be observed from these figures that as the amplitude ratio increases, the NFP also increases monotonically. Hence, it displays the hardening behavior characteristics. Further, the NFP decreases with the increase in the volume fraction grading index α. In analogous, the same trend can be observed in the literature[57].It can also be seen that NFP is maximum for the clamped case compared to simply supported boundary conditions of the plate.Moreover,each type of porosity distribution exhibits a different kind of impact on influencing the NFP.The Pctype of distribution has the highest,and the Ptbtype of distribution has the lowest influence on the NFP. The influence of porosity distributions on NFP follows the trend as Pc>Pe>Ptb. The change in the NFP attributes to the decrease and variation of the stiffness of the plate due to the presence of a different type of porosity distribution while it significantly decreases the NFP values as porosity volume increases.

Fig.9. Effect of skew angle on NFP of simply supported ZrO2/Ti-6Al-4V PFGS plate for Pe,Pc,and Ptb type of porosity distributions.(a/b=1,a/t=80,α=2,ep=0.3,and SSSS).

Fig.10. Effect of skew angle on NFP of clamped ZrO2/Ti-6Al-4V PFGS plate for Pe, Pc,and Ptb type of porosity distribution. (a/b = 1, a/t = 80, α = 2, ep = 0.3, and CCCC).

Fig. 11. (a) Effect of porosity parameter index for different skew angles on NFR for simply supported ZrO2/Ti-6Al-4V PFGS plate for Pe type of porosity distribution. (b)Effect of porosity parameter index for different skew angles on NFR for clamped ZrO2/Ti-6Al-4V PFGS plate for Pe type of porosity distribution.

The effect of the skew angle on the NFP of the PFGS plate for different porosity distributions Pe, Pc,and Ptbare shown in Figs. 9 and 10. It can be seen that with an increase in skew angle, there is a reduction of plate area,due to which the stiffness of the porous plate increases. Hence, the rise in NFP is observed. A significant increase in the NFP can be noticed for the skew angle φ = 45°as compared to lower skew angles(φ=0°,15°,and 30°).Besides,NFP is high for the clamped case compared to the corresponding values of the simply supported boundary conditions. However, there is a reduction of the plate’s stiffness due to the presence of porosity compared to the perfect FG skew plate.Figs.9 and 10 show that the Pctype of porosity distribution gives the highest NFP while the Ptbexhibits the lowest NFP.This indicates that the presence of porosity in the gradation region increases the NFP.

Fig. 12. (a) Effect of porosity parameter index for different skew angles on NFR for simply supported ZrO2/Ti-6Al-4V PFGS plate for Pc type of porosity distribution. (b)Effect of porosity parameter index for different skew angles on NFR for clamped ZrO2/Ti-6Al-4V PFGS plate for Pc type of porosity distribution.

Fig. 13. (a) Effect of porosity parameter index for different skew angles on NFR for simply supported ZrO2/Ti-6Al-4V PFGS plate for Ptb type of porosity distribution. (b)Effect of porosity parameter index for different skew angles on NFR for clamped ZrO2/Ti-6Al-4V PFGS plate for Ptb type of porosity distribution.

The influence of the porosity parameter index for various boundary conditions on NFR for Pe, Pc,and Ptbtype of porosity distributions are shown in Figs.11-13.The geometrical parameters used for PFGS plate are a/b=1,a/t=40,and α=1 for skew angles φ=0°,25°,and 45°.It can be seen that the NFR has higher values for SSSS boundary conditions compared to clamped boundary conditions for all types of porosity distributions.This signifies that the stiffness of the plate increases for the clamped plate in the presence of porosities and leads to an increase in the linear frequency. Therefore, NFR decreases for the clamped case while it increases for the simply supported case. Further, the noticeable difference of NFR is discerned in clamped boundary conditions for variation of porosity parameter index and skew angles.Also,Pchas the lowest NFR values than Peand Ptbtype of porosity distributions irrespective of skew angles. This infers that the presence of porosities in the middle portion of the plate has a lower impact on the NFR.Besides,the NFR decreases steadily as the porosity parameter index and skew angle increases for both the boundary conditions.

8.2. Non-linear transient response analysis

In this section, the non-linear transient deflection of the PFGS plate is investigated. The influence of skew angle, porosity distribution, porosity parameter index, and volume fraction grading index on the non-linear transient deflection is studied. The geometrical parameters of the plate a/t = 40, a/b = 1 under a uniform step load of q0= 1 N/mm2with a time step of Δti= 1×10-5being considered for the simply supported Si3N4/SUS304 PFGS plate. The dimensionless central deflection parameter considered for the study is w︶c= w/t. The effect of Porosity Parameter indices ep= 0.1, 0.2,and 0.3 on various skew angles φ = 15°,30°, and 45°for Pe, Pc,and Ptbtype of porosity distributions are investigated. It can be observed from Figs. 14-16 that the non-linear transient deflection decreases with an increase in skew angle. This may be due to the increase in stiffness of the plate which increases with an increase in the skew angle besides, the porosity parameter index also influences the deflection. As a result, for an increase in the porosity parameter index,the non-linear deflection also increases.It can also be seen that for the Petype distribution, the transient deflection is highest, whereas the Pctype distribution exhibit the lowest non-linear transient deflection irrespective of skew angles.Analogously, for a given value of epand skew angle, the effect of porosity distributions on the transient deflections follows the trend as Pc< Ptb

Fig.14. Effect of porosity parameter index on central deflection(w/t)for various skew angles for evenly distributed porosity (Pe) on simply supported Si3N4/SUS304 PFGS plate.

Fig.15. Effect of porosity parameter index on central deflection(w/t)for various skew angles for centrally distributed porosity (Pc) on simply supported Si3N4/SUS304 PFGS plate.

Fig.16. Effect of porosity parameter index on central deflection(w/t)for various skew angles for Ptb type of porosity distribution on simply supported Si3N4/SUS304 PFGS plate.

The effect of various volume fraction grading index α=0,0.5,2,10, and ∞for Pe, Pc, and Ptbtype of distributions on a transient deflection of the PFGS plate are shown in Figs.17-19.It can be seen that the non-linear transient deflection increases with an increase in the volume fraction grading index (α). It is due to a decrease in the plate’s ceramic composition as α increases,hence the reduction in stiffness of the plate.Figs.17-19 reveals that the Petype of distribution exhibits the highest and Pctype presents the lowest nonlinear transient deflections for each value of α and follows the trend as Pe>Ptb>Pc.Hence,the distribution of porosity influences on the deflection of the PFGS plate.

Fig. 17. Effect of various volume fraction index on central deflection (w/t) for evenly distributed porosity on simply supported Si3N4/SUS304 PFGS plate.

Fig.18. Effect of various volume fraction index on central deflection(w/t)for centrally distributed porosity on simply supported Si3N4/SUS304 PFGS plate.

Fig. 19. Effect of volume fraction index on central deflection (w/t) for Ptb type of porosity distribution on simply supported Si3N4/SUS304 PFGS plate.

9. Conclusions

In this paper, the impact of porosity distributions for skewness on non-linear free vibration and transient responses of the porous functionally graded skew plates are investigated.The non-linear FE formulation is adopted in the framework of FSDT in conjunction with von Karman’s non-linear relations. The effective material properties are extracted from the modified power-law distribution taking into account the different types of porosity distributions in the plate. The governing equations of the PFGS plates are derived using the principle of virtual work incorporating a direct iterative technique to extract non-linear frequency parameters and the Newmark’s integration technique for the non-linear transient response. The reliability of the proposed model is checked by conducting various convergence and validation studies. From the detailed parametric analysis,the following observations are made.

1. In the presence of porosities, both NFR (ωnl/ω) and NFP (ωnl)increases with an increase in the amplitude ratio (Wmax/t).However, for the rise in skew angles, the NFR decreases;meanwhile, the NFP increases.

2. The NFR and NFP are more sensitive to the centrally distributed porosity (Pc) than evenly distributed (Pe) and Ptbtypes of porosity distributions for all skew angles. The effect of porosity distributions on the NFR follows the trend as Pc Pe>Ptb.

3. The NFR decreases for an increase in the skewness of the PFGS plate for both side to thickness ratio(a/t)and aspect ratio(b/a).

4. The NFR decreases with an increase in porosity parameter index for evenly distributed and centrally distributed porosities while increases for Ptbtype of porosity distribution.

5. The NFR value is minimum for the CCCF boundary condition and maximum for the simply supported case for any given type of porositydistributions.Itfollowsatrendas SSSS>SSSF>SCSC>CCCC>CCCF.Whereas NFP has the highest values for clamped plates and lowest for SSSS plates for all the kinds of porosity distributions.

6. The non-linear transient deflections of the PFGS plate increases with the increase in porosity parameter index, volume fraction grading index. In contrast, it decreases with the rise in skew angle. Besides, the distributions of porosities exhibit a significant impact on reducing the stiffness of the PFGS plates. Analogously, the effect of porosity distributions on transient deflection in decreasing pattern is Pe>Ptb>Pc.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

Authors would like to register their gratitude to Science and Engineering Research Board(SERB)ASEAN-India S&T Collaborative(AISTDF)Project No.IMRC/AISTDF/CRD/2019/000128,Govt.of India for providing all necessary grants and facilities to carry out the research.

Appendix A

The generalized nodal strain displacement matrices

whereas,[?tbn],[?1n],[?2n],[?rbn],[?tsn]and[?rsn](n=1,2,3,…,8)are the sub-matrices and are given by

Appendix B

Elemental Stiffness matrices

From Eq.(32),elemental stiffness matrices which correspond to bending and stretching deformations and transverse shear deformations are given by

The various rigidity vectors and the rigidity matrices are given by