Quasi-3D Refined Theory for Functionally Graded Porous Plates: Displacements and Stresses

This paper presents a higher-order shear and normal deformation theory for the static problem of functionally graded porous thick rectangular plates. The effect of thickness stretching in the functionally graded porous plates is taken into consideration. The functionally graded porous material properties vary through the plate thickness with a specific function. The governing equations are obtained via the virtual displacement principle. The static problem is solved for a simply supported plate under a sinusoidal load. The exact expressions for displacements and stresses are obtained. The influences of the functionally graded and porosity factors on the displacements and stresses of porous plates are discussed. Some validation examples are presented to show the accuracy of the present quasi-3D theory in predicting the bending response of porous plates. The effectiveness of the present model is evaluated by numerical results that include displacements and stresses of functionally graded porous plates. The field variables of functionally graded plates are very sensitive to the variation of the porosity factor.


INTRODUCTION
Porous material beams, plates, and/or shells are widely used in structural design problems. The permeability, tensile strength, and electrical conductivity depend on the properties of the matrix and fluid in pores, in so doing porosity is regarded as the main property of the porous material. Numerous theoretical and experimental studies have been carried out on porous structures, which operate as novel multifunctional ultralight structural forms with new mechanical properties.
Biot was the first who developed a 3D model of wave propagation in fluid-saturated porous elastic structures [1]. In the meantime, poroelasticity has been successfully used in studies of various problems in engineering structures [2,3]. Taber performed the quasi-static analysis of a poroelastic plate by applying the classical Kirchhoff plate theory [4]. As it is well known, Kirchhoffs assumptions are invalid for thick plates. Therefore, Busse et al. used the first-order Mindlin plate theory to model moderately thick plates based on Biots poroelastic theory without presenting numerical results [5]. Bîrsan used the porous plate theory to propose field equations for bending of thermoelastic plates [6]. Magnucki and Stasiewicz investigated elastic bending of isotropic porous beams with nonlinear hypotheses [7]. Nappa and Iesan presented thermoelastic problems of porous circular pipes and cylinders [8]. Magnucki et al. discussed bending and buckling of a rectangular porous plate taking into account the effect of shear deformation [9]. Magnucka-Blandzi discussed the nonlinear dynamic stability, deflection, and buckling of porous plates [10]. Yang et al. discussed quasi-static and dynamic bending of porous beams under step loads at their free ends [11]. Bîrsan and Altenbach presented the theory of porous elastic and thermoelastic thin rods modeled by the direct approach [12,13]. Kumar and Devi presented transient problems for thermoelastic porous materials taking into account the temperature dependence of material properties [14]. Ghiba considered bending of thermoelastic porous plates of the Mindlin type [15]. Sladek et al. presented the first-order Mindlin theory for the bending response of porous plates according to Biots poroelastic theory [16]. Lyapin and Vatulyan developed a ZENKOUR PHYSICAL MESOMECHANICS Vol. 23 No. 1 2020 mathematical model to study deformation of porous plates [17].
In recent years, functionally graded materials (.GMs) have found other applications in electrical appliances, energy transformation, biomedical engineering, optics, etc. However, in the manufacture of .GMs, porosities may occur in the materials during the sintering process. Chen et al. performed the buckling and bending analyses of functionally graded porous beams using the first-order beam theory [18]. Bensaid and Guenanou [19] applied the nonlocal Timoshenko beam model to present deflection and buckling of functionally graded nanoscale beams with porosity. Akbas dealt with nonlinear static deflections of functionally graded porous beams under the thermal effect with position-and temperature-dependent material properties [20]. Behravan Rad presented the static response of porous multidirectional heterogeneous structures resting on developed gradient elastic foundations [21]. Barati et al. [22] and Barati and Zenkour [23] discussed the electromechanical vibration of smart piezoelectric functionally graded plates with porosities. Barati and Zenkour presented the postbuckling behavior of beams reinforced with graphene platelets with porosities and geometrical imperfection [24]. Barati and Zenkour discussed the nonlocal strain gradient elasticity for wave propagation in graded nanoporous double nanobeam systems on the elastic foundation [25,26].
The present paper obtains displacements and stresses in functionally graded porous plates subjected to a sinusoidal distributed load. It is assumed that the material properties of the porous plate are changed through the plate thickness. The plate is graded according to a novel type of the polynomial law. The plate faces are perfectly homogeneous while the whole plate has a perfect porous homogeneous shape according to the volume fraction of voids (porosity) or the graded factors. The variational principle is used to derive the governing equations based on the quasi-3D theory. Several important aspects that affect displacements and stresses are discussed in details.

Structural Model
Consider an functionally graded thick rectangular plate with thickness h, length a, and width b, as depicted in .ig. 1. The Cartesian coordinate system is Let the plate be under a distributed load q(x, y) at the upper face 2. z h = + Youngs modulus E(z) may vary continuously through the plate thickness by means of the polynomial material law [27]: h where α represents the porosity volume function and p is the scalar parameter that defines gradation of material properties in the through-thickness direction, m E and c E are Youngs moduli of the lower (as metal) and upper (as ceramic) faces of a functionally graded plate, respectively. In the case of p = 0 or p → ∞, the plate becomes purely homogeneous (ceramic or metal) without any porosity.

Quasi-3D Theory
The quasi-3D shear deformation theory for plates is suitable for the following displacements , .
The stress-strain constitutive equations for the present porous plate are represented as where i δ = 1 for i = 1, 2, 3, i δ = 0 for i = 4, 5, 6, λ(z) and µ(z) are the Lamé coefficients given by and E(z) varies in the through-thickness direction of the porous plate as discussed previously in Eq. (1) while ν represents a fixed value of the Poisson ratio.

EQUILIBRIUM EQUATIONS
The static equations can be obtained by applying the principle of virtual displacements. It can be stated in its analytical form as where j N and j M ( j = 1, 2, 6) are the stress resultants and stress couples, j S is the additional stress couples, and l Q (l = 4, 5) and 3 N are the transverse and normal shear stress resultants. They are introduced in Eq. (6) as Then, by integrating Eq.
In addition, the natural boundary conditions may be represented by Substituting Eqs. (14) and (15) into Eq. (11) yields a system of algebraic equations expressed in a compact form as where [k] represents the symmetric matrix of differential operators and denotes the generalized force vector while The elements ij ji k k = of the symmetric matrix [k] are written in Appendix A.
The stress resultants can be obtained in terms of the total strains by substituting Eq. (6) into Eq. (10): The following simply supported boundary conditions are set at the side edges of the functionally graded plate: The external force according to the Navier solution can be expressed in the sinusoidal form as and 0 q denotes the load intensity in the plate centre. According to the Navier technique, one assumes the following forms for 0 0 , , u L 0 1 1 , , , w u L and 1 w that satisfy the above conditions: where U, V, W, X, Y, and Z are the arbitrary parameters. Equation (11) can be combined with Eqs. (14) and (15)

NUMERICAL RESULTS AND DISCUSSION
The bending response of simply-supported functionally graded porous thick rectangular plates subjected to a sinusoidally distributed load is investigated. The effect of thickness stretching 3 ε of the function-ally graded porous plates is taken into account in the present quasi-3D theory. The displacements and stresses determined here are reported according to the following dimensionless forms: The lower surface of the functionally graded plate is metal m (E = 70 GPa) while the upper surface is ceramic c (E = 380 GPa). Poissons ratio is fixed at ν = 0.3. As a validation example, the present deflection and in-plane stress will be compared with the classical plate theory (CPT), the first-order shear deformation theory (.SDT), that uses a shear correction factor 5 6 K = and those from Carrera et al. [33,34] and Neves et al. [35,36], which account for 3 0 ε ≠ and use the Carrera unified formulation (CU.). Note that displacement fields of the classical plate theory and first-order shear deformation theory are given from Eq. (2), respectively, by setting ψ(z) = 0 and ψ(z) = z,   Tables 1 and 2 contain dimensionless deflection and in-plane stress in nonporous (α = 0) functionally graded square plates for different side-to-thickness ratio a h and graded parameter p. The inclusion of the porosity factor α is represented in these tables. The present nonporous results (α = 0) are more accurate than those generated by the classical plate theory and first-order shear deformation theory. The present results are comparable with those of other quasi-3D theories even for thicker plates. This points to the use of the new assumption given in Eq. (3), which has a maximal effect   Table 1 shows that w decreases as a h increases and the graded parameter p decreases. Table 2 shows that 1 σ increases as a h increases. Note that 1 σ no longer increases as p increases and 1 σ has its minimum values for homogeneous ceramic and metal materials.
Note that displacements of the sinusoidal shear deformation theory [37] are given from Eq. (2) by putting 1 ( ) sin ( ) and 0. z h z h w ψ = π π = (24) Displacements of the higher-order shear deformation theory [39] are given by putting Displacements of the third-order shear deformation theory [40] are given by putting .inally, displacements of the inverse trigonometric shear deformation theory [41] are given by putting where r is the arbitrary parameter. The present results reported in Table 3 are in close agreement with various shear deformation theories. We can also say that the present theory is more accurate than those theories since it includes the effect of transverse normal strain 3 ( 0). ε ≠ Table 3 shows that the displacements u and w increase as the graded parameter p increases. The in-plane normal stress 1 σ and transverse shear stress 5 σ no longer increase as p increases. In addition, the tangential stress 6 σ no longer decreases as p increases.
In fact, the present quasi-3D theory gives results comparable with the inclusion of the porosity factor  (α = 0.1). It is clear that this factor has a significant effect on displacements and stresses. The difference in deflections w obtained by the present quasi-3D theory with porosity (α = 0.1) and nonporosity (α = 0) may increase as p increases and a h decreases. The porosity normal stress 1 σ is greater than the nonporosity stress for functionally graded plates with 0 < p < 4 and vice versa with 4 ≤ p < ∞. In addition, the porosity in-plane displacement u is greater than the nonporosity in-plane displacement u for functionally graded plates for all p. .inally, the porosity shear stress 5 σ {tangential stress 6 } σ is greater {smaller} than the nonporosity shear stress 5 σ {tangential stress 6 } σ for functionally graded plates for all p.
The inclusion of the porosity factor will be investigated in .igs. 29. In these figures we assume, unless otherwise stated, that α = 0.1, p = 5, a h = 5, and b a = 2. All figures illustrate the variable quantities through the plate thickness versus the thickness ratio a h and the aspect ratio . a b .igure 2a shows that the deflection w increases as p increases for the fixed α = 0.1, while .ig. 2b shows that the deflection increases as α increases for the fixed p = 5. A porosity deflection value at α = 0.25 may triple as compared to nonporosity one. .igure 2c shows that the deflection decreases slightly as a h increases, while .ig. 2d shows that the deflection decreases rapidly as a b increases. The differences between deflections may be fixed at the variation of a h while these differences are larger for small aspect ratios.
.igure 3a shows that the porosity metallic in-plane displacement u at the lower surface z = 0.5 is the greatest one while the porosity ceramic in-plane displacement u is the smallest one. The porosity .G inplane displacement u increases as p increases. The inverse occurs at the upper surface z = 0.5. The same occurs in .ig. 3b. It is seen from .ig. 3b that the variation of the porosity factor has no effect on the in-plane displacement u at the level of 1 3. z = .igures 3c and 3d show that the in-plane displacement u decreases as a h and a b increase. The differences between the in-plane displacements u are larger for small thickness a h and aspect a b ratios. The variation of the porosity factor may have no effect on u for larger aspect ratios ( 3). a b > .igure 4a shows that the porosity metallic in-plane normal stress 1 σ is the same as the porosity ceramic in-plane normal stress. .igure 4b shows that the porosity  σ increases as a h increases and a b decreases. The normal stress 1 σ increases as the porosity factor α increases. The differences between the in-plane normal stresses 1 σ are larger for a large thickness ratio a h and a small aspect ratio . a b .igure 5 shows that the in-plane longitudinal stress 2 σ behaves similarly to the normal stress 1 .
σ The maximum in-plane longitudinal stress 2 σ occurs at a b = 0.6 as shown in .ig. 5d. Once again .ig. 6a shows that the porosity metallic out-of-plane transverse normal stress 3 σ is the same as the porosity ceramic one. .igure 6b shows that the porosity functionally graded transverse normal stress 3 σ is very sensitive to the variation of the porosity factor α at the level z = 0.38. .igure 6c shows that 3 σ increases as a h increases for α = 0.25. At α = 0.2, 3 σ no longer decreases as a h increases and the minimum value of 3 σ occurs at a h = 6.8. However, for α ≤ 0.15 the out-of-plane transverse normal stress 3 σ directly decreases as a h increases. .igure 6d shows that 3 σ slightly increases as a b increases only for nonporous plates (α = 0). When α = 0.1, 3 σ is not affected by the variation of the aspect ratio . a b However, at α ≥ 0.15 3 σ decreases as a b increases. .igures 7 and 8 show that the transverse shear stresses 4 σ and 5 σ have the same behavior through the thickness of the functionally graded porous and nonporous plates. The shear stresses are the same for ceramic and metallic homogeneous plates. The maximum shear stresses have different positions according the value of the graded parameter p. However, the maximum shear stresses occur at the position ( 1 3) z = for different porosity factors α. In this position, the maximum shear stress increases as α increases. The transverse shear stresses are affected by the variation of . a h .inally, 5 σ directly decreases as a b increases while 4 σ no longer increases as a b increases and has its maximum value at a b = 1. In the meantime, the transverse shear stresses decrease as α increases.
.igure 9a shows that the porosity metallic in-plane tangential stress 6 σ is the same as the porosity ceramic one. .igure 9b shows that the porosity functionally graded tangential stress 6 σ increases as α increases in the interval 0.16 < z < 0.38. Otherwise, 6 σ in- (d) (c) creases as α decreases. .igure 9c shows that 6 σ increases as a h increases. In addition, the tangential stress 1 σ increases as the porosity factor α decreases.
The differences between the in-plane tangential stresses 6 σ are larger for a large thickness ratio a h and small aspect ratio . a b Once again, the maximum in-plane tangential stress 6 σ occurs at a b = 0.6, as shown in .ig. 9d.

CONCLUSIONS
A refined quasi-3D shear and normal deformation theory is developed for functionally graded porous plates. The effect of thickness stretching in functionally graded porous plates is presented. The governing equations are derived and analytical solutions for simply supported functionally graded porous rectangular plates are obtained. The inclusion of the graded and porosity parameters is investigated. Many validation examples are reported, and numerical results of the present quasi-3D theory are accurate in predicting the bending response of nonporous plates. All stresses for porous or nonporous homogeneous plates are the same.
That is because the stresses are independent of Youngs modulus. The displacements and stresses are very sensitive to the variation of the porosity factor for functionally graded plates. It should be noted that the inclusion of the porosity factor is the main property of the porous materials, especially for functionally graded structures.

Appendix A
The elements of the symmetric matrix [k] presented in Eq. (17)