Nonlocal Strain Gradient Theory for the Bending of Functionally Graded Porous Nanoplates

Many investigators have become interested in nanostructures due to their outstanding mechanical, chemical, and electrical properties. Two-dimensional nanoplates with higher mechanical properties compared with traditional structural applications are a common structure of nanosystems. Nanoplates have a wide range of uses in various sectors due to their unique properties. This paper focused on the static analysis of functionally graded (FG) nanoplates with porosities. The nonlocal strain gradient theory is combined with four-variable shear deformation theory to model the nanoplate. The proposed model captures both nonlocal and strain gradient impacts on FG nanoplate structures by incorporating the nonlocal and strain gradient factors into the FG plate’s elastic constants. Two different templates of porosity distributions are taken into account. The FG porous nanoplate solutions are compared with previously published ones. The impact of nonlocal and strain gradient parameters, side-to-thickness ratio, aspect ratio, and porosity parameter, are analyzed in detail numerically. This paper presents benchmark solutions for the bending analysis of FG porous nanoplates. Moreover, the current combination of the nonlocal strain gradient theory and the four-variable shear deformation theory can be adapted for various nanostructured materials such as anisotropic, laminated composites, FG carbon nanotube reinforced composites, and so on.

The local (classical) elasticity theory cannot predict the minuscule size influence on nanostructures because of the lack of a nonlocal elasticity theory. Unfortunately, because the material length scale lacks a nonlocal parameter, the local (classical) elasticity theory cannot anticipate the small size influence on nanostructures. Experiments and molecular dynamics (MDs) simulations can be costly, time-consuming, and difficult to model nanostructure systems. Several nonlocal elasticity theories, including Eringen's nonlocal theory, couple stress theory, strain gradient theory, surface stress theory, and the modified couple stress theory [18][19][20][21][22][23][24][25] have been used to build such atypical plate models to overcome this issue. Using Eringen's nonlocal relations, Pradhan and Phadikar [26] utilized the classic and first-order shear deformation theories. Via Eringen's nonlocal linear theory that depends on third-order theory, Aghababaei and Reddy [27] examined the vibration and bending of nanoplates. Shen et al. [28] discussed the vibration of a one-layered graphene sheet via a nanomechanical sensor via the first-order theory. Malekzadeh and Shojaee [29] used and according to uneven porosity distribution, the material properties can be written as (Daikh and Zenkour [53]) where c and m are abbreviations for ceramic and metal, respectively. ζ (0 ≤ ζ 1) denotes the porosity coefficient and setting α = 0 yields the mechanical characteristics for a perfect FG porous nanoplate plate. k denotes the power law exponent (k ≥ 0). nanoplate for porosities with even distribution are described based on the modified power law function as (Daikh and Zenkour [53]) and according to uneven porosity distribution, the material properties can be written as (Daikh and Zenkour [53]) where and are abbreviations for ceramic and metal, respectively. (0 ≤ ≪ 1) denotes the porosity coefficient and setting = 0 yields the mechanical characteristics for a perfect FG porous nanoplate plate. denotes the power law exponent ( ≥ 0).

The Nonlocal Strain Gradient Theory for FG Porous Nanoplate
The nonlocal strain gradient theory states that the stress field takes into account both the nonlocal elastic stress field and the strain gradient stress field. The constitutive equation for the FG porous nanoplate can be written as (Askes and Aifantis [54]) where ∇ = + is the Laplacian operator. and represent the nonlocal and strain gradient length scale parameters. Setting the strain gradient parameter to zero ( = 0) yields Eringen's nonlocal model, whereas setting the nonlocal parameter to zero ( = 0) yields Kirchhoff's strain gradient model. The plate stiffness coefficients are written as where ( ) is young's modulus and ( ) is Poisson's ratio.

Displacement Model
The displacement model for the FG porous nanoplate is introduced in this study as (Shimpi and Patel [55])

The Nonlocal Strain Gradient Theory for FG Porous Nanoplate
The nonlocal strain gradient theory states that the stress field takes into account both the nonlocal elastic stress field and the strain gradient stress field. The constitutive equation for the FG porous nanoplate can be written as (Askes and Aifantis [54]) where ∇ 2 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is the Laplacian operator. η and λ represent the nonlocal and strain gradient length scale parameters. Setting the strain gradient parameter to zero (λ = 0) yields Eringen's nonlocal model, whereas setting the nonlocal parameter to zero (η = 0) yields Kirchhoff's strain gradient model. The plate stiffness coefficients c ij are written as where E(z) is young's modulus and ν(z) is Poisson's ratio.

Displacement Model
The displacement model for the FG porous nanoplate is introduced in this study as (Shimpi and Patel [55]) where u and v are the mid-plane displacements. The transverse displacement u 3 is divided into bending (w b ) and shear (w s ) components. The shape function that specifies the variation in transverse shear stresses across the thickness of the nanoplate is taken as (Thai and Kim [56] According to the linear elasticity theory, the strain-displacement relations are provided by The strain relations based on the mentioned displacement field in Equation (5) can be written as

Governing Equations
Using Hamilton's principle, the following governing equations and associated boundary conditions are derived as follows where q is the distributed transverse load. After replacing the components of strains from Equation (6a) into Equation (7), and thereafter integrating through the z-direction, Equation (7) can be written as where the stress resultants N ij , M ij , S ij , and Q iz are defined as The governing equations can be derived by the integration of Equation (8) and then collecting the coefficients of δu, δv, δw b , and δw s as follows Inserting the constitutive equations from Equation (3) into Equation (9), the stress resultants for the nanoplate based on the nonlocal strain gradient theory can be expressed as where the quantities mentioned in the above equations are defined as follows

Closed-Form Solution
The solution is found for a rectangular FG porous nanoplate. The FG nanoplate is assumed to be fully simply supported. The next boundary conditions are imposed on the nanoplate four edges. Following Navier method, assume the solution of the displacement components in the following form (Reddy [57]) where α = π/a, β = π/b (15) in which (U, V, W b , W s ) are the unknowns to be determined. A trigonometric development is used for the mechanical load as where q 0 denote the concentration of the distributed load at the nanoplate center. Substituting Equations (14) and (16) in Equation (10), reveals where {∆} and {F} denote the following and the nonzero elements a ij = a ji of the symmetric matrix [A] are in the Appendix A.

Results and Discussions
The obtained results are introduced to demonstrate the influence of nonlocal (η) and length scale (λ) parameters on the bending of porous FG plates with even and uneven porosity distributions. The FG porous nanoplate has a length a = 10 nm and the constituent materials have the following characteristics

Verification Analysis
The accuracy of the present model under the nonlocal effect and without considering strain gradient effect (λ = 0) is verified. In this case the nonlocal effect of FG nanoplates is investigated. A validation example compared with Sobhy [58] and Hoa et al. [59] is provided using the following non-dimensional displacement and stresses Table 1 shows the non-dimensional deflection and stresses of a local (η = 0) and nonlocal (η = 2) FG square nanoplate without the effect of porosities for two values of inhomogeneity parameter k. According to this case, the present numerical results are in good agreement to those of Sobhy [59] and Hoa et al. [59], for all cases of nonlocal coefficient and inhomogeneity parameters. Table 2 exhibits the variation in the non-dimensional deflection w in a square FG nanoplate (without considering the porosity factor) in terms of the nonlocal coefficient and length-to-thickness ratio. The current findings are consistent with those presented by Hoa et al. [59].

Parametric Analysis
This subsection presents numerical results for the FG porous nanoplate according to the nonlocal strain gradient theory. The influence of different values of the nonlocal parameter (η), length scale parameter (λ), porosity coefficient (ζ), length-to-thickness ratio (a/h), and aspect ratio (a/b) on the bending of the FG porous nanoplate plates are investigated. The dimensionless displacement and stresses used for the present results are The parameters considered (except otherwise clarified) are k = 2, a/h = 10, a/b = 1, λ = 1, η = 2. The variation in the non-dimensional deflection and stresses in a square  Even and uneven types of porosity distribution with two values of porosity coefficients are discussed in addition to the perfect case. It can be observed from these tables that the center deflection and stresses increase by the decreasing in length scale parameter λ and by the increasing in the nonlocal parameter η, whatever the FG nanoplates type and porosity coefficient ζ are. Furthermore, it can be concluded that as the porosity coefficient ζ increases, so do the deflection w and in-plane normal stress σ 1 as depicted in Tables 3 and 4. This is due to a decrease in plate stiffness caused by the presence of porosity. Tables 5 and 6 reveal that the transverse shear stress σ 5 and the in-plane shear stress σ 6 are decreased with the existence of porosity factor ζ. It can be concluded that the deflection and stresses results of FG nanoplates with uneven porosities have lower values than the ones with even porosities except for the in-plane stress σ 6 .
The center deflection w variation for three types of FG nanoplates versus the length scale parameter λ and the nonlocal parameter η are displayed in Figure 2a and 2b, respectively. The center deflection variation for three types of FG nanoplates versus the length scale parameter and the nonlocal parameter are displayed in Figure 2a and 2b, respectively. According to Figure 2, the presence of porosity increases deflection significantly when compared with nonporous FG nanoplates. The nanoplates with porosities that are unevenly distributed deflect less than those that are evenly distributed. Moreover, as previously mentioned, for the three types of FG nanoplates, the center deflection increase by the decreasing in and by the increasing in . In Figure 3a,b, the center deflection variation for five types of FG nanoplates in terms of aspect ratio / and length-to-thickness ratio /ℎ is demonstrated, respectively. The aspect ratio impact on the central deflection is much larger than that of the length-to-thickness ratio for all FG nanoplates types. The FG nanoplates with even porosities ( = 0.25) have the largest deflections of all types.  According to Figure 2, the presence of porosity increases deflection significantly when compared with nonporous FG nanoplates. The nanoplates with porosities that are unevenly distributed deflect less than those that are evenly distributed. Moreover, as previously mentioned, for the three types of FG nanoplates, the center deflection increase by the decreasing in λ and by the increasing in η. In Figure 3a,b, the center deflection w variation for five types of FG nanoplates in terms of aspect ratio a/b and length-to-thickness ratio a/h is demonstrated, respectively. The aspect ratio impact on the central deflection is much larger than that of the length-to-thickness ratio for all FG nanoplates types. The FG nanoplates with even porosities (ζ = 0.25) have the largest deflections of all types. The center deflection variation for three types of FG nanoplates versus the length scale parameter and the nonlocal parameter are displayed in Figure 2a and 2b, respectively. According to Figure 2, the presence of porosity increases deflection significantly when compared with nonporous FG nanoplates. The nanoplates with porosities that are unevenly distributed deflect less than those that are evenly distributed. Moreover, as previously mentioned, for the three types of FG nanoplates, the center deflection increase by the decreasing in and by the increasing in . In Figure 3a,b, the center deflection variation for five types of FG nanoplates in terms of aspect ratio / and length-to-thickness ratio /ℎ is demonstrated, respectively. The aspect ratio impact on the central deflection is much larger than that of the length-to-thickness ratio for all FG nanoplates types. The FG nanoplates with even porosities ( = 0.25) have the largest deflections of all types.    Figure 4a. Figure 4b shows that the in-plane normal stress σ 1 have the lowest values at the FG nanoplate upper surface with the uneven porosity distribution and λ = 2 while the even porosity distribution and λ = 2 cause the lowest stresses at the lower surface. For FG nanoplates, the variation in shear stress σ 5 through the thickness is not parabolic, and maximum values do not happen in the center of the plates as shown in Figure 4c.
Furthermore, the FG nanoplates with even and uneven porosities are identical in the upper 15% of the thickness. It can be found from Figure 4d that the in-plane stress σ 6 for the uneven type are equal to zero at z ∼ = 0.18 while for even type are equal to zero at z ∼ = 0.19.
With fixed length scale parameter (λ = 1) and two different values of the nonlocal parameter (η = 0, 2), the variation in center deflection and stresses are depicted in Figure 5 for FG nanoplate with even and uneven porosities. The lowest deflections of all types happened in the case of uneven porosities and η = 0 as shown in Figure 5a. The lowest values of in-plane normal stress σ 1 at the nanoplate upper surface occur with the case of uneven porosity distribution (η = 0) while the lowest values of stresses at the nanoplate lower surface happened with the even porosity distribution case (η = 0). The maximum values of shear stress σ 5 occur at z ∼ = 0.24 for all types of FG nanoplates as shown in Figure 5c.  Figure 4a. Figure 4b shows that the in-plane normal stress have the lowest values at the FG nanoplate upper surface with the uneven porosity distribution and = 2 while the even porosity distribution and = 2 cause the lowest stresses at the lower surface. For FG nanoplates, the variation in shear stress through the thickness is not parabolic, and maximum values do not happen in the center of the plates as shown in Figure 4c. Furthermore, the FG nanoplates with even and uneven porosities are identical in the upper 15% of the thickness. It can be found from Figure 4d that the in-plane stress for the uneven type are equal to zero at ≅ 0.18 while for even type are equal to zero at ≅ 0. 19.
With fixed length scale parameter ( = 1) and two different values of the nonlocal parameter ( = 0, 2), the variation in center deflection and stresses are depicted in Figure 5 for FG nanoplate with even and uneven porosities. The lowest deflections of all types happened in the case of uneven porosities and = 0 as shown in Figure 5a. The lowest values of in-plane normal stress at the nanoplate upper surface occur with the case of uneven porosity distribution ( = 0) while the lowest values of stresses at the nanoplate lower surface happened with the even porosity distribution case ( = 0). The maximum values of shear stress occur at ≅ 0.24 for all types of FG nanoplates as shown in Figure 5c. It is apparent that the in-plane stresses do not depend on the nonlocal parameter for both even and uneven cases at ≅ 0.19 as illustrated in Figure 5d. Figure 6 shows the variation in center deflection and stresses across the thickness direction in a square FG nanoplate with uneven and even porosity distribution and two values of . The smallest deflections happened for the uneven case ( = 0.15) as demonstrated in Figure 6a. From Figure 6b, we observe that the even porosity case ( = 0.25) causes the smallest values of normal stress at the plate's lower surface and the highest values at the plate's upper surface. The maximum value of shear stress at ≅ 0.23 5 for FG nanoplate with even and uneven porosities. The lowest deflections of all types happened in the case of uneven porosities and = 0 as shown in Figure 5a. The lowest values of in-plane normal stress at the nanoplate upper surface occur with the case of uneven porosity distribution ( = 0) while the lowest values of stresses at the nanoplate lower surface happened with the even porosity distribution case ( = 0). The maximum values of shear stress occur at ≅ 0.24 for all types of FG nanoplates as shown in Figure 5c. It is apparent that the in-plane stresses do not depend on the nonlocal parameter for both even and uneven cases at ≅ 0.19 as illustrated in Figure 5d. Figure 6 shows the variation in center deflection and stresses across the thickness direction in a square FG nanoplate with uneven and even porosity distribution and two values of . The smallest deflections happened for the uneven case ( = 0.15) as demonstrated in Figure 6a. From Figure 6b, we observe that the even porosity case ( = 0.25) causes the smallest values of normal stress at the plate's lower surface and the highest values at the plate's upper surface. The maximum value of shear stress at ≅ 0.23 It is apparent that the in-plane stresses σ 6 do not depend on the nonlocal parameter η for both even and uneven cases at z ∼ = 0.19 as illustrated in Figure 5d. Figure 6 shows the variation in center deflection and stresses across the thickness direction in a square FG nanoplate with uneven and even porosity distribution and two values of ζ. The smallest deflections happened for the uneven case (ζ = 0.15) as demonstrated in Figure 6a. From Figure 6b, we observe that the even porosity case (ζ = 0.25) causes the smallest values of normal stress σ 1 at the plate's lower surface and the highest values at the plate's upper surface. The maximum value of shear stress σ 5 at z ∼ = 0.23 are caused by the even porosity case with ζ = 0.25 as shown in Figure 6c. It can be observed from Figure 6d that the in-plane stresses σ 6 are not dependent on the porosity coefficient ζ and the porosity type at z ∼ = −0.28 and z ∼ = 0.28. are caused by the even porosity case with = 0.25 as shown in Figure 6c. It can be observed from Figure 6d that the in-plane stresses are not dependent on the porosity coefficient and the porosity type at ≅ −0.28 and ≅ 0.28. The two-dimensional (2D) distribution of the deflection and stresses of FG rectangular nanoplate ( / = 1/2) with even porosities in terms of and are examined in Figure 7. It can be observed from Figure 7a,b that the increasing in the porosity coefficient and the nonlocal parameter led to an increment of central deflection and normal stress . It can be found from Figure 7c that shear stress is increased with the decrease in porosity coefficient and the increase in the nonlocal parameter . Furthermore, we can observe that the influence of on the in-plane stress is more significant with the increment of as shown in Figure 7d. The two-dimensional (2D) distribution of the deflection and stresses of FG rectangular nanoplate (a/b = 1/2) with even porosities in terms of ζ and η are examined in Figure 7. It can be observed from Figure 7a,b that the increasing in the porosity coefficient and the nonlocal parameter led to an increment of central deflection and normal stress σ 1 . It can be found from Figure 7c that shear stress σ 5 is increased with the decrease in porosity coefficient ζ and the increase in the nonlocal parameter η. Furthermore, we can observe that the influence of ζ on the in-plane stress σ 6 is more significant with the increment of η as shown in Figure 7d.     In Figure 9, the variation in center deflection and stresses across the thickness direction in a square FG nanoplate with even porosity distribution and different values of the power law exponent is plotted. Figure 9a shows that the deflection increases as increases for fixed = 0.15, = 1 and = 2. The in-plane normal stresses of the FG porous nanoplate are tensile at the upper surface and compressive at the lower surface, as depicted in Figure 9b. The homogeneous FG porous nanoplate yields the highest compressive stresses at the lower surface and the lowest tensile stresses at the upper surface. Figure 9c captures the non-dimensional shear stress along the plate thickness of various FG porous nanoplates for different power law exponent k values. For the homogeneous FG porous nanoplate, the maximum results occur at a point on the mid-plane. The maximum values for the other FG porous nanoplate, on the other hand, are at different locations of the FG porous nanoplate. For = 4, the highest magnitude of the FG porous nanoplate is obtained.
Finally, Figure 9d illustrates the through thickness distribution of in-plane tangential stresses of FG porous nanoplate for various exponential factors. The in-plane tangential stresses, unlike in-plane normal stresses , are tensile at the lower surface and compressive at the upper surface. The highest compressive stress is found at the upper surface of the FG porous nanoplate for = 10. In Figure 9, the variation in center deflection and stresses across the thickness direction in a square FG nanoplate with even porosity distribution and different values of the power law exponent is plotted. Figure 9a shows that the deflection increases as k increases for fixed ζ = 0.15, λ = 1 and η = 2. The in-plane normal stresses σ 1 of the FG porous nanoplate are tensile at the upper surface and compressive at the lower surface, as depicted in Figure 9b. The homogeneous FG porous nanoplate yields the highest compressive stresses at the lower surface and the lowest tensile stresses at the upper surface. Figure 9c captures  Finally, Figure 9d illustrates the through thickness distribution of in-plane tangential stresses σ 6 of FG porous nanoplate for various exponential factors. The in-plane tangential stresses, unlike in-plane normal stresses σ 1 , are tensile at the lower surface and compressive at the upper surface. The highest compressive stress is found at the upper surface of the FG porous nanoplate for k = 10. , (c) , (d) .

Conclusions
This paper is interested in developing the nonlocal strain gradient theory for the analysis of the bending of FG porous nanoplates. A four-variable shear deformation theory is used for the modeling of the FG porous nanoplate. Two different porosity distributions are considered in this paper. The equilibrium equations are described in detail and derived utilizing the virtual work concept and Navier's procedure. The nonlocal and strain gradient parameters, porosity factor, length-to-thickness ratio, and aspect ratio are all explored. Comparison studies are provided. Additional results are provided to serve comparison purposes. The major findings are as follows: • The presence of a porosity factor has a significant impact on the static response of the FG nanoplate and FG nanoplates with unevenly distributed porosities deflect less than those with evenly distributed porosities; • The impact of the nonlocal parameter on the central deflection and stresses is opposite to that of the response of strain gradient parameter under the same conditions; • The impact of aspect ratio / on the central deflection is much larger than that of the side-to-thickness ratio /ℎ;

Conclusions
This paper is interested in developing the nonlocal strain gradient theory for the analysis of the bending of FG porous nanoplates. A four-variable shear deformation theory is used for the modeling of the FG porous nanoplate. Two different porosity distributions are considered in this paper. The equilibrium equations are described in detail and derived utilizing the virtual work concept and Navier's procedure. The nonlocal and strain gradient parameters, porosity factor, length-to-thickness ratio, and aspect ratio are all explored. Comparison studies are provided. Additional results are provided to serve comparison purposes. The major findings are as follows: • The presence of a porosity factor has a significant impact on the static response of the FG nanoplate and FG nanoplates with unevenly distributed porosities deflect less than those with evenly distributed porosities; • The impact of the nonlocal parameter on the central deflection and stresses is opposite to that of the response of strain gradient parameter under the same conditions; • The impact of aspect ratio a/b on the central deflection is much larger than that of the side-to-thickness ratio a/h;

•
The results in the tables and figures indicate that the decreasing in the length scale parameter λ increases the center deflection and stresses; • The center deflection and stresses increase by the increase in the nonlocal parameter η for all types of FG porous nanoplates. It implies that the nonlocal parameter may be capable of reducing the stiffness of FG porous nanoplates.