Exact solution for bending analysis of functionally graded micro-plates based on strain gradient theory

1 Literature survey

Requirements of new high technology devices, which are used in different fields of engineering such as aeronautical or biomechanical applications, expand wide subjects for researchers. Recent advances in the Micro electro mechanical systems (MEMS) and Nano electro mechanical systems (NEMS) have necessitated special studies in these fields. NEMS and MEMS are usually made out of different parts which can be modeled as beams or plates. Therefore, it is important to survey an accurate comprehending of their behavior.

Experimental studies on the plates and beams show that structures in small scales are influenced by their dimensions [1], [2], [3], [4], [5]. Hence, classical theories that were used for analyzing plates are modified [6]. For investigating micro-plates, different theories have been proposed by researchers such as strain gradient theory [7], [8], couple stress theory [9] and micro polar theory [10], [11]. Considering the effect of micro-structures to capture the size effects in these theories, constitutive equations are defined.

Among the theories, strain gradient theory is one of the most usable ones which is a higher-order generalized form of classical elasticity theory. According to the general form of Mindlin theory [7], strain energy density is composed of three terms, i.e. displacements, gradients of strain and gradients of rotation. In the couple stress theory, it is assumed that the energy density contains expressions of gradient of rotation. Mindlin and Eshel considered the first-order gradient of strains in the energy density and proposed that a form contains five elastic constants in addition to two usual Lamé constants for an isotropic linear elastic micro-structured solid [8].

A review of the above mentioned higher-order theories of elasticity can be found in the references [12], [13], [14], [15], [16]. According to the modified couple stress theory, anti-symmetric part of gradient of rotation tensor is considered as the higher-order terms in the constitutive equations. In the modified strain gradient theory second-order deformations are considered with three material length scale parameters. It is important to note that in the modified strain gradient theory governing equations have higher order in comparison with modified couple stress theory. In the simple form of strain gradient theory, a gradient of strains is considered for capturing the small-scale effects in the micro-structure [12].

A size-dependent model for thin micro-plates (Kirchhoff micro-plates) was presented by Farahmand et al. [17], [18], [19] and Wang et al. [20]. They presented length scale parameters to capture the effect of size on the solution and improved the modified strain gradient theory in order to predict accurate results for micro-plates. Lazopoulos investigated bending of strain gradient elastic thin plates based on the Kirchhoff plate theory [21]. The variational approach was utilized in determination of the governing equations and boundary conditions. Subsequently, the effect of cross-sectional area on bending of thin micro-plates was discussed. A gradient strain elasticity theory of plates is developed for the study of non-linear problems by Lazopoulos [22]. The theory is applied to the study of the buckling behavior of a long rectangular plate under uniaxial compression and small lateral load, supported on a rigid plane foundation.

Since MEMS and NEMS components are usually subjected to thermal effects (because of electrical resistance), applying heat-resistant materials in construction of their components is an important issue for engineers. Recent developments in the metallurgy engineering lead to suggest composites with continuous distribution of material properties. Consequently, functionally graded (FG) distribution of materials was the solution for the problem [23]. The most usable functionally graded materials (FGMs) are usually combinations of ceramic and metal. Besides the important mechanical properties of the metal part, simultaneously the ceramic component in FGMs improves the thermal capacity of the composite. Therefore, FGMs can be considered as appropriate materials for micro-structures in a variety of applications, especially in MEMS. Several studies on the application of FGMs for classical plate models were carried out by Mohammadi et al. [24], [25]. Several studies were done on micro-beams made out of FGMs, e.g. Ke and Wang investigated the dynamic stability of micro-beams made out of FGMs [26]. They used the modified couple stress theory for analyzing a Timoshenko beam. It was assumed that material properties of the FG micro-beam fluctuate through the thickness, and also material properties were estimated by Mori-Tanaka homogenization technique. Consequently, the effects of length scale parameter and material properties on the dynamic stability of the beam were presented. Based on the modified couple stress theory, a new size-dependent formulation for the FG Timoshenko beam was derived by Asghari et al. [27]. In that study, material properties were considered to vary through the thickness by power law distribution. The static and vibration response of cantilever and simply supported beams were determined for different length scale parameters. Furthermore, nonlinear vibration of size-dependent FG Timoshenko micro-beams was employed by Ke et al. [28]. Modified couple stress theory of FG Timoshenko micro-beams, in addition to von-Kármán geometric nonlinearity, was used in their study to obtain higher-order nonlinear governing equations. Moreover, different boundary conditions were surveyed by using Hamilton principle in their work. Static analysis of thin FG rectangular plate with arbitrary boundary conditions were conducted by Pradhan and Chakraverty [29]. They used Rayleigh-Ritz method for determining the static response of plate subjected to uniformly distributed load and hydrostatic pressure. Sitar et al. [30] studied the large deflection of nonlinearly elastic FGM beam. It was supposed that the beam is made of finite number of laminae, and material properties vary arbitrarily through the thickness. Free vibration analysis of micron-scaled annular sector and sector graphene is firstly investigated by Civalek and Akgoz [31]. They studied the effect of the surrounded elastic matrix via two-parameter elastic foundation models. Also, sector shape is transformed via geometric transformation based on the nonlocal continuum theory.

Akgoz and Civalek presented analytical solutions for buckling, bending and vibration of micro-sized plates resting on elastic foundations using the modified couple stress theory [32]. Governing equations for bending, buckling and vibration were determined via Hamilton’s principles together with the modified couple stress and Kirchhoff plate theories. The surrounding elastic medium is modeled as the Winkler elastic foundation for micro-plate with four edges simply supported. Natural frequencies of FG nano-plates were analyzed for different combinations of boundary conditions by Zare et al. [33]. They presented an analytical solution for solving the governing equations of motion. Zhang et al. [34] develop a novel size-dependent plate model for the axisymmetric bending, buckling and free vibration analysis of FG circular/annular micro-plates based on the strain gradient elasticity theory. Refined third-order shear deformation theory which assumes that the in-plane and transverse displacements are partitioned into bending and shear components and satisfies the zero traction boundary conditions on the top and bottom surfaces of micro-plate is utilized for analysis.

As it was reviewed briefly, most of the studies on the FG micro-structures have been limited to beams and not plates. Hence, the present study is emphasised on bending analysis of thin FG rectangular micro-plates. In accordace to the strain gradient theory along with inclusion of one length scale parameter, governing equations are obtained for flextural rectangular micro-plate. Since micro-plate is made out of FGMs, there is a coupling in the governing equations which is removed by introducing a new method. Decoupled equations are solved analyticaly for a micro-plate with Levy boundary conditions. Finally, the effect of length scale parameter, material properties and boundary conditions on the micro-plate deflection are investigated in detail.

2 Strain gradient elasticity theory

In small-scale structures such as micro-structures, the size and micro-structural effects are important and affect mechanical behaviors. In strain gradient theory, the size effects are involved in the stress-strain relations by implementing gradient of strains, which in general form is written as

where σ₀ is initial stress, ε is strain, g is length scale parameter and η is strain gradient. The simplest possible version of strain gradient elasticity theory based on the Mindlin studies [7], [8] (which contains five length scale constants in addition to the two Lamé constants) is a model with just one length scale constant in addition to the Lamé constants. The constitutive equations for this model are given as [35]

In the above equations, σ¯ and τ¯ are the total and the classical Cauchy stress tensors, respectively. Also, parameter I is the unit tensor, ε¯ and tr(ε¯) are the strain tensor and its trace which are expressed in terms of the displacement vector u as:

In Equation (2), g² is the volumetric strain energy gradient coefficient or simply gradient coefficient where g (length scale parameter) is defined as the internal or characteristic length of a micro-structure. Also, λ and μ are the two classical Lamé constants.

Imposing the strain gradient term with length scale parameter in conventional elasticity as a constraint was first discussed by Farahmand and Arabnejad [36]. Comparison of experimental results from torsion and bending tests of beams with the theoretical ones obtained from the study and other higher-order elasticity models reveal that the magnitude of the gradient coefficient g (length scale parameter) is of the same order as the diameter of the basic building block in a micro-structure, e.g. the grain in metals or ceramics, the osteon in bones or the cell in foams [16].

3 Governing equations of functionally graded micro-plate

3.1 Fundamental equations

Consider a thin flat rectangular micro-plate subjected to arbitrary transverse distributed load q(x, y) as shown in Figure 1, where z coordinate is measured through the thickness direction and x and y are in-plane coordinates. Based on the classical (Kirchhoff) plate theory, the following displacement field is expressed as

Figure 1:

Rectangular micro-plate subjected to distributed load.

(4)ux(x, y, z)=u−zw,x   uy(x, y, z)=v−zw,y    uz(x, y, z)=w(x, y)

where u_x and u_y are in-plane displacement in x and y directions, respectively and u_z is the transverse displacement. Also, u and v are midplane displacements and (,) indicates differentiation with respect to the variables. Therefore, the strain-displacement relations of micro-plate (considering Von-Kármán hypothesis for linear terms) are shown in Equation (5a) where strain and curvature components are defined in Equation (5b).

(5a)εx=ε¯x+zkx   εy=ε¯y+zky   2εxy=ε¯xy+zkxy

(5b)ε¯x=u,xε¯y=v,yε¯xy=u,y+v,xkx=−w,xxky=−w,yykxy=−2w,xy

Also, the strain energy for a continuum medium is defined as

(6)U=12∫VσijεijdV   (ij=x, y, xy)

where V is the volume of media. In order to obtain the equilibrium equations, principle of minimum total potential energy is used. Thus, variation of strain energy is simplified as

(7)δU=12∫∬A(σxxδεxx+σyyδεyy+2σxyδεxy)dAdz=12∫∬A(σxxδε¯x+zσxxδkx+σyδε¯y+zσyδky+ 2σxyδε¯xy+2σxyδkxy)dAdz

In addition, the work done by the transverse distributed load per unit area q(x, y) is

(8)δW=∫A(q(x, y)δw)dA

Upon substituting the relations (5b) into the Equation (7) and also using Equation (8), the principle of minimum total potential energy is simplified as

(9)δΠ=δW+δU=∫A(q(x, y)δw)dA+12∫∬A(σxxδu,x−zσxxδw,xx+σyδv,y−zσyδw,yy+ 2σxyδ(u,y+v,x)−4σxyδw,xy)dAdz

Consider the force and moment resultants as

(10)(Nx, Ny, Nxy)=∫−h/2h/2(σx, σy, σxy)dz(Mx, My, Mxy)=∫−h/2h/2z(σx, σy, σxy)dz

Therefore, using the divergence theorem and simplifying Equation (9) leads to

(11)δΠ=∬A((Nx,x+Nxy,x)δu+(Ny,y+Nxy,x)δv +(Mx,xx+My,yy+2Mxy,xy)δw)dA+B.C.'s=0

Consequently, equilibrium equations in terms of force and moment resultants are

(12)δu:Nx,x+Nxy,y=0δv:Nxy,x+Ny,y=0δw:Mx,xx+My,yy+2Mxy,xy+q(x, y)=0

In the following, the constitutive relations for FG micro-plates are determined. In order to investigate the micro-structures effects, strain gradient theory with one length scale parameter is used for capturing the size effects. Hence, it is assumed that stresses are related to strains and also gradient of strains as

(13)σx=E(z)1−ν2(z)(εx+νεy)−g2E(z)1−ν2(z)∇2(εx+νεy)σy=E(z)1−ν2(z)(εy+νεx)−g2E(z)1−ν2(z)∇2(εy+νεx)σxy=E(z)1+ν(z)(εxy)−g2E(z)1+ν(z)∇2(εxy)

where ∇² is the two-dimensional Laplacian operator in Cartesian coordinate (∇2=∂2∂x2+∂2∂y2). Furthermore, in Equation (13), the parameter E(z) represents the elastic modulus and ν(z) is the Poisson’s ratio. Since it is supposed that the micro-plate is made out of FGMs, the material properties vary as functions of coordinates, especially thickness variable. Hence, the elasticity modulus is defined according to the power law function as

(14)E(z)=Em+Ecm(1/2−z/h)n   Ecm=Ec−Em

where subscripts c and m refer to the ceramic and metal components, respectively. Moreover, h is the plate’s thickness and n is known as the FGM index. In addition, it was shown that variation of Poisson’s ratio with respect to the coordinates is not significant in FGMs, so ν(z)=ν is supposed to be constant through the thickness [24].

Substituting Equations (13) and (14) into equilibrium Equations (12) and simplifying the results yields the following relations for force and moment resultants

(15)[NxNyNxyMxMyMxy]=[A11A120B11B120A12A220B12B22000A3300B33C11C120D11D120C12C220D12D22000C3300D33]([u,xv,yu,y+v,xw,xxw,yyw,xy]−g2∇2[u,xv,yu,y+v,xw,xxw,yyw,xy])

where the components of the matrix, shown in Equation (15), are

(16)(A11=A22, A12, A33)=∫−h/2h/2(1, ν, 1−ν2)E(z)1−ν2dz(B11=B22, B12, B33)=−∫−h/2h/2(1, ν, (1−ν))zE(z)1−ν2dz(C11=C22, C12, C33)=(−B11, −B12, −B332)(D11=D22, D12, D33)=−∫−h/2h/2(1, ν, (1−ν))z2E(z)1−ν2dz

Hence, substituting the relations (15) into equilibrium Equations (12) leads to the governing equilibrium equations as

(17a)A11{u,xx+v,xy−g2(u,xxxx+v,xxxy+u,xxyy+v,xyyy)} +B11{w,xxx+w,xyy−g2(w,xxxxx+2w,xxxyy+w,xyyyy)} +A33{u,yy−v,xy−g2(u,xxyy−v,xyyy+u,yyyy−v,xxxy)}=0

(17b)A11{u,xy+v,yy−g2(u,xxxy+v,xxyy+u,xyyy+v,yyyy)} +B11{w,xxy+w,yyy−g2(w,xxxxy+2w,xxyyy+w,yyyyy)} +A33{−u,xy+v,xx−g2(−u,xxxy+v,xxyy−u,xyyy+v,xxxx)}=0

(17c)−B11{u,xxx+v,xxy+u,xyy+v,yyy−g2(u,xxxxx+v,xxxxy +2u,xxxyy+2v,xxyyy+u,xyyyy+v,yyyyy)} −D11{−∇4w+g2∇6w}+q(x, y)=0

It should be noted that governing Equations (17) are simplified, using relation between constants (A₁₂=A₁₁−2A₃₃, B₁₂=B₁₁−2B₃₃, D₁₂=D₁₁−2D₃₃). Equations (17) show three governing equations for bending analysis of FG rectangular micro-plates, which are coupled in terms of displacement field. As it was explained before, since the micro-plate is made out of FGMs and in definition of FGMs, material properties are functions of coordinates, thereby a coupling exists in the equations. In order to solve the governing equations analytically, it is necessary to simplify and decouple them.

4 Bending analysis

Consider a micro-plate as shown in Figure 1 which is subjected to arbitrary distributed load. As it was explained, Levy boundary conditions are supposed for this micro-plate. In order to satisfy the simply supported boundary conditions, the transverse components of displacement field is written as

It is obvious that Equation (28) satisfies simply supported conditions of Equation (27a). Also, it is necessary to write Fourier series of arbitrary distributed load as

where

Substituting Equations (28) and (29) in Equation (23) leads to

Therefore, the analytical solution of ordinary differential Equation (31) is obtained as

where the constants (C_i, i=1, 2, …, 6) are determined by imposing boundary conditions (27) on the solution at x=0 and x=a, separately. It should be noted that in the above relation it is assumed that micro-plate is subjected to uniform distributed load q (x, y)=q.

5 Results and discussion

In order to investigate the numerical results, it is assumed that FG micro-plate is made out of silicon carbide as the ceramic part (E_c=420 GPa) and aluminum as metal part (E_m=70 GPa). In addition, the Poisson’s ratio is constant and equal to ν=0.3.

To keep the generality of study, results are presented in non-dimensional form. In the following, classical results with overhead bar (such as W̅) refer to the isotropic plate with fully ceramic material (n=0) and also zero length scale parameter ((g/b)=0).

In Figures 2 and 3, maximum deflection of micro-plate is plotted. The corresponding coordinate for maximum deflection is obtained by maximizing the deflection function in each case.

Figure 2:

Non-dimensional deflection vs. variation of aspect ratio.

(A) SCSC micro-plate, (B) SSSS micro-plate, (C) SFSF micro-plate, (D) SSSC micro-plate, (E) SSSF micro-plate, and (F) SFSC micro-plate.

Figure 3:

Non-dimensional deflection vs. variation of non-dimensional length scale parameter.

(A) SCSC micro-plate, (B) SSSS micro-plate, (C) SFSF micro-plate, (D) SSSC micro-plate, (E) SSSF micro-plate, and (F) SFSC micro-plate.

In Figure 2, non-dimensional maximum deflection is depicted vs. variation of aspect ratio (a/b), different boundary conditions and several FGM indices. Non-dimensional length scale parameter is assumed to be (g/b)=0.33. According to the figures, it is clear that increasing the aspect ratio increases the non-dimensional deflection. Obviously, depending on the boundary conditions, the rate of increasing is more apparent for aspect ratios <3. This variation is not so sharp for SSSF and SFSF, but it is so steep for SCSC. Also, in this figure, the effect of material properties is shown. It is clear that increasing the index of FGM drops the load carrying capacity.

In Figure 3, effect of micro-structures on the non-dimensional deflection is presented for different material properties and boundary conditions. As seen, increasing the length scale parameters decreases the non-dimensional stress of micro-plate that shows the rising rigidity of the micro-plate, regardless of boundary conditions. Also, it is inferred that larger values of FGM index leads to more range of variations in the deflections, while the micro-structure parameter increased.

In Figures 4 and 5, non-dimensional stresses are depicted in the thickness direction for some typical micro-plates. As it was explained before, in FG micro-plate a new neutral plane was introduced where stresses on this plane are singular. Therefore, stresses are calculated for one edge to the new neutral plane. According to Figure 4, it is obvious that increasing the index of FGM increases the value of non-dimensional stresses (or higher bending rigidity).

Figure 4:

Variation of non-dimensional stress through the thickness for SCSC and SFSC micro-plate.

(A) SCSC micro-plate (n=1), (B) SFSC micro-plate (n=1), (C) SCSC micro-plate (n=2), and (D) SFSC micro-plate (n=2).

Figure 5:

Comparison of non-dimensional stress through the thickness for different boundary conditions.

(A) Symmetric boundary conditions. (B) Asymmetric boundary conditions.

Figure 5 presents non-dimensional stresses for a square micro-plate with linear variation of change in material properties through the thickness for different boundary conditions. According to this figure, it is inferred that SCSC micro-plate indicates the highest stiffness in comparison with the other cases. In Figure 6 different mode shapes for all Levy boundary conditions are illustrated.

Figure 6:

Typical deflected micro-plate.

(A) SCSC micro-plate, (B) SSSS micro-plate, (C) SFSF micro-plate, (D) SSSC micro-plate, (E) SSSF micro-plate, and (F) SFSC micro-plate.

In Tables 1 and 2, the non-dimensional deflections for FG micro-plate with different material properties, different boundary conditions and various (g/b) and aspect ratio are presented. As it is tabulated, relatively higher rigidity is obtained for maximum values of length scale parameter and increasing aspect ratio causes rapid growing rate of non-dimensional deflections.

6 Conclusion

In the present study, by considering strain gradient theory, bending analysis of thin rectangular FG micro-plates is surveyed. Using the variational approach, governing equations containing higher-order terms are obtained for rectangular micro-plates. Since micro-plate is made out of FGM, there was a coupling in the governing equations which has been removed by introducing a new method. Finally, simplified equations were solved analytically for micro-plates with two opposite edges simply supported.

Consequently, deflections and stresses of the micro-plate were obtained and the effects of boundary condition, material properties, aspect ratio and micro-structures on the solution were investigated.

Comparison of non-dimensional micro-plates’ results with macro ones reveals gross differences between classical theory predictions and strain gradient results. Therefore, applying classical plate theory for micro-plates leads to overestimated results, especially for large values of FGM index (n) and length scale parameter (g).

Additionally, it was concluded that increasing the index of FGM increases the non-dimensional deflections. So the bending rigidity of micro-plate diminishes as a result of increasing the index of FGM, which corresponds to bigger portion of the metal part in comparison with the ceramic part.