## Abstract

A non-local plate model is proposed based on Eringen's theory of non-local continuum mechanics. The basic equations for the non-local Kirchhoff and the Mindlin plate theories are derived. These non-local plate theories allow for the small-scale effect which becomes significant when dealing with micro-/nanoscale plate-like structures. As illustrative examples, the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non-local solutions are discussed in relation to their corresponding local solutions.

## 1. Introduction

Size-dependent theories of continuum mechanics have received increasing attention in recent years due to the need to model and analyse very small-sized mechanical structures and devices in the rapid developments of micro-/nanotechnologies. One of the well-known models is the non-local elasticity theory (Kroner 1967; Eringen 1983, 2002). This non-local theory has been applied to solve wave propagation, dislocation and crack problems. The theory includes scale effects and long-range atomic interactions so that it can be used as a continuum model for atomic lattice dynamics. Therefore, this continuum theory on one hand is suitable for modelling submicro- or nanosized structures, while on the other hand it avoids enormous computational efforts when compared with discrete atomistic or molecular dynamics simulations (Sun & Zhang 2003; Zhang & Sun 2004). Owing to the aforementioned advantages, several researchers have applied the non-local continuum theory for the mechanical analysis of micro- and nanostructures in more recent years (Peddieson *et al.* 2003; Sudak 2003; Wang & Hu 2005; Zhang *et al.* 2005; Lu *et al.* 2006*a*; Xu 2006; Wang *et al.* 2006; Reddy 2007). However, most of these studies focused on one-dimensional beam-like structures.

In modelling micro- or nanoelectromechanical systems (MEMS or NEMS) and devices, some mechanical components—such as thin film elements (Freund & Suresh 2003), nanosheet resonators (Bunch *et al.* 2007), paddle-like resonators (Evoy *et al.* 1999; Lobontiu *et al.* 2006) and two-dimensional suspended nanostructures (Tighe *et al.* 1997; Zalalutdinov *et al.* 2006)—have to be modelled as a two-dimensional plate-like structure. For this purpose, the non-local plate theories are studied herein. Based on the non-local elasticity model, pioneered by Eringen (1983, 2002), the general governing equations for a thin plate can be derived by integrating the equations of motion for the non-local linear elasticity through the thickness. With the proper assumptions for displacement components, specific plate theories can be further obtained. Considered herein are two well-known plate theories: the Kirchhoff plate theory and the Mindlin plate theory. The Kirchhoff plate theory is a thin-plate theory that neglects the effect of transverse shear deformation, whereas the Mindlin plate theory is a first-order shear-deformable plate theory that incorporates this effect which becomes significant in thick plates and shear-deformable plates. Based on these two non-local plate model versions, the bending and vibration problems of a rectangular plate with simply supported edges are solved in order to examine the effect of small scale on the bending and vibration solutions.

## 2. Non-local elastic plate model

### (a) Review of non-local elasticity theory

For non-local linear elastic solids, the equations of motion have the form (Eringen 1983, 2002)(2.1)where *ρ* and *f*_{i} are, respectively, the mass density and the body (and/or applied) forces; *u*_{i} is the displacement vector; and *t*_{ij} is the stress tensor of the non-local elasticity defined by(2.2)in which ** x** is a reference point in the body; is the non-local kernel function; and

*σ*

_{ij}is the local stress tensor of classical elasticity theory at any point

**′ in the body and satisfies the constitutive relations(2.3)for a general elastic material, in which**

*x**c*

_{ijkl}are the elastic modulus components with the symmetry properties

*c*

_{ijkl}=

*c*

_{jikl}=

*c*

_{ijlk}=

*c*

_{klij}, and

_{kl}is the strain tensor. It should be emphasized here that the boundary conditions involving tractions are based on the non-local stress tensor

*t*

_{ij}and not on the local stress tensor

*σ*

_{ij}.

The properties of the non-local kernel have been discussed in detail by Eringen (1983). When *α*(|** x**|) takes on a Green's function of a linear differential operator , i.e.(2.4)the non-local constitutive relation (2.2) is reduced to the differential equation(2.5)and the integro-partial differential equation (2.1) is correspondingly reduced to the partial differential equation(2.6)By matching the dispersion curves with lattice models, Eringen (1983, 2002) proposed a non-local model with the linear differential operator defined by(2.7)where

*a*is an internal characteristic length (lattice parameter, granular size or molecular diameters) and

*e*

_{0}is a constant appropriate to each material for adjusting the model to match some reliable results by experiments or other theories. Therefore, according to (2.3), (2.5) and (2.7), the constitutive relations with this kernel function may be simplified to(2.8)

For simplicity and to avoid solving integro-partial differential equations, the non-local elasticity model, defined by the relations (2.5)–(2.8), has been widely adopted for tackling various problems of linear elasticity and micro-/nanostructural mechanics.

### (b) Plate equations of non-local elastic model

The foregoing non-local elastic model can be extended to two-dimensional thin-plate structures. Consider a thin plate with a constant thickness *h*. A Cartesian coordinate system *x*_{i} (*i*=1, 2, 3) is introduced so that the axes *x*_{1} and *x*_{2} lie in the mid-plane of the plate. Since the thickness of the plate is very small when compared with the other two dimensions, it is assumed that *σ*_{33}=0 in the considered plate theories. The constitutive relations (2.3) can thus be reduced to(2.9)where(2.10)are the reduced elastic modulus components.

The non-local resultant forces *N*_{ij} and the non-local resultant moments *M*_{ij} are defined as(2.11)The global governing equations of the plate structures can be derived by integrating the equations of motion (2.1) through the thickness (Lu *et al.* 2006*b*). By multiplying equation (2.1) by d*x*_{3}, then integrating through the thickness and noting (2.11)_{1}, we have(2.12)where . Furthermore, multiplying equation (2.1) by *x*_{3} d*x*_{3} followed by integrating through the thickness and noting (2.11)_{2}, we have(2.13)Since the equation with *i*=3 in equation (2.13) has no physical application, it is omitted in the remaining part of the derivations.

By applying the linear differential operator (2.7) and the differential equations (2.5) to equation (2.11), we have(2.14)where and are the local (classical) resultant forces and the local resultant moments defined by(2.15)Furthermore, by applying the operator to equations (2.12) and (2.13), we obtain the general equations of motion for the non-local plate model as(2.16)

The differential operator ∇^{2} in (2.16) is the three-dimensional Laplace operator in general. For thin-plate models, it may be reduced to the two-dimensional Laplace operator by ignoring the differential component with respect to *x*_{3}, i.e. . With this approximation, the equations of motion (2.16) become(2.17)and the non-local resultant force and moment tensors, *N*_{ij} and *M*_{ij}, respectively, in (2.11) can be simplified as(2.18)where the integrals are taken along the mid-plane *A* of the plate, and are given in (2.15). The two-dimensional non-local kernel in equation (2.18) can be defined to satisfy the relation (2.4), in which the differential operator is as given in equation (2.7) instead of a two-dimensional Laplace operator, i.e. . This approximation is acceptable for plates with very small thickness–span ratios. For thick-plate models, the ‘exact’ expressions (2.11) and (2.16) may be required.

The later derivations for the thin-plate models are based on the simplified equations (2.17) and (2.18). Beginning from equations (2.11) and (2.16), the derivations can be shown to arrive at the same formulations, but the non-local resultant force and moment tensors are defined by equation (2.11) and not by equation (2.18).

## 3. Basic equations for two plate theories

Equations (2.9)–(2.18) are the general equations of the non-local plate model. For different plate theories, the related equations of motion can be obtained by substituting the assumed displacement components *u*_{i} into these equations. There are a number of plate theories, of which the most commonly used are the Kirchhoff and the Mindlin plate theories. The basic equations of these two plate theories are derived in this section based on the foregoing non-local relations.

### (a) Kirchhoff plate theory

In the Kirchhoff plate theory, the displacement components are assumed to have the form(3.1)where is the displacement components of the mid-plane at time *t*.

The strain components for the plate theory can be obtained by substituting equation (3.1) into equation (2.3)_{2} as(3.2)with(3.3)The equations of motion for the plate theory can be obtained by substituting equation (3.1) into equations (2.12) and (2.13), i.e.(3.4)where(3.5)The boundary conditions are given by either one of each of the following pairs of conditions being specified:(3.6)in which *Q*_{α} is the effective shear forces as defined by *Q*_{1}=*M*_{1β,β}+*M*_{12,2} and *Q*_{2}=*M*_{2β,β}+*M*_{21,1}.

The local resultant forces and the local resultant moments for the Kirchhoff plate theory can be obtained by substituting equations (2.9), (2.10), (3.2) and (3.3) into equation (2.15) as(3.7)where(3.8)are the extensional, the coupling and the bending stiffnesses, respectively. For a symmetric composite plate, .

By substituting equations (3.7) and (2.14) into equation (3.4), the equations of motion for the non-local Kirchhoff plate theory can be expressed in terms of the displacements as(3.9)in which the mass inertia *I*_{2} defined in equation (3.5) is neglected for the Kirchhoff plate theory. Using the Voigt notation, the plate constants , and can be converted to the conventional form expressed by two indices as *A*_{IJ}, *B*_{IJ} and *D*_{IJ}.

### (b) Mindlin plate theory

In the Mindlin plate theory, the displacement components are assumed to have the form(3.10)where are independent variables. The strain components for the plate theory can be obtained by substituting equation (3.10) into equation (2.3)_{2} as(3.11)with(3.12)The equations of motion of the Mindlin plate theory can be obtained by substituting equation (3.10) into equations (2.12) and (2.13), thus yielding(3.13)The boundary conditions are given by either one of each of the following pairs of conditions being specified:(3.14)

The local resultant forces and the local resultant moments for the Mindlin plate theory can be obtained by substituting equations (2.9), (2.10), (3.11) and (3.12) into equation (2.15) as(3.15)where the constants , and are given in equation (3.8), and .

By substituting equations (3.15) and (2.14) into equation (3.13), the equations of motion for the non-local Mindlin plate theory can be expressed in terms of displacements as(3.16)

## 4. Bending and free vibrations of symmetrically orthotropic plates

In order to illustrate the applications of the foregoing non-local plate theories, we consider the case of symmetrically orthotropic plates for Kirchhoff and Mindlin plate models. For such plates, the in-plane and the out-of-plane variables are uncoupled, and only flexural deformations are considered in the examples for the sake of simplicity. The bending and free vibration solutions of a simply supported, rectangular plate based on both Kirchhoff and Mindlin non-local plate models are then derived, and are compared with the results based on local (classical) plate theories.

### (a) Solutions based on Kirchhoff plate theory

For a symmetrical orthotropic plate, the coupling stiffnesses in equation (3.7) are zero. The constitutive relations for the local bending moments are thus reduced to(4.1)in which the subscripts of the bending stiffness components have been written with two-index Voigt notation. The equation of motion (3.9)_{2} for bending becomes(4.2)In the case of cylindrical bending, equation (4.2) reduces to Euler beam-type equations (Lu *et al.* 2006*a*)(4.3)

The boundary conditions for the simply supported edges of the rectangular plate are defined by(4.4)From (2.14), it follows that these conditions are equivalent to:(4.5)

Consider the static bending problem of a simply supported plate subjected to a transverse sinusoidally distributed load given by(4.6)where *P*_{3nm} is a known constant, and(4.7)with *n* and *m* being positive integers. The deflection solution that satisfies the boundary conditions (4.4) or (4.5) can be assumed to take the form(4.8)in which *ς*_{n} and *η*_{m} are defined in equation (4.7), and *U*_{3nm} is the constant to be determined. By substituting equations (4.8) and (4.6) into equation (4.2), one obtains *U*_{3nm} as(4.9)where(4.10)is the non-local effect-related parameter, and(4.11)is the value of the maximum transverse displacement based on the local Kirchhoff plate theory. Since *H*_{nm}>1, it is clear that the transverse displacements predicted by the non-local plate theories are generally larger than those predicted by the classical plate theories as the non-local effect makes the plate models more flexible.

For the free transverse vibration problem of the simply supported, rectangular plate, the time-dependent displacement solution satisfying the boundary conditions (4.4) or (4.5) can be assumed to take the form(4.12)where *ω*_{nm} is the related order natural frequency of the transverse vibration, and *ς*_{n} and *η*_{m} are defined in equation (4.7). By substituting equation (4.12) into equation (4.2) with *p*_{3}=0, *ω*_{nm} can be obtained as(4.13)where(4.14)is the natural frequency based on the Kirchhoff classical plate theory, and *H*_{nm} is the non-local effect-related parameter defined in equation (4.10). The free vibration and natural frequencies of rectangular plates based on the local Kirchhoff plate theory were discussed in detail by Leissa (1973).

### (b) Solutions based on Mindlin plate theory

For the symmetrical orthotropic plate, the coupling stiffnesses in equation (3.15) are also zero. The constitutive relations for the uncoupled local bending components are thus reduced to(4.15)in which the strain components and _{3α} have been written in the displacement components according to equations (3.11) and (3.12). The equations of motion for bending can be obtained from equation (3.16) as(4.16)For the case of cylindrical bending, the equations in (4.16) reduce to the equations of the non-local Timoshenko beam model (Lu *et al.* 2007).

The boundary conditions for the simply supported edges of the rectangular plate are defined by(4.17)In view of equation (2.14), it follows that these conditions are equivalent to(4.18)

For the free transverse vibration problem, the solutions satisfying the boundary conditions (4.17) or (4.18) can be assumed to take the form(4.19)where and are defined in equation (4.7).

By substituting equation (4.19) into equation (4.16) with *p*_{3}=0, we have(4.20)where(4.21)and *H*_{nm} is as given in equation (4.10). By setting the determinant of the coefficient matrix in equation (4.20) to be zero, one obtains the corresponding characteristic equation as(4.22)where(4.23)By solving the characteristic equation (4.22), the frequencies for the fixed values *n* and *m* are obtained as(4.24)where(4.25)are the natural frequencies based on the local Mindlin plate theory. It can be seen that, for each combination of *n* and *m*, we obtain three natural frequencies. The lowest of these corresponds to the mode where the transverse deflection dominates, whereas the other two frequencies are much higher and correspond to shear modes (Soedel 1993).

The static bending problem of a simply supported rectangular plate under a sinusoidally distributed transverse load (4.6), based on the non-local Mindlin plate theory, can be solved similarly. Assume the static displacement components to take the forms as shown in equation (4.19), but omitting the time-dependent terms, i.e. by letting . By substituting the static displacement components into the governing equations (4.16), one obtains the maximum values of the displacement components as(4.26)where(4.27)are the values of the maximal generalized displacement components based on the local Mindlin plate theory, and *k*_{ij} and *Δ* are defined in equations (4.21) and (4.23), respectively. Again, it can be seen that the displacements predicted by the non-local Mindlin plate theory are larger than those predicted by the local Mindlin plate theory.

### (c) Discussions

For a simply supported rectangular plate, it can be seen from equations (4.13) and (4.24) that, for given *n* and *m*, the ratio between the non-local and the local frequencies is 1/*H*_{nm} for both Kirchhoff and Mindlin plate theories. By defining and to be the non-local and the local natural frequencies obtained in equations (4.13), (4.14), (4.24) and (4.25), the ratios can be written as(4.28)in which is a non-dimensional non-local parameter, and *l*_{1}/*l*_{2} is the aspect ratio of the rectangular plate.

The properties of the natural frequencies of the simply supported rectangular plates based on the local Kirchhoff and Mindlin plate theories have been well studied (see, for instance, Leissa (1973) and Soedel (1993)). The corresponding non-local Kirchhoff and Mindlin plate models modify the frequency results by the factor *R*_{nm}. Therefore, the properties of *R*_{nm} are of interest for the examples presented herein. Figure 1 shows the variations of *R*_{11} with respect to and *l*_{1}/*l*_{2}. It can be seen that *R*_{11} decreases rapidly with increasing for all aspect ratios *l*_{1}/*l*_{2}. This means that, for very small-sized plate-like structures in MEMS or NEMS, in which the size effect becomes significant, the frequency properties predicted using the local plate theories are considerably overestimated. On the other hand, it can be seen from figure 2 that the decreasing rate of *R*_{11} is slightly increased with increasing aspect ratios *l*_{1}/*l*_{2}. For higher order frequencies, the changes of the corresponding parameters *R*_{nm} have similar trends as shown in figure 1, and are plotted in figure 2 for aspect ratios *l*_{1}/*l*_{2}=1 and 0.4. Some numerical values are given in table 1. It can be observed that the non-local effects have more significant influences on the higher order frequencies. For instance, for , the frequency drops by approximately 20% while drops by approximately 60% when compared with the frequencies obtained from the local plate theories.

On the other hand, the solutions for the simply supported plates given in equations (4.9) and (4.26) show that the displacements obtained by the non-local plate models are larger than those predicated by the local plate theories. This implies that the non-local effects ‘soften’ the structures, and make them more flexible. These mechanical properties for the structures in micro- and nanoscales should be taken into consideration in design and fabrication of MEMS/NEMS components.

The foregoing simple examples show that one can apply the non-local plate models to predict the mechanical properties of micro- and nanoscale plate-like structures. For complex boundary value problems, analytical solutions are generally not available and numerical treatments are required.

## 5. Concluding remarks

In this paper, the general equations and relations of non-local elastic plate models have been presented, and the governing equations of two non-local plate theories modified from their corresponding local Kirchhoff plate theory and local Mindlin plate theory have been derived. The non-local theories can be applied for the analysis of micro-and nanoscale plate-like structures, in which the small-scale effects become significant. As illustrative examples, the bending and free vibration problems of a simply supported rectangular plate based on both the non-local Kirchhoff and Mindlin plate models have been studied. The results show that, for very small-sized plates, the influences of the non-local effects on the mechanical properties are considerable.

For general boundary value problems, like non-zero boundary force conditions, the method for solution of the non-local plate theories will be more complicated than those of the local plate theories. It is known that the force boundary conditions for the non-local plate models are based on the non-local components *N*_{ij} and *M*_{ij} defined in equation (2.11) or equation (2.18). Since these generalized force components are coupled by a set of the second-order differential equations (2.14), it is very difficult to obtain their explicit expressions as in the case of one-dimensional non-local beam models (Lu *et al.* 2006*a*). Therefore, the non-local generalized force components of the non-local plate theories for general boundary value problems should be determined by the relations (2.11) or (2.18), in which the three-dimensional non-local kernel satisfying the relations (2.4) and (2.7) is given by(5.1)and the two-dimensional non-local kernel satisfying the relations is given by(5.2)where *K*_{0} is the modified Bessel function and *l* is a characteristic length of the considered structure. For more examples of different boundary value problems based on the non-local elasticity models, refer to Eringen (2002).

## Footnotes

- Received March 20, 2007.
- Accepted September 4, 2007.

- © 2007 The Royal Society