Non-local elastic plate theories

Pin Lu, P.Q Zhang, H.P Lee, C.M Wang, J.N Reddy

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.

Keywords:

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. 2006a; 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)Embedded Image(2.1)where ρ and fi are, respectively, the mass density and the body (and/or applied) forces; ui is the displacement vector; and tij is the stress tensor of the non-local elasticity defined byEmbedded Image(2.2)in which x is a reference point in the body; Embedded Image is the non-local kernel function; and σij is the local stress tensor of classical elasticity theory at any point x′ in the body and satisfies the constitutive relationsEmbedded Image(2.3)for a general elastic material, in which cijkl are the elastic modulus components with the symmetry properties cijkl=cjikl=cijlk=cklij, and Embedded Imagekl is the strain tensor. It should be emphasized here that the boundary conditions involving tractions are based on the non-local stress tensor tij and not on the local stress tensor σij.

The properties of the non-local kernel Embedded Image have been discussed in detail by Eringen (1983). When α(|x|) takes on a Green's function of a linear differential operator Embedded Image, i.e.Embedded Image(2.4)the non-local constitutive relation (2.2) is reduced to the differential equationEmbedded Image(2.5)and the integro-partial differential equation (2.1) is correspondingly reduced to the partial differential equationEmbedded Image(2.6)By matching the dispersion curves with lattice models, Eringen (1983, 2002) proposed a non-local model with the linear differential operator Embedded Image defined byEmbedded Image(2.7)where a is an internal characteristic length (lattice parameter, granular size or molecular diameters) and e0 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 toEmbedded Image(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 xi (i=1, 2, 3) is introduced so that the axes x1 and x2 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 σ33=0 in the considered plate theories. The constitutive relations (2.3) can thus be reduced toEmbedded Image(2.9)whereEmbedded Image(2.10)are the reduced elastic modulus components.

The non-local resultant forces Nij and the non-local resultant moments Mij are defined asEmbedded Image(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. 2006b). By multiplying equation (2.1) by dx3, then integrating through the thickness and noting (2.11)1, we haveEmbedded Image(2.12)where Embedded Image. Furthermore, multiplying equation (2.1) by x3 dx3 followed by integrating through the thickness and noting (2.11)2, we haveEmbedded Image(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 haveEmbedded Image(2.14)where Embedded Image and Embedded Image are the local (classical) resultant forces and the local resultant moments defined byEmbedded Image(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 asEmbedded Image(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 x3, i.e. Embedded Image. With this approximation, the equations of motion (2.16) becomeEmbedded Image(2.17)and the non-local resultant force and moment tensors, Nij and Mij, respectively, in (2.11) can be simplified asEmbedded Image(2.18)where the integrals are taken along the mid-plane A of the plate, Embedded Image and Embedded Image are given in (2.15). The two-dimensional non-local kernel Embedded Image 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. Embedded Image. 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 ui 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 formEmbedded Image(3.1)where Embedded Image 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 asEmbedded Image(3.2)withEmbedded Image(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.Embedded Image(3.4)whereEmbedded Image(3.5)The boundary conditions are given by either one of each of the following pairs of conditions being specified:Embedded Image(3.6)in which Qα is the effective shear forces as defined by Q1=M1β,β+M12,2 and Q2=M2β,β+M21,1.

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

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 asEmbedded Image(3.9)in which the mass inertia I2 defined in equation (3.5) is neglected for the Kirchhoff plate theory. Using the Voigt notation, the plate constants Embedded Image, Embedded Image and Embedded Image can be converted to the conventional form expressed by two indices as AIJ, BIJ and DIJ.

(b) Mindlin plate theory

In the Mindlin plate theory, the displacement components are assumed to have the formEmbedded Image(3.10)where Embedded Image are independent variables. The strain components for the plate theory can be obtained by substituting equation (3.10) into equation (2.3)2 asEmbedded Image(3.11)withEmbedded Image(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 yieldingEmbedded Image(3.13)The boundary conditions are given by either one of each of the following pairs of conditions being specified:Embedded Image(3.14)

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

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 asEmbedded Image(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 Embedded Image in equation (3.7) are zero. The constitutive relations for the local bending moments are thus reduced toEmbedded Image(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 becomesEmbedded Image(4.2)In the case of cylindrical bending, equation (4.2) reduces to Euler beam-type equations (Lu et al. 2006a)Embedded Image(4.3)

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

Consider the static bending problem of a simply supported plate subjected to a transverse sinusoidally distributed load given byEmbedded Image(4.6)where P3nm is a known constant, andEmbedded Image(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 formEmbedded Image(4.8)in which ςn and ηm are defined in equation (4.7), and U3nm is the constant to be determined. By substituting equations (4.8) and (4.6) into equation (4.2), one obtains U3nm asEmbedded Image(4.9)whereEmbedded Image(4.10)is the non-local effect-related parameter, andEmbedded Image(4.11)is the value of the maximum transverse displacement based on the local Kirchhoff plate theory. Since Hnm>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 formEmbedded Image(4.12)where ωnm is the related order natural frequency of the transverse vibration, and ςn and ηm are defined in equation (4.7). By substituting equation (4.12) into equation (4.2) with p3=0, ωnm can be obtained asEmbedded Image(4.13)whereEmbedded Image(4.14)is the natural frequency based on the Kirchhoff classical plate theory, and Hnm 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 Embedded Image in equation (3.15) are also zero. The constitutive relations for the uncoupled local bending components are thus reduced toEmbedded Image(4.15)in which the strain components Embedded Image and Embedded Image3α 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) asEmbedded Image(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 byEmbedded Image(4.17)In view of equation (2.14), it follows that these conditions are equivalent toEmbedded Image(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 formEmbedded Image(4.19)where Embedded Image and Embedded Image are defined in equation (4.7).

By substituting equation (4.19) into equation (4.16) with p3=0, we haveEmbedded Image(4.20)whereEmbedded Image(4.21)and Hnm 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 asEmbedded Image(4.22)whereEmbedded Image(4.23)By solving the characteristic equation (4.22), the frequencies for the fixed values n and m are obtained asEmbedded Image(4.24)whereEmbedded Image(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 Embedded Image. By substituting the static displacement components into the governing equations (4.16), one obtains the maximum values of the displacement components asEmbedded Image(4.26)whereEmbedded Image(4.27)are the values of the maximal generalized displacement components based on the local Mindlin plate theory, and kij 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/Hnm for both Kirchhoff and Mindlin plate theories. By defining Embedded Image and Embedded Image 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 asEmbedded Image(4.28)in which Embedded Image is a non-dimensional non-local parameter, and l1/l2 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 Rnm. Therefore, the properties of Rnm are of interest for the examples presented herein. Figure 1 shows the variations of R11 with respect to Embedded Image and l1/l2. It can be seen that R11 decreases rapidly with increasing Embedded Image for all aspect ratios l1/l2. 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 R11 is slightly increased with increasing aspect ratios l1/l2. For higher order frequencies, the changes of the corresponding parameters Rnm have similar trends as shown in figure 1, and are plotted in figure 2 for aspect ratios l1/l2=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 Embedded Image, the frequency Embedded Image drops by approximately 20% while Embedded Image drops by approximately 60% when compared with the frequencies obtained from the local plate theories.

Figure 1

Variations of frequency ratio, R11, with respect to non-local parameter Embedded Image and aspect ratio l1/l2.

Figure 2

(a,b) Variations of frequency ratios Rnm with respect to non-local parameter Embedded Image for different aspect ratios l1/l2.

View this table:
Table 1

Frequency ratios Rnm in terms of non-local parameter Embedded Image and aspect ratio l1/l2.

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 Nij and Mij 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. 2006a). 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 byEmbedded Image(5.1)and the two-dimensional non-local kernel satisfying the relations is given byEmbedded Image(5.2)where K0 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.

References

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