## Abstract

A non-classical model for a Mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory, a surface elasticity theory and a two-parameter Winkler–Pasternak foundation model. It includes all five kinematic variables possible for a Mindlin plate. The equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on Hamilton's principle, and the microstructure, surface energy and foundation effects are treated in a unified manner. The newly developed model contains one material length-scale parameter to describe the microstructure effect, three surface elastic constants to account for the surface energy effect, and two foundation parameters to capture the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the new model includes the Mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases, recovers the Kirchhoff plate model incorporating the microstructure, surface energy and foundation effects, and degenerates to the Timoshenko beam model including the microstructure effect. To illustrate the new Mindlin plate model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived.

## 1. Introduction

Thin beams and plates resting on elastic foundations have been widely used in nano- and micro-scale devices and systems. Different types of elastic foundation models have been proposed, which include those by Winkler [1], Filonenko-Borodich [2], Pasternak [3], Kerr [4] and Vlasov [5]. The Winkler foundation model contains only one parameter and is the simplest elastic foundation model, while the Winkler–Pasternak foundation model uses two parameters and can better capture the foundation effect (e.g. [6–8]). As size effects play a significant role in nano- and micro-scale applications and classical continuum theories are unable to capture such effects, higher order continuum theories, which contain material length-scale parameters and can account for the microstructure and surface energy effects, have recently been used to develop new models for thin beams and plates resting on elastic foundations.

Khajeansari *et al.* [8] studied the bending deformation of an Euler–Bernoulli beam lying on a Winkler–Pasternak foundation by using a surface elasticity theory (e.g. [9,10]) to incorporate the surface energy effect. Şimşek & Reddy [11] proposed a model for a functionally graded micro-beam embedded in an elastic medium by applying a modified couple stress theory (e.g. [12,13]) and the Winkler–Pasternak foundation model. Limkatanyu *et al.* [14] provided a model for an Euler–Bernoulli beam resting on a Winkler–Pasternak foundation by including both the microstructure and surface energy effects, which extends the non-classical beam model of Gao & Mahmoud [15].

However, very few models have been developed for thin plates that incorporate the elastic foundation effect in addition to the microstructure and surface energy effects. Akgöz & Civalek [16] proposed a non-classical Kirchhoff plate model by using a modified couple stress theory and the Winkler foundation model, but the surface energy effect is not incorporated in their model. Recently, a general non-classical model that considers the microstructure, surface energy and foundation effects was provided by Gao & Zhang [17] for Kirchhoff plates using a variational formulation.

The objective of the current paper is to develop a new model for Mindlin plates in a general form involving all five possible kinematic variables (rather than three for Kirchhoff plates) by including the microstructure, surface energy and foundation effects in a unified manner.

The rest of the paper is organized as follows. In §2, a new non-classical model for a Mindlin plate resting on an elastic foundation is developed using a modified couple stress theory [12,13], a surface elasticity theory [9,18] and a two-parameter Winkler–Pasternak foundation model (e.g. [19,20]) through a variational formulation based on Hamilton's principle. It is shown that the new Mindlin plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the new model includes the Mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases, recovers the counterpart non-classical model for Kirchhoff plates and degenerates to the Timoshenko beam model incorporating the microstructure effect. In §3, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the new model. The numerical results are also presented there to quantitatively show the differences between the current non-classical Mindlin plate model and its classical counterpart. The paper concludes in §4 with a summary.

## 2. Formulation

The Mindlin plate theory, also known as the first-order shear deformation plate theory, is the simplest plate theory including transverse shear strains. By using the Cartesian coordinate system (*x*, *y*, *z*) shown in figure 1, where the *xy*-plane is coincident with the geometrical mid-plane of the undeformed plate, the displacement field in a Mindlin plate of uniform thickness *h* can be written as (e.g. [21,22])
*a*
*b*
*c*where *u*_{1}, *u*_{2} and *u*_{3} are, respectively, the *x*-, *y*- and *z*-components of the displacement vector ** u** of a point (

*x*,

*y*,

*z*) in the plate at time

*t*;

*u*,

*v*and

*w*are, respectively, the

*x*-,

*y*- and

*z*-components of the displacement vector of the corresponding point (

*x*,

*y*, 0) on the plate mid-plane at time

*t*; and

*ϕ*

_{x}and

*ϕ*

_{y}are, respectively, the rotation angles of a transverse normal about the

*y*- and

*x*-axes (figure 1). Note that, in equations (2.1a–

*c*), there are five independent kinematic variables, i.e.

*u*,

*v*,

*w*,

*ϕ*

_{x}and

*ϕ*

_{y}, which will need to be determined in order to fully describe the displacement field in the Mindlin plate.

In figure 1, *S*^{+} and *S*^{−} are two surface layers of zero thickness that are taken to be perfectly bonded to the bulk plate material at *z*=±*h*/2, respectively. The bulk material satisfies a modified couple stress theory [12,13], while the two surface layers are governed by a surface elasticity theory [9,18].

Figure 2 shows a Mindlin plate resting on a Winkler–Pasternak foundation. The Winkler–Pasternak foundation model contains two parameters, namely the Winkler foundation modulus *k*_{w} for the spring elements and the Pasternak foundation modulus *k*_{p} for the shear layer (e.g. [19,20]). The effect of this two-parameter elastic foundation on the plate can be treated as a vertical body force *q* (in N m^{−2}) given by [19]
^{2} is the Laplacian, and *w* is the plate mid-plane deflection first introduced through equation (2.1c).

According to the modified couple stress theory [12,13], the constitutive equations for an isotropic linear elastic material read
*σ*_{ij}, *m*_{ij} and *δ*_{ij} are, respectively, the components of the Cauchy stress tensor, the components of the deviatoric part of the couple stress tensor and the Kronecker delta, *λ* and *μ* are the Lamé constants in classical elasticity, *l* is a material length-scale parameter measuring the couple stress effect (e.g. [23,24]), and *ε*_{ij} and *χ*_{ij} are, respectively, the components of the infinitesimal strain tensor and the symmetric curvature tensor given by
*u*_{i} being the displacement components and *θ*_{i} being the components of the rotation vector defined by

According to the surface elasticity theory (e.g. [9,10,18,25–27]), the zero-thickness surface layer of a bulk elastic material has distinct elastic properties and satisfies the following governing equations (e.g. [9,28,29]):
*a*
*b*where *κ*_{αβ} are the components of the surface curvature tensor, *n*_{i} are the components of the outward-pointing unit normal **n**(=*n*_{i}**e**_{i}) to the surface (with *n*_{β} being the two in-plane components of **n**), *τ*_{αβ} are the in-plane components of the non-symmetric surface stress tensor given by [9]
*a*and *τ*_{3β} are the out-of-plane components of the surface stress tensor expressed as [9,18]
*b*where *μ*_{0} and *λ*_{0} are the surface elastic constants, and *τ*_{0} is the residual surface stress (i.e. the surface stress at zero strain). These three constants *μ*_{0}, *λ*_{0} and *τ*_{0} can be determined from either atomistic simulations or experimental measurements (e.g. [30–32]).

Note that, in equations (2.3)–(2.9a,*b*) and throughout the paper, the summation convention and standard index notation are used, with the Greek indices running from 1 to 2 and the Latin indices from 1 to 3 unless otherwise indicated.

From equations (2.1a–*c*) and (2.5)–(2.7), it follows that in the bulk of the current Mindlin plate:

The total strain energy in the elastically deformed Mindlin plate is given by
*Ω* is the region occupied by the plate, *S*^{+} and *S*^{−} represent, respectively, the bottom and top surface layers at *z*=±*h*/2 of the Mindlin plate (figure 1), *R* denotes the area occupied by the mid-plane of the plate, d*V* is the volume element and d*A* is the area element. In equation (2.13), *U*_{B} is the strain energy in the bulk of the plate which is governed by the modified couple stress theory, *U*_{S} is the strain energy in the surface layers *S*^{+} and *S*^{−} satisfying the surface elasticity theory, and *U*_{F} is the strain energy representing the effect of the two-parameter Winkler–Pasternak foundation.

The first variation of the total strain energy in the Mindlin plate over the time interval [0,*T*] can be obtained from equations (2.9a,*b*)–(2.13) as
*R* is the boundary curve enclosing the area *R*, d*s* is the differential element of arc length along ∂*R*, and *S*^{+}) and top (*S*^{−}) surfaces.

Note that the volume integral of a sufficiently smooth function *D*(*x*,*y*,*z*,*t*) over the region *Ω* can be written as
*h* is the (uniform) plate thickness and *R* is the area occupied by the mid-plane of the plate.

From equations (2.10) and (2.15), it follows that, with the help of Green's theorem,

Similarly, the use of equations (2.12) and (2.15) and Green's theorem yields

Also, it follows from equations (2.10) and (2.15) and Green's theorem that
*S*^{+}=*R*=*S*^{−}, ∂*S*^{+}=∂*R*=∂*S*^{−} for the uniform-thickness plate under consideration in order to facilitate the integral evaluations.

In addition, using Green's theorem gives

The kinetic energy of the Mindlin plate can be written as (e.g. [22])
*ρ* is the mass density of the plate material. Note that, here and in the sequel, the overhead ‘⋅’ and ‘⋅⋅’ denote, respectively, the first and second time derivatives (e.g.

It should be mentioned that the kinetic energy expression in equation (2.22) is in the simple form which is standard in classical elasticity. For a microstructured material that undergoes micro-deformations in addition to macro-deformations, there could be a second term related to the micro-deformations in the kinetic energy expression [33]. However, in the modified couple stress theory adopted in the current formulation, no micro-deformation is considered. As a result, the kinetic energy takes the simple form shown in equation (2.22).

The first variation of the kinetic energy, over the time interval [0,*T*], can be obtained from equations (2.1a–*c*), (2.15) and (2.22) as
*t*=0) and final (*t*=*T*) configurations of the plate are prescribed so that the virtual displacements vanish at *t*=0 and *t*=*T*. In addition, *ρ* is taken to be constant along the plate thickness and over the time interval [0,*T*] such that *m*_{2}, as defined in equation (2.24), accounts for the rotary inertia of the plate.

From the general expression of the work done by external forces in the modified couple stress theory (e.g. [13,24]) and in the surface elasticity theory (e.g. [9,18]), the virtual work done by the forces applied on the current Mindlin plate over the time interval [0,*T*] can be expressed as
*f*_{i} and *c*_{i} (*i*=*x*,*y*,*z*) are, respectively, the components of the body force resultant (force per unit area) and the body couple resultant (moment per unit area) through the plate thickness acting in the area *R* (i.e. the plate mid-plane); *R* (i.e. the boundary of *R*); and *y*- and *x*-axes acting on ∂*R*. Note that the positive directions of *ϕ*_{x} and *ϕ*_{y} (figure 1). Also, in reaching equation (2.25), use has been made of Green's theorem. It should be mentioned that the last two terms in the virtual work expression in equation (2.25) account for the contribution of the normal stress on the top and bottom plate surfaces

According to Hamilton's principle (e.g. [17,21,22,35–38]),
*δu*, *δv*, *δw*, *δϕ*_{x} and *δϕ*_{y} and the relations *S*^{+}=*R*=*S*^{−}, ∂*S*^{+}=∂*R*=∂*S*^{−} due to the uniform thickness of the plate,
*a*
*b*
*c*
*d*
*e*as the equations of motion of the current Mindlin plate for any (*x*,*y*)∈*R* and *t*∈(0,*T*), and

Note that the integrand of the line integral in equation (2.28) is expressed in terms of the Cartesian components of the resultants and displacements that are functions of the Cartesian coordinates (*x*, *y*, *z*) with the unit base vectors {**e**_{1},**e**_{2},**e**_{3}}. This is convenient for a rectangular plate whose edges are parallel to the *x*- and *y*-axes. However, for a more general case of a plate whose boundary is not aligned with the *x*- or *y*-axis, as shown in figure 3, it is more convenient to use a Cartesian coordinate system (*n*,*s*,*z*) with the unit base vectors {**e**_{n},**e**_{s},**e**_{3}}, where **e**_{n} (=*n*_{x}**e**_{1}+*n*_{y}**e**_{2}) and **e**_{s} (=−*n*_{y}**e**_{1}+*n*_{x}**e**_{2}) are, respectively, the unit normal and tangent vectors on the plate boundary ∂*R*.

It can be shown that the components in the coordinate system (*x*,*y*,*z*) are related to those in the coordinate system (*n*,*s*,*z*) through the following transformation expressions:
*a*and
*b*with

Using equations (2.29) and (2.30a,*b*) in equation (2.28) yields, after some lengthy algebra,

Note that on the closed boundary ∂*R* the following identity:
*D* and *g* are two smooth functions. With the help of equation (2.32), equation (2.31) becomes
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*

Using the fundamental lemma of the calculus of variations in equation (2.33) then gives
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*as the BCs for any (*x*,*y*)∈∂*R* and *t*∈(0,*T*), where the overhead bar indicates the prescribed value.

From equations (2.3), (2.10) and (2.17), the Cauchy stress resultants can be expressed in terms of the five kinematic variables *u*, *v*, *w*, *ϕ*_{x} and *ϕ*_{y} as
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*where *k*_{s} is a shear correction factor introduced to account for the non-uniformity of the shear strain components *ε*_{xz} and *ε*_{yz} over the plate thickness (e.g. [42,43]).

From equations (2.4), (2.12) and (2.19), the couple stress resultants through the plate thickness can be expressed in terms of *u*, *v*, *w*, *ϕ*_{x} and *ϕ*_{y} as
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*

Also, from equations (2.1a–*c*) and (2.9a,*b*), it follows that the surface stress components can be expressed in terms of *u*, *v*, *w*, *ϕ*_{x} and *ϕ*_{y} as
*a*
*b*
*c*
*d*
*e*
*f*

The equations of motion of the Mindlin plate in terms of the five kinematic variables *u*, *v*, *w*, *ϕ*_{x} and *ϕ*_{y} can then be obtained by using equations (2.36a–*h*), (2.37a–*h*) and (2.38a–*f*) in equations (2.27a–*e*) as
*a*
*b*
*c*
*d*
*e*for any (*x*,*y*)∈*R* and *t*∈(0,*T*). Note that the rotary inertia represented by *m*_{2} is explicitly involved in the governing equations listed in equations (2.39d,*e*).

The differential equations in equations (2.39a–*e*), the BCs in equations (2.35a–*h*) (along with equations (2.34a–*h*), (2.29) and (2.30a,*b*)), and the given initial conditions at *t*=0 and *t*=*T* define the boundary-initial value problem for determining *u*, *v*, *w*, *ϕ*_{x} and *ϕ*_{y}. It is seen from equations (2.39a–*e*) that the in-plane displacements *u* and *v* are uncoupled with the out-of-plane displacement *w* and the rotations *ϕ*_{x} and *ϕ*_{y}. Therefore, *u* and *v* can be obtained separately from solving equations (2.39a,*b*) subject to prescribed BCs of the form in equations (2.35a,*b*,*g*) and suitable initial conditions.

When *k*_{w}=*k*_{p}=0, equations (2.39a–*e*) will reduce to the governing equations for the Mindlin plate without the foundation (but incorporating the microstructure and surface energy effects).

When *l*=0 and *c*_{i}=0, equations (2.39a–*e*) will become the governing equations for the Mindlin plate in the absence of the microstructure (or couple stress) effect (but including the surface energy and foundation effects).

When *λ*_{0}=*μ*_{0}=*τ*_{0}=0, equations (2.39a–*e*) will degenerate to the governing equations for the Mindlin plate without considering the surface energy effect (but accounting for the microstructure and foundation effects).

When *l*=0, *c*_{i}=0 and *λ*_{0}=*μ*_{0}=*τ*_{0}=0, equations (2.39a–*e*) will be simplified as the classical elasticity-based governing equations for the Mindlin plate resting on the two-parameter Winkler–Pasternak elastic foundation.

When *k*_{w}=*k*_{p}=0 and *λ*_{0}=*μ*_{0}=*τ*_{0}=0, equations (2.39a–*e*) become
*a*
*b*
*c*
*d*
*e*which are the governing equations for the Mindlin plate incorporating the microstructure effect alone. These equations are identical to those derived in Ma *et al.* [22] using the same modified couple stress theory. By setting *l*=0 and *c*_{i}=0, equations (2.40a–*e*) will be further reduced to the governing equations for the Mindlin plate based on classical elasticity, as shown in Ma *et al.* [22].

When *ϕ*_{x}=∂*w*/∂*x* and *ϕ*_{y}=∂*w*/∂*y*, equations (2.39a–*e*) are simplified as
*a*
*b*
*c*which are the same as the governing equations first derived in Gao & Zhang [17] for the non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects and involving three independent kinematic variables *u*, *v* and *w*. That is, the newly developed Mindlin plate model reduces to the non-classical Kirchhoff plate model when the normality assumption is reinstated.

When *v*=0, *ϕ*_{y}=0, *u*=*u*(*x*,*t*), *w*=*w*(*x*,*t*), *ϕ*_{x}=*ϕ*_{x}(*x*,*t*), *f*_{y}=0, *c*_{x}=0 and *c*_{z}=0, the Mindlin plate considered here becomes a Timoshenko beam. For this case, equations (2.39a–*e*) reduce to, by setting *k*_{w}=*k*_{p}=0 and *λ*_{0}=*μ*_{0}=*τ*_{0}=0 additionally,
*a*
*b*
*c*which are identical to the equations of motion for a Timoshenko beam with a unit width and a height *h* derived in Ma *et al.* [35]. That is, the current Mindlin plate model recovers the non-classical Timoshenko beam model based on the same modified couple stress theory as a special case.

## 3. Examples

To further demonstrate the new non-classical Mindlin plate model developed in §2, the static bending and free vibration problems of a simply supported rectangular plate (figure 4) are analytically solved in this section by directly applying the general forms of the governing equations and BCs of the new model.

In view of the general form of the BCs in equations (2.35a–*h*), the BCs for this simply supported plate can be identified as
*x*, *y*) on the boundaries *x*=0, *a* and *y*=0,*b*. Also, the following applied traction resultants vanish on these boundaries:

For the boundaries *x*=0 and *x*=*a*, *n*_{y}=0 and *n*_{x}=−1 (on *x*=0) or *n*_{x}=1 (on *x*=*a*), and equation (3.1) becomes, with the help of equations (2.29), (2.30a), (2.34b,*d*,*f*–*h*) and (3.2),
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*

Using equations (2.36b,*f*), (2.37b,*c*,*e*,*g*) and (2.38a–*c*) in equations (3.3d–*h*) yields
*a*
*b*
*c*
*d*
*e*on *x*=0 and *x*=*a*.

For the boundaries *y*=0 and *y*=*b*, *n*_{x}=0 and *n*_{y}=−1 (on *y*=0) or *n*_{y}=1 (on *y*=*b*), and equation (3.1) becomes, with the help of equations (2.29), (2.30a), (2.34b,*d*,*f*–*h*) and (3.2),
*a*
*b*
*c*
*d*
*e*
*f*
*g*
*h*

Using equations (2.36b,*h*), (2.37b,*c*,*e*,*h*) and (2.38b–*d*) in equations (3.5d–*h*) gives
*a*
*b*
*c*
*d*
*e*

on *y*=0 and *y*=*b*.

### (a) Static bending

For static bending problems, *u*, *v*, *w*, *ϕ*_{x} and *ϕ*_{y} are independent of time *t* so that all of the time derivatives involved in equations (2.39a–*e*) vanish.

The boundary value problem (BVP) for the static bending of the simply supported plate shown in figure 4 is defined by the governing equations in equations (2.39a–*e*) and the BCs in equations (3.3a–*c*), (3.4a–*e*), (3.5a–*c*) and (3.6a–*e*), with *u*=*u*(*x*,*y*), *v*=*v*(*x*,*y*), *w*=*w*(*x*,*y*), *ϕ*_{x}=*ϕ*_{x}(*x*,*y*) and *ϕ*_{y}=*ϕ*_{y}(*x*,*y*).

As mentioned in §2, the in-plane displacements *u* and *v* are uncoupled with *w*, *ϕ*_{x} and *ϕ*_{y}. Therefore, *u* and *v* can be obtained from solving the BVP defined by equations (2.39a,*b*), (3.3a), (3.4a,*d*), (3.5a) and (3.6a,*d*). For the current case with *f*_{x}=*f*_{y}=0 and *c*_{z}=0, the solution of this BVP gives *u*=*v*=0 for any (*x*, *y*)∈*R*.

The out-of-plane displacement *w* and rotations *ϕ*_{x} and *ϕ*_{y} can be obtained from solving the BVP defined by equations (2.39c–*e*), (3.3b,*c*), (3.4b,*c*,*e*), (3.5b,*c*) and (3.6b,*c*,*e*), as shown next.

Consider the following Fourier series solutions for *w*, *ϕ*_{x} and *ϕ*_{y}:
*W*_{mn}, *m* and *n*. It can be readily shown that *w*, *ϕ*_{x} and *ϕ*_{y} in equation (3.7) satisfy the BCs in equations (3.3b,*c*), (3.4b,*c*,*e*) at *x*=0,*a* and in equations (3.5b,*c*), (3.6b,*c*,*e*) at *y*=0, *b* for any *W*_{mn},

Note that the double Fourier series solution for *w* given in equation (3.7) was first explored for a simply supported rectangular Kirchhoff plate based on classical elasticity, which is known as the Navier solution (e.g. [44,45]). The Navier solution has been extended to a rectangular Mindlin plate based on classical elasticity [46]. Also, the solutions in the form listed in equation (3.7) have been successfully obtained for a simply supported rectangular Mindlin plate based on the modified couple stress theory that accounts for the microstructure effect [22].

The resultant force *f*_{z} (*x*, *y*) can also be expanded as a Fourier series,
*Q*_{mn} is given by

In the current case (figure 4), *f*_{z}(*x*,*y*)=*Pδ*(*x*−*a*/2)*δ*(*y*−*b*/2), where *δ*(⋅) is the Dirac delta function. Using this *f*_{z} in equation (3.9) yields

Substituting equations (3.7) and (3.8) into equations (*c*–*e*) results in, with *c*_{x}=*c*_{y}=0,
*C*] is a 3-by-3 matrix whose components are

Solving the linear algebraic equation system in equation (3.11) will yield *W*_{mn}, *w*, *ϕ*_{x} and *ϕ*_{y} based on the current non-classical Mindlin plate model for the simply supported plate subjected to the concentrated force at the centre of the plate (figure 4).

Figures 5 and 6 display, respectively, the variations of the plate deflection *w* and the rotation *ϕ*_{x} along the line *y*=*b*/2 predicted by the newly developed Mindlin plate model (with or without the Winkler–Pasternak foundation) and by the classical Mindlin plate model. The numerical results for the plate with the Winkler–Pasternak foundation (solid lines) are directly obtained from equations (3.7) and (3.10)–(3.12), while those for the plate without the foundation (dashed lines) are computed using the same equations but with *k*_{w}=*k*_{p}=0. The values for the classical Mindlin plate (dotted lines) are determined from equations (3.7) and (3.10)–(3.12) by setting *l*=0, *λ*_{0}=*μ*_{0}=*τ*_{0}=0, and *k*_{w}=*k*_{p}=0.

For illustration purposes, in the numerical analysis presented herein, the plate material is taken to be aluminium with the following properties [15,47]: *E*=90 GPa, *v*=0.23, *l*=6.58 *μ*m for the bulk properties and *μ*_{0}=−5.4251 N m^{−1}, *λ*_{0}=3.4939 N m^{−1}, *τ*_{0}=0.5689 N m^{−1} for the surface layers, where Young's modulus *E* and Poisson's ratio *v* are related to the Lamé constants *λ* and *μ* by (e.g. [48])
*k*_{s} is taken to be 0.8 (e.g. [22,42,43]). In addition, the shape of the plate is fixed by letting *a*=*b*=20 *h*, while the plate thickness *h* is varying.

In figures 5 and 6, the foundation moduli are non-dimensionalized and taken to be *D*=*Eh*^{3}/[12(1−*v*^{2})] being the plate flexural rigidity. The number of terms included in equation (3.7) is controlled by adjusting *m* and *n*. The numerical results for *w*, *ϕ*_{x} and *ϕ*_{y} obtained with *m*=30 and *n*=30 are found to be the same as those computed with larger *m* and *n* values (up to *m*=60, *n*=60) to the third decimal place. This indicates that using *m*=30, *n*=30 in the expansions is sufficient for the convergent numerical solutions of *w* and *ϕ*_{x} displayed in figures 5 and 6. Note that the values of *ϕ*_{y} along the line *x*=*a*/2 are the same as those of *ϕ*_{x} along the line *y*=*b*/2 due to the loading and geometrical symmetry of the square plate under consideration. Hence, *ϕ*_{y} is not plotted here.

From figures 5 and 6, it is clearly seen that both the deflection *w* and the rotation *ϕ*_{x} predicted by the current Mindlin plate model with or without the foundation are always smaller than those predicted by the classical model in all cases considered. It is also observed that the differences between the values predicted by the new model (with or without the foundation) and those predicted by the classical model are very large when the plate thickness *h* is small (with *h*=*l*=6.58 *μ*m here), but the differences diminish when *h* becomes large (with *h*=5*l*=32.9 *μ*m here). This predicted size effect agrees with the general trend observed experimentally (e.g. [49]). In addition, it is observed from figures 5 and 6 that the presence of the elastic foundation does reduce the plate deflection and rotation, as expected. The foundation effect on the deflection of the simply supported plate (figure 4) is further shown in figure 7, where more cases with different values of *k*_{w} and *k*_{p}, including the case without the foundation (as the top curve with *k*_{w}=*k*_{p}=0) and the case with the Winkler foundation (as the red dash curve with *k*_{p}=0), are compared. Note that the values of the other parameters remain the same as those used in obtaining the numerical results shown in figure 5.

Both the microstructure and surface energy effects are included in the numerical results shown in figures 5–7. To illustrate the surface energy effect separately, additional numerical results are presented in figure 8 for the deflection of the simply supported plate, which are obtained from equations (3.7) and (3.10)–(3.12) by letting *l*=0. For comparison purposes, the results predicted by the classical elasticity-based Mindlin plate model are also plotted in figure 8, which are computed using equations (3.7) and (3.10)–(3.12) with *l*=0 and *λ*_{0}=*μ*_{0}=*τ*_{0}=0. In addition, the foundation moduli are set to be *k*_{w}=*k*_{p}=0 to examine only the surface energy effect. Note that the plate material properties used here remain the same as those employed earlier, and the plate shape is kept to be the same by letting *a*=*b*=20*h* (figure 4) for all cases (with different values of *h*).

From figure 8, it is clearly seen that the plate deflection predicted by the current model including the surface energy effect alone is always smaller than those predicted by the classical model in all cases considered here. Figure 8 also illustrates that the differences between the two sets of predicted values are significant only when the plate thickness *h* is very small, but they diminish as *h* increases. This indicates that the surface energy effect is important only when the plate is sufficiently thin.

### (b) Free vibration

For free vibration problems, the boundary-initial value problem for the simply supported plate shown in figure 4 is defined by equations (2.39a–*e*), (3.3a–*c*), (3.4a–*e*), (3.5a–*c*) and (3.6a–*e*), with all external forces vanished (i.e. *f*_{x}=*f*_{y}=*f*_{z}=0 and *c*_{x}=*c*_{y}=*c*_{z}=0).

For the current case with *f*_{x}=*f*_{y}=0 and *c*_{z}=0, equations (2.39a,*b*), (3.3a), (3.4a,*d*), (3.5a) and (3.6a,*d*) give *u*=*u*(*x*,*y*,*t*)=0, *v*=*v*(*x*,*y*,*t*)=0 for any (*x*, *y*)∈*R* and *t*∈[0,*T*].

For *w*, *ϕ*_{x} and *ϕ*_{y}, consider the following Fourier series expansions:
*ω*_{n} is the *n*th natural frequency of vibration of the plate, *i* is the imaginary unit satisfying *i*^{2}=−1. It can be readily shown that the expressions of *w*, *ϕ*_{x} and *ϕ*_{y} in equation (3.14) satisfy the BCs in equations (3.3b,*c*), (3.4b,*c*,*e*) at *x*=0,*a* and in equations (3.5b,*c*), (3.6b,*c*,*e*) at *y*=0, *b* for any *t*∈[0,*T*].

Using equation (3.14) in equations (2.39c–*e*) gives
*C*] is the 3-by-3 matrix whose components are defined in equation (3.12), and *m*_{0} and *m*_{2} are given in equation (2.24). Note that the rotary inertia represented by *m*_{2} is explicitly included in equation (3.15) for the current free vibration problem.

Let
*I*] is the 3-by-3 identity matrix. For a non-trivial solution of *D*] matrix whose components *D*_{ij} are defined in equation (3.16).

Equation (3.19) is a cubic equation in *n*th natural frequency, *ω*_{n}, for the free vibration of the plate. Note that the rotary inertia effect has been incorporated in equation (3.19).

With *ω*_{n} determined from equation (3.19), *w*(*x*, *y*, *t*), *ϕ*_{x}(*x*,*y*,*t*) and *ϕ*_{y}(*x*,*y*,*t*) through equation (3.14), and thereby completing the solution.

Figure 9 shows the variation of the first natural frequency *ω*_{1}, obtained from equation (3.19) (with *m*=1, *n*=1 in equation (3.12)), with the plate thickness, which is predicted by the current Mindlin plate model with the Winkler–Pasternak *k*_{p}=0) or no foundation (*k*_{w}=*k*_{p}=0) and by the classical model (i.e. with *l*=0, *λ*_{0}=*μ*_{0}=*τ*_{0}=0, and *k*_{w}=*k*_{p}=0). The material properties and geometry of the aluminium plate used here are the same as those employed earlier to obtain the numerical results shown in figures 5–8. In addition, the density for the aluminium plate is taken to be *ρ*=2.7×10^{3} kg m^{−3}, which is needed in equation (3.19).

From figure 9, it is clearly seen that the natural frequency predicted by the current model with or without the foundation is always higher than that predicted by the classical elasticity-based model. The difference between the predictions by the current model without the foundation and the classical model is significant when the plate thickness *h* is very small (with *h*<2*l*=13.16 *μ*m here), while the difference is diminishing as *h* becomes large (with *h*>6*l*=39.48 *μ*m here for the case with *k*_{w}=*k*_{p}=0). This shows that the size effect on the natural frequency is important only when the plate thickness is very small. In addition, it is observed from figure 9 that the presence of the elastic foundation indeed increases the natural frequency, and this effect can be significant when the plate thickness is small but diminishes as the thickness becomes large. The effect of the foundation on the natural frequency of the simply supported plate (figure 4) is further illustrated in figure 10, where more cases with different values of *k*_{w} and *k*_{p}, including the case with the Winkler foundation (as the bottom curve with *k*_{p}=0), are compared. Note that the values of the other parameters remain the same as those used to obtain the numerical results shown in figure 9.

Clearly, figure 10 shows that the larger the value of *k*_{w} or *k*_{p}, the larger the natural frequency *ω*_{1}, which supports what is observed from figure 9.

## 4. Summary

A new non-classical Mindlin plate model is developed in a general form by using a modified couple stress theory, a surface elasticity theory and a two-parameter Winkler–Pasternak elastic foundation model and by including all five kinematic variables possible for a Mindlin plate. The equations of motion and the complete BCs are determined simultaneously through a variational formulation based on Hamilton's principle, and the microstructure, surface energy and foundation effects are treated in a unified manner. The newly developed model can capture the size effects exhibited by thin plates at the micrometre scale.

As direct applications of the new Mindlin plate model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved, with the solutions compared with those based on the classical Mindlin plate theory. For the static bending problem, the numerical results reveal that the deflection and rotations of the simply supported plate with or without the Winkler–Pasternak elastic foundation predicted by the current model are smaller than those predicted by the classical model, and the differences are very large when the plate thickness is sufficiently small but diminish as the thickness of the plate increases. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical model, and the difference is significant for very thin plates. These predicted size effects at the micrometre scale agree with the general trends observed in experiments. Finally, the numerical results show quantitatively that the plate deflection is reduced and the plate natural frequency is increased in the presence of the elastic foundation, as expected.

## Competing interests

There are no conflicts or competing interests in this work.

## Funding

The work reported here is funded by a grant from the US National Science Foundation (NSF), Mechanics of Materials and Structures Program, with Dr Kara Peters as the programme manager.

## Acknowledgements

The support from NSF is gratefully acknowledged. The authors also would like to thank two anonymous reviewers for their encouragement and helpful comments on an earlier version of the paper.

- Received April 20, 2016.
- Accepted June 21, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.