1 Introduction

Miniaturized electro-mechanical devices are gaining a great attention by virtue of their remarkable features such as rapid and high sensitivity, low-power consumption and cost, and integrability in small packages. Notably, nanoplates are promising structural components of small-scale devices such as actuators [23], gas sensing systems [20], supercapacitors [36], bionsensors [16, 25], monitoring devices [9], nanosensors [35], resonators [15], and mass sensors [10, 14]. It is well acknowledged that when dealing with nanoscale structures, refined tools are required to account for complex small-scale phenomena. An effective strategy to model long-range interactions arising in nano-structures is provided by nonlocal continuum theories. Starting from pioneering works contributed in [22, 27, 28], a nonlocal model of elasticity based on a strain-driven integral approach was proposed by Eringen [12]. Efficiently applied in the framework of screw dislocation and wave propagation, Eringen model leads instead to ill-posed problems when applied to structural elements [32].

During the last decades, several theories have been formulated to overcome the above-mentioned issues and to provide consistent methodologies to model size-dependent mechanical behaviors. Among these theories, strain-driven two-phase model based on convex combination of local and nonlocal responses was first introduced in [13] and resorted to by various authors (e.g., [8, 11, 21]). However, two-phase theory in the strain-driven formulation provides well-posed structural problems only for non-vanishing local fraction [31]. Effective remedy to singularities emerged from strain-driven-based formulations can be provided if the two-phase model is formulated as convex combination of local and stress-driven nonlocal responses [30, 31], as successfully applied in recent contributions such as [4, 5, 33]. Alternative strategies to model small-scale structures are based on the strain gradient theory [1] which has been exploited to examine laminated composite nanoplate with piezo-magnetic face sheets [24] and to investigate microstructure-dependent plates adopting sinusoidal shear deformation theories [3]. Thermo-mechanical buckling of functionally graded microbeams has been examined in [2] exploiting the modified couple stress theory [37]. Strain-driven approaches of nonlocal elasticity have been adopted for free vibration analyses of nanoplates with auxetic honeycomb core in [18].

In the present paper, a consistent stress-driven nonlocal methodology based on a two-phase approach is proposed to investigate size-dependent behavior of Kirchhoff nanoplates. The plan is the following. Kinematics and equilibrium of Kirchhoff axisymmetric plate theory are first illustrated in Sect. 2. In Sect. 3, stress-driven two-phase theory is generalized to elastic nanoplates with the aim to model size-dependent elastostatic behaviors. Then, the relevant integro-differential problem of elastic equilibrium is conveniently converted into a differential form. In Sect. 4, parametric analyses are performed for selected structural problems to show influence of nonlocal and mixture parameters on transverse displacement field. Numerical benchmark solutions are finally provided and commented upon.

2 Axisymmetric plates: mathematical formulation

Let us consider a Kirchhoff axisymmetric annular plate with internal and external radii \(\,R_i\,\) and \(\,R_e\,\), respectively, and a uniform thickness \(\,h\,\). A cylindrical coordinate system \(\,r, \theta , z\,\) is conveniently introduced and the two-dimensional domain of plate cross-section in the \(\,r\,\theta \,\)-plane is defined by \(\,\Omega =[\,R_i,\, R_e] \times [0,\, 2 \pi ]\,\). Boundary of domain is denoted by \(\,\partial \Omega _k\) for \(\,r = R_k\,\) with \(\,k = \{i, e\}\,\). The base vectors in the \(\,r\,\theta \,\) coordinate plane are radial \(\,\mathbf{e} _r\,\) and circumferential \(\,\mathbf{e} _\theta \,\) unit vectors.

According to linearized Kirchhoff theory, the geometric curvature tensor is defined as \(\,\varvec{\chi }:=\nabla \nabla u\,\), being \(\,u:[\,R_i,\, R_e] \rightarrow \Re \,\) the transverse displacement field and \(\,\nabla \,\) the gradient operator. Denoting by \(\,\partial _r\,\) derivative along the radial axis and by \(\,\otimes \,\) tensor product, the curvature can be explicitly expressed as follows:

$$\begin{aligned} \varvec{\chi }= \partial _r^2 u\,\mathbf{e} _r\otimes \mathbf{e} _r+ \frac{\partial _ru}{r}\,\mathbf{e} _\theta \otimes \mathbf{e} _\theta , \end{aligned}$$
(1)

where the eigenvalues \(\,\partial _r^2 u\,\) and \(\,\displaystyle \frac{\partial _ru}{r}\,\) are radial \(\,\chi _r\,\) and circumferential \(\,\chi _{\theta }\,\) bending curvatures, respectively. By duality, stress tensor field is the bending interaction \(\,\mathbf{M } = M_r\,\mathbf{e} _r\otimes \mathbf{e} _r+ M_\theta \,\mathbf{e} _\theta \otimes \mathbf{e} _\theta \,\), being \(\,M_r\,\) and \(\,M_\theta \,\) the radial and circumferential moments, respectively. Denoting by \(\,:\,\) the scalar product, equilibrium is expressed by the variational condition that the external virtual power of the loading system is equal to the internal virtual power

$$\begin{aligned} \int _{\Omega }\,\mathbf{M }:\varvec{\chi }_{\delta u}\, \mathrm{d}A, \end{aligned}$$
(2)

for all virtual displacement fields \(\,\delta u\,\) fulfilling homogeneous kinematic boundary conditions, being \(\,\varvec{\chi }_{\delta u}\,\) the bending curvature kinematically compatible with \(\,\delta u\,\). Integration by parts and localization procedures lead to the following differential equation of equilibrium:

$$\begin{aligned} \frac{1}{r}\bigg {(}\partial _r^2(M_r\,r)-\partial _rM_\theta \bigg {)} = q, \quad \, r \,\in \Omega , \end{aligned}$$
(3)

equipped with boundary conditions

$$\begin{aligned} \left\{ \begin{aligned}&M_r\,\partial _r\delta u = -\,{\bar{M}}_i \, \partial _r\delta u, \quad r \in \partial \Omega _i\\&M_r\,\partial _r\delta u = \,{\bar{M}}_e \, \partial _r\delta u, \quad r \in \partial \Omega _e\\&\bigg {(}M_\theta - \partial _r(M_r\,r)\bigg {)}\, \delta u = - {\bar{Q}}_i \,r\,\delta u, \quad r \in \partial \Omega _i\\&\bigg {(}M_\theta - \partial _r(M_r\,r)\bigg {)}\, \delta u = {\bar{Q}}_e \,r\,\delta u, \quad r \in \partial \Omega _e, \end{aligned} \right. \end{aligned}$$
(4)

with \(\,q\,\) transverse distributed loading, \(\,\{{\bar{M}}_i, \,\bar{M}_e\}\,\) distributed edge bending couples and \(\{\,{\bar{Q}}_i,\, \bar{Q}_e\}\,\) distributed edge transverse forces. From Eq. (4), it can be observed that the shear force is consequently defined as \(\,Q_r := \displaystyle \frac{M_\theta - \partial _r(M_r\,r)}{r}\,\).

Finally, it is worth noting that circular plates can be derived as a particular case of annular plates free on internal boundary for vanishing internal radius \(\,R_i\rightarrow 0\,\).

3 Stress-driven two-phase elasticity

According to classical local continuum theory of linearly elastic homogeneous isotropic plate [26, 29, 34], elastic radial and circumferential flexural curvatures \(\,\chi _r^{el},\,\chi _{\theta }^{el}\,\) are related to bending interaction fields as follows:

$$\begin{aligned} \begin{vmatrix} \chi _r^{el} \\ \chi _{\theta }^{el} \end{vmatrix} = \frac{1}{D\,(1-\nu ^2)} \begin{bmatrix} 1 &{} - \nu \\ -\nu &{} 1 \end{bmatrix} \begin{vmatrix} M_r\\ M_\theta \end{vmatrix} \end{aligned}$$
(5)

being \(\,\nu \,\) the Poisson ratio and \(\,D:=\displaystyle \frac{E h^3}{12(1-\nu ^2)}\) the plate flexural rigidity. Since no inelastic effects are involved subsequently, hereinafter the apex \(\,el\,\) will be suppressed.

To properly capture small-scale phenomena, the stress-driven two-phase theory of elasticity [31] is here extended to axisymmetric annular nanoplates. Motivated by axisymmetry, the key idea is to express the radial curvature as convex combination of local and nonlocal responses by means of the mixture parameter \(\,\alpha \in [0, 1]\,\), id est

$$\begin{aligned} \chi _r(r)= & {} \alpha \,\frac{M_r(r)-\nu M_\theta (r)}{D\,(1-\nu ^2)} + (1-\alpha )\,\int _{R_i}^{R_e} \phi _\lambda (r-\xi ) \,\nonumber \\&\frac{M_r(\xi )-\nu M_\theta (\xi )}{D\,(1-\nu ^2)} \,\mathrm{d}\xi , \end{aligned}$$
(6)

where the second term is the convolution integral between the local radial curvature and the averaging kernel \(\,\phi _\lambda \,\) described by a positive nonlocal parameter \(\,\lambda \,\). To enable explicit inversion of the integral equation (6), the averaging kernel in Eq. (6) is chosen to be the bi-exponential Helmholtz’s function [12]

$$\begin{aligned} \begin{aligned}&\phi _\lambda (r) = \frac{1}{2\,l_c}\,\exp \bigg {(}-\frac{|r|}{l_c}\bigg {)}, \quad l_c:=\lambda \,(R_e-R_i)\,, \end{aligned} \end{aligned}$$
(7)

fulfilling the following properties on the real axis:

  • Positivity and symmetry

    $$\begin{aligned} \phi _\lambda (r-\xi ) = \phi _\lambda (\xi -r) \ge 0. \end{aligned}$$
    (8)

  • Normalisation

    $$\begin{aligned} \int _{-\infty }^{\infty } \phi _\lambda (r)\, \mathrm{d}r= 1. \end{aligned}$$
    (9)

  • Impulsivity

    $$\begin{aligned} \lim _{\lambda \rightarrow 0^+}\int _{-\infty }^{\infty } \phi _\lambda (r-\xi )\,f(\xi ) \mathrm{d}\xi = f(r) \end{aligned}$$
    (10)

for any continuous test field \(\,f:\Re \rightarrow \Re \,\).

By extending the equivalence theorem in [31], it can be proven that the integral equation (6) equipped with the Helmholtz’s averaging kernel in Eq. (7) is equivalent to the following differential equation:

$$\begin{aligned} \frac{\chi _r}{l_c^2} -\,\partial _r^2\chi _r=\frac{M_r-\nu M_\theta }{l_c^2\,D\,(1-\nu ^2)} - \alpha \,\frac{\partial _r^2(M_r-\nu M_\theta )}{D\,(1-\nu ^2)}, \end{aligned}$$
(11)

equipped with constitutive boundary conditions

$$\begin{aligned} \left\{ \begin{aligned}&\partial _r \chi _r\bigg {|}_{r=R_i}-\frac{1}{l_c}\,\chi _r(R_i) \\&\quad = \alpha \, \bigg {(} \frac{\partial _r (M_r-\nu M_\theta )}{D\,(1-\nu ^2)} - \frac{M_r-\nu M_\theta }{l_c\,D\,(1-\nu ^2)}\bigg {)}\bigg {|}_{r=R_i}, \\&\partial _r \chi _r\bigg {|}_{r=R_e}+\frac{1}{l_c}\,\chi _r(R_e) \\&\quad = \alpha \, \bigg {(} \frac{\partial _r (M_r-\nu M_\theta )}{D\,(1-\nu ^2)} + \frac{M_r-\nu M_\theta }{l_c\,D\,(1-\nu ^2)}\bigg {)}\bigg {|}_{r=R_e}. \end{aligned} \right. \end{aligned}$$
(12)

By virtue of the choice of the Helmholtz’s kernel, the equivalent constitutive differential law in Eqs. (11)–(12) can be adopted to solve the relevant elastostatic problem of Kirchhoff axisymmetric nanoplates without involving integro-differential formulations.

4 Elastostatic analyses

This section is devoted to solve elastic equilibrium problems of axisymmetric nanoplates exploiting the stress-driven two-phase model illustrated in the previous section. A graphene nanoplate is considered with Euler–Young modulus \(\,E = 1.06\) [TPa] and Poisson ratio \(\,\nu = 0.25\) [14]. External and internal radii are \(R_e= 20\) [nm] and \(\,R_i= 2.5\) [nm], respectively, with a thickness of \(\,0.34\) [nm]. Parametric elastostatic analyses are carried out for selected case studies to investigate the influence of nonlocal and mixture parameters on structural responses.

4.1 Plate with clamped edges under distributed transverse loading

As a first case study, let us consider an annular nanoplate clamped at boundary and subjected to a uniformly distributed loading \(\, q = - 10^{-3}\) [nN/nm\(^2\)]. Kinematic boundary conditions are written as follows:

$$\begin{aligned} \left\{ \begin{aligned}&u(R_i) = \partial _ru\big {|}_{r = R_i} = 0,\\&u(R_e) = \partial _ru\big {|}_{r = R_e} = 0. \end{aligned} \right. \end{aligned}$$
(13)

Then, from Eq. (4), differential condition of equilibrium in Eq. (3) is prescribed without any natural boundary condition. Constitutive laws are provided by Eq. (5)\(_2\) and differential system of Eqs. (11)–(12) where compatibility conditions must be introduced, i.e., \(\,\chi _r = \partial _r^2 u\,\) and \(\,\chi _{\theta } = \displaystyle \frac{\partial _ru}{r}\,\).

Parametric plots in Fig. 1 show transverse displacement fields \(\,u(r)\,\) for increasing nonlocal parameter \(\,\lambda \,\), with a fixed mixture parameter \(\,\alpha \,\). Figure 2 shows influence of nonlocal and mixture parameters on non-dimensional maximum displacement \(\,{\bar{u}}_{\max }:=u_{\max }/u^\mathrm{loc}_{\max }\,\), being  \(u^\mathrm{loc}_{\max }\,\) the maximum displacement of the corresponding local case. Obtained numerical solutions are shown in Table 1 as functions of mixture and nonlocal parameters. It is worth noting that the response stiffens for increasing nonlocal parameter and exhibits a softening behavior for increasing mixture parameter.

Fig. 1
figure 1

Nanoplate under uniformly distributed loading with clamped edges: transverse displacement \(\,u\) [nm] for \(\,\alpha = 0.3\,\) and \(\,\lambda = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Fig. 2
figure 2

Nanoplate under uniformly distributed loading with clamped edges: non-dimensional maximum transverse displacement \(\,{\bar{u}}_{\max }\,\) versus nonlocal parameter \(\,\lambda \,\) for \(\,\alpha = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Table 1 Nanoplate under uniformly distributed loading with clamped edges: maximum transverse displacement \(\,u_{\max } \,[10^{-2}\) nm] versus nonlocal and mixture parameters
Full size table

4.2 Plate with pinned-clamped edges under distributed boundary couples

Now, let us consider an annular nanoplate pinned at boundary \(\,\partial \Omega _e\,\) and clamped at boundary \(\,\partial \Omega _i\,\), subjected to uniformly distributed boundary couples \(\, {\bar{M}}_e = 2\cdot 10^{-2}\) [nN]. Kinematic boundary conditions are written as follows:

$$\begin{aligned} \left\{ \begin{aligned}&u(R_i) = \partial _ru\big {|}_{R_i} = 0, \\&u(R_e) = 0. \end{aligned} \right. \end{aligned}$$
(14)

Thus, from Eq. (4), differential condition of equilibrium in Eq. (3) is equipped with the following natural boundary condition:

$$\begin{aligned} M_r(R_e) = {\bar{M}}_e. \end{aligned}$$
(15)

Constitutive laws are provided by Eq. (5)\(_2\) and by differential system in Eqs. (11)–(12) where compatibility conditions must be enforced. Parametric plots of transverse displacement field \(\,u(r)\,\) are shown in Fig. 3 for a fixed mixture parameter \(\,\alpha \,\) and for increasing nonlocal parameter \(\,\lambda \,\).

Non-dimensional maximum displacement \(\,{\bar{u}}_{\max }\,\) as function of nonlocal and mixture parameters is represented in Fig. 4. Numerical results for the examined case are collected in Table 2 showing a stiffening or softening behavior for increasing nonlocal or mixture parameter, respectively.

Fig. 3
figure 3

Nanoplate under uniformly distributed boundary couples with pinned-clamped edges: transverse displacement \(\,u\) [nm] for \(\,\alpha = 0.3\,\) and \(\,\lambda = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Fig. 4
figure 4

Nanoplate under uniformly distributed boundary couples with pinned-clamped edges: non-dimensional maximum transverse displacement \(\,{\bar{u}}_{\max }\,\) versus nonlocal parameter \(\,\lambda \,\) for \(\,\alpha = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Table 2 Nanoplate under uniformly distributed boundary couples with pinned-clamped edges: maximum transverse displacement \(\,u_{\max } \,[10^{-2}\) nm] versus nonlocal and mixture parameters
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4.3 Plate with clamped-free edges under distributed boundary forces

Let us consider an annular nanoplate clamped at boundary \(\,\partial \Omega _e\,\) and free at boundary \(\,\partial \Omega _i\,\), subjected to uniformly distributed boundary forces \(\, {\bar{Q}}_i = - 10^{-3}\) [nN/nm]. Kinematic boundary conditions write as follows:

$$\begin{aligned} u(R_e) = \partial _ru\big {|}_{R_e} = 0. \end{aligned}$$
(16)

Thus, differential condition of equilibrium in Eq. (3) is equipped with the following natural boundary conditions:

$$\begin{aligned} \left\{ \begin{aligned}&M_r(R_i) = 0,\\&\bigg {(}M_\theta - \partial _r(M_r\,r)\bigg {)}\bigg {|}_{R_i} = - {\bar{Q}}_i\,R_i. \end{aligned} \right. \end{aligned}$$
(17)

Constitutive laws are given by the set (5)\(_2\)–(11)–(12) where compatibility conditions must be introduced. Transverse displacement field \(\,u(r)\,\) is shown in Fig. 5 for increasing nonlocal parameter \(\,\lambda \,\), with a fixed mixture parameter \(\,\alpha \,\). Figure 6 shows non-dimensional plots of maximum displacement \(\,{\bar{u}}_{\max }\,\) as function of nonlocal and mixture parameters. Numerical solutions are shown in Table 3, showing that transverse displacement exhibits stiffening or softening behavior for increasing nonlocal or mixture parameter, respectively.

Fig. 5
figure 5

Nanoplate under uniformly distributed boundary forces with clamped-free edges: transverse displacement \(\,u\) [nm] for \(\,\alpha = 0.3\,\) and \(\,\lambda = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Fig. 6
figure 6

Nanoplate under uniformly distributed boundary forces with clamped-free edges: non-dimensional maximum transverse displacement \(\,{\bar{u}}_{\max }\,\) versus nonlocal parameter \(\,\lambda \,\) for \(\,\alpha = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Table 3 Nanoplate under uniformly distributed boundary forces with clamped-free edges: maximum transverse displacement \(\,u_{\max } \,[10^{-2}\) nm] versus nonlocal and mixture parameters
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4.4 Plate with simply supported edges under distributed loading

Let us consider a simply supported annular nanoplate under uniformly distributed loading \(\, q = - 10^{-4}\) [nN/nm2]. Kinematic boundary conditions write as follows:

$$\begin{aligned} u(R_i) = u(R_e) = 0. \end{aligned}$$
(18)

Differential equilibrium condition Eq. (3) is then equipped with the following natural boundary conditions:

$$\begin{aligned} M_r(R_i) = M_r(R_e) = 0. \end{aligned}$$
(19)

Constitutive laws are given by relation (5)\(_2\) and by differential system Eqs. (11)–(12) where compatibility conditions must be introduced. Figure 7 shows transverse displacement field \(\,u(r)\,\) for increasing nonlocal parameter \(\,\lambda \,\) and for a fixed mixture parameter \(\,\alpha \,\). Non-dimensional maximum displacements \(\,{\bar{u}}_{\max }\,\) as function of nonlocal and mixture parameters are represented in Fig. 8 and numerical solutions are finally shown in Table 4.

Fig. 7
figure 7

Nanoplate under uniformly distributed loading with simply supported edges: transverse displacement \(\,u\) [nm] for \(\,\alpha = 0.3\,\) and \(\,\lambda = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Fig. 8
figure 8

Nanoplate under uniformly distributed loading with simply supported edges: non-dimensional maximum transverse displacement \(\,{\bar{u}}_{\max }\,\) versus nonlocal parameter \(\,\lambda \,\) for \(\,\alpha = \{0.1, 0.2, 0.3, 0.4\}\,\)

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Table 4 Nanoplate under uniformly distributed loading with simply supported edges: maximum transverse displacement \(\,u_{\max } \,[10^{-2}\) nm] versus nonlocal and mixture parameters
Full size table

In the present section, the proposed mixture local/stress-driven nonlocal methodology has been exploited to examine the size-dependent behavior of Kirchhoff axisymmetric nanoplates. The relevant integro-differential elastostatic formulation has been converted into a differential problem by means of the Helmholtz’s averaging kernel. Solution strategies recently adopted for local elastic Kirchhoff plates of arbitrary geometry (see, e.g., [17]) and based on machine learning methods (see, e.g., [19, 38]) can be further extended to small-scale two-dimensional continua. It is worth noting that limiting elastostatic solutions of the examined case studies for vanishing mixture parameter are coincident with those obtained in [7] exploiting the purely nonlocal stress-driven methodology.

5 Closing remarks

In the paper, stress-driven two-phase theory developed in [6] for elastic nanobeams has been generalized to axisymmetric nanoplates to capture the size-dependent mechanical behavior. Kinematics and equilibrium of axisymmetric Kirchhoff plates have been first illustrated. Then, mixture local/nonlocal theory of elasticity has been conceived for nanoplates, extending the previous contribution in [7].

It has been shown that the relevant elastostatic problem is governed by a set of integro-differential equations. Helmholtz’s averaging kernel has been advantageously adopted to get explicit inversion of the integral constitutive law. Therefore, the nonlocal elastic equilibrium problem has been reformulated into a differential form.

Examplar schemes of nanomechanical interest have been examined and solved. A parametric study has been performed, and benchmark numerical solutions have been detected to show influence of mixture and nonlocal parameters on structural responses. The presented mixture local/nonlocal methodology has been proven to be advantageously able to simulate both softening and stiffening mechanical behaviors, since structural responses are driven by two parameters. Therefore, the proposed approach can be efficiently adopted for modeling and optimization of a wide class of electro-mechanical nanodevices.