Two-phase elastic axisymmetric nanoplates

In the present work, the two-phase integral theory of elasticity developed in Barretta et al. (Phys E 97:13–30, 2018) for nano-beams is generalized to model two-dimensional nano-continua. Notably, a well-posed mixture local/stress-driven nonlocal elasticity is proposed to accurately predict size effects in Kirchhoff axisymmetric nanoplates. The key idea is to express the elastic radial curvature as a convex combination of local and nonlocal integral responses, that is a coherent choice motivated by virtue of the plate axisymmetry. The relevant structural problem is shown to be governed by a set of integro-differential equations, whose solution is computationally onerous. Thus, Helmholtz’s averaging kernel is advantageously adopted, since it enables explicit inversion of the integral constitutive law by virtue of an equivalence property. Specifically, the elastostatic problem of axisymmetry nanoplates is equivalently formulated in a differential form whose solution in terms of transverse displacement field is governed by nonlocal and mixture parameters. A parametric study is performed for case studies of applicative interest, and numerical solutions are finally provided and discussed. The presented methodology can be adopted to design and optimization of plate-based nano-electro-mechanical-systems (NEMS).


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 straindriven-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 sizedependent elastostatic behaviors. Then, the relevant integrodifferential 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.

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, , z is conveniently introduced and the two-dimensional domain of plate cross-section in the r -plane is defined by . Boundary of domain is denoted by Ω k for r = R k with k = {i, e} . The base vectors in the r coordinate plane are radial r and circumferential unit vectors.
According to linearized Kirchhoff theory, the geometric curvature tensor is defined as ∶= ∇∇u , being u ∶ [ R i , R e ] → ℜ the transverse displacement field and ∇ the gradient operator. Denoting by r derivative along the radial axis and by ⊗ tensor product, the curvature can be explicitly expressed as follows: where the eigenvalues 2 r u and r u r are radial r and circumferential bending curvatures, respectively. By duality, stress tensor field is the bending interaction = M r r ⊗ r + M ⊗ , being M r and M the radial and circumferential moments, respectively. Denoting by ∶ the scalar product, equilibrium is expressed by the 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 → 0.

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 el r , el are related to bending interaction fields as follows: being the Poisson ratio and D ∶= Eh 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 stressdriven 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 ∈ [0, 1] , id est where the second term is the convolution integral between the local radial curvature and the averaging kernel described by a positive nonlocal parameter . 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] fulfilling the following properties on the real axis: -Positivity and symmetry -Normalisation -Impulsivity for any continuous test field f ∶ ℜ → ℜ.
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: equipped with constitutive boundary conditions 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 integrodifferential formulations.

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: 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., r = 2 r u and = r u r .
Parametric plots in Fig. 1 show transverse displacement fields u(r) for increasing nonlocal parameter , with a fixed mixture parameter . Figure 2 shows influence of nonlocal and mixture parameters on non-dimensional maximum displacement ū max ∶= u max ∕u loc max , being u 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.

Plate with pinned-clamped edges under distributed boundary couples
Now, let us consider an annular nanoplate pinned at boundary Ω e and clamped at boundary Ω i , subjected to uniformly distributed boundary couples M e = 2 ⋅ 10 −2 [nN]. Kinematic boundary conditions are written as follows: Thus, from Eq. (4), differential condition of equilibrium in Eq. (3) is equipped with the following natural boundary condition: 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 and for increasing nonlocal parameter . Non-dimensional maximum displacement ū 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.

Plate with clamped-free edges under distributed boundary forces
Let us consider an annular nanoplate clamped at boundary Ω e and free at boundary Ω i , subjected to uniformly distributed boundary forces Q i = −10 −3 [nN/nm]. Kinematic boundary conditions write as follows: Thus, differential condition of equilibrium in Eq. (3) is equipped with the following natural boundary conditions: 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 , with a fixed mixture parameter . Figure 6 shows non-dimensional plots of maximum displacement ū 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.

Plate with simply supported edges under distributed loading
Let us consider a simply supported annular nanoplate under uniformly distributed loading q = −10 −4 [nN/nm 2 ]. Kinematic boundary conditions write as follows: Differential equilibrium condition Eq. (3) is then equipped with the following natural boundary conditions:  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 and for a fixed mixture parameter . Non-dimensional maximum displacements ū max as function of nonlocal and mixture parameters are represented in Fig. 8 and numerical solutions are finally shown in Table 4.
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.

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. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.