Elasticity solutions for nano-plane structures under body forces using lattice elasticity, continualised nonlocal model and Eringen nonlocal model

Abstract

This paper presents exact elasticity solutions for nano-plane structures subjected to any distribution of inplane body forces. In deriving the plane stress solutions, three different models are used. They are a lattice elasticity model called the Hencky bar-grid model (eHBM), the continualised nonlocal plane model (CNM) and Eringen’s nonlocal plane model (ENM). eHBM is a physical structural model comprising a system of rigid bar grids with bars connected by axial and torsional springs. CNM is a nonlocal model derived by continualising the governing discrete equations of the eHBM. ENM is a stress gradient nonlocal model. The use of three models allows independent confirmation of the solutions as well as providing a better understanding of the phenomenological similarities between them. Based on the exact solutions for a nano-plane structure under a partial uniformly inplane body force, it is found that by setting the bar grid length of eHBM to be equal to the characterisitc length $\ell$ of CNM and ENM, the maximum inplane displacements predicted by eHBM and CNM are in exact agreement when CNM small length scale coefficient $c_{{0}}=1/\sqrt{12}$. However, the ENM maximum inplane displacements are in agreement with eHBM solutions only when ENM’s small length scale coefficient $e_{{0}}$ lies between $1/\sqrt{50}$ and $1/\sqrt{10} .$ These results confirm some phenomenological similarities among eHBM, CNM and ENM; with CNM being closely related to the eHBM physical structure model.

Appendices

Appendix A: Comparison between Gazis et al. lattice model, Born-Karman (or Suiker et al.) lattice model and Henky bar-grid model

The physical representation of Gazis et al. lattice model [33] for isotropic elasticity case is shown in Fig. 9. In Fig. 9, $\alpha$ is an axial spring’s stiffness corresponding to the axial forces between two nearest lattices, $\beta$ is a diagonal spring’s stiffness simulating the interaction forces between two second-nearest lattices and $\gamma$ is a rotational spring’s stiffness modelling the shear forces between lattices.

Fig. 9

Physical representation of Gazis et al. lattice model [33] for plane stress elasticity. Here u and v are inplane displacements in x- and y- directions, respectively. $\alpha$, $\beta$ and $\gamma$ are the stiffnesses of axial springs, diagonal springs and rotational springs, respectively

For plane stress case, the potential energy function associated to the Gazis et al. lattices may be given by

$$\begin{aligned} {\overline{U}}= & {} \sum \nolimits _i \sum \nolimits _j \frac{\alpha }{4}\left[ \left( u_{i+1,j}-u_{i,j} \right) ^{2}+\left( u_{i+1,j+1}-u_{i,j+1} \right) ^{2}+\left( v_{i,j+1}-v_{i,j} \right) ^{2}+\left( v_{i+1,j+1}-v_{i+1,j} \right) ^{2} \right] \nonumber \\&+\frac{\beta }{4}\left[ \left( u_{i+1,j+1}-u_{i,j}+v_{i+1,j+1}-v_{i,j} \right) ^{2}+\left( u_{i+1,j}-u_{i,j+1}+v_{i,j+1}-v_{i+1,j} \right) ^{2} \right] \nonumber \\&+\frac{\gamma }{2}\left[ \left( u_{i,j+1}-u_{i,j}+v_{i+1,j}-v_{i,j} \right) ^{2}+\left( u_{i+1,j+1}-u_{i+1,j}+v_{i+1,j}-v_{i,j} \right) ^{2}\right. \nonumber \\&\left. +\left( u_{i,j+1}-u_{i,j}+v_{i+1,j+1}-v_{i,j+1} \right) ^{2}+\left( u_{i+1,j+1}-u_{i+1,j}+v_{i+1,j+1}-v_{i,j+1} \right) ^{2} \right] . \end{aligned}$$

(A1)

The kinetic energy of each lattice is given by

$$\begin{aligned} {\overline{T}}=\sum \nolimits _i \sum \nolimits _j {\frac{1}{2}\rho h\ell ^{2}\left( {\dot{u}}_{i,j}^{2}+{\dot{v}}_{i,j}^{2} \right) } . \end{aligned}$$

(A2)

where $\rho$ is the density. According to the Hamilton’s principle,

$$\begin{aligned} \delta \int _{t_{1}}^{t_{2}} \left( {\overline{U}}-{\overline{T}} \right) \mathrm {dt}=0, \end{aligned}$$

(A3)

where $t_{\mathrm {1}}$ and $t_{{2}}$ are the initial and final times, we can deduce the following difference equations of this model (see also Eq. (34) of Gazis et al. [33])

$$\begin{aligned}&\alpha \left( u_{i+1,j}-2u_{i,j}+u_{i-1,j} \right) +\frac{\beta }{2}\left( u_{i+1,j+1}+u_{i-1,j+1}+u_{i+1,j-1}+u_{i-1,j-1}-4u_{i,j} \right) \nonumber \\&\quad +4\gamma \left( u_{i,j+1}-2u_{i,j}+u_{i,j-1} \right) \nonumber \\&\quad +\left( \gamma +\frac{\beta }{2} \right) \left( v_{i+1,j+1}+v_{i-1,j-1}-v_{i+1,j-1}-v_{i-1,j+1} \right) =\ell ^{2}h\rho \ddot{u}_{i,j}, \end{aligned}$$

(A4)

and

$$\begin{aligned}&\alpha \left( v_{i,j+1}-2v_{i,j}+v_{i,j-1} \right) +\frac{\beta }{2}\left( v_{i+1,j+1}+v_{i-1,j+1}+v_{i+1,j-1}+v_{i-1,j-1}-4v_{i,j} \right) \nonumber \\&\quad +4\gamma \left( v_{i+1,j}-2v_{i,j}+v_{i-1,j} \right) +\left( \gamma +\frac{\beta }{2} \right) \left( u_{i+1,j+1}+u_{i-1,j-1}-u_{i+1,j-1}-u_{i-1,j+1} \right) =\ell ^{2}h\rho \ddot{v}_{i,j}. \nonumber \\ \end{aligned}$$

(A5)

By using Taylor’s series expansion of Eqs. (45)–(50) and Eqs. (A4) and (A5), we obtain

$$\begin{aligned}&\alpha \left( \ell ^{2}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}u}{\partial x^{4}} \right) +\frac{\beta }{2}\left( 2\ell ^{2}\frac{\partial ^{2}u}{\partial x^{2}}+2\ell ^{2}\frac{\partial ^{2}u}{\partial y^{2}}+\ell ^{4}\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}u}{\partial x^{4}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}u}{\partial y^{4}} \right) \nonumber \\&\quad +4\gamma \left( \ell ^{2}\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}u}{\partial y^{4}} \right) +\left( \gamma +\frac{\beta }{2} \right) \left( 4\ell ^{2}\frac{\partial ^{2}v}{\partial x\partial y}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}v}{\partial y\partial x^{3}} \right) +O\left( \ell ^{6} \right) =\ell ^{2}h\rho \ddot{u}, \nonumber \\ \end{aligned}$$

(A6)

and

$$\begin{aligned}&\alpha \left( \ell ^{2}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}v}{\partial y^{4}} \right) +\frac{\beta }{2}\left( 2\ell ^{2}\frac{\partial ^{2}v}{\partial x^{2}}+2\ell ^{2}\frac{\partial ^{2}v}{\partial y^{2}}+\ell ^{4}\frac{\partial ^{4}v}{\partial x^{2}\partial y^{2}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}v}{\partial x^{4}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}v}{\partial y^{4}} \right) \nonumber \\&\quad +4\gamma \left( \ell ^{2}\frac{\partial ^{2}v}{\partial x^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}v}{\partial x^{4}} \right) +\left( \gamma +\frac{\beta }{2} \right) \left( 4\ell ^{2}\frac{\partial ^{2}u}{\partial x\partial y}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}u}{\partial x\partial y^{3}}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}u}{\partial y\partial x^{3}} \right) +O\left( \ell ^{6} \right) =\ell ^{2}h\rho \ddot{v}. \nonumber \\ \end{aligned}$$

(A7)

The continuum equations of elastodynamics (sometimes called Navier’s equations of elastodynamics) in the plane stress case are given by

$$\begin{aligned} \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}u}{\partial y^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}v}{\partial x\partial y}=\rho \ddot{u}, \end{aligned}$$

(A8)

and

$$\begin{aligned} \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}v}{\partial x^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}u}{\partial x\partial y}=\rho \ddot{v}. \end{aligned}$$

(A9)

By comparing Eqs. (A6) and (A8) or (A7) and (A9) and neglecting higher-order derivatives, we obtain

$$\begin{aligned} \alpha =\frac{Eh}{1+\upsilon }, \beta =\frac{Eh\upsilon }{1-\upsilon ^{2}}, \gamma =\frac{Eh\left( 1-3\upsilon \right) }{8\left( 1-\upsilon ^{2} \right) }. \end{aligned}$$

(A10)

Noting that the special case $\gamma =0$, means any two adjacent lattices are subjected only to central interaction forces (Poisson’s or Cauchy’s molecular assumption). For plane stress elasticity problems, this special case $\gamma =0$ implies $\upsilon =1/3$. Same result had been found by McHenry [24] and Hrennikoff [25, 26] when they solved the plane stress elasticity problems by using discrete truss or frame model.

Fig. 10

Physical representation of Born Karman or Suiker et al lattice model [32, 36] for plane stress elasticity. u and v are inplane displacements in x and y directions, respectively. ${\overline{\alpha }}$ and ${\overline{\beta }}$ are longitudinal and diagonal normal spring stiffnesses, respectively. ${\overline{\delta }}$ and ${\overline{\chi }}$ are longitudinal and diagonal shear spring stiffnesses, respectively

The Born-Karman lattice model [32] comprises a system of monatomic simple cubic lattice with same distance l between two nearest lattices. This model considers the axial and shear interaction forces between any two adjacent lattices. The physical representation of Born-Karman lattice model is similar to a lattice model used by Suiker et al. [36] which comprises a system of lattices connected by longitudinal and diagonal springs as shown in Fig. 10. The distance between any two nearest lattices is ${\ell }$. The longitudinal normal springs and diagonal normal springs have the value of stiffnesses  ${\overline{\alpha }}$ and ${\overline{\beta }}$, respectively. The longitudinal shear springs and diagonal shear springs have the value of stiffnesses ${\overline{\delta }}$ and ${\overline{\chi }}$, respectively. Poisson’s effect is modelled by diagonal normal springs.

The difference equations for Born-Karman (or Suiker) lattice model are given by [32, 36]

$$\begin{aligned}&\frac{1}{2}{\overline{\alpha }}\left( 2u_{i+1,j}-4u_{i,j}+2u_{i-1,j} \right) +\frac{1}{2}{\overline{\beta }}\left( u_{i+1,j+1}+u_{i-1,j+1}-4u_{i,j}\right. \nonumber \\&\quad \left. \quad +u_{i-1,j-1}+u_{i+1,j-1}+v_{i+1,j+1}+v_{i-1,j-1}-v_{i-1,j+1}-v_{i+1,j-1} \right) \nonumber \\&\quad +\frac{1}{2}{\overline{\delta }}\left( -{4u}_{i,j}+2u_{i,j+1}+2u_{i,j-1} \right) \nonumber \\&\quad +\frac{1}{2}\overline{\chi }\left( u_{i+1,j+1}+u_{i-1,j+1}-4u_{i,j}+u_{i-1,j-1}+u_{i+1,j-1}-v_{i+1,j+1}\right. \nonumber \\&\left. \quad -v_{i-1,j-1}+v_{i-1,j+1}+v_{i+1,j-1} \right) =\ell ^{2}h\rho \ddot{u}_{i,j}, \end{aligned}$$

(A11)

and

$$\begin{aligned}&\frac{1}{2}{\overline{\alpha }}\left( 2v_{i,j+1}-4v_{i,j}+2v_{i,j-1} \right) +\frac{1}{2}{\overline{\beta }}\left( v_{i+1,j+1}+v_{i-1,j+1}-4v_{i,j}+v_{i-1,j-1}+v_{i+1,j-1}+u_{i+1,j+1}\right. \nonumber \\&\left. \quad +u_{i-1,j-1}-u_{i-1,j+1}-u_{i+1,j-1} \right) +\frac{1}{2}{\overline{\delta }}\left( -{4v}_{i,j}+2v_{i+1,j}+2v_{i-1,j} \right) \nonumber \\&\quad +\frac{1}{2}{\overline{\chi }}\left( v_{i+1,j+1}+v_{i-1,j+1}-4v_{i,j}+v_{i-1,j-1}+v_{i+1,j-1}-u_{i+1,j+1}\right. \nonumber \\&\left. \quad -u_{i-1,j-1}+u_{i-1,j+1}+u_{i+1,j-1} \right) =\ell ^{2}h\rho \ddot{v}_{i,j}. \end{aligned}$$

(A12)

By using Taylor’s series expansion of Eqs. (45)–(50) and Eqs. (A11) and (A12), we obtain

$$\begin{aligned}&{\overline{\alpha }}\left( \ell ^{2}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}u}{\partial x^{4}} \right) +\frac{{\overline{\beta }}+{\overline{\chi }}}{2}\left( 2\ell ^{2}\frac{\partial ^{2}u}{\partial x^{2}}+2\ell ^{2}\frac{\partial ^{2}u}{\partial y^{2}}+\ell ^{4}\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}u}{\partial x^{4}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}u}{\partial y^{4}} \right) \nonumber \\&\quad +{\overline{\delta }}\left( \ell ^{2}\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}u}{\partial y^{4}} \right) +\frac{{\overline{\beta }}+{\overline{\chi }}}{2}\left( 4\ell ^{2}\frac{\partial ^{2}v}{\partial x\partial y}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}v}{\partial y\partial x^{3}} \right) +O\left( \ell ^{6} \right) =\ell ^{2}h\rho \ddot{u},\nonumber \\ \end{aligned}$$

(A13)

and

$$\begin{aligned}&{\overline{\alpha }}\left( \ell ^{2}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}v}{\partial y^{4}} \right) +\frac{{\overline{\beta }}+{\overline{\chi }}}{2}\left( 2\ell ^{2}\frac{\partial ^{2}v}{\partial x^{2}}+2\ell ^{2}\frac{\partial ^{2}v}{\partial y^{2}}+\ell ^{4}\frac{\partial ^{4}v}{\partial x^{2}\partial y^{2}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}v}{\partial x^{4}}+\frac{\ell ^{4}}{3}\frac{\partial ^{4}v}{\partial y^{4}} \right) \nonumber \\&\quad +{\overline{\delta }}\left( \ell ^{2}\frac{\partial ^{2}v}{\partial x^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}v}{\partial x^{4}} \right) +\frac{{\overline{\beta }}+{\overline{\chi }}}{2}\left( 4\ell ^{2}\frac{\partial ^{2}u}{\partial x\partial y}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}u}{\partial x\partial y^{3}}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}u}{\partial y\partial x^{3}} \right) +O\left( \ell ^{6} \right) =\ell ^{2}h\rho \ddot{v}.\nonumber \\ \end{aligned}$$

(A14)

By comparing Eqs. (A13) and (A8) or (A14) and (A9) and neglecting higher-order derivatives, we obtain

$$\begin{aligned} {\overline{\alpha }}=\frac{Eh\left( 3-\upsilon \right) }{4\left( 1-\upsilon ^{2} \right) },\, \, \, \overline{\beta }=\frac{Eh}{4\left( 1-\upsilon \right) }\, ,\, \, {\overline{\delta }}=\frac{Eh\left( 1-3\upsilon \right) }{4\left( 1-\upsilon ^{2} \right) },\, \, \, \overline{\chi }=0. \end{aligned}$$

(A15)

The special case ${\overline{\delta }}=0$ for Born-Karman lattice (no shear interaction) implies $\upsilon =1/3$. In the case of the assumption of pure central forces, Born-Karman lattice coincides with Gazis et al. lattice with only axial and diagonal springs, which is equivalent to the McHenry truss.

Figure 11 shows the physical representation of eHBM where $k^{xx}$, $k^{xy}$ and $k^{S}$ are the spring stiffnesses for the primary axial springs, secondary axial springs and torsional springs, respectively. Poisson’s effect is simulated by the secondary axial springs. By applying Hamilton’s principle, the difference equations are given by

$$\begin{aligned}&-k^{xx}\left( u_{i+1,j}-2u_{i,j}+u_{i-1,j} \right) -\frac{1}{4}\left[ \left( k^{xy}+k^{yx} \right) +\frac{k^{S}}{\ell ^{2}} \right] \left( v_{i+1,j+1}-v_{i-1,j+1}-v_{i+1,j-1}+v_{i-1,j-1} \right) \nonumber \\&\quad -\frac{k^{S}}{\ell ^{2}}\left( u_{i,j+1}-2u_{i,j}+u_{i,j-1} \right) =-\ell ^{2}h\rho \ddot{u}_{i,j}, \end{aligned}$$

(A16)

and

$$\begin{aligned}&-k^{yy}\left( v_{i,j+1}-2v_{i,j}+v_{i,j-1} \right) -\frac{1}{4}\left[ \left( k^{xy}+k^{yx} \right) +\frac{k^{S}}{\ell ^{2}} \right] \left( u_{i+1,j+1}-u_{i-1,j+1}-u_{i+1,j-1}+u_{i-1,j-1} \right) \nonumber \\&\quad -\frac{k^{S}}{\ell ^{2}}\left( v_{i+1,j}-2v_{i,j}+v_{i-1,j} \right) =-\ell ^{2}h\rho \ddot{v}_{i,j}. \end{aligned}$$

(A17)

Fig. 11

Physical representation of Hencky bar-grid model for plane stress elasticity. u and v are inplane displacements in x- and y-directions, respectively. $k^{xx}$, $k^{xy}$ and $k^{s}$ are spring stiffnesses for primary axial springs, secondary axial springs and torsion springs, respectively

By expanding Eqs. (A16) and (A17) using Taylor’s series, we obtain

$$\begin{aligned}&k^{xx}\left( \ell ^{2}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}u}{\partial x^{4}} \right) +\frac{k^{S}}{\ell ^{2}}\left( \ell ^{2}\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}u}{\partial y^{4}} \right) \nonumber \\&\quad +\frac{1}{4}\left[ \left( k^{xy}+k^{yx} \right) +\frac{k^{S}}{\ell ^{2}} \right] \left( 4\ell ^{2}\frac{\partial ^{2}v}{\partial x\partial y}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}v}{\partial y\partial x^{3}} \right) +O\left( \ell ^{6} \right) =\ell ^{2}h\rho \ddot{u},\nonumber \\ \end{aligned}$$

(A18)

and

$$\begin{aligned}&k^{yy}\left( \ell ^{2}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}v}{\partial y^{4}} \right) +\frac{k^{S}}{\ell ^{2}}\left( \ell ^{2}\frac{\partial ^{2}v}{\partial x^{2}}+\frac{\ell ^{4}}{12}\frac{\partial ^{4}v}{\partial x^{4}} \right) +\frac{1}{4}\left[ \left( k^{xy}+k^{yx} \right) +\frac{k^{S}}{\ell ^{2}} \right] \nonumber \\&\quad \left( 4\ell ^{2}\frac{\partial ^{2}u}{\partial x\partial y}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}u}{\partial x\partial y^{3}}+\frac{2}{3}\ell ^{4}\frac{\partial ^{4}u}{\partial y\partial x^{3}} \right) +O\left( \ell ^{6} \right) =\ell ^{2}h\rho \ddot{v}. \end{aligned}$$

(A19)

By comparing Eqs. (A18) and (A19) to the continuum equations of motion for plane stress elasticity Eqs. (A8) and (A9), we obtain

$$\begin{aligned} k^{xx}=\frac{Eh}{1-\upsilon ^{2}},\, \, \, k^{xy}=k^{yx}=\frac{Evh}{2\left( 1-\upsilon ^{2} \right) },\, \, \, k^{S}=\frac{Eh{\ell }^{2}}{2\left( 1+\upsilon \right) }. \end{aligned}$$

(A20)

By substituting Eqs (A10), (A15) and (A20) into (A6), (A7), (A13), (A14) and (A18), (A19), we obtain the governing differential equation of Gazis et al’s lattice model up to second order:

$$\begin{aligned} \left\{ {\begin{array}{l} \underline{\left( \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}u}{\partial y^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}v}{\partial x\partial y}-\rho \ddot{u} \right) }+\ell ^{2}\left[ \frac{E}{12\left( 1+\upsilon \right) }\frac{\partial ^{4}u}{\partial x^{4}}+\frac{E\left( 1-3\upsilon \right) }{24\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}u}{\partial y^{4}} \right] \\ \quad \quad + \ell ^{2}\left[ \frac{E\upsilon }{1-\upsilon ^{2}}\left( \frac{1}{2}\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{1}{6}\frac{\partial ^{4}u}{\partial x^{4}}+\frac{1}{6}\frac{\partial ^{4}u}{\partial y^{4}} \right) +\frac{E}{12\left( 1-\upsilon \right) }\left( \frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{\partial ^{4}v}{\partial y\partial x^{3}} \right) \right] =0 \\ \underline{\left( \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}v}{\partial x^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}u}{\partial x\partial y}-\rho \ddot{v} \right) }+\ell ^{2}\left[ \frac{E}{12\left( 1+\upsilon \right) }\frac{\partial ^{4}v}{\partial y^{4}}+\frac{E\left( 1-3\upsilon \right) }{24\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}v}{\partial x^{4}} \right] \\ \quad \quad + \ell ^{2}\left[ \frac{E\upsilon }{1-\upsilon ^{2}}\left( \frac{1}{2}\frac{\partial ^{4}v}{\partial x^{2}\partial y^{2}}+\frac{1}{6}\frac{\partial ^{4}v}{\partial y^{4}}+\frac{1}{6}\frac{\partial ^{4}v}{\partial x^{4}} \right) +\frac{E}{12\left( 1-\upsilon \right) }\left( \frac{\partial ^{4}u}{\partial y\partial x^{3}}+\frac{\partial ^{4}u}{\partial x\partial y^{3}} \right) \right] =0 \\ \end{array}} \right. , \end{aligned}$$

(A21)

the governing differential equation of Born-Karman’s (or Suiker et al.’s) lattice model up to second order:

$$\begin{aligned} \left\{ {\begin{array}{l} \underline{\left( \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}u}{\partial y^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}v}{\partial x\partial y}-\rho \ddot{u} \right) }+\ell ^{2}\left[ \frac{E\left( 3-\upsilon \right) }{48\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}u}{\partial x^{4}}+\frac{E\left( 1-3\upsilon \right) }{48\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}u}{\partial y^{4}} \right] \\ \qquad + \ell ^{2}\left[ \frac{E}{4\left( 1-\upsilon \right) }\left( \frac{1}{2}\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{1}{6}\frac{\partial ^{4}u}{\partial x^{4}}+\frac{1}{6}\frac{\partial ^{4}u}{\partial y^{4}} \right) +\frac{E}{12\left( 1-\upsilon \right) }\left( \frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{\partial ^{4}v}{\partial y\partial x^{3}} \right) \right] =0 \\ \underline{\left( \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}v}{\partial x^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}u}{\partial x\partial y}-\rho \ddot{v} \right) }+\ell ^{2}\left[ \frac{E\left( 3-\upsilon \right) }{48\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}v}{\partial y^{4}}+\frac{E\left( 1-3\upsilon \right) }{48\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}v}{\partial x^{4}} \right] \\ \qquad + \ell ^{2}\left[ \frac{E}{4\left( 1-\upsilon \right) }\left( \frac{1}{2}\frac{\partial ^{4}v}{\partial x^{2}\partial y^{2}}+\frac{1}{6}\frac{\partial ^{4}v}{\partial y^{4}}+\frac{1}{6}\frac{\partial ^{4}v}{\partial x^{4}} \right) +\frac{E}{12\left( 1-\upsilon \right) }\left( \frac{\partial ^{4}u}{\partial y\partial x^{3}}+\frac{\partial ^{4}u}{\partial x\partial y^{3}} \right) \right] =0 \\ \end{array}} \right. , \end{aligned}$$

(A22)

and the governing differential equation of Hencky bar-grid model up to second order:

$$\begin{aligned} \left\{ {\begin{array}{l} \underline{\left( \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}u}{\partial x^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}u}{\partial y^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}v}{\partial x\partial y}-\rho \ddot{u} \right) }+\ell ^{2}\left[ \frac{E}{12\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}u}{\partial x^{4}}+\frac{E}{24\left( 1+\upsilon \right) }\frac{\partial ^{4}u}{\partial y^{4}} \right] \\ \qquad \qquad + \ell ^{2}\left[ \frac{E}{12\left( 1-\upsilon \right) }\left( \frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{\partial ^{4}v}{\partial y\partial x^{3}} \right) \right] =0 \\ \underline{\left( \frac{E}{1-\upsilon ^{2}}\frac{\partial ^{2}v}{\partial y^{2}}+\frac{E}{2\left( 1+\upsilon \right) }\frac{\partial ^{2}v}{\partial x^{2}}+\frac{E}{2\left( 1-\upsilon \right) }\frac{\partial ^{2}u}{\partial x\partial y}-\rho \ddot{v} \right) }+\ell ^{2}\left[ \frac{E}{12\left( 1-\upsilon ^{2} \right) }\frac{\partial ^{4}v}{\partial y^{4}}+\frac{E}{24\left( 1+\upsilon \right) }\frac{\partial ^{4}v}{\partial x^{4}} \right] \\ \qquad \qquad + \ell ^{2}\left[ \frac{E}{12\left( 1-\upsilon \right) }\left( \frac{\partial ^{4}u}{\partial y\partial x^{3}}+\frac{\partial ^{4}u}{\partial x\partial y^{3}} \right) \right] =0 \\ \end{array}} \right. . \end{aligned}$$

(A23)

In view of Eqs (A21)–(A23), the governing differential equations of Gaizs et al.’s lattice model, Born-Karman’s (or Suiker et al,’s) lattice model and Hencky bar-grid model all converge towards Navier’s partial differential equations (under line term) when $\ell \rightarrow 0$. However, their second-order terms are indeed different. Only for some specific cases such as considering only pure central forces, the governing equations of the compared models can be identical.

The definite positiveness of the discrete lattice energy imposes that the equivalent Poisson’s ratio should be lower than $\upsilon =1/3$ for Born-Karman lattice (${\overline{\delta }}\ge 0)$ and Gazis et al. lattice (${\overline{\gamma }}\ge 0)$, whereas for the eHBM lattice, larger values of the Poisson’s ratio can be considered ($\upsilon$ could be greater than 1/3).

In conclusion, the physical representation, the governing difference equations and the higher-order continualised equations given by each model are different. However, by matching the second-order derivatives of the contiualised equations to the continuum equations of elastodynamics or Navier’s equations of elastodynamics in the case of plane stress, the force constants and the spring stiffnesses of each model can be calibrated.

It is worth to noting that by multiplying Eqs. (A18) and (A19) by ${1-}\frac{{\ell }^{{2}}}{{12}}\left( \frac{\partial ^{{2}}}{\partial x^{{2}}}{+}\frac{\partial ^{{2}}}{\partial y^{{2}}} \right)$ and then neglecting the higher-order terms, the governing differential equations for CNM (60) and (61) can be restored.

Appendix B: Continualised nonlocal model for one-dimensional discrete model

The governing difference equations given by all discrete lattice or bar-springs models discussed in “Appendix A”, will be the same for one-dimensional elasticity problems. Their difference equation is given by

$$\begin{aligned} E\left( u_{i+1}-2u_{i}+u_{i-1} \right) +b^{x}\ell ^{2}=0. \end{aligned}$$

(B1)

The continualised nonlocal model (CNM) for one-dimensional elasticity problems could be obtained by truncating Eq. (B1) by using a Taylor asymptotic expansion Eq. (45). In this way, the resulting governing equation of CNM is given by

$$\begin{aligned} E\left( \frac{\partial ^{2}u}{\partial x^{2}}+\frac{\ell ^{2}}{12}\frac{\partial ^{4}u}{\partial x^{4}} \right) +b^{x}+O\left( \ell ^{4} \right) =0. \end{aligned}$$

(B2)

Alternatively, the governing equation of CNM can be derived from truncating Eq. (B1) by using a rational expansion of the pseudo-differential operator [61] based on a Padé approximant of order [2, 2] [62, 63] and by neglecting higher-order terms, we can have

$$\begin{aligned} E\frac{\frac{\partial ^{2}}{\partial x^{2}}}{1-\frac{\ell ^{2}}{12}\frac{\partial ^{2}}{\partial x^{2}}}u+b^{x}=0. \end{aligned}$$

(B3)

By multiplying Eq. (B3) by $1-\frac{\ell ^{2}}{12}\frac{\partial ^{2}}{\partial x^{2}}$, one obtains

$$\begin{aligned} E\frac{\partial ^{2}u}{\partial x^{2}}+\left( 1-\frac{\ell ^{2}}{12}\frac{\partial ^{2}}{\partial x^{2}} \right) b^{x}=0. \end{aligned}$$

(B4)

It can be seen that the governing equation given by later continualisation approach (B4) can avoid higher-order space operator. Furthermore, Eq. (B4) can ensure a definite positive elastic potential function as discussed by Challamel et al. [62, 63]. In views of Eqs. (B2) and (B4), it can be seen that Eq. (B4) can also be derived by multiplying Eq. (B2) by $\left( 1-\frac{\ell ^{2}}{12}\frac{\partial ^{2}}{\partial x^{2}} \right)$ and neglecting any higher-order terms. This approach is applied to obtain the governing equations of CNM for plane stress elasticity problems.

Appendix C: Hencky bar-grid model for an orthotropic plane structure

In this section, we shall present an example of using the Hencky bar-grid model (eHBM) to model a special type of orthotropic plane structure. According to the Hooke’s law, the continuum equations of elastodynamics (sometimes called Navier’s equations of elastodynamics) in the orthotropic plane stress case are given by [64]

$$\begin{aligned} \frac{E_{x}}{1-\upsilon _{xy}\upsilon _{yx}}\frac{\partial ^{2}u}{\partial x^{2}}+\left( \frac{v_{yx}E_{x}}{1-v_{xy}v_{yx}}+G_{xy} \right) \frac{\partial ^{2}v}{\partial x\partial y}+G_{xy}\frac{\partial ^{2}u}{\partial y^{2}}=\rho \ddot{u}, \end{aligned}$$

(C1)

and

$$\begin{aligned} \frac{E_{y}}{1-\upsilon _{xy}\upsilon _{yx}}\frac{\partial ^{2}v}{\partial y^{2}}+\left( \frac{v_{xy}E_{y}}{1-v_{xy}v_{yx}}+G_{xy} \right) \frac{\partial ^{2}u}{\partial x\partial y}+G_{xy}\frac{\partial ^{2}v}{\partial x^{2}}=\rho \ddot{v}, \end{aligned}$$

(C2)

where $E_{x}$ and $E_{y}$ are the Young’s moduli along x- and y-directions, respectively. $\upsilon _{xy}$ and $\upsilon _{yx}$ are the Poisson’s ratios with primary strain along x- and y-directions, respectively. $G_{xy}$ is the shear modulus.

By comparing the continuum equations of motion for orthotropic plane stress elasticity Eqs. (C1) and (C2) to the governing differential equations of eHBM Eqs. (A18) and (A19) with neglecting higher-order terms and assume $k^{xy}=k^{yx}$, we obtain

$$\begin{aligned} \left\{ {\begin{array}{l} k^{xx}=\frac{E_{x}h}{1-\upsilon _{xy}\upsilon _{yx}}, \\ k^{yy}=\frac{E_{y}h}{1-\upsilon _{xy}\upsilon _{yx}}, \\ \end{array}} \right. \, \, \, k^{xy}=k^{yx}=\frac{\upsilon _{xy}E_{y}h}{2\left( 1-\upsilon _{xy}\upsilon _{yx} \right) }=\frac{\upsilon _{yx}E_{x}h}{2\left( 1-\upsilon _{xy}\upsilon _{yx} \right) },\, \, \, k^{S}=G_{xy}h{\ell }^{2}, \end{aligned}$$

(C3)

if $\upsilon _{xy}E_{x}=\upsilon _{yx}E_{y}$ or $E_{x}/E_{y}=\upsilon _{yx}/\upsilon _{xy}$.