A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material

Abstract

In this paper, we present a new united approach to formulate the equivalent micropolar constitutive relation of two-dimensional (2-D) periodic cellular material to capture its non-local properties and to explain the size effects in its structural analysis. The new united approach takes both the displacement compatibility and the equilibrium of forces and moments into consideration, where Taylor series expansion of the displacement and rotation fields and the extended averaging procedure with an explicit enforcement of equilibrium are adopted in the micromechanical analysis of a unit cell. In numerical examples, the effective micropolar constants obtained in this paper and others derived in the literature are used for the equivalent micropolar continuum simulation of cellular solids. The solutions from the equivalent analysis are compared with the discrete simulation solutions of the cellular solids. It is found that the micropolar constants developed in this paper give satisfying results of equivalent analysis for the periodic cellular material.

Graphic abstract

Appendix

Details about the equilibrium conditions of a joint.

Let us consider the equilibrium conditions of joint D in the first oriented cell in Fig. 3. The displacements and rotations at joints E, F, and D in the unit cell are expressed by Taylor approximations of corresponding kinematic variables at the joint A. When the explicit enforcement of equilibrium is not used, the forces and moments on midpoints , , and shown in Fig. 3a can be obtained as below.

At midpoint :

(35)

(36)

(37)

The forces and moments at midpoints and can be evaluated similarly.

According to Eq. (9), the displacement and rotations at joints D, E, and F can be expressed through a Taylor approximation of corresponding kinematic variables at the joint A, e.g., at joint F:

$$\begin{aligned} {\varvec{u}}^{\mathrm{F}}\approx & {} {\varvec{u}}+L_{\mathrm{AF}} \frac{\partial {\varvec{u}}}{\partial t_{\mathrm{AF}} }+\frac{L_{\mathrm{AF}}^2 }{2}\frac{\partial ^{2}{\varvec{u}}}{\partial t_{\mathrm{AF}}^2 },\nonumber \\ \varphi ^{\mathrm{F}}\approx & {} \varphi +L_{\mathrm{AF}} \frac{\partial \varphi }{\partial t_{\mathrm{AF}} }+\frac{L_{\mathrm{AF}}^2 }{2}\frac{\partial ^{2}\varphi }{\partial t_{\mathrm{AF}}^2 }. \end{aligned}$$

(38)

$L_\mathrm{AF} =\sqrt{3}L$ and

$$\begin{aligned} \frac{\partial }{\partial t_{\mathrm{AF}} }= & {} \frac{\sqrt{3}}{2}\frac{\partial }{\partial x_1 }-\frac{1}{2}\frac{\partial }{\partial x_2 },\nonumber \\ \frac{\partial }{\partial t_{\mathrm{AF}}^2 }= & {} \frac{3}{4}\frac{\partial }{\partial x_1^2 }+\frac{1}{4}\frac{\partial }{\partial x_2^2 }-\frac{\sqrt{3}}{2}\frac{\partial ^{2}}{\partial x_2 \partial x_1 }. \end{aligned}$$

(39)

The expansions for displacement and micro-rotations at joints D and E can be obtained similarly. By substituting the kinematic variables from these Taylor approximations into the expressions of forces and moments at midpoints , , and , we can obtain the expressions of these forces and moments in terms of the displacement u, v and micro-rotations $\varphi$ of joint A and their derivatives.

Then let us consider the equilibrium conditions at joint D, the resulting forces and moment are

(40)

(41)

(42)

Inserting Eqs. (35)–(37) at midpoint and the expressions of forces and moments at midpoints and into Eq. (40)–(42) after using Taylor approximations of displacements and rotations of joint A, we obtain the resulting forces and moment

(43)

(44)

(45)

Equilibrium of forces and moments at joint D is satisfied only if $f_1,\, f_2$, and m equal zero. Unfortunately, it is not the case here.

By substituting these forces and moments acting on the boundary of the unit cell without explicit equilibrium enforcement into Eqs. (10) and (11), we can obtain the constitutive constants (46)–(51), which are the same as in Chen et al. [42] except for the constants for the couple stress. However, they are too stiff. Thus, this point also proves that it is very necessary in equivalent analysis of cellular structures to enforce the explicit equilibrium of forces and moments besides maintaining compatibility of displacements. Our new united approach actually realizes this requirement and obtains the micropolar constitutive relationship described in Eq. (25)–(30).

$$\begin{aligned} \sigma _{11}= & {} \varepsilon _{11} \left( {\frac{\sqrt{3}E^{{\prime }}}{4}+\frac{\sqrt{3}k}{L^{2}}} \right) +\varepsilon _{22} \left( {\frac{\sqrt{3}E^{{\prime }}}{12}-\frac{\sqrt{3}k}{L^{2}}} \right) , \nonumber \\\end{aligned}$$

(46)

$$\begin{aligned} \sigma _{22}= & {} \varepsilon _{11} \left( {\frac{\sqrt{3}E^{{\prime }}}{12}-\frac{\sqrt{3}k}{L^{2}}} \right) +\varepsilon _{22} \left( {\frac{\sqrt{3}E^{{\prime }}}{4}+\frac{\sqrt{3}k}{L^{2}}} \right) , \nonumber \\\end{aligned}$$

(47)

$$\begin{aligned} \sigma _{12}= & {} \frac{\left[ {\left( {36k+E^{{\prime }}L^{2}} \right) \varepsilon _{12} +\left( {-12k+E^{{\prime }}L^{2}} \right) \varepsilon _{21} } \right] }{4\sqrt{3}L^{2}}, \nonumber \\\end{aligned}$$

(48)

$$\begin{aligned} \sigma _{21}= & {} \frac{\left[ {\left( {-12k+E^{{\prime }}L^{2}} \right) \varepsilon _{12} +\left( {36k+E^{{\prime }}L^{2}} \right) \varepsilon _{21} } \right] }{4\sqrt{3}L^{2}}, \nonumber \\\end{aligned}$$

(49)

$$\begin{aligned} m_{13}= & {} \frac{k}{\sqrt{3}}\chi _{13}, \nonumber \\\end{aligned}$$

(50)

$$\begin{aligned} m_{23}= & {} \frac{k}{\sqrt{3}}\chi _{23}. \end{aligned}$$

(51)