## Abstract

In this paper, a set of expressions for the characterization of circular cell honeycombs as micropolar elastic solids is derived using a combination of non-dimensional analysis and numerical analysis. Closed-form expressions for the four in-plane (i.e. the plane normal to the generators of the cells) micropolar compliances are derived in terms of the cell size, cell thickness and the linear elastic properties of the cell wall material. Independent analyses are conducted to verify the accuracy of the derived constants.

## 1. Introduction

Materials with cellular microstructure are increasingly used in several industrial sectors driven by the necessity for reducing weight. In order to design structures made of these materials, it is necessary to understand and develop methods to characterize mechanical response. Classical continuum theories, while still applicable to solids with microstructure, present difficulties when it is needed to capture macroscopic (volumes of material that contain several representative repeating units of the material microstructure) mechanical response. When a solid with microstructure is replaced by an equivalent continuum solid (through an appropriate homogenization procedure), it becomes necessary to include higher order effects, such as microrotations (Ostoja Starzewski 2008). These effects are not included in classical continuum mechanics that characterizes deformation at a point using three independent displacement degrees of freedom.

Cosserat & Cosserat (1909) developed a theory of materials with microstructure using interactions of forces and couple stress. Mindlin (1963) introduced a restricted couple stress theory, while Eringen (1983) developed a more general higher order theory, which also includes non-local micropolar effects. Bažant & Christensen (1998) computed the stress field in a micropolar orthotropic solid. Such a representation was obtained after ‘homogenizing’ a rectangular grid structure. Mora & Waas (2007) studied the micropolar elastic representation of a honeycomb structure using the configuration of a thick plate with a rigid circular inclusion, and in conjunction with experimental measurements of the deformation response. Several other studies have been conducted on hexagonal honeycombs with a view to understand macroscopic representations for the purpose of mechanical modelling. These include the effects due to curved cell edges and the effects of plateau borders, for example, as reported in Huang & Chang (2005), Yang & Huang (2005) and Yang *et al*. (2008*a*,*b*).

Ostoja-Starzewski & Jasiuk (1995) examined planar Cosserat elasticity with respect to certain features that are absent in two-dimensional classical elasticity. They introduced the following stress–strain relation related to planar problems of micropolar elasticity:(1.1)

In relations (1.1), *A*, *S*, *P* and *M*_{i}, (*i*=1,2) are the plane strain bulk compliance, the shear compliance, the micropolar compliance and the bending compliance, respectively. Also, *σ*_{ij} and *μ*_{ij} are the stress and the couple stress tensors, respectively, while *γ*_{ij} and *κ*_{ij} are the strain and the torsion tensors, respectively, expressed as follows:

In the above equations, *u*_{i}(*i*=1, 2) and *φ*_{3} are the in-plane displacements and the microrotation in the 1–2 plane, respectively.

Chung & Waas (1999, 2001) have studied the behaviour of perfectly circular cell polycarbonate honeycombs subjected to uniaxial and biaxial loadings. They showed analytically that perfectly circular cell honeycombs, when hexagonally packed, displayed isotropic behaviour, macroscopically, in the 1–2 plane (Chung & Waas 2000). Their analysis and method was limited to representing a hexagonally packed circular cell honeycomb, macroscopically, as a transversely isotropic classical continuum. They derived the following closed-form expressions for the macroscopic in-plane elastic constants of the honeycombs:

In the above equations, *R* is the cell radius of the honeycomb and *t* is the thickness of the cell wall. Furthermore, *E* and *ν* indicate the Young modulus and the Poisson ratio of the honeycomb wall solid material, respectively. In the above expressions, an asterisk is used to indicate macroscopic honeycomb properties.

In the present paper, closed-form expressions for the macroscopic1 micropolar elastic constants of a perfectly circular cell hexagonally packed honeycomb in the plane that is normal to the plane of the cell generators are derived. The honeycomb structure is shown in figure 1. The 1–2 axes are the in-plane axes of the honeycomb and the 3 axis is the out-of-plane axis that is parallel to the cell generators. Starting with the analytically derived elastic constants of the honeycombs (Chung & Waas 2000), closed-form expressions for the in-plane micropolar elastic constants of the honeycombs are derived through numerical analyses. The derived expressions for these constants are validated by considering different independent geometries.

## 2. In-plane micropolar constants of perfectly circular cell hexagonally packed honeycombs

The Abaqus commercial finite-element (FE) code was used for the numerical analysis and the numerical model is a 3×2 circular cell honeycomb structure made of B22 finite elements in the Abaqus library (figure 2). These are beam elements, which are three-noded quadratic elements, and they include transverse shear deformation and axial membrane deformation. The transverse shear deformation in these elements are treated as if the responses were linear elastic, independent of the axial and bending responses.

### (a) The closed forms of *A* and *S* using normal loading conditions

When the honeycomb structure is only subjected to normal loading, which produces macroscopically homogeneous deformation, the stress–strain relation of the structure (equation (1.1)) is reduced as follows:(2.1)

From Chung & Waas (2000), the macroscopic elastic normal properties of the honeycomb structure in the 1–2 plane are (figure 1)

Then, from dimensional analysis, it follows that the micropolar constants, *A* and *S*, must have the following closed-form structure, in terms of the four variables *E*, *ν*, *R* and *t*, as:(2.2)(2.3)where *m*_{1} and *m*_{2} are constants. If we know the values of *A* and *S* through a single numerical analysis of a particular problem configuration, with specific values of *E*, *ν*, *R* and *t*, then we can obtain the values of *m*_{1} and *m*_{2}, using the assumption of linearity. Figure 2 shows a schematic of the honeycomb structure under normal loads. In figure 2, cases 1 and 2 are two load cases corresponding to normal loading in the 1 and 2 directions, respectively. In these displacement control loading scenarios, only a single component of strain is activated at the macroscopic level, for each load case, as indicated by the deformed shapes of the honeycomb structure. The values of *A* and *S* obtained through the numerical experiments depicted as case 1 must be the same as the values obtained from case 2, because the honeycomb structure is isotropic in the 1–2 plane macroscopically. To perform the numerical analysis using the FE method, we used measured material data2 for the specific values of *E*, *ν*, *R* and *t*.

The displacements (*u*_{1}, *u*_{2}) and the microrotation (*φ*_{3}) corresponding to case 1 in figure 2 can be expressed as

In the above equation, *k*_{1} is a constant that is chosen in the numerical analysis. Hence, from equation (2.1),

In a similar vein, in case 2 (figure 2), the displacements (*u*_{1}, *u*_{2}) and the microrotation (*φ*_{3}) are

Again, *k*_{2} is a constant that is chosen in the numerical analysis. Hence, from equation (2.1),

In order to realize the behaviour of the honeycomb structure under normal loads as seen in cases 1 and 2 of figure 2, the boundary and loading conditions in figure 3 are applied in the numerical model of the honeycomb structure. As shown in figure 3, the honeycomb structure is under displacement control loading. By substituting for the non-zero strain in cases 1 and 2 and solving for the stresses, one obtains, in each case of loading, two known stress components and two unknowns, *A* and *S*. The macroscopic stresses are computed in the numerical model by defining them to be the sum of the reaction forces in a particular direction divided by the macroscopic area over which these forces act (i.e. the side areas that are perpendicular to the stress component being defined). Also, the sum of the reaction shear forces and moments at the boundaries of the numerical model is found to be zero. The values of *A* and *S* computed through the numerical analysis using case 1 are

These values were found to be identical to those computed in case 2. Hence, we can apply the values of *A* and *S* computed through the numerical analysis to equations (2.2) and (2.3) to obtain the following closed-form expressions for the micropolar constants *A* and *S*:

Figure 4 shows the deformed shapes of the honeycomb structure corresponding to cases 1 and 2 in figure 2, respectively.

### (b) Closed-form expressions for *P* and *S* using shear loading conditions

Closed-form expressions for the micropolar compliance (*P*) and the shear compliance (*S*) can be obtained by performing numerical analyses in which the honeycomb structure is subjected to displacement control shear loading. In the case of the shear compliance, *S*, we have already obtained the closed-form expression using the load cases (cases 1 and 2) earlier. If that expression is accurate for *S*, then we can verify its validity by performing a shear loading numerical analysis as described below.

The stress–strain relation of the structure (equation (1.1)) for shear loading, with zero normal strains, reduces to(2.4)

The macroscopic shear elastic constant of the honeycomb structure in the 1–2 plane is (Chung & Waas 2000)

Then, using dimensional analysis, it follows that the micropolar constants (*P* and *S*) can be predicted as(2.5)

(2.6)

In equations (2.5) and (2.6), *n*_{1} and *n*_{2} are constants. If we know the values of *P* and *S* through a suitable numerical analysis, then we can obtain the values of *n*_{1} and *n*_{2}. Also, in the case of the shear compliance (*S*), the value of *n*_{2} in equation (2.6) must be the same as that of *m*_{2} in equation (2.3) because the value of S obtained through the normal loading cases must be the same as the one through the shear loading cases. Figure 5 shows the honeycomb structure under simple shear deformation. In figure 5, cases 3 and 4 correspond to simple shear applied in the 1 and 2 directions, respectively. The values of *P* and *S* obtained through the numerical analysis corresponding to case 3 must be the same as the values obtained from case 4. The displacements (*u*_{1}, *u*_{2}) and the microrotation (*φ*_{3}) corresponding to case 3 in figure 5 can be expressed as

In the above equation, *k*_{3} is a constant that is chosen in the numerical analysis. Hence, from equation (2.4),

In case 4 (figure 5), the displacements (*u*_{1}, *u*_{2}) and the microrotation (*φ*_{3}) are

In the above equation, *k*_{4} is a constant that is chosen in the numerical analysis. Hence, from equation (2.4),

The boundary and loading conditions for cases 3 and 4 in figure 5 are indicated in figure 6. As shown in figure 6, ‘fix 1’ and ‘fix 2’ indicate that, during the loading condition, these nodes are constrained in the 1 and 2 directions, respectively, but free to move in the other direction.

Using a procedure similar to that in §2*a*, one can obtain the values of *P* and *S* computed through the numerical analysis. In other words, by substituting for the known non-zero strains in cases 3 and 4, the macroscopic stresses can be computed through defining them to be the sum of the reaction forces divided by the macroscopic area. Also, the sum of the reaction normal forces and moments at the boundaries of the numerical model is found to be zero. Then, substituting the macroscopic strains and stresses in equation (2.4), the two unknowns, *P* and *S*, can be computed. The values of *P* and *S* computed through the numerical analysis corresponding to case 3 are

The values of *P* and *S* for case 4 were found to be identical to those obtained from case 3. Also, the value of *S* obtained through the shear loading cases is found to be the same as that obtained through the normal loading cases. Hence, we can apply the values of *P* and *S* computed through the numerical analysis to equations (2.5) and (2.6) to obtain the following closed-form expressions for the micropolar constants *P* and *S*:

Figure 7 shows the deformed shapes of the honeycomb structure related to cases 3 and 4 in figure 5, respectively.

### (c) The closed-form expressions for *M*_{1} and *M*_{2} using pure bending curvature conditions

The closed-form expressions for the bending compliances, *M*_{1} and *M*_{2}, can be obtained through a numerical analysis in which the honeycomb structure is subjected to pure bending deformation conditions. When the honeycomb structure is under pure bending, the stress–strain relation of the structure (equation (1.1)) reduces to(2.7)

The moment–curvature relation from classical beam-bending theory is given by

From the above equation, we can obtain the following equations for the micropolar constants, *M*_{1} and *M*_{2}:(2.8)

In the above equations, and are the macroscopic areas normal to the 1 and 2 axes, respectively. We can predict the general form for and as follows:(2.9)

In the above equations, *p*_{1} and *p*_{2} are constants and *b* is the dimension of the honeycomb structure in the out-of-plane (‘3’) direction.

In equation (2.8), and are the macroscopic area moments of inertia normal to the 1 and 2 axes, respectively, and they are expressed in the following general way:(2.10)

In the above equations, *q*_{1} and *q*_{2} are constants. If equations (2.9) and (2.10) are substituted into equation (2.8), then we obtain the following general expressions for the bending compliances *M*_{1} and *M*_{2}:(2.11)

(2.12)

In equations (2.11) and (2.12), *α*_{1} and *α*_{2} are constants that can be obtained using the values of *M*_{1} and *M*_{2} computed through the numerical analysis.

Figure 8 shows the honeycomb structure under pure bending deformation. In figure 8, cases 5 and 6 correspond to the honeycomb structure subjected to pure bending deformation in the direction 3 on the surface normal to the directions 1 and 2, respectively. As explained earlier, the honeycomb structure is isotropic macroscopically. This implies that the honeycomb structure must satisfy the following two conditions. First, the dimension of the honeycomb material in the vertical direction to the neutral plane of the honeycomb structure does not change during the static macroscopic deformation ( in case 5 of figure 8 and in case 6 of figure 8). Second, the angle between the neutral plane and the vertical direction to the plane of the honeycomb structure is always 90° during the static macroscopic deformation. In order to apply these constraint conditions in the numerical model, the multi-point constraint (MPC) beam elements in Abaqus have been used. These MPC beam elements do not elongate or contract. However, the element can translate and rotate. For the first condition, as shown in figure 9, nodes (A, C, E, F) on the centroidal line of the honeycomb structure and nodes (A_{1}, A_{2}, C_{1}, C_{2}, E_{1}, E_{2}, F_{1}, F_{2}) at the boundaries of the honeycomb structure are connected through the MPC beam elements. The nodes at the boundaries are forced to follow the movements of the nodes on the centroidal line of the honeycomb structure (not included in the honeycomb structure). To apply the second condition, the constraint equation among rotations of nodes on the centroidal line of the honeycomb structure is used. Also, MPC slider elements are used because the nodes B_{1}, B_{2} and D_{1}, D_{2} in figure 9 are free to move on the inclined line composed of A_{1} and A_{2} and the inclined line composed of C_{1} and C_{2}, respectively.

In equation (2.7),

Substituting for the non-zero macroscopic curvatures that are known by the displacement control loading in cases 5 and 6, the macroscopic couple stresses can be numerically computed. The macroscopic couple stresses are the sum of the reaction moments in a particular direction divided by the macroscopic area. Also, the sum of the reaction normal and shear forces at the boundaries of the numerical model is found to be zero. Substituting the macroscopic curvatures and couple stresses in equation (2.7), the values of *M*_{1} and *M*_{2} can be obtained. The values of *M*_{1} and *M*_{2} computed through the numerical analyses using cases 5 and 6 are

Hence, we can apply the values of *M*_{1} and *M*_{2} computed through the numerical analyses to equations (2.11) and (2.12) to obtain the following closed-form expressions for the bending compliances *M*_{1} and *M*_{2}:

Figure 10 shows the deformed shapes of the honeycomb structure related to cases 5 and 6 in figure 8, respectively.

## 3. Discussion and concluding remarks

In order to validate the derived closed-form expressions, a comparison of two different honeycomb structures having different numbers of cells was conducted. The closed-form expressions for these constants must be independent of the number of cells that are used to represent the honeycomb. To confirm this independence of the cell number, we recomputed the values for the micropolar constants using a honeycomb model that has 4×2 circular cells and compared the results against the values obtained with the 3×2 circular cell model (both honeycombs have the same values on *E*, *ν*, *t* and *R* that were measured). The comparison of micropolar constants between the numerical analyses using the 3×2 and 4×2 circular cells is shown in table 1. As can be seen in table 1, the differences in the values of the computed micropolar constants are negligible between the two cases, suggesting that the values of the micropolar constants obtained are independent of the number of cells used in representing the honeycomb structure. Hence, it appears that the closed-form expressions derived for the micropolar constants are independent of the size of the honeycomb structure.

Further validation of the derived closed-form expressions was conducted through a comparison of numerical values obtained by considering the numerical analyses that use honeycomb representations having different cell radii and cell wall thicknesses. Two cases were compared: one is a case of a honeycomb structure that has a cell radius that is twice the cell radius used in cases 1–6, and the other is of a structure that has twice the cell radius and 1.5 times the cell wall thickness. Table 2 shows the results, in which the numerical results show a good agreement with the derived closed-form expressions for all these different cases, suggesting the accuracy of the closed-form expressions presented in this paper. It is noted that the micropolar compliance, *P*, for a hexagonally packed circular cell honeycomb is zero. Hence, the stress field of such a honeycomb structure depends only on four micropolar constants *A*, *S*, *M*_{1} and *M*_{2}, for which we have provided the closed-form expressions.

In summary, general closed-form expressions for the macroscopic micropolar constants (*A*, *S*, *P*, *M*_{1} and *M*_{2}) of a hexagonally packed circular cell honeycomb structure were derived through a combination of dimensional and numerical analyses. The constants *A* and *S* were obtained using normal loading conditions, the constant *P* using shear loading conditions and the constants *M*_{1} and *M*_{2} using pure bending curvature conditions. In deriving these expressions, a numerical honeycomb model that has 3×2 circular cells was used. The derived expressions were validated for honeycomb representations with different numbers of cells and for different combinations of geometric parameters.

## Footnotes

↵By macroscopic, we mean a volume of solid that occupies several tens of cells.

↵The Young modulus of the polycarbonate material,

*E*=2421.872 MPa, the Poisson ratio of the polycarbonate material,*v*=0.3, the average thickness of the honeycomb cell,*t*=0.0660 mm, the average radius of the honeycomb cell,*R*=2.0394 mm (Chung & Waas 1999, 2001).- Received June 1, 2008.
- Accepted July 28, 2008.

- © 2008 The Royal Society