Auxetic materials exhibit a unique characteristic due to the altered microstructure. Different structures have been used to model these materials. This paper treats a development of finite element model and theoretical formulation of 3D star honeycomb structure of these materials. Various shape parameters of the structural cell were evaluated with respect to the basic mechanical properties of the cell. Finite element and analytical approach for various geometrical parameters were numerically used to formulate the characteristics of the material. The study aims at quantifying mechanical properties for any domain in which auxetic material is of interest for variations in geometrical parameters. It is evident that mechanical properties of the material could be controlled by changing the base wall angle of the configuration. The primary outcome of the study is a design guideline for the use of 3D star honeycomb auxetic cellular structure in structural applications.
For decades, several geometrical structures representing auxetic behaviour have been introduced, fabricated, and tested particularly to obtain their mechanical properties. These geometrical structures are beneficial since they make contribution to researchers to comprehend on how auxetic behaviour can be obtained and how auxetic materials can be fabricated. Computational procedure is the main contribution in predicting their properties.
Reentrant structures which have been investigated by various researchers [
In the present study, a simple 3D star honeycomb cellular structure was modelled by adapting a 2D cellular structure. Subsequently, the influence of structure shape on basic properties has also been evaluated. In particular, the basic mechanical properties discussed in this study are elastic modulus, density ratio, and Poisson’s ratio which were examined for different values of shape parameters, namely, the wall length, the cell wall thicknesses, and the cellular structure angle. Consisting of eight elastic beams, the 2D cellular structure has a symmetrical configuration. Analytical formulation of mechanical properties for the structure has eventually been developed. This investigation highlights finite element discretization and the developed formulation of 3D star honeycomb structures for auxetic materials that have not been proposed previously. From this study, it is evident that mechanical properties of auxetic material are controllable, thus facilitating the fabrication technique used in preparing samples in the laboratory.
Figure
(a) Star honeycomb structure in 2D and 3D. (b) An element of auxetic material made of 3D star honeycomb structures.
An elastic cellular structure has numerically been developed using finite element technique to determine the basic properties of the star honeycomb structures. ABAQUS finite element code has been used to perform these analyses. An element of the auxetic material shown in Figure
Meanwhile, the Young’s modulus was calculated by dividing the averaged uniaxial stresses by the uniaxial strains while the Poisson’s ratios were obtained by dividing the averaged transverse strain by the uniaxial imposed ones. Several finite element models were consecutively developed for different values of the length, cross section, and angles. The Poisson’s ratios and elastic modulus including
Von Misses stress of Model number 1 in the case of uniaxial imposed displacement along
Von Misses stress of Model number 1 in the case of uniaxial imposed displacement along
Using analytical approach, Poisson’s ratios, elastic modulus, and density ratios of all models may also be predicted. From mathematics and mechanics point of views, there is relationship between the geometrical parameters of the cell, mechanical properties of the base material, and mechanical properties of the star honeycomb structure including Poisson’s ratio, elastic modulus, and density ratio. The relationships can be expressed as the following functions:
For the validity of a set of (
Values of
Model number 





0.8727  0.02  0.8 
2  0.8727  0.04  0.9 
3  0.8727  0.06  0.9 
4  0.9599  0.02  0.8 
5  0.9599  0.02  0.9 
6  0.9599  0.04  0.8 
7  0.9599  0.04  0.9 
8  0.9599  0.06  0.8 
9  0.9599  0.06  0.9 
10  1.0472  0.02  0.8 
11  1.0472  0.06  0.9 
12  1.1345  0.02  0.8 
13  1.1345  0.04  0.8 
14  1.1345  0.04  0.9 
15  1.1345  0.06  0.8 
16  1.309  0.02  0.8 
17  1.309  0.02  0.9 
18  1.309  0.04  0.8 
19  1.309  0.06  0.8 
20  1.309  0.06  0.9 
21  1.3963  0.02  0.8 
22  1.3963  0.02  0.9 
23  1.3963  0.04  0.9 
24  1.3963  0.06  0.8 
25  1.4835  0.02  0.8 
26  1.4835  0.02  0.9 
27  1.4835  0.04  0.8 
By using a set of (
Obtaining
Also, the coefficients of
As illustrated, Poisson’s ratio and elastic modulus of all defined models have been determined by using finite element approach. In terms of density ratio of the models, the formula of density for auxetic material was used to calculate density ratio. In particular, dependency of material density on geometrical parameters is presented as follows:
This present study has focused on Poisson’s ratio, Young’s modulus, and the density of a 3D element of auxetic material made of star honeycomb structure using numerical method in conjunction with analytical approach. Details of the results obtained for the models consisting of cells in 27 cases are also presented in Table
Finite element results of Poisson’s ratios, elastic modulus, and density ratio of models.
Model 








1 






0.001 
2 






0.004 
3 






0.009 
4 






0.0009 
5 






0.0007 
6 






0.0035 
7 






0.0028 
8 






0.0082 
9 






0.0064 
10 






0.0007 
11 






0.0047 
12 






0.0005 
13 






0.002 
14 






0.0016 
15 






0.0044 
16 







17 






0.0003 
18 






0.0012 
19 






0.0027 
20 






0.0023 
21 






0.0002 
22 






0.001 
23 






0.0009 
24 






0.0022 
25 






0.0002 
26 






0.0002 
27 






0.0008 
Mechanical properties versus
Mechanical properties versus
The primary outcomes of the finite element analyses could be remarked as follows.
From
From
From
In terms of elastic modulus and density,
Dimensionless geometrical parameters used in this research are
As illustrated, the aim of this study is to introduce a method for formulating beam like star honeycomb structures. Although the result of that is reasonable for limited domain of geometrical parameters, it offers a method for formulating mechanical properties for any desired domain in which the auxetic material is about to be designed and fabricated.
Although the shape of the defined structure seems to exhibit auxetic behaviour in all directions, it has interestingly found that it does not exhibit auxetic behaviour in some cases depending on the values of geometrical parameters. For instance, from
Coefficients of Poisson’s ratios, elastic modulus, and density ratio functions obtained from (

























































































































































































































































































A symmetrical configuration for the star honeycomb cellular structure of an auxetic material has been proposed and subsequently its analytical principle has also been formulated. The topology allows more degrees of freedom to the designer in controlling the mechanical properties of auxetic material by varying the wall lengths, cell wall thickness, and angle. Manipulation of stiffness properties including Poisson’s ratio and fracture toughness can be performed by modification of the cellular structure’s structural parameters. The overall mechanical properties can be controlled by modification of the base wall angle of the configurations, with the consequent changes, in particular, of the inplane Poisson’s ratio. This research provides a guideline for applications to 3D star honeycomb auxetic structures and also provides a basis for experimental additive manufacturing processes such as electron beam melting or selective laser sintering.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Funding by the Ministry of Education (MOE), Government of Malaysia through Universiti Teknologi Malaysia, research Grants R.J130000.7824.4L105 and R.J130000.7824.4F248 are sincerely acknowledged.