Variation of Poisson ’ s Ratio with Axial Strain for Three-Dimensional Reentrant Auxetic Structures

This study presents numerical investigations of the in ﬂ uence of structural parameters on the variation of the Poisson ratio with axial strain for auxetic structures. Three-dimensional reentrant structures were considered with variable strut thickness to length ratio and reentrance angles. The variation of the volume, relative density, and relative sti ﬀ ness with axial strain was also studied.


Introduction
Mechanical metamaterials consist of designed structures that have no natural counterpart, that exhibit superior mechanical properties with respect to conventional materials [1,2]. The scope of the engineered cells and patterns is to overcome the issue of mechanical property scaling with density that occurs for regular structures: E str /E sol ∝ ðρ str /ρ sol Þ n (where E str and ρ str are the structure stiffness and density, and E sol and ρ sol are the stiffness and density of the solid) [2]. Practical applications of mechanical metamaterial cellular plastics are scarce due to the high expenses associated with their manufacturing, usually consisting of rapid prototyping (RP) [3][4][5]. However, if proven to possess outstanding properties, this class of materials can be used in highend applications, where the cost of production is not considered a priority.
Negative Poisson ratios (that determine an expansion in tension and contraction in compression) were experimentally observed in pyrites and single crystals in the beginning of the 20 th century [6], and later in biological tissues [7] and composite laminates [8]. The first theoretical auxetic structures were proposed by Alexander Kolpakov [9] and Robert Almgren [10] while Roderic Lakes obtained the first polymer foams that exhibit such characteristics [11]. The term "auxetic," referring to materials and structures that possess nega-tive Poisson ratios, was first introduced by Ken Evans in 1991 [12].
A negative Poisson ratio for a structure can be achieved through a variety of shapes. There are several categories in which auxetic structures can be classified: reentrant structures, rolling-up structures [23], and star/triangle-shaped structures [30] represent some of the basic 2D auxetic geometries. Out of the abovementioned structures, reentrant geometries can be patterned into 3D geometries with the least of effort [31].

The 3D Reentrant Structure
The structures analyzed in this work consisted of tessellated regular auxetic cells (struts of equal length) with square cross-section walls and three parameters: the reentrance angle α, the strut thickness 2t, and the strut length l ( Figure 1). Considering that the theoretical structures have no defects, size effect is nonexistent; thus, the two variable parameters of the designed structures are the reentrance angle α and the strut thickness to strut length ratio t/l.
For this study, the reentrance angle α was varied between 65°(the lowest angle value that allowed clearance between the vertical struts) and 75°while the t/l ratio was varied between 0:3 and 0:7.
2.1. Relative Density. The relative density of the structures ρ/ρ s is expressed as the ratio between the density of the structure and the density of the solid. This relation can be extrapolated for any base material through expressing the relative density as the ratio between the volume of the structure V structure and the volume of the circumscribed prism V prism [3].ρ The variation of the relative density with the re-entrance angle α and the t/l ratio is depicted in Figure 2 while Equation (2) presents the surface equation that was fitted for the obtained data (R 2 = 0:9997).

Results
Numerical analyses were performed on the designed structures that were generated using the parameter variation presented in Table 1. A total of 50 analyses were performed (25 for tensile loadings and 25 for compressive loadings). For the material model, a isotropic linear elastic formulation was used with Young's modulus E solid = 1500 MPa and a Poisson's ratio ν solid = 0:3, corresponding an ABS compound (VeroWhitePlus RGD835) used in photopolymerization rapid prototyping [3]. The analyses were performed in Abaqus using the Standard solver. The models were meshed with second order tetrahedral elements (C3D10) with varying sizes to ensure a number of at least 4 elements per cross section.
During the analyses, the displacements variation with step time of the most outer nodes of the structure was recorded (d 1 and d 2 , respectively) for all three directions, the length of the sides of the circumscribed prism (xðtÞ, yðtÞ, zðtÞ) resulting from the subtraction of the two values Subsequently, the nominal strains along the three directions were calculated with the relations: Poisson's ratios with respect to the xOy pane (ν xy ) and the zOy plane (ν zy ) were observed to be equal and are expressed with the relation 3.1. Size Effect. In order to reduce computational time, a study was performed in order to determine the smallest structure that produces accurate results. For this purpose, structures varying from one cell up to 8 × 8 × 7 cells were investigated ( Figure 3). The results of Poisson's ratio variation with strain ( Figure 4) show a slight difference between the structure sizes, more noticeable in compression. Considering the relative errors and simulation times, it was decided that for the rest of the study, a 3 × 3 × 7 structure will be used.

Poisson's Ratio Variation with Structural Parameters.
The variation of Poisson's ratio with strain is due to the modification of the reentrance angle α with the axial strain ε y ðtÞ. For compressive loadings, Poisson's ratio increases with the increase in strain (the reentrance angle diminishing) while for tensile loadings, Poisson's ratio decreases with the increase in strain (due to the augmentation of the reentrance angle). When the reentrance angle reaches values larger than 90°, Poisson's ratio becomes positive, and the structures ceases to be auxetic. A typical variation of Poisson's ratio with axial strain for various values of the reentrance angle is presented in Figure 5 for a t/l ratio of 0:5.
For design purposes, the variation of Poisson's ratio with the initial reentrance angle and with strain can be expressed as a surface for each t/l ratio. The surfaces were fitted for Equation (6) with the coefficients being presented in Table 2.
A graphical depiction of the resulted surfaces is presented in Figure 6. 3.3. Volume and Relative Density Change. The variation of the volume of a structure with strain can be defines as

Material Design & Processing Communications
Considering the initial circumscribed volume of the structure V 0 = x 0 •y 0 •z 0 , the relative volume of the structure at a given strain can be expressed as   (5), the strains along the x and z directions will become Introducing Equation (9) in Equation (8) yields An example for the variation of the relative volume of the circumscribed prism with axial strain is presented in Figure 7(a)) for a t/l ratio of 0.7.
Considering the definition of the relative density (Equation (1)) and the variation of the circumscribed volume with strain (Equation (7)), the former's variation with strain is presented in Equation (11).    Material Design & Processing Communications determined. The true stress s y is expressed as the recorded force in the fixed nodes F divided by the instantaneous area: The logarithmic (Hencky) strain e y is expressed as e y t ð Þ = ln ε y + 1 À Á : An example of a true stress-logarithmic strain curve is presented in Figure 8 for the structure with α = 65°and t/l = 0:7. A nonlinear response is observed due to the volume change. In addition, for strain below0:08 mm/mm, a sudden increase in the absolute stress values is recorded, due to the contact between the struts of the structure.
The stiffness of the structureẼ corresponds to the slope of the stress-strain curve and is evaluated as the variation of the true stress divided by the variation of the corresponding logarithmic strain (for small strain values).
The relative stiffness of the structure is expressed as the stiffness of the structure divided by the stiffness of the solid.
The variation of the relative stiffness of the structures with the reentrance angle and the t/l ratio is presented in Figure 9.
The surface depicted in Figure 9 was fitted for the Equation (16)

Discussions and Conclusions
This work presents the influence of the structural parameters (reentrence angle α and the thickness to lenght ratio t/l) on the stiffness, Poisson's ratio and density of auxetic structures. Due to the change in values of the reentrance angle value during deformation, a variation of the Poisson's ratio and of the relative density with strain is observed. Considering the increase in both values with the increase in compressive strain, this class of materials is expected to perform better than conventional cellular structures in energy absorption applications, as the material becomes denser and stronger as it is compressed. Equations describing the variation of relative density and stiffness with the structural parameters are presented, which can be useful in the design stage of auxetic structures for specific applications. Future work will focus on the investigation of the influence of structural parameters on the energy absorption of the investigated structures, incorporating more advanced constitutive models, which account for damage and failure.

Data Availability
The data supporting the results can be provided at request.

Conflicts of Interest
The author declares that there is no actual or potential conflict of interest in relation to this article.