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

Authors: Dan Andrei Şerban
Journal: Material Design & Processing Communications
 (2022)

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## Abstract

This study presents numerical investigations of the influence 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 stiffness with axial strain was also studied.


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## Body

## 1. 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: Estr/Esol∝ρstr/ρsoln (where Estr and ρstr are the structure stiffness and density, and Esol and ρsol 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) [3–5]. However, if proven to possess outstanding properties, this class of materials can be used in high-end 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 20th 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 negative Poisson ratios, was first introduced by Ken Evans in 1991 [12].Over the years, auxetic structures were predicted, observed, and designed for various sizes, from nanomaterials (such as carbon allotropes [13, 14], liquid crystal polymers [15], or metals [16]) to microstructured foams [17–19] and macroscale lattices [20–22].Considering their unique structural characteristics, in comparison with conventional materials, auxetic structures exhibit higher shear and fracture resistance as well as improved hardness and indentation resistance [23, 24] . Consequently, auxetic structures have a very wide range of potential application fields, such as aerospace industry (nose cones, wing panels) [25], automotive industry (bumpers, cushions) [26], biomedical applications (wound pressure pads, artificial blood vessels and skins) [27], military applications (helmets, bullet-proof vests) [23], textile industry (functional fabrics, color-change straps) [28], or sports industry (footwear applications, protective equipment) [29].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].


## 2. 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 angleα, 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.Figure 1
Auxetic cell (a) and structural parameters (b).
(a)(b)For this study, the reentrance angleα 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 structuresρ~/ρs 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 Vstructure and the volume of the circumscribed prism Vprism [3].
(1)ρ~ρs=VstructureVprism−.The variation of the relative density with the re-entrance angleα and the t/l ratio is depicted in Figure 2 while Equation (2) presents the surface equation that was fitted for the obtained data (R2=0.9997).
(2)ρrelα,tl=e184.53α+1.84lntl−4.405.Figure 2
Variation of relative density with the reentrance angleα and the t/l ratio.


## 2.1. Relative Density

The relative density of the structuresρ~/ρs 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 Vstructure and the volume of the circumscribed prism Vprism [3].
(1)ρ~ρs=VstructureVprism−.The variation of the relative density with the re-entrance angleα and the t/l ratio is depicted in Figure 2 while Equation (2) presents the surface equation that was fitted for the obtained data (R2=0.9997).
(2)ρrelα,tl=e184.53α+1.84lntl−4.405.Figure 2
Variation of relative density with the reentrance angleα and the t/l ratio.


## 3. Results

Numerical analyses were performed on the designed structures that were generated using the parameter variation presented in Table1. A total of 50 analyses were performed (25 for tensile loadings and 25 for compressive loadings).Table 1
Auxetic structure parameters.
α°t/l−650.367.50.4700.572.50.6750.7For the material model, a isotropic linear elastic formulation was used with Young’s modulusEsolid=1500MPa and a Poisson’s ratio νsolid=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 (d1 and d2, respectively) for all three directions, the length of the sides of the circumscribed prism (xt,yt,zt) resulting from the subtraction of the two values
(3)xt=d1xt−d2xtyt=d1yt−d2ytzt=d1zt−d2zt.Subsequently, the nominal strains along the three directions were calculated with the relations:(4)εxt=xt−x0x0εyt=yt−y0y0εzt=zt−z0z0.Poisson’s ratios with respect to thexOy pane (νxy) and the zOy plane (νzy) were observed to be equal and are expressed with the relation
(5)νxyt=νzyt=νt=−εxtεyt=−εztεyt.


### 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 to8×8×7 cells were investigated (Figure 3).Figure 3
Auxetic structures of3×3×7 cells (a) and 8×8×7 (b).
(a)(b)The results of Poisson’s ratio variation with strain (Figure4) 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.Figure 4
Variation of Poisson’s ratio with axial strain for various structure sizes.


### 3.2. Poisson’s Ratio Variation with Structural Parameters

The variation of Poisson’s ratio with strain is due to the modification of the reentrance angleα with the axial strain εyt. 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.Figure 5
Variation of Poisson’s ratio with the reentrance angle and axial strain fort/l=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 eacht/l ratio. The surfaces were fitted for Equation (6) with the coefficients being presented in Table 2.
(6)να,εy=a+bα+cεy+dα2+fεy2+gαεy.Table 2
Surface equation coefficients.
t/labcdfg0.30.1935−0.0211−0.29060.0002−0.04220.00910.40.4715−0.0284−0.30640.0003−0.03660.0090.50.3231−0.0238−0.31950.0002−0.03230.0090.60.4775−0.0275−0.35620.0002−0.02750.00920.70.6844−0.0326−0.39370.0003−0.02420.0095A graphical depiction of the resulted surfaces is presented in Figure6.Figure 6
Variation of Poisson’s ratio with the reentrance angle and axial strain for varioust/l ratios.


### 3.3. Volume and Relative Density Change

The variation of the volume of a structure with strain can be defines as(7)Vεx,εy,εz=x01+εxt∙y01+εyt∙z01+εzt.Considering the initial circumscribed volume of the structureV0=x0∙y0∙z0, the relative volume of the structure at a given strain can be expressed as
(8)Vrelεx,εy,εz,t=Vεx,εy,εzV0=1+εxt∙1+εyt∙1+εzt.Considering Equation (5), the strains along the x and z directions will become
(9)εx=εz=−νεy∙εy.Introducing Equation (9) in Equation (8) yields
(10)Vrelεy=1−νεy∙εy2∙1+εy.An example for the variation of the relative volume of the circumscribed prism with axial strain is presented in Figure7(a)) for a t/l ratio of 0.7.Figure 7
Relative volume of the circumscribed prism (a) and relative density (b) variation with strain fort/l=0.7.
(a)(b)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).
(11)ρrelεx,εy,εz=Vstrx01+εx∙y01+εy∙z01+εz.Figure7(b) depicts the variation of the relative density with strain for a t/l ratio of 0.7 showing an exponential increase in compression and decrease in tension.


### 3.4. Relative Stiffness

In order to evaluate the stiffness of the structures, the true stress and logarithmic strain must be determined. The true stresssy is expressed as the recorded force in the fixed nodes F divided by the instantaneous area:
(12)syt=Ftx01+εxt∙z01+εzt.The logarithmic (Hencky) strainey is expressed as
(13)eyt=lnεy+1.An example of a true stress-logarithmic strain curve is presented in Figure8 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 below0.08mm/mm, a sudden increase in the absolute stress values is recorded, due to the contact between the struts of the structure.Figure 8
True stress variation with logarithmic strain for the structure withα=65° and t/l=0.7.The stiffness of the structureE~ 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).
(14)E~=dsydey=syt1−syt0eyt1−eyt0.The relative stiffness of the structure is expressed as the stiffness of the structure divided by the stiffness of the solid.(15)Erel=E~Es.The variation of the relative stiffness of the structures with the reentrance angle and thet/l ratio is presented in Figure 9.Figure 9
Variation of the relative stiffness with the re-entrance angle andt/l ratio.The surface depicted in Figure9 was fitted for the Equation (16) with R2=0.9992:
(16)Erelα,tl=e109.66α+3.95lntl−7.109.


## 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 to8×8×7 cells were investigated (Figure 3).Figure 3
Auxetic structures of3×3×7 cells (a) and 8×8×7 (b).
(a)(b)The results of Poisson’s ratio variation with strain (Figure4) 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.Figure 4
Variation of Poisson’s ratio with axial strain for various structure sizes.


## 3.2. Poisson’s Ratio Variation with Structural Parameters

The variation of Poisson’s ratio with strain is due to the modification of the reentrance angleα with the axial strain εyt. 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.Figure 5
Variation of Poisson’s ratio with the reentrance angle and axial strain fort/l=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 eacht/l ratio. The surfaces were fitted for Equation (6) with the coefficients being presented in Table 2.
(6)να,εy=a+bα+cεy+dα2+fεy2+gαεy.Table 2
Surface equation coefficients.
t/labcdfg0.30.1935−0.0211−0.29060.0002−0.04220.00910.40.4715−0.0284−0.30640.0003−0.03660.0090.50.3231−0.0238−0.31950.0002−0.03230.0090.60.4775−0.0275−0.35620.0002−0.02750.00920.70.6844−0.0326−0.39370.0003−0.02420.0095A graphical depiction of the resulted surfaces is presented in Figure6.Figure 6
Variation of Poisson’s ratio with the reentrance angle and axial strain for varioust/l ratios.


## 3.3. Volume and Relative Density Change

The variation of the volume of a structure with strain can be defines as(7)Vεx,εy,εz=x01+εxt∙y01+εyt∙z01+εzt.Considering the initial circumscribed volume of the structureV0=x0∙y0∙z0, the relative volume of the structure at a given strain can be expressed as
(8)Vrelεx,εy,εz,t=Vεx,εy,εzV0=1+εxt∙1+εyt∙1+εzt.Considering Equation (5), the strains along the x and z directions will become
(9)εx=εz=−νεy∙εy.Introducing Equation (9) in Equation (8) yields
(10)Vrelεy=1−νεy∙εy2∙1+εy.An example for the variation of the relative volume of the circumscribed prism with axial strain is presented in Figure7(a)) for a t/l ratio of 0.7.Figure 7
Relative volume of the circumscribed prism (a) and relative density (b) variation with strain fort/l=0.7.
(a)(b)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).
(11)ρrelεx,εy,εz=Vstrx01+εx∙y01+εy∙z01+εz.Figure7(b) depicts the variation of the relative density with strain for a t/l ratio of 0.7 showing an exponential increase in compression and decrease in tension.


## 3.4. Relative Stiffness

In order to evaluate the stiffness of the structures, the true stress and logarithmic strain must be determined. The true stresssy is expressed as the recorded force in the fixed nodes F divided by the instantaneous area:
(12)syt=Ftx01+εxt∙z01+εzt.The logarithmic (Hencky) strainey is expressed as
(13)eyt=lnεy+1.An example of a true stress-logarithmic strain curve is presented in Figure8 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 below0.08mm/mm, a sudden increase in the absolute stress values is recorded, due to the contact between the struts of the structure.Figure 8
True stress variation with logarithmic strain for the structure withα=65° and t/l=0.7.The stiffness of the structureE~ 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).
(14)E~=dsydey=syt1−syt0eyt1−eyt0.The relative stiffness of the structure is expressed as the stiffness of the structure divided by the stiffness of the solid.(15)Erel=E~Es.The variation of the relative stiffness of the structures with the reentrance angle and thet/l ratio is presented in Figure 9.Figure 9
Variation of the relative stiffness with the re-entrance angle andt/l ratio.The surface depicted in Figure9 was fitted for the Equation (16) with R2=0.9992:
(16)Erelα,tl=e109.66α+3.95lntl−7.109.


## 4. Discussions and Conclusions

This work presents the influence of the structural parameters (reentrence angleα 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.



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