Complex Analysis of an Auxetic Structure under Compressive Loads

Abstract

Cellular structures subjected to compressive loads provide a reliable solution for improving safety. As a member of cellular material, auxetic metamaterials can enhance performance according to the definition of the negative Poisson ratio. In conjunction with Rapid Prototyping by Additive Manufacturing methods, complex structures can be manufactured using a wide range of materials. This paper debuts the development process of a reliable material model that is useful for the numerical simulation, and further details and investigates the performance indicators of an auxetic structure, namely anti-tetra-chiral. These indicators are related to the force developed during the plateau stage, the length of the plateau stage, and the nominal dimensions of the structure to avoid buckling during compression. Two new indicators discussed in this paper aim to provide a complete set of performance indicators. The first analytical solution provides the displacement of the circular nodes during the compression. The second analytical solution estimates the strain developed in the ligaments. Considering the performance of the processed material, this analysis aims to determine whether the structure can develop the complete plateau stage or whether premature failure will occur.

1. Introduction

Technological progress requires modern, improved materials that are as low-density, durable, cost-effective, and sustainable as possible [1]. The development of such materials is also in line with the trends of Industry 4.0, where we can find the concept of smart materials [2]. Smart materials have the unique ability to modify their properties in response to specific stimuli such as temperature [3], light, pressure [4], and magnetic or electric fields [5]; this versatility and adaptability of smart materials [6] make them increasingly popular in various fields compared to conventional materials [7,8,9].

This study analyzed the mechanical metamaterials mainly characterized by the value of the Poisson’s ratio. Poisson’s ratio expresses the ratio of the deformation in the normal direction to the other orthogonal directions. Depending on how this relative deformation occurs, we can speak of natural materials $\nu \in (0;0.5)$ or materials with auxetic character in which Poisson’s ratio has negative values [10,11]. Owing to their counterintuitive deformation characteristics, the auxetic materials [12] present numerous extraordinary mechanical properties like energy absorption [13], plane-strain fracture toughness [14], fracture resistance [15], indentation resistance [16], impact resistance [17], hardness, shear resistance [18], blast resistance and other excellent properties, and potential to form synclastic curvatures under out-of-plane bending [19,20,21]. The mechanical performance of the cellular structures depends on the loading direction, and these cellular structures are usually anisotropic [22]. Because of these exceptional characteristics established for auxetic structures [6], applications can be fulfilled in a wide range of applications: aerospace, mechanical engineering [23,24,25,26,27], medical application [28,29], acoustic engineering [30,31], seismic engineering [32,33], smart sensors [34,35], tubular structures [36,37,38,39], defense [40,41], sports [42], textile industry [43], smart filters, and protective engineering [6,23,44,45,46]. The performance of the negative Poisson’s ratio is given by the shape and geometrical arrangement of the basic units: re-entrant, with the unit defined by an irregular polygon with an interior angle greater than ${180}^{°}$ [23,24], chiral, with the unit comprised of a central cylinder with tangentially attached ligaments [47,48], and rotating polygonal models, which use a complex unit formed of square and equilateral triangular shapes connected by hinges at their vertices [49].

Using Rapid Prototyping (RP), parts can be obtained by two different methods of manufacturing: traditional method or Additive Manufacturing (AM) [50]. Both can allow obtaining the sample directly from the CAD model using additive layer manufacturing techniques. In the past, RP equipment was used to produce models and prototype parts, but now, it is widely used even for small production parts [51,52]. Furthermore, Additive Manufacturing is fully supporting the sustainable manufacturing process as it can minimize the waste of raw materials [53]. This process allows the manufacturing of fully customized parts [54,55].

From a structural point of view, the performance of a part increases as gaps in the outer surfaces are eliminated. While voids in the outer surfaces are ideally removed, the inner voids, hereafter referred to as porosity, are ideally present in the structure of a product. Additive Manufacturing (AM), according to the ISO/ASTM 52900 standard, is defined as “the process of joining materials to make parts from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing and formative manufacturing methodologies” [56]. Additive Manufacturing can produce auxetic structures with complex designs that cannot be achieved using conventional manufacturing methods, leading to improved material properties and new functionalities.

Industry assimilation of AM [57] requires efforts in three main areas: price, the comparative advantages of AM over traditional manufacturing methods of the same sample, and the rate at which such advantages can occur. One of the most important factors is represented by the cost. AM machine costs constitute between 50% and 75% of the total production cost. Cutting these costs may considerably affect the adoption of AM technologies in terms of quality, performance, and expanding dimensional capabilities [58]. AM technology can manufacture various materials, including metal alloys, ceramics, polymers, and composite materials [59,60].

For AM technology, the literature [8,52,61,62,63,64] presents an ample field of applications in various industries, such as the automotive, aerospace, medical, architecture, tooling and devices, chemical, and biomedical industries. This fact gives this technology its due importance.

According to experimental determinations, the main advantages of AM technology are [52,65]:

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flexible design—unlimited possibilities from the point of view of shape/geometry (samples with complex geometry, geometry that would be hard or impossible to achieve using conventional technology);

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minimizing waste [2];

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time manufacturing reduction by eliminating operations and planning the production process [53];

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improved efficiency and increased productivity for prototypes and spare parts on a small or medium scale;

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flexibility and agility;

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reduced costs by the removal of traditional process tools and procedures [66].

These AM technologies have a series of drawbacks, but the most important are [58]:

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reduced surface finish;

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limited range of materials that can be used;

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limited dimensional tolerances (can be caused by raw material’s dimensions);

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reproducibility/repeatability.

The properties of the parts obtained by these methods have not yet been standardized. There are several parameters that should be considered by the design process, the raw material, and the printing and validation processes that can lead to different final products. At this point, it is impossible to ensure good repeatability of the process [67].

Therefore, AM is a rapid prototyping method that allows the manufacturing of complex geometrical shapes, and gives access to many options regarding surface adjustments. Laser and electron-based techniques are the most promising AM technologies for manufacturing [68].

In this study, we developed and studied two configurations defined and manufactured using Fused Filament Fabrication (FFF).

FFF technology is part of the material extrusion technologies of AM, and it is one of the most famous, low-cost, safe 3D printing techniques with wide material options to process in AM technologies for extruding materials [69]. FFF technology is probably the most accessible method of Additive Manufacturing. For an accurate modeling process, a proper material modeling procedure is needed. An experimental plan was developed, and the results allowed a complex characterization of FFF structures considering the particular stress state using the concept of triaxiality [70].

In the case of FFF technology, the raw material is supplied into a heated nozzle and then deposited layer by layer at constant pressure and uninterrupted flow. After every layer, the platform moves vertically on the Z-axis, and the nozzle moves horizontally in two directions (the X and Y axes) [71]. The nozzle temperature and speed, infill pattern and density, orientation angle, layer thickness, and bed temperature can influence the strength and precision of the manufactured sample. These characteristics lead to changes in the final mechanical characteristics of the part in terms of tensile strength, flexural strength, and compressive strength [72]. The current literature confirms that the performance of cellular material significantly depends on the loading direction and geometrical characteristics [69].

According to past studies, the main drawbacks of FFF are poor surface finish and weak mechanical properties. The main issues in the FFF process are surface roughness, sample shrinkage, warping from the edges, voids and porosity, misaligned sample geometry, lack and loss of adhesion, and sample deformation [72].

2. Theoretical Background

The deformation process of the anti-tetra-chiral structures is generated by the rotation of nodes producing ligament bending under out-of-plane loading conditions [73,74,75] (tension/compression). Euler–Bernoulli’s small linear elastic bending deformation theory [76,77,78,79] is used for the derivation of the analytical solution [80,81]. Following previous research, these hypotheses are formulated:

−

the shape of the nodes connecting the ligaments does not change under the applied loads [20,48,80,82];

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Bernoulli’s model of the beam under bending load is applied to study the deformation process of the ligament [77,81,83,84];

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the rotation of the rigid nodes generates internal forces and moments;

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internal forces that are oriented in a direction perpendicular to the externally applied stress vanish;

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shear deformations of the ligaments are neglected;

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the reaction force is determined by the dimensions of the unit structure and the number along the horizontal axis;

−

the deformation range (plateau length) is dependent on the number of units in the vertical direction [85];

−

the buckling load is derived from the classical solution (Euler’s formula) [86] applied for a column with a centric load [87].

2.1. Poisson’s Ratio

Poisson’s ratio is defined as [82,88]:

$${\nu }_{HV}=-\frac{{\epsilon }_H}{{\epsilon }_V}$$

(1)

For an auxetic tetra-anti-chiral structure, Poisson’s ratio can be estimated using the following equation:

$${\nu }_{HV}=-\frac{\frac{1}{2·l_H}}{\frac{1}{2·l_V}}·\frac{\left[\frac{l_V}{I_V}·{\left(I_V·l_H\right)}^2+\frac{l_H}{I_H}·{\left(I_H·l_V\right)}^2\right]·\frac{2·R_i^2}{{\left({I_V·l}_H+{I_H·l}_V\right)}^2}}{\frac{{2·l}_H}{A_H}+\left[\frac{l_V}{I_V}·{\left(I_V·l_H\right)}^2+\frac{l_H}{I_H}·{\left(I_H·l_V\right)}^2\right]·\frac{2·R_i^2}{{\left({I_V·l}_H+{I_H·l}_V\right)}^2}}$$

(2)

where:

$$A_H=t_H·w;A_V=t_V·w;I_H=\frac{t_H^3·w}{12};I_V=\frac{t_V^3·w}{12}$$

(3)

and:

$$l_H=L_H/2;l_V=L_V/2$$

(4)

2.2. Plateau Force

During the compressive load process, the reaction force developed in a horizontal cross-section (in truth, the reaction force) is determined by [85]:

$$F_p=\left(2·n_H\right)·\frac{3}{R_i+t/2}·{\sigma }_y·\frac{t^2·w}{4}$$

(5)

where $n_H$ is the number of cells in the horizontal direction, and $t=t_H=t_V$.

2.3. Plateau Length

Using the geometrical features of the cell, namely the gap $\left(\delta \right)$ and the number of cells along the vertical direction $\left(n_V\right)$, the minimum plateau length $\left(d_p\right)$ before densification can be determined:

$$d_p=\left(n_V-1\right)·\delta$$

(6)

where the gap $\left(\delta \right)$ to be closed is determined using the equation:

$$\begin{gathered} \delta =\frac{1}{2}·\left[L-2·\left(R_i+t\right)\right] \\ L=L_H=L_V \end{gathered}$$

(7)

The compression process is correctly developed if the plateau length does not exceed the critical buckling load.

2.4. Geometrical Features of the Structure (Stable Axial Compression Process)

For an auxetic, tetra-anti-chiral structure, the critical buckling load $F_{cr}$ is defined as:

$${F_{cr}=F}_{cr}·{\left(\frac{2·L_V-R_i}{2·L_V+R_i}\right)}^{1/2}·\frac{1}{2}·\left[1+\left(1-2·\frac{2}{3}·\frac{\left(R_i+t\right)}{L_V}·\sin \alpha \right)\right]$$

(8)

Therefore, the condition of axial compression is fulfilled if:

$$\frac{F_p}{F_{cr}}<1$$

(9)

The maximum number of units that can be stacked is:

$$n_V\le \left[{\left(\frac{\frac{3·{\sigma }_y}{R_i+t/2}}{\frac{{\pi }^2·E·t}{{\left(k_b·L_V\right)}^2}·w·{\left(\frac{2·L_V-R_i}{2·L_V+R_i}\right)}^{1/2}}\right)}^{\frac{1}{2}}\right]$$

(10)

where [ ] gives the integer.

By a convenient manipulation of the terms, the ratio between the thickness $\left(t\right)$ and width $\left(w\right)$ can be determined for a structure to display axial crushing:

$$\frac{F_p}{F_{cr}}=1$$

(11)

which yields:

$$\frac{\frac{3·{\sigma }_y·\frac{t^2·w}{4}}{R_i+t/2}}{\frac{{\pi }^2·E·\frac{w^3·t}{12}}{{\left(k_b·L_V·n_V\right)}^2}·{\left(\frac{2·L_V-R_i}{2·L_V+R_i}\right)}^{1/2}}=1$$

(12)

or:

$$\frac{t}{w}=\frac{\frac{1}{2}·\frac{{\pi }^2·E}{{\left(k_b·L_V·n_V\right)}^2}·{\left(\frac{2·L_V-R_i}{2·L_V+R_i}\right)}^{1/2}}{\frac{3·{\sigma }_y}{R_i+t/2}}$$

(13)

Using the complete definition Equation (13) becomes [87]:

$$\frac{t}{w}=\frac{\frac{1}{2}·\frac{{\pi }^2·E}{{\left(k_b·L_V·n_V\right)}^2}·{\left(\frac{2·L_V-R_i}{2·L_V+R_i}\right)}^{1/2}·\frac{1}{2}·\left[1+\left(1-2·\frac{2}{3}·\frac{\left(R_i+t\right)}{L_V}·\sin \alpha \right)\right]}{\frac{3·{\sigma }_y}{R_i+t/2}}$$

(14)

2.5. Model of the Nodal’ Vertical Displacements

To develop the entire theoretical plateau, the ligaments connecting the nodes should not fail; thus, an investigation of the strains recorded in the ligaments is required.

The structure’s deformation process depends on the constraints applied to the boundaries of the structure. Therefore, the nodes do not rotate at the same angle during this process. Thus, the theoretical deformation mode is altered, and the contraction is higher at the vertical half-length of the structure.

Considering the vertical imposed displacement $\left(V\right)$ the nodal displacement $\left(∆v_i\right)$ is defined by:

$$∆v_i=\frac{0.5·V}{n_V+ratio·{\sum }_{j=1}^{n_V}j}·\left(1+i·ratio\right)$$

(15)

The value of parameter $ratio$, capable of accurately describing the deformation process, was found to be the value of Poisson’s ratio (modulus) determined for the analyzed configuration.

$$∆v_i=\frac{0.5·V}{n_V+\left|{\nu }_{HV}\right|·{\sum }_{j=1}^{n_V}j}·\left(1+i·\left|{\nu }_{HV}\right|\right)$$

(16)

Equation (16) can provide the relative displacements of the nodes. The absolute displacements of the nodes $\left(∆V_i\right)$ are obtained using the assumption that the vertical displacement of the top node $\left(∆v_1\right)$ is identical to the imposed displacement of the structure $\left(V\right)$. The absolute vertical displacements $\left(∆V_i\right)$ are determined using the following relation:

$$∆V_i=V-∆v_1+∆v_i$$

(17)

2.6. Strain Analysis

Strain analysis [79] can be applied for the determination of the ultimate load that can be applied to a structure prior to failure (or densification). Using the nodal displacement defined by Equation (16), the nodal rotation $\left({\phi }_i\right)$ is determined:

$${\phi }_i={\sin }^{-1}\frac{0.5·∆v_i}{R_i+t}$$

(18)

The differential equations used for the investigation of the beams under bending loads state the following:

$$\begin{gathered} \frac{\mathrm{d}\phi }{\mathrm{d}x}=-\frac{M}{E·I} \\ \frac{{\mathrm{d}}^{2}v}{{\mathrm{d}x}^2}=-\frac{M}{E·I} \end{gathered}$$

(19)

Considering the hypothesis of small displacements:

$$\tan {\phi }_i\approx \sin {\phi }_i\approx {\phi }_i$$

(20)

Rotation angle $\left({\phi }_i\right)$ can be expressed as a function of the curvature radius $\left(\mathsf{\rho }\right)$:

$${\phi }_i=\frac{\mathrm{d}x}{\mathsf{\rho }}=\frac{M}{E·I}$$

(21)

Investigating the deformed shape of the ligaments, the following can be determined (see Appendix A):

$$\begin{gathered} L_{1,i}=\frac{L_m·{\phi }_{2,i}}{{\phi }_{1,i}+{\phi }_{2,i}} \\ {L}_{2,i}=\frac{L_m·{\phi }_{1,i}}{{\phi }_{1,i}+{\phi }_{2,i}} \end{gathered}$$

(22)

The strain developed in the ligament can be generally expressed thus:

$$\epsilon =\frac{t}{2·\mathsf{\rho }}$$

(23)

By replacing with the terms defined by Equations (19) and (20) the strain developed in the ligament can be determined thus:

$$\begin{gathered} {\epsilon }_{1,i}=\frac{t·{\phi }_{1,i}}{2·L_{1,i}} \\ {\epsilon }_{2,i}=\frac{t·{\phi }_{2,i}}{2·L_{2,i}} \end{gathered}$$

(24)

Additionally, the vertical displacement is defined by the following:

$$\begin{gathered} v_{1,i}=L_{1,i}·{\phi }_{1,i} \\ {v}_{2,i}=L_{2,i}·{\phi }_{2,i} \end{gathered}$$

(25)

3. Material Model

This section details the material model used for the numerical simulation. The standard material model used for the numerical simulation was defined as a piece-wise material model reconstructed using experimental tests and numerical simulation.

3.1. General Information

In the present paper, the study was conducted on ABS (acrylonitrile butadiene styrene) [89], a material that is part of the family of thermoplastic polymers. The main advantages of ABS material are:

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at a temperature equal to 105 °C, the material becomes fluid;

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it is cheap/raw material is available;

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it has excellent characteristics in terms of malleability, being easily transformed into parts of simple or complex shapes;

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it has a low weight, much lower than ordinary metals or even aluminum;

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it is very rigid, resistant to shocks and pressure;

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it can be colored;

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it resists ultraviolet radiation thanks to the added additives, which makes it suitable for the production of plastic objects that are in direct sunlight;

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it has excellent performance, in a wide range of temperatures, without visible degradation;

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it is easy to glue by using common substances;

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it can be printed and painted thanks to its glossy surface;

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it insulates electrically at various temperatures and even in the presence of moisture;

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it sustains a certain amount of deformation before yielding, returning to its original shape when the pressure is stopped.

Of course, this material also presents a few drawbacks [90], and the most important are:

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it is not environmentally friendly;

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it does not resist certain solvents (there are solvents that can dissolve it, and because of this, it cannot be used in certain fields, such as the chemical industry);

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ABS is more expensive than other types of plastic, such as polystyrene and polyethylene;

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burning ABS plastic can produce dangerous gases, which is not desirable;

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the chemical composition of this type of plastic makes it non-biodegradable;

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there is limited use of ABS in the food industry because there are not enough studies to demonstrate the long-term effects of ABS plastic packaging on human health.

Table 1. General information (ABS).

Parameter	Value	Units
$\mathrm{Density}\left(\rho \right)$	1.00–1.22	$\left[\mathrm{g}/{\mathrm{c}\mathrm{m}}^3\right]$
$\mathrm{Young}’\mathrm{s}\mathrm{Modulus}\left(E\right)$	1.12–2.87	$\left[\mathrm{M}\mathrm{P}\mathrm{a}\right]$
Elongation at break	3–75	$\left[\mathrm{\%}\right]$
Melting Temperature	88–128	$\left[\mathrm{℃}\right]$
Glass Transition Temperature	100	$\left[\mathrm{℃}\right]$
$\mathrm{Yield}\mathrm{Stress}\left({\sigma }_{\mathrm{y}}\right)$	18.5–51	$\left[\mathrm{M}\mathrm{P}\mathrm{a}\right]$
$\mathrm{Ultimate}\mathrm{Tensile}\mathrm{Strength}\left({\sigma }_u\right)$	33–110	$\left[\mathrm{M}\mathrm{P}\mathrm{a}\right]$

summarizes the main structural properties of the ABS material [91].

3.2. Standard Material Model

The material model assigned to the deformable structures is *MAT_PIECEWISE_LINEAR_ PLASTICITY. The Von Mises yield function ($\mathsf{\Phi }$) is implemented thus:

$$\mathsf{\Phi }=\frac{1}{2}·s_{ij}·s_{ij}-\frac{{\sigma }_y^2}{3}\le 0$$

where $s_{ij}$ is the deviatoric stress and ${\sigma }_y$ is the current radius of yield surface, defined by the following:

$${\sigma }_y=\beta ·\left[{\sigma }_0+f_h\left({\epsilon }_{eff}^p\right)\right]$$

(26)

$f_h\left({\epsilon }_{eff}^p\right)$ is the hardening function defined by tabulated data considering the effective plastic strain $\left({\epsilon }_{eff}^p\right)$. It is extracted from the true stress-strain curve. As the simulations and experimental tests were performed under quasi-static conditions, the strain rate effect is not included in the present work.

3.3. Material Failure Mechanism

A number of samples were manufactured using a 3D printer (MakerBot Experimental 2x, MakerBot Industries, Brooklyn, NY, USA) with the process parameters listed in

Table 2. 3D printing parameters.

Parameter	Value	MakerBot Slicer’s Settings
Extruder temperature	230 °C
Bead temperature	110 °C
Layer thickness	0.3 mm	maxSparseFillThickness
Raft	on	raftSurfaceLayers
Infill	100%	infillDensity
Infill pattern	linear	sparseInfillPattern
Speed of the head	50 mm/min	feed rate
Air gap	default	spurOverlap
Pattern alignment	0°	solidFillOrientationInterval: 0°

.

Figure 1a presents the sample’s dimensions (ASTM D638—Type I) and the printing direction aligned with the longitudinal axis. This printing direction is also representative of the manufacturing process of the tetra-anti-chiral structures, considering the actual configuration defined by circular nodes and joining ligaments.

The specimens were tested in traction (Figure 1b) with the displacement rate set to 2 mm/min, using a computerized electromechanical universal testing machine (WDW-50E, class 0.5).

Figure 2 presents the measured force–displacement and true stress–strain curves obtained from the tensile test.

The force–displacement data analysis reveals that a peak value is present, followed by a descending set of values. This is a characteristic of a softening mechanism produced by accumulating the damage in the specimen’s cross section. The damage process is mainly defined by the void formation in the material, causing the decrease of the true area of the cross section. Therefore, the measured stress–strain data requires correction prior to application for the numerical simulation process.

The mechanical description of material’s behavior is completed by a damage model that can capture the failure of the specimens under specific loads.

In Ls-Dyna, the mechanism for failure as a consequence of damage induced in the material is based on the use of the *MAT_ADD_DAMAGE_GISSMO card. The GISSMO model—or generalized incremental stress-state dependent damage model—[17,92] is based on the use of the triaxiality measure. Triaxiality is the measure of the ratio of the hydrostatic mean stress $\left({\sigma }_h\right)$ to the equivalent von Mises stress $\left(\stackrel{-}{\sigma }\right)$, and provides a solution to define the loading states [93,94].

$$\eta =\frac{{\sigma }_h}{\stackrel{-}{\sigma }}$$

(27)

where ${\sigma }_h$ is the mean stress or the hydrostatic pressure and $\stackrel{-}{\sigma }$ is the equivalent or von Mises stress. These terms are defined in terms of the principal stresses $\left({\sigma }_1,{\sigma }_2,{\sigma }_3\right)$ as below:

$${\sigma }_h=\frac{1}{3}·\left({\sigma }_1+{\sigma }_2+{\sigma }_2\right)$$

(28)

and:

$$\stackrel{-}{\sigma }=\frac{1}{3}·\sqrt{\frac{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2}{2}}$$

(29)

The GISSMO model is based on the incremental formulation of the damage accumulation in the form of the following:

$$∆D=\frac{\epsilon }{{\epsilon }_f\left(\eta \right)}·D^{\left(1-1/\mathrm{D}\mathrm{M}\mathrm{G}\mathrm{E}\mathrm{X}\mathrm{P}\right)}·∆\epsilon$$

(30)

where ${\epsilon }_f\left(\eta \right)$ is the equivalent plastic strain to failure determined from the input curve (as a function of the triaxiality parameter), $∆\epsilon$ is the equivalent plastic strain increment, and $\mathrm{D}\mathrm{M}\mathrm{G}\mathrm{E}\mathrm{X}\mathrm{P}$ is a specific parameter. The failure model is engaged by the parameter $\mathrm{D}\mathrm{C}\mathrm{R}\mathrm{I}\mathrm{T}$, which defines the minimum damage that must accumulate to couple the stress tensor with damage ($\mathrm{F}\mathrm{A}\mathrm{D}\mathrm{E}\mathrm{X}\mathrm{P}$ is the exponent for damage-related stress fadeout).

$$\sigma =\stackrel{-}{\sigma }·\left[1-{\left(\frac{D-DCRIT}{1-DCRIT}\right)}^{\mathrm{F}\mathrm{A}\mathrm{D}\mathrm{E}\mathrm{X}\mathrm{P}}\right]$$

(31)

$DCRIT$ is the critical damage when instability parameter $F=1$.

The instability parameter defines the initial state of the failure process. Once the critical value is reached, the structure is likely to fail under the prescribed load.

$$F={\left(\frac{\epsilon }{{\epsilon }_{crit}\left(\eta \right)}\right)}^{DMGEXP}$$

(32)

where ${\epsilon }_{crit}\left(\eta \right)$ is the equivalent plastic strain to initiate the instability process.

Figure 3 presents the stress–strain curve of the material with the strain hardening section defined as flat [95], and the force–displacement results obtained from the simulation $\left(\rho =1 \mathrm{g}/{\mathrm{c}\mathrm{m}}^3;E=1500 \mathrm{M}\mathrm{P}\mathrm{a};{\sigma }_y=18 \mathrm{M}\mathrm{P}\mathrm{a};\upsilon =0.30\right)$.

Results presented in Figure 3 show a reasonable agreement between the experimental and reconstructed datasets using numerical simulation.

The triaxiality curve [95] is presented in Figure 4. The results were previously investigated using a series of experiments and numerical simulations to define the strain to fracture under various loading states (compression $\eta =-0.333$; pure shear $\eta =0$; tensile $\eta =0.333$).

4. Experimental Analysis

4.1. Geometrical Definition of the Experimental Parts

Equations (2), (5), (6) and (14) can be used to properly design a structure capable of sustaining compressive forces. The set of equations can be represented as parametric, considering the radius of the node (Figure 5a). Subsequently, the force (stress normalized force: ${\sigma }_y=1.0 \mathrm{M}\mathrm{P}\mathrm{a}$), presented in Figure 5b, and the length of the plateau (Figure 5c) can be estimated to provide an initial estimate of the compression behavior of the structure. The minimum width of the structure was determined from the stability analysis presented in Figure 5d).

Following the results listed in Figure 5, two configurations were defined and manufactured using the fused filament fabrication method. The 3D printing process of the specimen used to characterize the material is representative for the slicing solution applied to the finished product (Figure 6).

Table 3. Parameters of the tested structures.

Parameter	Configuration A	Configuration B
$\mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{node}\left(R_i\right)$	$2.7 \mathrm{m}\mathrm{m}$	$1.7 \mathrm{m}\mathrm{m}$
$\mathrm{The}\mathrm{thickness}\mathrm{of}\mathrm{the}\mathrm{ligament}\left(t\right)$	$1.25 \mathrm{m}\mathrm{m}$	$1.25 \mathrm{m}\mathrm{m}$
$\mathrm{Lengths}\mathrm{of}\mathrm{the}\mathrm{ligaments}\left(l\right)$	$10 \mathrm{m}\mathrm{m}$	$10 \mathrm{m}\mathrm{m}$
$\mathrm{Width}\mathrm{of}\mathrm{the}\mathrm{structure}\left(w\right)$	$15 \mathrm{m}\mathrm{m}$	$15 \mathrm{m}\mathrm{m}$
Poisson’s ratio (theoretical)	$-0.9615$	$-0.9173$
$\mathrm{Number}\mathrm{of}\mathrm{cells}\left(\mathrm{horizontal}\right)\left(n_H\right)$	3	3
$\mathrm{Number}\mathrm{of}\mathrm{cells}\left(\mathrm{vertical}\right)\left(n_V\right)$	3	3
$\mathrm{Predicted}\mathrm{plateau}\mathrm{force}\left(F_p\right)$	$0.60\mathrm{k}\mathrm{N}$	$0.82\mathrm{k}\mathrm{N}$
$\mathrm{Predicted}\mathrm{plateau}\mathrm{length}\left(d_p\right)$	$2.1\mathrm{m}\mathrm{m}$	$4.1\mathrm{m}\mathrm{m}$

presents the main dimensions of the structure and principal features.

4.2. Compression Test and Data Processing

The structures were tested under compressive loads by setting a displacement speed of $5.0 \mathrm{m}\mathrm{m}/\mathrm{m}\mathrm{i}\mathrm{n}$ to the mobile part of the universal testing machine [18]. The digital images were captured at a speed of 25 fps. The tested structures are presented in Figure 7.

The images were processed using a custom code application developed in MATLAB to identify the nodes and trace the centers. Figure 8 presents the workflow of digital image processing.

5. Numerical Modeling and Simulation

Numerical simulation is a versatile solution to extract characteristic information related to the behavior of structures under various loading conditions. In this section, the methods used for the definition of the numerical model are presented, including the convergence study, followed by a detailed discussion of the material model applied for the numerical analysis.

5.1. Numerical Modeling

The numerical models used in this paper were developed according to Ls-Dyna nomenclature. The element model is 3D using an eight-node fully integrated solid (hexahedral finite element) [96]. A series of MATLAB procedures were developed to allow a quick process for the construction of the numerical model. For the interaction effect between contacting elements, *CONTACT_AUTOMATIC_SINGLE_SURFACE [95] were implemented. The numerical models were solved using the implicit solver *CONTROL_IMPLICIT (LSOLVR = 2; NSOLVER = 12) and the *CONTROL_IMPLICIT_DYNAMICS control card. The structure was loaded by an imposed displacement using *RIGIDWALL_GEOMETRIC_FLAT_MOTION, proportional to the simulation time (*DEFINE_CURVE).

5.2. Convergence Analysis

For the convergence evaluation under compressive loads, the width of structures was set to $10 \mathrm{m}\mathrm{m};$ the ligaments were of equal lengths $\left(20 \mathrm{m}\mathrm{m}\right),$ the inner radius of the node was $4 \mathrm{m}\mathrm{m}$, the overall thickness was $2 \mathrm{m}\mathrm{m},$ and the width was $3 \mathrm{m}\mathrm{m}$. Figure 9 presents the discretization solution of the structures for the convergence analysis.

The results of the numerical simulations for the compression analysis are presented in Figure 9. During the plateau stage, the reaction force computed for C1 and C2 was, on average, about $0.69 \mathrm{k}\mathrm{N}$, while for C3 and C4 was on average about $0.62 \mathrm{k}\mathrm{N}$.

5.3. Analysis of the Results

The results of the numerical simulation are detailed in this section. For the first iteration of the numerical analysis, the material damage mechanism was disabled (see *MAT_ADD_DAMAGE_GISSMO in Section 3.3). This procedure was applied to obtain an evaluation of the plateau force and plateau length. The initial and the final deformed shapes of the tested auxetic structures are presented in Figure 10.

The results obtained from the numerical simulation were compared to the analytical estimation of these parameters (Figure 11).

The results presented in Figure 11 show that the numerical simulation results correlate with the analytical evaluation of both plateau force and plateau length. The experimental data reveal a similar path to the numerical analysis. The experimental results are presented to outline the similarities between numerical and experimental datasets.

6. Discussion

In this section, the deformation process of the structure under compressive loads is investigated and discussed. The local strain is a result of the local rotation and expansion of the elementary building elements (e.g., nodes) of the structure [79,97].

6.1. Analysis of the Vertical Displacements of the Node

The model developed for estimating the strain developed in the ligament uses the vertical displacements of the nodes as inputs. Thus, a first analysis investigates the results obtained for vertical displacement of the center of the nodes. Figure 12 presents the nodes investigated by this analysis.

The simulation data was extracted from the numerical model, the theoretical displacements were computed using Equation (17), and the experimental data were scaled to displacement units. Figure 13 presents the cumulated results.

The results listed in Figure 13 display a good correlation between the datasets. However, it should be noted that the analytical model uses the assumption of a rigid node; this, explains the small deviations between the reported datasets.

6.2. Strain Analysis

The set of results obtained from the numerical simulation was extended using the specific card *DATABASE_EXTENT_BINARY to estimate the total strain of the elements (maximum principal strain).

The results of the experimental, numerical, and analytical analyses are further investigated. A set of elements was selected for each numerical for a detailed strain analysis (Figure 14).

The theoretical evaluation model developed in Section 2.6 was applied to evaluate the strain developed in the ligaments during the compression process. Figure 15 presents the results of the analytical solution and the numerical simulation.

The upper and lower bounds (dotted line) represent the strain values computed for ${\epsilon }_{1,i}$ and ${\epsilon }_{2,i}$, while the solid line represents the average strain ${\epsilon }_i=\left({\epsilon }_{1,i}+{\epsilon }_{2,i}\right)/2$.

The results show that there is a good approximation between the numerical and theoretical results, and that the upper and lower bound can accurately predict the evolution of the strain.

The numerical value of the strain displays, in some cases, a peak followed by a decrease in the value. This behavior results from the gap closing and the contact between the ligaments.

To capture this effect, the theoretical model evaluates the distance between the nodes to provide an estimate of the gap between the ligaments. Thus, once the gap is closed, the strain value is limited.

The distance between ligaments is estimated using the vertical displacements and the rotation of the nodes. Thus, the current gap is defined by:

$${\delta }_i=0.5·\left[\left(L_V-2·\left(R_i+t\right)\right)-\left(∆v_i-∆v_{i-1}\right)-\left(L_{1,i}·{\phi }_{1,i}+L_{2,i}·{\phi }_{2,i}\right)\right]$$

(33)

When the value becomes negative, the ligaments are in contact, and the strain value is limited:

$$\begin{array}{c}{\epsilon }_{1,i}={\epsilon }_{1,i-1}\\ {\epsilon }_{2,i}={\epsilon }_{2,i-1}\end{array}$$

(34)

The conventional material model (e.g., *MAT_PIECEWISE_LINEAR_ PLASTICITY) does not include specific failure limits considering the element strain state. Therefore, the element will fail under compression or tension once the ultimate strain is reached. The compressive strain that can be developed before failure is higher than the failure stain developed under tractive forces. The use of the GISSMO model provides the solution for differentiating between strain states.

Additive manufacturing technologies allow the fabrication of complex structures. However, the material performances are altered; thus, it is necessary to evaluate the behavior of the structures, considering their potential application.

To exemplify the strain analysis’ functionality, the strain-to-failure value was amended to produce premature fractures of the ligaments. The following values of the strain to failure were updated in the triaxiality map.

$$\begin{array}{c}\left\{\begin{array}{c}{\eta }_{0.333}=0.020\\ {\eta }_{0.395}=0.020\end{array}\right.\\ \left\{\begin{array}{c}{\eta }_{0.333}=0.050\\ {\eta }_{0.395}=0.050\end{array}\right.\end{array}$$

(35)

The results for this analysis, in the case of Structure A, are presented in Figure 16.

The maximum strain can be roughly estimated as the sum of the yield strain $\left({\epsilon }_y=0.012\right)$ and strain at failure:

$${\epsilon }_i={\epsilon }_y+{\epsilon }_f$$

(36)

Results presented in Figure 16b indicate that for a vertical displacement of $\approx 3 \mathrm{m}\mathrm{m}$ the strain at failure in tension is reached, and the ligaments fractures. The vertical displacement imposed for the compression process is directly proportional, by a factor of $1,$ to the simulation time. Figure 16a shows the fractures developed in the structure under compressive loads.

The same investigation procedure was applied for Structure B. The results presented in Figure 17 capture the behavior of the structure using the second dataset listed under Equation (35). The vertical displacement required to produce a failure of the ligaments is $\approx 4 \mathrm{m}\mathrm{m}$, according to the analytical solution. A similar value is displayed by the numerical simulation, showing the agreement between the numerical and analytical solutions.

It should be noted that following the investigation of the numerical results and the predictions of the analytical model, the rotations of the first and second nodes (top and following on vertical direction nodes) were amended as follows:

$$\begin{array}{c}{\phi }_{1,u}=0.5·{\phi }_1\\ {\phi }_{2,u}={\phi }_2+0.5·{\phi }_1\end{array}$$

(37)

This amendment was imposed by the constrained motion of the top node, and the results presented in Figure 16b and Figure 17b capture this effect.

7. Conclusions

The paper presents analytical, experimental, and numerical analyses of an auxetic metamaterial. The parts investigated are subjected to compressive loads.

The introductory section presents recent findings on the current structural design trends, pointing to metamaterials and the existing manufacturing methods.

The use of AM methods allows for a wide range of materials to be used for the manufacturing of complex shapes, although, in some cases, the structural performances of the raw or processed material, or the fabrication process itself, cannot fully match the application.

Thus, it is necessary to investigate and develop a reliable material model that can evaluate a structure’s behavior under various loads. A comparison between models developed using two different material models is presented. The numerical simulation results outline the influence of the use of the triaxiality concept implemented in the extended material model.

The analysis is complex and points to a series of parameters defining the structure’s performance, namely:

−

Poisson’s ratio—that is the main parameter of an auxetic structure;

−

plateau force—which displays the structural response of the structure under compressive loads;

−

plateau length—which outlines the compression performance;

−

buckling—which provides a required dimensional verification of the structure subjected to compression;

−

ligament strain—which investigates the capability of the structure to develop the predicted plateau force during the plateau stage.

For these parameters, a set of analytical solutions was developed and discussed. These analytical solutions are simple to implement and can be successfully used to evaluate the preliminary of a complex structure. This paper fills this gap by providing an analytical solution for strain analysis.

The experimental data fully support the analytical and numerical results. The analytical and numerical models accurately capture the true behavior of structures under applied loads.