Computational Study of Non-Porous Auxetic Plates with Diamond Shape Inclusions

Abstract

Creating non-porous structures that offer auxetic behavior can have a variety of industrial applications, especially when the porosity impairs the functionality of the auxetic structures. This study presents the design and finite element analysis of architected bi-material auxetic plates consisting of repeating unit cells that comprise rigid rotary units and soft inclusions. The change in the design parameters of unit cells produces a variety of mechanical properties, such as different levels of Poisson’s ratio and stiffness for the architected plates that can result in specific static or dynamic responses. The natural frequencies and deflection under uniform lateral loading of the architected plates with clamped boundary conditions were investigated. Furthermore, the effectiveness of the homogenization technique based on the mechanical properties obtained from finite element analysis in predicting the dynamic and static response of the architected plate was also studied.

1. Introduction

Poisson’s ratio is a dimensionless measure that describes the deformation of a material in the perpendicular direction of the loading [1]. Materials with negative Poisson’s ratio (NPR) properties are usually referred to as auxetic materials [2,3]. The term auxetic in the scientific literature was introduced by Evans [4]. Auxetic materials have been known for more than 150 years, and in 1882, a study revealed that they could be found in nature as iron pyrite nanocrystals with an estimated Poisson’s ratio of −1/7 [5]. There are other natural examples such as α-cristobalite, cow teat skin, pyrolytic graphite, polymorphic silicones, zeolites, silicates, and crystal cadmium [6]. The first artificial auxetic material was synthesized as a foam shape in 1987 by Lakes [7], which was followed by a significant number of studies to develop materials with NPR properties along with cost-effectiveness and manufacturing controllability [8,9,10,11,12,13,14,15,16,17,18].

Based on the classical theory of elasticity, all three-dimensional isotropic materials can have a Poisson’s ratio ranging from −1 to 0.5, while for two-dimensional isotropic materials, this ratio is between −1 to 1 [19,20]. Materials that show a negative Poisson’s ratio in all directions are called auxetic or completely auxetic materials [21,22]. On the other hand, some materials behave as auxetic in some directions and non-auxetic in others, and are known as partially auxetic materials [23,24].

One of the first mechanical structures with auxetic behavior was proposed by Almgren [25] and the first 2D molecular models (spontaneously forming auxetic phases) were studied by Wojciechowski [19]. Auxetic structures can provide a variety of improved properties such as high shear and overall stiffness [26,27,28], fracture toughness [29], and indentation resistance [9]. In addition, the damping capacity of auxetic structures [10], their sound absorption [30,31], and crashworthiness [32] were also found to be enhanced compared to conventional materials. These remarkable properties made auxetic materials a perfect choice for some engineering applications, including biomedical materials [24], energy absorbers [33], and applications in the automotive [34] and military and aerospace industries [35].

Conventional materials show a saddle shape (anticlastic curvature) when a bending force acts on them due to the transverse shrinkage. However, auxetic materials demonstrate a dome shape (synclastic curvature) [7,27] in the transverse direction, which has the same curve trend as the bending direction. Therefore, the synclastic curvature property makes an auxetic material a better fit for creating structures with complex curvatures without excessive machining and forcing materials to take up a specific shape (e.g., conical shape) that could result in possible damage [36,37].

The auxetic effect is not only an intrinsic property of the material, but can also be achieved by designing the structure and the deformation mechanism [38]. Auxetic materials can have different structural shapes, such as chiral structures, fibril/nodule structures, Miura-folded structures, buckling-induced structures, helical auxetic yarn structures, and crumpled structures [39]. The re-entrant structures that comprise rigid rotary units with square or triangle inclusions can also exhibit a negative Poisson’s ratio [40,41,42,43,44,45]. The unit cells with orthogonal geometrical shapes can only offer an auxetic behavior in certain symmetry axes. However, it has been shown that inclusions in the form of re-entrant triangles can offer an NPR effect along all symmetry axes due to the nonorthogonal symmetrical feature of the structure [46]. The auxetic behavior both in tension and compression was observed in structural sheets containing either diamond- or star-shaped perforations [47]. The auxetic behavior of the re-entrant structures at the molecular level has also received great attention [48,49].

Manufactured auxetic materials, like other periodic structures, usually possess some porosities in their structures [50,51,52], since the formation of the porosity is crucial to the auxetic properties. The porose auxetic structures typically contain sharp corners or wedge angles. In regards to fracture mechanics, the stress concentration around the holes and corners may compromise the structural integrity under loading conditions [53,54]. This phenomenon may limit the scope of application of the auxetic material. Having porosity in the structure may also cause other disadvantages, such as exposing the interior part of the structure to erosive and corrosive environments that can damage the structure and impair its function [55,56,57,58]. To eliminate the porosity, some continuous auxetic structures are proposed in the literature, such as smooth curve sheets [59], dimpled sheets [60], re-entrant hollow skeletons [61], wrinkled paper sheets [18], and origami sheets [62]. However, all of these structures are made from single materials. Creating structures from different materials can offer some benefits, such as producing certain structural properties at exact locations and tailoring them for specific applications [63].

Numerous studies have found analytical models for calculating natural frequencies of isotropic plates as well as their lateral deflection under uniform loading [64,65]. The effects of different geometrical parameters, boundary conditions, and material properties on an isotropic plate’s natural frequency and lateral deflection have been well established [66,67]. Due to the complex shapes of auxetic materials, computational methods are primarily used in the literature to characterize the behavior of auxetic structures [68,69,70]. The dynamic and static response of auxetic plates has been investigated by using finite element analysis (FEA) [71,72,73,74].

In this study, the effect of design parameters on the mechanical properties and auxetic behavior of bi-material non-porous plates is investigated; then, finite element simulation is employed to determine the natural frequency of the architected plates and their lateral deflection under static loading. Moreover, the accuracy of the homogenization technique in the prediction of dynamic and static responses of the architected plates is investigated.

2. Materials and Methods

The NPR effect can be achieved by the rigid rotary units in 2D structures wherein internal void spaces enable the rigid units to rotate freely under loading conditions, as illustrated in Figure 1. After rotation, void spaces between the rigid units significantly alter their shapes, and as a result, the area of the unit cells decreases, which can produce an NPR effect.

Similarly, a plate comprises a hard matrix, and soft inclusions can offer an NPR effect. A unit cell that includes a hard matrix and soft diamond shape inclusions is shown in Figure 2. Both matrix and inclusions are isotropic elastic materials.

Multi-material additive manufacturing is one of the techniques that can be employed to create architected bi-material plates [63,75]. Matrix and inclusions should be selected from materials that develop a strong adhesion to each other [76,77]. Figure 3 shows an example of an architected plate fabricated by multi-material additive manufacturing. The hard matrix is from carbon fiber reinforced PLA, and the soft inclusions are from PLA.

In practice, plates having a finite thickness will tend to deform out of plane under compression, a phenomenon that will affect the Poisson’s ratio in the plane. Therefore, the plate needs to have a considerable thickness to work effectively. In this regard, the scope of this study is limited to the design and computational analysis of architected bi-material plates.

Poisson’s ratio and Young’s modulus of architected bi-material auxetic plates can be determined by finite element analysis. The finite element analysis was performed using Ansys 2019 version (ANSYS Inc., Canonsburg, PA, USA), and 3D solid tetrahedral and hexahedral elements with quadratic element order were used to mesh the architected plates in this study. The length and width of the architected plate are L and L’, as shown in Figure 4a. The left side of the plate is fixed, and a longitudinal deformation of δ is applied on the right side of the plate, as shown in Figure 4b. The maximum total lateral deformation, δ’, is determined from finite element simulation. Therefore, the strains along the length of the plate, ε, and along the width, ε’, are defined as

$$\epsilon =\frac{\delta }{L} \epsilon \prime =-\frac{\delta \prime }{L\prime }$$

(1)

The Poisson’s ratio v is determined as

$$v=-\frac{\epsilon \prime }{\epsilon }$$

(2)

The auxetic behavior of the architected plate can be observed in Figure 4b.

In order to find Young’s modulus of the plate, E, first, the reaction force, F, on the left (fixed) side of the plate is determined from finite element simulation. The amount of stress σ developed in the plate with the lateral cross-sectional area of A is determined as

$$\sigma =\frac{F}{A}$$

(3)

Then, Young’s modulus is obtained as

$$E=\frac{\sigma }{\epsilon }$$

(4)

It should be noted that each unit cell, as shown in Figure 2, is a representative unit of the architected plate.

The angular frequencies, ω_n, of an isotropic elastic square plate with clamped edges are given as [66]

$${\omega }_n=2\pi f_n=k_n{\left[\frac{E{t}^2}{\rho L^4\left(1-v^2\right)}\right]}^{0.5}, k_1=10.40, k_2=21.21, k_3=31.29,$$

(5)

where $f_n$ are natural frequencies, E is Young’s modulus, v is the Poisson’s ratio, n is a mode of vibration, ρ is density, L is the length of the sides of the plate, and t is the plate thickness.

The lateral deflection at the center of an isotropic elastic rectangular plate under uniform lateral loading, as shown in Figure 5, is [67]

$$W_{x=0, y=0}=\alpha \frac{p{b}^4}{D}$$

(6)

where α is a constant depending on (b/a) and Poisson’s ratio, b and a are half the length and width of the plate ($a\ge b$), p is the intensity of lateral pressure, and D is the constant flexural rigidity defined as

$$D=\frac{E{t}^3}{12\left(1-v^2\right)}$$

(7)

where E is Young’s modulus, v is the Poisson’s ratio, and t is the plate thickness.

3. Results and Discussion

3.1. Effect of Design Parameters on Mechanical Properties

The design parameters of unit cells may significantly change the mechanical properties of the architected bi-material plate. a, s, θ, g, and t are defined as geometric design parameters of the cellular structure with diamond shape inclusions, as shown in Figure 6. t is the thickness of the unit cell.

The ratio of Young’s modulus of the soft inclusions, E_s, to Young’s modulus of the hard matrix, E_h, introduces another design parameter, λ, defined as

$$\lambda =\frac{ E_s}{E_h}$$

(8)

For example, the geometric design parameters can be taken as a = g = 5 mm, s = t = 1 mm, θ = 30°, which yields the plate side length of L = L’ = 56.99 mm. Young’s modulus of hard and soft materials can be taken as E_h = 1 GPa, E_s = 0.01 GPa, which yields λ = 0.01. Both the hard matrix and soft inclusions have the Poisson’s ratio of 0.25. By applying δ = 1 mm, as shown in Figure 4, δ^’ is computed as 0.473 mm, which yields the Poisson’s ratio of $v=$ −0.473. The area at the fixed end is A = 56.99 mm². The reaction force at the fixed end is computed as F = 184.76 N, which creates longitudinal stress of $\sigma =$ 3.24 MPa in the plate. The longitudinal strain in the plate is $\epsilon =$ 0.01754. Therefore, Young’s modulus of the plate is $E$ = 184.76 MPa.

In this study, dimensionless values are used. Here, E_h takes the value of 1, and both the hard matrix and soft inclusions have the same Poisson’s ratio $(v_h=v_s=$ 0.25). The effects of design parameters on Poisson’s ratio and Young’s modulus of the bi-material plate are given in Figure 7a–e. In Figure 7a, θ is a variable (2.5° ≤ θ ≤ 90°), and other design parameters are set to a = g = 5, s = t = 1, and λ = 0.01. The Poisson’s ratio and Young’s modulus reach their minimum values of −0.47 and 0.18, respectively, at θ = 30°. In Figure 7b, g is a variable (0.1 ≤ g ≤ 5), and other parameters are set to a = 5, s = t = 1, θ = 30°, and λ = 0.01. The Poisson’s ratio reaches its minimum value of −0.54 at g = 4. The Young’s modulus reaches its maximum value of 0.53 at g = 1.5. In Figure 7c, s is a variable (0.1 ≤ s ≤ 8), and other parameters are set to a = g = 5, θ = 30°, t = 1, and λ = 0.01. The Poisson’s ratio and Young’s modulus obtain their minimum values of −0.52 and 0.1 at the smallest value of s (i.e., s = 0.1). In Figure 7d, t is a variable (0.1 ≤ t ≤ 6), and other design parameters are set to a = g = 5, s = 1, and λ = 0.01. It can be seen that the variation of the plate thickness has a relatively small effect on the Poisson’s ratio and Young’s modulus of the plate. The Poisson’s ratio and Young’s modulus obtain their minimum values of −0.476 and 0.183 at the smallest value of t (i.e., t = 0.1)

In Figure 7e, λ is a variable (0.00001 ≤ λ ≤ 1), and other parameters are set to a = g = 5, s = t = 1, and θ = 30°. The Poisson’s ratio and Young’s modulus obtain their minimum values of −0.92 and 0.08 at the smallest value of λ (i.e., λ = 0.00001). This shows that having porosity instead of soft inclusions increases the auxetic behavior of the plate and yields a more flexible plate. The Poisson’s ratio obtains a zero value at λ = 0.058.

Depending on the application of the plate, the design parameters can be adjusted to render a specific mechanical property. It can be observed that only by changing the geometric design parameters, the Poisson’s ratio with a range of −0.54 to 0.78 and Young’s modulus with a range of 0.10 to 0.69 are produced merely by the use of two distinct materials with certain moduli (i.e., E_h = 1, E_s = 0.01) and the same Poisson’s ratio $(v=$ 0.25). This is significant because we can create a variety of mechanical properties for plates with a limited number of materials.

3.2. Effect of Design Parameters on Dynamic Response

In order to investigate the dynamic response of the bi-material architected plate, λ was selected as a variable, while geometric design parameters were taken as constants; a = g = 5, s = t = 1, and θ = 30°. The variation in the non-geometric design parameter λ does not alter the size of the plate, and the plate keeps its square shape with the same number of inclusions, as shown in Figure 8a. The first three mode shapes of vibration of the architected plate with clamped boundary conditions are presented in Figure 8b–d. It can be observed that the plate shows the same mode shapes of vibration as a homogenous rectangular plate, though with different natural frequencies.

Figure 9 presents the natural frequencies of the first three modes of vibration of the architected plate with different values of λ. All frequencies are normalized to the first natural frequency of the plate with λ = 1. It is assumed that both the hard matrix and soft inclusions have the same density (${\rho }_h={\rho }_s$). For example, the first natural frequency for the homogenous plate (i.e., λ = 1) with E_h = E_s = 1 GPa,$v_h=v_s=$ 0.25, ${\rho }_h={\rho }_s=$ 1000 kg/m³, a = g = 5 mm, s = t = 1 mm, and θ = 30° is 526.34 Hz, which is the frequency that all frequencies are normalized to in the chart. It can be observed that by reducing λ, the natural frequencies decrease as well.

Figure 10 presents the natural frequencies of the first three modes of vibration that are normalized to the natural frequency of the plate with λ = 1 of the corresponding vibration mode. The comparison of these natural frequencies to the natural frequencies of a homogenized plate is also given in Figure 10. The natural frequencies of a homogenized plate were calculated based on Equation (5), using the Young’s modulus and Poisson’s ratio that were obtained from finite element analysis of the architected plate at each λ. The natural frequencies of a homogenized plate are then normalized to the natural frequency of the plate with λ = 1 of the corresponding vibration mode, which gives the same value at each λ for all modes of vibration, as can be seen in Figure 10.

According to Equation (5), the natural frequencies of an isotropic homogeneous square plate depend on its Young’s modulus and Poisson’s ratio, which vary by the change in the design parameter λ, as demonstrated in Figure 7e. It can be observed that the homogenization technique provides a good estimate of natural frequencies, primarily for the first and second modes of vibration.

3.3. Effect of Design Parameters on Static Response

For evaluation of the static response of the architected plate, similarly to the dynamic response analysis, λ was selected as a variable, while geometric design parameters were taken as constants; a = g = 5, s = t = 1, and θ = 30°. The deformation of the architected plate under uniform lateral loading with clamped edges is shown in Figure 11. It can be observed that the maximum deflection occurs at the center of the plate.

Figure 12 presents the lateral deflection at the center of the architected plate under lateral pressure that is normalized to the lateral deflection of the center of a plate with λ = 1 under the same loading condition. As stated in Equation (6), the lateral deflection of an isotropic homogeneous rectangular plate under lateral pressure depends on its Young’s modulus and Poisson’s ratio, which vary by the change in λ. The comparison of the results to the lateral deflection at the center of an isotropic elastic rectangular plate with the same Young’s modulus and Poisson’s ratio is also presented in Figure 12. As shown in the dynamic response, the homogenization technique provides a reasonable estimate for the static response of the architected plate over the high and low range of λ. Contrarily, it gives a poor estimate for λ values in between, as demonstrated in Figure 12. This is due to the fact that under direct loading, soft inclusions cannot freely deform, as they are bounded by the hard matrix. However, this condition does not apply to the equivalent homogenized plate, and therefore, a larger deformation can be observed. The effect of this condition decreases when the inclusions are made of harder materials, or they can be considered porosity when they are extremely soft.

4. Conclusions

In this study, a non-porous auxetic plate was developed by embedding soft diamond shape inclusions within the hard matrix. It was shown that a variety of mechanical properties, such as different levels of Young’s modulus and negative Poisson’s ratio, could be achieved by changing the geometric design parameters of cellular structures. It was observed that the Poisson’s ratio with a range of −0.54 to 0.78 and Young’s modulus with a range of 0.10 to 0.69 were produced by only the use of two unique materials with certain moduli (i.e., E_h = 1, E_s = 0.01) and the same Poisson’s ratio (v = 0.25). Furthermore, the effect of Young’s modulus ratio, λ, as a design parameter on the mechanical properties of the architected plate and its natural frequencies with clamped boundary conditions as well as its deflection under uniform lateral loading were presented. Soft inclusions reduced the auxetic behavior of the architected plate as compared to porosities. It was demonstrated that the homogenization technique based on the mechanical properties obtained by finite element analysis could well estimate the architected plate’s dynamic response for different λ values, specifically for the first and second modes of vibration. The homogenization technique also provides a reasonable estimate of the static response over the high and low range of λ.