A mechanical metamaterial is a multiscale system of materials with engineered mechanical properties that can vary dramatically from those of the base material. Mechanical materials can therefore exhibit some global unusual deformation mechanisms. Lattice structures are a popular example of mechanical metamaterials because of their high strength to density ratio, compared to traditional structural materials (Ashby, 2006). Another important aspect of lattice structures is their tailorable mechanical response, both at global and hierarchical scale (Sun and Pugno, 2013).

A subset of mechanical metamaterials is represented by auxetics. While conventional cellular foams and rubber-like materials have a Poisson's ratio varying from ν≈0.5 (incompressible) to ν = 0 (e.g., cork) with decreasing density, auxetics possess instead a negative Poisson's ratio. Negative ν means that the elongation in one direction is accompanied by an elongation in the transverse direction. On the opposite, the auxetic material will shrink under compression. Auxetic structures can be deployed to provide a required indentation resistance (Evans and Alderson, 2000), high compressive and shear stiffness (Sanami et al., 2014), and high energy absorption (Mohsenizadeh et al., 2015). In biomedical applications of auxetics have been sought in bone implants (Warmuth et al., 2017; Kolken et al., 2018). Auxetics can be also developed by applying patterns of cuts and holes (Grima et al., 2005). Perforated auxetic composite plates have been used in a hybrid flexible cushioning support for multiple sclerosis patients (Mohanraj et al., 2016).

The mechanical properties of lattices can be changed dramatically (Ashby, 2006) by varying the connectivity of the struts and the angles between the same struts at joints, therefore creating auxeticity in essentially bending-dominated lattices (Grima et al., 2010; Bouakba et al., 2012; Saxena et al., 2016). Auxetic structures exhibit synclastic behavior when bent, i.e., they deform into dome shapes (Alderson and Alderson, 2007), rather than saddle ones, which are typical of structures with positive ν. This makes auxetics useful in applications such as pressure vessels (Adachi, 1968) and anti-blast protections (Sahu and Gupta, 2015). Auxetic domes designs could therefore be suitable for protective equipment applications. A lower cell density in an auxetic lattice also leads to lower stiffness and higher bending and rotation motion of the struts (Scarpa et al., 2000; Carneiro et al., 2016), and this increases the effective negative effective Poisson's ratio ν_a (Carneiro et al., 2016). The energy absorption of chiral auxetic topologies is greater than that of the re-entrant dome as the strut thickness increased and the ν_a reduced (Scarpa et al., 2000). This suggests that there is a smaller dependence on cell geometry configurations like the tri-chiral lattice.

The energy absorption of a foam increases with density, because the indenter/impactor force is distributed over a larger number of cells (Lakes and Elms, 1993; Fleck and Qiu, 2007). The increase of the cell density in a structure like an auxetic dome should therefore lead to a greater amount of strut deformation, and thus to higher energy absorption.

In the arrow-head topology ν_a grows from a negative value to zero with increasing strain (Yang et al., 2018). The arrow-head and re-entrant topology has higher energy dissipation than the hexagonal lattice due to the negative in-plane ν_a. The re-entrant topology has the highest energy absorption (Yang et al., 2018). Both re-entrant and arrow-head topologies are significantly anisotropic (Zied et al., 2015).

Plasticity and non-linear geometry have a significant effect on FE modeling of deformation of auxetic structures (Zhang et al., 2018). Structures ranging from a unit cell to full scale geometrical models have been modeled successfully with beam elements and 3D solid elements (Blachut and Galletly, 1988).

Although auxetic foams have received a lot of attention (Lakes and Elms, 1993; Zied et al., 2015; Yang et al., 2018), little research has been done on auxetic dome-shaped structures. The work presented in this paper focuses on the buckling and post-buckling behavior of novel auxetic lattice domes, such as those shown in Figure 1.

In 1946 Reissner showed that the term (1 − ν²)^1/2, where ν is the Poisson's ratio of the material, appears in the linear differential equations for normal displacements, w, in thin shallow spherical shells (Reissner, 1946). Reissner considered a spherical shell segment being shallow if the ratio of its height to the base diameter did not exceed 1/8. Ashwell extended Reissner's theory to large deflections w and showed that a spherical shell segment loaded by an inward point load, F, does not exhibit instability (Ashwell, 1960). Ashwell also demonstrated that the linear theory predicts a linear relationship between F and w, while his non-linear theoretical model predicts a non-linear F(w) response, but still with no loss of elastic stability. A snap-through instability (buckling) is predicted when distributed load (external pressure), p, is applied to a spherical shell, with the same term (1 − ν²)^1/2 appearing in the expression for the critical pressure, p_c (Budiansky, 1960), see Equation (1), where E is the Young's modulus, and R and t are the shell radius of curvature and thickness respectively.

Leckie showed that a snap through instability does indeed occur under the application of an inward radial force thorough a rigid “boss” of a finite diameter, if plastic deformation is allowed (Leckie, 1969). However, as a plastic deformation is incompressible, no dependence of the critical load on ν was found. This aspect is reinforced also by the particular case we consider in the following example. The loading case is somewhat intermediate between an inward radial force and external pressure on a simply supported spherical shell segment (hereafter–a dome) compressed between a pair of rigid plates. A preliminary non-linear elastic FE study using full 3D solid theory and an isotropic elastic homogeneous material shows that, although a snap-through behavior is clearly present, no sensitivity to ν is also apparent (see Figure 2). In this figure w has been scaled by the dome thickness t; F^* is the applied load scaled as shown in Equation (2). The parameters h and b correspond to the dome height and the base radius, respectively.

In this particular example the following dimensions have been considered: R = 50 mm, b = 40 mm, h = 20 mm, t = 2 mm. The Young's modulus E = 2, 774 MPa represents a polymeric material (typical PLA used in 3D printing). The elements considered in this simulations were 2nd order tetra elements C3D10 (Dassault Systèmes, 2014). The models feature 4 elements through the thickness. The Abaqus eigenvalue (Dassault Systèmes, 2014) results for the “boss” loading of a simply supported dome are shown in Figure 3. Note that the predicted critical loads are 3 order of magnitude higher than from plate compression. However, the shape of the curve is the one given by Equation (1).

Figures 2, 3 suggest that the issue of the stability of domes with different Poisson's ratio values is complex, and cannot be simply resolved by using existing analytical solutions. In this work we tackle this particular problem with a program of experimental and numerical experiments on lattice domes, which possess different lattice geometries that mimic varying Poisson's ratios.

The Abaqus FE package was used (Dassault Systèmes, 2014) to model the compression of the auxetic cellular domes. Both implicit and explicit solvers were used, depending on the analysis. Continuum 3D 10-node quadratic tetrahedral elements (Abaqus type C3D10) were used and isotropic homogeneous material properties were assumed, with isotropic hardening. An analysis of the material's Poisson's ratio concluded that it has a negligible effect on the buckling load or post-buckling behavior. Therefore, the reference value of ν = 0.3 was used in all simulations (Farah et al., 2016). While a mesh sensitivity analysis (provided as a Supplementary Material) shows that a global mesh size of 1.5 mm was acceptable for post-buckling analysis.

The static analysis was performed using Newton-Raphson and Riks methods. The Riks method is typically used to model the snap through buckling in solid domes. However, since lattice domes (and auxetic domes in particular) exhibit a much weaker snap through effect, the Newton-Raphson method with displacement control was found in this work to be sufficiently stable. Element deletion was used for modeling fracture propagation with the explicit dynamic solver (Abaqus/Explicit). Fracture simulations used the Johnson-Cook (JC) hardening and the JC failure criterion (Johnson and Cook, 1985) due to their simplicity and ready availability in the Abaqus explicit dynamic solver (tuning JC is described in Supplementary Material).

Schematics of the FE compression test is shown in Figure 6, where the bottom plate was fixed and the top plate was moved vertically down under displacement control. Both plates were modeled as rigid bodies and the interaction between them and the dome was defined as friction contact. the friction coefficient, μ, of the interaction was found to have a significant affect on buckling in the FE simulations, with higher friction increasing the buckling load and the post-buckling stiffness. It was found that μ = 0.4 gives the best agreement between the FE results and the DIC data regarding the change in dome base diameter.

Once the FE model was calibrated, several parametric studies were performed (presented as a Supplementary Material) showcasing their affect on the dome performance.

Figure 7 shows a very good agreement between the FE and the experimental results for the arrow-head and re-entrant domes. The experimental data error bars represent here the standard deviation from 5 experiments. For the FE, the error bars represent the mean and the upper and the lower bounds of the flow curve. Note that the re-entrant topology exhibits virtually no snap through behavior, and even the arrow-head topology shows a much lower snap though compared to the one exhibited by a solid dome (Figure 2).

The 3D printing using PLA produces struts of uneven thickness. Our measurements showed that the mean strut thickness was 2.14 mm with a standard deviation of 0.07 mm. The FE strut also thickness varied due to the limitations of converting a CAD model into a FE mesh. The mean thickness of the struts in the Finite Element models was in reality equal to 2.12 mm, with a standard deviation of 0.14 mm. This small systematic error might explain the higher experimental loads observed in Figure 7.

The Finite Element results have been computed by considering both large geometric deformations and the non-linear mechanical properties of the PLA material. A question may arise about the importance in the mechanical response of the architecture of the domes vs. the core material behavior. To this end we have performed simulations related to arrow, re-entrant and trichiral configurations under large geometric deformations, but with linear elastic material only (see Figure S5 right). The equivalent Young's modulus of the PLA has also been varied to assess any material scaling effect on the response. The results clearly show that up to a scaled displacement of 0.4–0.5 the response of the dome is negligibly dependent of the mechanical properties (both linear and non-linear—Figure 7). The first post-buckling occurrence in the response is scarcely dependent upon the linearity or non-linearity of the core materials properties, and appear at the same scaled displacement value. These are clear indications that the force/displacement behavior of these cellular domes under indentation is essentially governed by the architecture of the lattice.

Figure 8 shows a very good qualitative agreement of the displacement contour plots between DIC and FE, and an excellent quantitative agreement is seen in Figure 9, which shows vertical (z) displacement at the center of the dome (G1), and points located 2 struts away from the top (G2), and 4 struts away from the top (G3). The DIC error bars are standard deviations from 5 experiments.

Figure 10 shows a very good match between the DIC and FE engineering strains in the struts of the central cell, up to the end of the simulation/experiment, (Figure 7). The time variable in the graphs corresponds to the displacement steps of the indentor surface. One can also notice a dramatic change in the strain histories between the 3 different auxetic topologies.

Figure 11 compares FE vs. DIC ϵ₁ and Figure 12 compares FE vs. DIC ϵ₁₃, or ϵ_xz, where z is the vertical direction. Note that the LaVision's “normal” in Figure 11 means principal (LaVision, 2017). All contour plots show the onset of buckling, and the matching color bars are used for FE and DIC for ease of comparison. Regions of particular interest are circled in these figures.

Two-dimensional (flat) lattice specimens were also printed and tested under tension to measure the lateral contraction and determine an equivalent Poisson's ratio, ν for the lattice structures (Table 1). Analytical models of 2D tri-chiral lattices report ν = −1 (Prall and Lakes, 1997), whereas our 2D tests give ν = −0.72. However, Prall and Lakes (1997) assumed zero specimen thickness; for higher thicknesses or lower slenderness ratios of the struts the shear deformation of the cross section becomes important (Scarpa et al., 2000).

The coefficient a for the domes is defined as the ratio between the diameter extension along x to the diameter contraction in y under a compression along z. Thus, a is an indicator of anisotropy, as well as of the Poisson's ratio of the dome lattices. The hexagonal lattice has a≈−1, because it is an isotropic topology, which expands equally in x and y under compression. Note that the 3 auxetic lattices show similar trends for ν and a.

Fracture was observed experimentally during the testing of the re-entrant, hexagonal, and tri-chiral domes. An example is shown in Figure 13B, which is a DIC shear strain map. The region circled in red, on the outer rim of the dome, was where the ductile fracture first appeared. Soon after, the fracture propagated fast toward the center of the dome along the path indicated by the black dashed line. These later fractures were all brittle.

Explicit dynamic FE simulations of fracture were then carried out using the JC failure criterion, on the 3 topologies (Figure 13). In all cases the fracture initiation was predicted at the outer rim. Fracture propagation was simulated using the element removal technique. The solid and the arrow-head domes showed a significant plastic strain prior to fracture, whereas, the tri-chiral dome fractured with little plasticity. This could be attributed to the rotation of the tri-chiral cells, which represents another deformation mechanism to absorb energy, aside from the bending/stretching/shear of the struts.

The SEM images of the PLA fracture surfaces are provided as Supplementary Material. The amount of plastic deformation in the 3D printed PLA depends on the print direction. Voids and coalescence are signs of ductile fracture in 3D printed polymers (Torrado Perez et al., 2014; Gao and Qiang, 2017). Brittle fracture is seen as the presence of “river lines,” which are a sign of two surfaces being rapidly torn apart, like those on a cleavage fracture in crystalline materials. There is evidence of both ductile and brittle fractures in the SEM images.

Figure 14 compares compression response of domes with 4 lattice topologies (3 auxetic and hexagonal). The tri-chiral topology shows a load which is monotonically increasing (within the data uncertainty), with little indication of buckling. The other 3 topologies show similar response, even though the hexagonal lattice is not auxetic.

An accurate and optimized DIC method was produced and used to successfully validate the FE models of auxetic cellular domes. Very good agreement was achieved between DIC and FE for the onset of buckling and post-buckling response, including strain, displacement, and loads. Experimental results are highly dependent on the errors and inaccuracies in 3D printing.

Auxetic cellular domes offer less indentation resistance under compressive loading than solid domes. They also show reduced snap-through in comparison with conventional lattices and solid domes. Tri-chiral topologies show higher resistance to compression than conventional lattices, which in turn show marginally higher resistance than the re-entrant topologies. The smallest snap-through effect is observed with the re-entrant topology. The cell density and dome curvature (contact size) were shown to affect the compressive resistance and the severity of snap-through.

We are currently working on assessing the dynamic response of auxetic lattices, using a similar experimental and modeling setup. These results will be presented in a follow-up publication.

NE and DC made the test rig and conducted all experimental work. WM and AB run the elasto-plastic FE simulations. YD created the dome CAD models. AS did the eigenvalue analysis and directed the work. FS directed the work. YD, AS, and FS wrote the paper.