Abstract

The mechanics of phase transforming cellular materials (PXCMs) with three different chiral honeycomb architectures, viz., hexachiral, tetra-anti-chiral, and tetra-chiral, are investigated under quasi-static loading/unloading. Each PXCM comprises interconnected unit cells consisting of tape springs rigidly affixed to circular nodes that can rotate and/or translate. The phase change is associated with snap-through instability due to bending of the tape springs and corresponds to sudden changes in the geometry of the unit cells from one stable configuration to another stable (or metastable) configuration during loading/unloading. When compared with similar chiral materials with flat ligaments, the chiral PXCMs exhibit a significantly higher energy dissipation in quasi-static experiments. The hexachiral PXCM was selected for detailed parametric analysis with finite element simulations including 21 models constructed to investigate the effects of PXCM geometry on phase change and energy dissipation. An analytical formalism is developed to predict the minimum compressive load required to induce phase transformation and snap-through. The formalism predictions are compared with those from finite element simulations. An Ashby plot is developed in which the energy dissipated per unit volume versus work conjugate plateau stress of the H-PXCM is compared with other energy absorbing materials.

1 Introduction

Materials used in many structural applications must be formed or shaped into a specific geometry prior to assembly. Examples are aircraft wings, energy absorbing components in transportation industries, and structural members in buildings and bridges. The forming or shaping process is dependent upon the underlying material microstructure which represents a length scale that is several orders of magnitude smaller than that of the structural component. For example, in moving load applications that require low levels of noise and vibration, a thin metallic sheet panel may be too flexible due to low stiffness requiring additional support structures. Forming a bead pattern in the flat sheet via stamping increases its stiffness and mitigates excessive vibration and noise [1]. Stamping requires suitable ductility and hardenability which are largely controlled by the metal microstructure and processing conditions. Incorporation of metal microstructure into a processing simulation is challenging and preference is for homogenization schemes based upon a representative volume element that is intended to be statistically representative of the entire microstructure [2]. Structural optimization of the formed part may suggest bead patterns that are difficult or impossible to produce because of limited ductility or tooling constraints. In general, applications that require high strength and ductility [3] or both high strength and toughness [4] often require significant compromise in material selection since these properties are not directly correlated. Similar observations apply to various nonmetallic materials [4].

Architectured materials offer the potential to address these and other limitations in functionality by effectively “bridging” material behavior across a broad range of disparate length scales [5]. The unusual properties of architectured materials (e.g., negative Poisson’s ratio [6], simultaneous high strength, and toughness [7]) are achieved by combining or “blending” geometrical designs at different length scales together with individual material combinations, forming a single architecture or hybrid material with properties that differ from those of the constituent materials [810]. This can be accomplished with a unit cell design that allows the material to either exploit periodicity or randomness. For example, three-dimensional cellular materials, such as microlattice structures [11], are architectured materials with properties that exploit periodicity based upon a unit cell geometry [5,12,13] while metallic foams used for absorbing and dissipating energy often have random architectures [14]. Other examples of note are found in Ashby [8]. Interest in architectured materials has been stimulated not only by their potential to address some of the complex issues that face component designs with traditional materials and manufacturing and assembly methods but also from the abundant examples in nature [5]. The advent of additive manufacturing continues to accelerate the development of new architectured materials [15,16].

Restrepo et al. [17] recently explored a new class of architectured materials known as phase transforming cellular materials (PXCMs) that have the potential for numerous applications in energy dissipation, shape morphing, and wave filtering [1828]. Restrepo et al.’s [17] unit cell consisted of two sinusoidal beams connected by stiffening walls. The cells were assembled in chains connected in series. This material exhibits reversible (i.e., no plasticity) solid state-like energy dissipation associated with first-order phase transformations which correspond to sudden changes in unit cell geometry from one stable configuration to another stable (or metastable) configuration during loading or unloading. Specifically, each stable or metastable configuration defines a phase at the unit cell level, and the transitions between these unit cell configurations can be interpreted as phase transformations. During the transition from the first stable configuration (regime 1 of the mechanical response, as shown in Ref. [17]) to the second stable (or metastable) configuration (regime 3 in Ref. [17]), the topology of the unit cells remains unchanged but there is a cooperative rearrangement of the beams in each of the cells. This is similar to solid state displacive phase transformations that occur in other materials [29,30]. In many cases, phase changes are associated with a discontinuous change in the first derivative of a state variable. In Restrepo’s PXCM, there is a step change in the specific volume of the ensemble because the composing beams are packed more closely together in the second (meta-)stable configuration. This analogy leads to the following interpretation for PXCMs: when all bent beams in a part of a PXCM sample are in regimes 1 or 3, that part of the PXCM is said to be in either phase 1 or 2, respectively [17]. Any intermediate stage (i.e., when some beams in a part of a PXCM are in phase 1 and others are in phase 2), that part of the PXCM is deemed to be in a mixture of phases. During loading and unloading, a serrated force–displacement behavior was observed in Restrepo et al.’s [17] PXCM and attributed to the progressive phase transformation of rows of PXCM unit cells. Moreover, this serrated behavior was associated with snap-through behavior, which is a reduction in the equilibrium force (indicative of phase transformation) and a reversal of displacement [31] through bending of the sinusoidal beams. Snap-through caused a hysteretic behavior between loading and unloading paths corresponding to energy dissipation in the material. As demonstrated by Restrepo et al. [17], snap-through and therefore energy dissipation in their PXCMs depend upon the number of unit cells forming a chain.

The early work of Mansfield [31] and Seffen and Pellegrino [32] provided a foundation for recent studies of unit cell based architectured materials with thin strip cylindrical ligaments or tape springs connected to nodes that can rotate and/or translate. Mansfield investigated the mechanics of infinitely long curved strips or ligaments [31] and reported “snap-through flexural buckling” under certain conditions. Seffen and Pellegrino [32], and later Soykasap [33] and Guinot et al. [34], analytically and experimentally investigated the mechanics of curved ligaments with a finite length. Prall and Lakes [35] investigated a honeycomb cellular structure based on a unit cell consisting of flat tape spring ligaments rigidly connected to circular nodes. Axial loads are converted into bending of the ligaments and rotation of the nodes with the ligaments essentially wrapping around the nodes during loading. An interesting feature of these materials is that they exhibit negative Poisson’s ratio. Recent studies have investigated small strain (∼5%) behavior of similarly inspired honeycomb cellular structures under compressive loading in Refs. [36,37]. However, phase change, as observed in Restrepo et al. [17], and the underlying mechanisms for energy dissipation have not been investigated for honeycomb chiral structures with curved tape spring ligaments connected to translating and/or rotating nodes under large strain (e.g., >5%) loading.

Inspired by the work of Prall and Lakes [35], we investigated the mechanics of tape spring-based unit cell chiral architectures that induce bending and local rotations when they are subject to axial loads leading to phase transformation via snap-through instabilities. The tape springs have a constant (initial) transverse curvature. Experimental, computational, and analytical methods were applied to three PXCM architectures with different chiral honeycomb topologies (C-PXCMs) subject to large strain loading needed for energy dissipation. These topologies are referred to as hexachiral, tetra-chiral, and anti-tetra-chiral and each consists of tape spring ligaments connected to a central node that can rotate and/or translate [38,39]. These materials are intriguing from the standpoint that they dissipate energy per unit volume in loading and unloading cycles independent of the number of unit cells they contain. This behavior was not observed in the PXCMs considered in Ref. [17]. Several chiral PXCM models were constructed and tested in quasi-static compression in a load frame. Since it exhibits behavior under loading that is representative of that observed for the tetra- and anti-tetra-chiral PXCMs, the hexachiral PXCM was subsequently chosen for finite element simulations aimed at exploring the effects of initial ligament curvature as well as the radius of hollow-cylinder nodes to which the ligaments were rigidly attached. Analytical expressions that provide a means for the rapid preliminary design of PXCMs are then developed and are compared with the finite element simulation results.

Important questions addressed in this paper are as follows: (i) What are the key differences in experimentally measured loading/unloading curves, over two successive loading/unloading cycles, between the three C-PXCM architectures and what inferences can be drawn about phase transformation via snap-through instabilities and energy dissipation in these materials? (ii) How does the loading/unloading behavior of each C-PXCM differ from that for comparable structures with flat ligaments? (iii) What C-PXCM geometric parameters control energy dissipated per unit volume and the minimum compressive load required to induce phase transformation and snap-through? (iv) Can an analytical formalism be developed to predict the minimum compressive load required to induce phase transformation and snap-through in C-PXCMs such that it is in reasonable accord with finite element simulations? (v) How does the hexachiral-PXCM compare with other materials with respect to energy dissipation?

The remainder of this paper is organized as follows: Section 2 discusses the mechanics of the tape spring ligaments as well as the dimensionless groups and geometry of the C-PXCMs. Section 3 covers the experimental models, procedure, and results. Section 4 includes a discussion of the finite element models created to conduct the parametric analysis to investigate the dependence of energy dissipation and minimum compressive load required for phase transformation on unit cell geometry. Section 5 includes a derivation of the analytical formalism for the minimum compressive load (peak load) to induce phase transformation in the unit cells and compares these results to those obtained from the finite element simulations in the parametric analysis. Section 6 analyzes the energy dissipation capability of the hexachiral PXCM as compared with other architectured materials on an Ashby plot. Section 7 includes the concluding remarks.

2 Chiral Honeycomb Phase Transforming Cellular Materials (C-PXCMs)

A tape spring ligament, otherwise known as a carpenter’s tape, is a compliant structure with an initial transverse curvature. Tape springs can exhibit snap-through instability, and hence phase transformation between a stable (unbent) state and a metastable (bent) configuration. To specify, the stable (unbent) or metastable (bent) configuration defines the two phases of the tape spring ligament, and the transitions between these configurations can be interpreted as phase transformations. However, the ability of a tape spring to snap-through and phase transform depends upon the “sense” of the bending associated with the curvature [33]. This is better understood with the schematic depiction of the variation in the moment M as a function of the angle (ϕa) applied to the ends of the tape spring ligaments as shown in Fig. 1. The applied angle (ϕa) is between the axis running parallel along the ligament’s length in the unbent state and that of the ligament in the bent state (depicted by the small horizontal dashed line and the bent curve, respectively (Fig. 1)). The tape spring can be bent either in the same sense of its transverse curvature, as shown in Fig. 1(a), or in the opposite sense, as shown in Fig. 1(b). Note that the contours on the tape spring ligaments shown in Figs. 1(a) and 1(b) represent the minimum principle stress normalized by the largest value for minimum principle stress. When the ligaments are bent, the stresses are concentrated (shown in red in the Fig. 1 contours) near the fold.

The tape spring can only exhibit a snap-through instability, and hence phase transform, when the applied moment results in the opposite-sense bending of the tape spring, such as that shown in Fig. 1(b) [33]. The variation in M as a function of ϕa is schematically depicted by curves (a) and (b) in Fig. 1(c) [40]. Curve (a) corresponds to the moment-angle behavior of the ligament shown in Fig. 1(a). The “dip” in curve (a) corresponds to the formation of a transverse localized fold that occurs with the onset of same-sense bending which results in no instabilities characterized by sudden changes in the moment applied to the ligament. However, two snap-through instabilities (indicative of phase transformation), represented by the arrows at points (1) and (2) in Fig. 1(c), occur along curve (b), corresponding to the ligament in Fig. 1(b) subjected to opposite-sense bending. Under opposite-sense bending, the tape spring will bend until a transverse localized fold occurs along its curvature, at which point the tape spring will buckle and exhibit a snap-through instability denoted by the arrow at point (1) in Fig. 1(c) [32]. When the tape spring ligament is returned to its unbent state from the oppositely bent state, a second snap-through instability occurs at point (2) when the localized fold regains its curvature without any plastic deformation.

In this paper, the mechanics of the chiral phase transforming cellular materials or C-PXCMs, which are architectured materials consisting of tape spring ligaments interconnected by rigid cylindrical nodes, is investigated. Three different topologies, namely, the hexachiral, the tetra-anti-chiral, and tetra-chiral, each a representative C-PXCM and henceforth referred to as H-PXCM, TA-PXCM, and T-PXCM, respectively, are considered [38,39]. Each tape spring ligament in the C-PXCMs mentioned above had an initial transverse curvature. The tape spring ligaments were considered to be rigidly connected to the nodes at their point of contact. The nodes in each of these C-PXCMs were left free to translate and rotate only in response to the loading response of the tape spring ligaments.

Based upon insights from Restrepo et al. [17], each C-PXCM is expected to dissipate energy via snap-through instabilities that induce serrated load–displacement curves in each loading/unloading cycle due to its tape spring ligaments. Ligament deformation is typically, but not always, dominated by bending when the C-PXCM is subjected to a compressive load and snap-through instability. Hence, any phase transformations depend upon the “sense” of bending associated with the curvature of each ligament [33]. As discussed above in terms of Restrepo et al.’s [17] PXCM, a C-PXCM can transition between a stable (bent) and metastable (unbent) state such that each stable or metastable configuration defines a phase at the unit cell level of the C-PXCM, and the transitions between these unit cell configurations can be interpreted as phase transformations. C-PXCMs exhibit a step change in the specific volume of the ensemble (similar to Restrepo’s PXCM [17]) because the composing tape spring ligaments are packed more closely together in the second metastable configuration. This analogy leads to the following interpretation of phase transformation in C-PXCMs: when the tape spring ligaments in a part of a C-PXCM sample are in a particular configuration (bent or unbent), that part of the C-PXCM is said to be in a corresponding phase (Fig. 1). Thus, when the tape spring ligament in a C-PXCM undergoes a snap-through instability due to opposite-sense bending, the material is phase transforming from its stable unbent state to a metastable bent state.

The relevant geometric parameters of the H-PXCM, TA-PXCM, and the T-PXCM are detailed in Fig. 2(d): the radius of the nodes r, the tape spring ligament length L, the tape spring angle of curvature θ, tape spring thickness t, and the tape spring radius of curvature R.

By utilizing the Buckingham Pi theorem, the geometry of each C-PXCM topology can be described with just two dimensionless parameters: π1 = L/ρ (the slenderness ratio), and π2 = r/L (the support ratio) [17]. The slenderness ratio is the relationship between the length of a ligament and its radius of gyration, ρ=I/A, where I is the area moment of inertia of a tape spring ligament and A is its cross-sectional area. The circular support ratio is the relationship between the radius of the nodes and the length of the tape spring ligaments.

3 Loading/Unloading Experiments

3.1 Experimental Models.

To provide a proof of the concepts discussed in the previous section, several C-PXCM experimental samples of each topological variant were fabricated for quasi-static, cyclic loading tests. To create the C-PXCM variants, tape spring ligaments were made of steel and aluminum alloy AA6061 was used to fabricate the hollow cylindrical nodes whose circular faces were covered with paper (Fig. 3). The tape spring ligaments were taken from a Stanley power lock 5 m tape measure with the following dimensions: L = 80 mm, R = 11.25 mm, θ = 106 deg, and t = 0.17 mm, with the dimensionless groups π1 = 15.4 and π2 = 0.15. While caution was taken when extracting the ligaments from the Stanley power lock, a cursory examination revealed that the ligaments were not perfectly smooth, i.e., some small geometric imperfections were noted. For comparison with the tape spring based PXCMs, hexachiral, tetra-anti-chiral, and tetra-chiral honeycombs were fabricated with flat ligaments cut from sheet spring steel with the following dimensions: L = 80 mm, t = 0.23 mm, and width w = 19 mm. The nodes for all samples were cut from hollow aluminum alloy AA6061 extrusions with inner and outer diameters of 20.32 mm and 25.4 mm, respectively. Each model was subjected to several quasi-static loading (compression) and unloading (tensile) cycles with a 10 kN load cell in an MTS INSIGHT load frame, and a cross-head velocity of 1 mm/min under displacement control from which the load–displacement curve for each model was obtained.

Ligaments were fastened to the nodes with screws at 30 deg intervals to create the H-PXCM and 90 deg intervals to create the TA-PXCM and T-PXCM chiral topologies (see Fig. 10 of the  Appendix). To obtain the experimental load–displacement curves, each experimental model was wedged between two horizontal AA8020 T-slotted extruded beams as shown in Fig. 3. The bottom, stationary extruded AA8020 beam was held in place by a grip which was itself rigidly attached to the MTS machine (not shown) and the top moving beam was connected to the 10 kN load cell (not shown) via a grip. Additionally, two AA8020 beams were vertically positioned to allow the top horizontal beam to move up and down without lateral buckling of a C-PXCM.

3.2 Experimental Results.

Figures 3(a), 3(d), and 3(g) show the unloaded H-PXCM, TA-PXCM, and T-PXCM models constructed for testing in the MTS load frame. Figures 3(b), 3(c), 3(e), 3(f), 3(h), and 3(i) show the H-PXCM, TA-PXCM, and T-PXCM experimental models at various stages of large strain loading (e.g., >5% engineering strain). It is important to note that all strains measured here are engineering based upon cross-head displacement. The H-PXCM, TA-PXCM, and the T-PXCM models were subjected to different net downward displacements of 150 mm (∼50%), 80 mm (∼25%), and 150 mm (∼50%), respectively.

Several snap-through instabilities (each denoting a phase transformation occurring in one of the tape spring ligaments in the C-PXCM samples) occurred in the experimental H-PXCM, TA-PXCM, and T-PXCM models (Fig. 3) during loading and unloading. These instabilities are denoted by the serrations in their respective load–displacement curves in Figs. 4(a)4(c) through two successive loading/unloading cycles. Each obvious serration that occurred under loading (compression) is numbered in each of Figs. 4(a)4(c) with the H-PXCM showing seven serrations, the TA-PXCM showing two serrations, and the T-PXCM showing eight serrations. Serrations that occur upon unloading tend to diminish in amplitude relative to those occurring during loading. The reason for this is that upon unloading, each of the C-PXCM samples is dissipating energy (indicated by the hysteresis in Figs. 4(a)4(c)), resulting in serrations with diminished amplitudes similar to the behavior suggested in Fig. 1(c) for an individual tape spring ligament at point (2). Additionally, it is important to note that the number of serrations occurring in the load–displacement curves in Figs. 4(a)4(c) varied for different applied cross-head displacements and resulting strains. For example, after an applied strain of 25% (associated with 40 mm cross-head displacement for the TA-PXCM and 75 mm for the T-PXCM and the H-PXCM) only one serration occurred in the load–displacement curve for the TA-PXCM, whereas three serrations occurred in the T-PXCM and H-PXCM curves. This suggests that the T-PXCM and H-PXCM have a greater potential for energy dissipation than the TA-PXCM due to additional snap-through instabilities.

The snap-through instabilities (serrations) are characterized by a sudden rise and drop in the cross-head load (schematically depicted in Fig. 1(c)). They are the result of the tape spring ligaments in the C-PXCM samples undergoing opposite-sense bending, to the point at which they exhibit a snap-through instability and phase transform (see Fig. 1(b)) in response to the global loading applied by the MTS machine, allowing each C-PXCM to exhibit hysteresis after a loading and unloading cycle with energy dissipation. Recall that phase transformation occurring in the tape spring ligaments of each C-PXCM is considered to be the transition of the tape spring ligaments, experiencing opposite-sense bending, between a stable (unbent) and a metastable (bent) configuration over the course of the loading cycle. There is no accumulation of plastic strain nor other defects as indicated by the generally coincident loading/unloading curves for each of the two cycles shown in Figs. 4(a)4(c).

The load–displacement curves for each chiral honeycomb topology with flat ligaments were also extracted for two loading cycles and are detailed in the Tape Spring-Node Connections in the Experimental Models section of the  Appendix. In these curves, no serrations, characterized by sudden drops in the cross-head load, indicating snap-through instabilities and phase transformation, were observed. Additionally, the energy dissipated by the experimental models was calculated from their respective load–displacement curves, from which the corresponding loss factors [17] (indicating how well the models dissipate energy) for the C-PXCM’s with tape spring ligaments and chiral honeycombs with flat ligaments were obtained. These results are detailed in the Experimental Details section of the  Appendix. The results suggest that the C-PXCMs with tape spring ligaments can dissipate more of the energy put into them upon unloading than the chiral honeycombs with flat ligaments are capable under the same loading/unloading conditions.

4 H-PXCM Parametric Analysis

4.1 Finite Element Models.

To investigate how the geometry of the tape spring ligaments and the nodes (dictated by π1 and π2) affected energy dissipation during loading, a parametric analysis was conducted on the H-PXCM unit cell. This unit cell geometry, shown below in Fig. 5(a), was chosen by observing how buckling occurred within the H-PXCM material in the experiments. Long wave (global) buckling modes were not observed to occur in the experimental samples (Figs. 3(a), 3(b), and 3(c)) during loading [41,42] (for extra details highlighting our reasoning for our choice of unit cell geometry, see the video3). However, in the case of the H-PXCM, buckling in the tape spring ligaments was observed to be localized in the experimental samples (Fig. 3), which provided reassurance (but not a guarantee) that the lowest mode is the one that was captured in the chosen unit cell. Additionally, the H-PXCM was observed to be y-periodic (i.e., buckling occurred in subsequent unit cells along the vertical axis (loading direction), see Fig. 3). No assumptions were made for the TA-PXCM and T-PXCM unit cell sizes since no further analyses were conducted on these architectures. The H-PXCM unit cell chosen for this parametric analysis provided insight on how to further maximize energy dissipation and it allowed us to study the peak load of the H-PXCM unit cell and therefore trends of its overall behavior. Twenty-one finite element (FE) models of the H-PXCM unit cell were created, each with different values of π1 and π2 which controlled the variation in θ and r, respectively, while holding R and L constant. The values of θ and r and the corresponding π1 and π2 are detailed in the Table of Parameter Values Used for Parametric Analysis section of the  Appendix. The models were run using an implicit solver in abaqus cae 2017.

The Gmsh [43] and abaqus cae 2017 preprocessors were used to create the H-PXCM unit cell FE models. Tape spring ligaments and nodes were meshed with shell elements. Ligaments were modeled with steel, with a 210 GPa Young’s modulus, Poisson’s ratio of 0.29, and density of 7860 kg/m3. The nodes were modeled as rigid bodies to ensure that only the ligaments would deform during loading. Periodic boundary conditions (detailed in Fig. 5(a)) were applied to the unit cell FE models with a dummy node (not part of the model geometry) in such a way that any boundary conditions imposed on this dummy node would have the same effect on the FE model. It is important to note that the use of periodic boundary conditions represents the assumption that all of the tape spring ligaments in one unit cell of the H-PXCM will buckle before the onset of deformation causes those ligaments composing the next unit cell to buckle. These boundary conditions are not necessarily the case for the H-PXCM in the experiments. This meant that a quantitative comparison between the experiments (where different tapes buckle in different cells before all of the tapes buckle in one single cell) and the finite element models with periodic boundary conditions was not possible. However, the use of periodic boundary conditions to model the H-PXCM unit cell provided a good first-order approximation to understand how the tape spring ligaments contributed to energy dissipation in the H-PXCM. Loading resulted from the application of a 10 mm/s strain rate applied to the dummy node. An example of one of the unit cell FE models under loading with the periodic boundary conditions can be viewed in Figs. 5(a)5(f): these resulted from sequentially applied strains.

Load–displacement curves were computed for each FE model in the (π1, π2) design space of the parametric analysis, in which several snap-through instabilities (indicative of phase transformation occurring in one or more ligaments within one or more unit cells) were observed. It was from these load–displacement curves that the energy dissipated per unit volume as well as the peak load (i.e., the minimum required load to induce the first-phase transformation in a tape spring ligament in a unit cell) achieved by each H-PXCM unit cell FE model was calculated.

The snap-through instabilities in the H-PXCM unit cell FE models caused premature termination of the FE simulations since an unrealistically small time increment (O(10−11 s)) was required. This was caused by ill-conditioning of the stiffness matrix just after a snap-through event since the post snap-through stiffness of a ligament was much smaller than its pre snap-through stiffness. To address this issue, geometric imperfections were incorporated in each tape spring ligament in each FE model [44,45]. Local regularization approaches such as this have been shown to significantly improve convergence rates in numerical approximations to solutions of singular partial differential equations (PDEs), for example [46]. An eigenfrequency analysis was conducted on each unit cell FE model to extract to the first 15 modes of vibration. Affine combinations of these modes with very small (O(L × 10−5)) coefficients were applied to the FE models as imperfections, such that a small distribution of displacements was applied normal to the surface of each tape spring ligament [45]. The differences between the load–displacement curve for simulations with and without geometric imperfections are discussed in the Validation of Imperfections section of the  Appendix.

4.2 Energy and Peak Loads.

The results of the parametric analysis are summarized in contour plots for the energy dissipated per unit volume Wvol and the peak loads Ppeak (i.e., minimum compressive load required to induce phase transformation and snap-through in Figs. 6(a) and 6(b), respectively). Note that Ppeak is the load at which the first serration occurs in the load–displacement curve of the H-PXCM (labeled 1 in Fig. 4(a)). We observe that the dimensionless parameters π1 and π2, also considered in Figs. 6(a) and 6(b), exert considerable influence over Wvol as well as Ppeak of the H-PXCM as indicated by the variations in different contours in Figs. 6(a) and 6(b). Each numbered circle in the contour plots in Fig. 6 corresponds to the model number of a FE model used in the parametric analysis and the corresponding π1 and π2 values, of that model (detailed in the Experimental Details section of the  Appendix). The values of π1 and π2 were varied between 55–200 and 0.2–0.4, respectively. The physical quantities that were varied between each of the finite element models were θ and r, which determined π1 and π2, respectively, while the rest of the parameters were held constant. The contour plot in Fig. 6(a) shows that Wvol is primarily controlled by π2 since contours show their greatest variation with π2, while that in Fig. 6(b) shows that Ppeak is primarily controlled by π1, since contours show their greatest variation with π1.

Higher π2 and lower π1 designs correspond to tape spring ligaments that are short and stubby, which naturally have the highest Ppeak. Conversely, lower π2 and higher π1 values correspond to long, spindly tape spring ligaments with many buckling modes (including twisting ones) that naturally have low buckling loads. Further, for a given low π2, as we increase π1 the number of available buckling modes increases, thereby reducing the energy absorbed by elastic deformation up to the point of buckling, and hence, the energy released during the buckling (snap-through) event. To understand why π2 controls energy dissipation in the H-PXCM, recall that π2 = r/L and that r (the radius of the nodes) is one of the physical parameters that changes in the parametric analysis. Changing π2 affects the inclination of the segment relative to the direction of the load. Additionally, to understand why π1 controls the peak load (Fig. 6(b)) of the H-PXCM, one can simply consider Euler Buckling (i.e., the peak load) for an individual tape spring ligament [47]:
Fcrit=4π2EIL2
(1)
Here, E is the elastic modulus of the base material composing the tape spring and I is the area moment of inertia of a tape spring, which is given as [48]
I=ωtR3(12(sin(ω)ω)2+(sin(2ω)2ω))
(2)
Note that ω is one-half of the tape spring angle of curvature θ (see Fig. 2(d)). The Euler Buckling equation for a tape spring can be rewritten in terms of π1
Fcrit=4π2EAπ12
(3)
Recall that π1 = L/ρ and θ (the angle of curvature of the tape spring) is the other physical parameter that varies in the parametric analysis.

5 Analytical Formalism

5.1 Motivation and Derivation.

The H-PXCM unit cell orientation with respect to the loading direction differed in the experiments from that of the orientation with respect to the loading direction in the FE simulations. This was done to accommodate the challenges of loading the unit cell both experimentally and computationally in the same way (Figs. 3(a) and 5(a)). A qualitative comparison between the initial peak load (required to induce the first snap-through instability and first-phase transformation) observed in the experiments and those observed in the simulations yielded dramatic differences. Thus, it was desirable to develop an analytical formalism from which an analytical expression can be derived for the initial peak load of the H-PXCM unit cell. This was done to provide a means for the rapid preliminary design of the unit cell based on its geometry and orientation with respect to the loading direction as well as to understand whether the H-PXCM material was isotropic. It is important to note that this is an analytical formulation in development, which will be addressed in future work.

To understand how the unit cell is oriented with respect to the loading direction in the experiments, a free body diagram of one-half of the unit cell that was chosen for the parametric analysis is detailed in Fig. 7(a). The free body diagram depicts the unit cell of the H-PXCM as it was orientated in the experiments detailed in Sec. 3. Because of the rotational symmetry of the H-PXCM, only three ligaments were used to represent the free body diagram of the unit cell. There are two coordinate systems in the free body diagram in Fig. 7(a): (1) the coordinate system described by axes x and y is the global coordinate system, in which a compressive load P acts along the vertical y-axis and (2) the unit cell coordinate system described by axes 1 and 2. The angles α, β, and γ describe the angles at which tape springs 3, 2, and 1, respectively, are attached to the node as measured from the global x axis. Forces F1, F2, and F3 are developed in ligaments 1, 2, and 3, respectively, and are oriented along the lengths of these ligaments: they are a consequence of P. The node is subjected to a shear force Fx, which counteracts the horizontal components of F1, F2, and F3. The forces F1, F2, and F3 also impose a moment on each tape spring, with moments M1, M2, and M3 associated with tape springs 1, 2, and 3, respectively. The moment felt by the H-PXCM unit cell, which is induced by P, is denoted as Mc.

The angle ψ (see Fig. 7(a)) is the angle between the global coordinate system and the unit cell coordinate system and describes the orientation of the H-PXCM unit cell with respect to the loading direction along the global y-axis. However, the value of ψ for which the H-PXCM unit cell has the same orientation as that which was used for the experiments needs to be determined. Figure 7(b) depicts two nodes connected by a single tape spring, the orientation of which is independent of the global or unit cell coordinate systems described by the free body diagram in Fig. 7(a).

The angle θ′ is the angle between the straight line connecting the centers of any two adjacent nodes (shown by the dashed line in Fig. 7(b)) and the tape spring ligament connecting those two adjacent nodes. The angle θ′ is purely dependent upon the geometry of the tape springs and nodes composing the H-PXCM and can be calculated via the following:
θ=tan1(2r/L)=tan1(2π2)
(4)
When the H-PXCM unit cell is oriented in the same way that it was for the experiments, ψ = θ′. Thus, the angle ψ describing the orientation of the H-PXCM unit cell with respect to the loading direction (global y-axis) is given by
ψ=θξ
(5)
Note that the angle ξ (shown in Fig. 7(a)) describes the deviation of the orientation of the unit cell away from the experimental orientation. For the unit cell to be in the experimental orientation, ξ = 0 deg; however, to be in the simulated orientation, ξ = 30 deg. The angles α, β, and γ can all be represented as functions of ψ
α=30deg+ψ
(6)
β=90deg+ψ
(7)
γ=150deg+ψ
(8)
It is assumed that any tape spring in Fig. 7(a) will snap-through (resulting in phase transformation) when the axial force in that spring is compressive and has a magnitude equal to the Euler Buckling load Fcrit given by Eqs. (1) and (3) for a fixed-fixed boundary condition. By summing the forces along the global y-axis of the free body diagram in Fig. 7(a), Ppeak was derived as a function of ψ and is given as
Ppeak=F1cos(30ψ)F2sin(ψ)+F3cos(30+ψ)
(9)
The tape springs in the free body diagram (Fig. 7(a)) do not buckle, snap-through, and phase transform simultaneously; hence, F1F2F3. However, one or more of these loads will equal the Euler Buckling load given by Eq. (1) when the first peak load in the unit cell occurs. As the unit cell represents a statically indeterminate structure, we cannot determine the values of F1, F2, and F3 through the equations for static equilibrium alone. We therefore obtained the values of F1, F2, and F3 from the FE simulations for a particular configuration of the unit cell, such that ξ = 30 deg. The values of F1, F2, and F3 obtained from the FE simulations were then normalized by Fmax = max(F1, F2, F3). At the point in the loading history, i.e., where the first peak load occurs, the force in the spring with the highest load equals the critical Euler Buckling load, while those in the other two springs have values that are given by the ratios F1/Fmax, F2/Fmax, and F3/Fmax. Thus, we can estimate the forces in the various springs by multiplying these ratios by the Euler Buckling load given by Eq. (1). Substituting these values for F1, F2, and F3 in Eq. (9) gives the following expression results for the initial peak load of the unit cell Ppeak
Ppeak=F1FmaxFcritcos(30ψ)F2FmaxFcritsin(ψ)+F3FmaxFcritcos(30+ψ)
(10)

5.2 Comparison of the Analytical Equations and Parametric Analysis.

Figure 8(a) compares results for the analytical Ppeak, derived in Eq. (10), to those of the initial Ppeak computed for each of the FE models in the parametric analysis as a function of π1. Additionally, the analytical Ppeak as a function of ξ is shown in Fig. 8(b). In order to compare the analytical equations with the results of the FE simulations (in Fig. 8(a)), the analytical Ppeak (decreasing dashed curve in Fig. 8(a)) was normalized by the maximum analytical Ppeak that was calculated by Eq. (10). Additionally, the results of the FE models were normalized by the maximum Ppeak observed from the different H-PXCM geometries as a consequence of different values of π2. The increasing dashed curve in Fig. 8(a) represents the average percentage error (values given by the vertical axis on the right).

The analytical formalism predicts the initial peak load reasonably well for low values of π1 (55–80), but the average error in this prediction (relative to that of the results from the finite element simulations) increases progressively for higher values of π1, ranging between a percentage error of 40% (at π1 = 103) and 85% (at π1 = 200). Figure 8(a) shows that while there are some compelling differences in the predictions of both the analytical formalism and the FE approach, the general trends are quite similar. Both the analytical formalism and the FE approach predict that as the value of π1 increases, the initial peak load of the unit cell decreases. Recall that Eq. (1) for the Euler Buckling load of a tape spring ligament can also be written in terms of π1 (Eq. (3)), thus the peak load of an H-PXCM given by Eq. (10), is also dependent upon π1, which was found to primarily control the peak load of the H-PXCM during the parametric analysis (see Fig. 6(b)). It is known that as π1 increases, additional complex deformation modes (e.g., twisting) that are not included in this model may occur, thereby limiting the accuracy of this analytical formalism. Additionally, the curves presented in Fig. 8(a) highlight that both the analytical formalism and the FE models predict that as the value of π1 increases, the initial peak load of the unit cell decreases. This suggests that H-PXCMs can be designed to achieve snap-through instabilities, phase transformation, and energy dissipation for a particular applied load. The values of π2, θ′, ξ, ψ, α, β, and γ as well as of F1, F2, F3, Fmax, and the ratios F1/Fmax, F2/Fmax, and F3/Fmax used to calculate the analytical peak load as a function of π1 are detailed in the Table of Values Used for the Parametric Analysis section of the  Appendix.

The curves in Fig. 8(b) show the variation in the analytical prediction of Ppeak as a function of ξ (see Fig. 7(a)) between that for the experimental unit cell orientation at ξ = 0 deg and the simulated unit cell orientation at ξ = 30 deg for the three π2 values used for the parametric analysis. Note that Ppeak is normalized by its value at ξ = 30 deg since the values of the ratios used in Eq. (9) for Ppeak were calculated for the H-PXCM unit cell in the simulated orientation, such that ξ = 30 deg. Consequently, the analytical model is most accurate at ξ = 30 deg. For any other value of ξ, the ratios in the analytical model should be reevaluated for more accurate values of Ppeak. However, for this study ξ = 30 deg was the only orientation considered for the FE models in the parametric analysis. The variation in Ppeak shown in Fig. 7(b) suggests that for large strains (>5%), such that the unit cells exhibit phase transformations induced by snap-through instabilities, the H-PXCM unit cell is anisotropic with respect to its response when loaded in different orientations.

6 Ashby Plot

To compare the H-PXCM with other energy absorbing architectured materials, the energy dissipated per unit volume was plotted against the work conjugate plateau stress for each H-PXCM unit cell FE model in the parametric analysis in an Ashby plot [8]. The work conjugate plateau stress is defined as follows:
σpl=Wvol/εd
(11)
Here, Wvol is the energy dissipated by the material during a complete load–unload cycle and ɛd is the densification strain of the material, which is given by [17]
εd=11.4ρr
(12)
Here, ρr is the relative density of the material, computed from [49]
ρr=ρ*ρs
(13)
where ρ* is the density of the material and ρs is the density of the composing base material (i.e., the material from which a structure is made). The region occupied by the H-PXCM and the regions occupied by other energy absorbing architectured materials are compared in Fig. 9 [17].

Unlike the other materials on this chart, the H-PXCM does not exhibit a nearly constant stress corresponding to the localization and propagation of an instability. Instead, the energy dissipation in H-PXCMs occurs through discrete and discontinuous snap-through events at the ligament level, that are characterized by serrations in the load–displacement curve. The work conjugate plateau stress is a generalization of the plateau stress that is typically used to compare the energy dissipation capacity of materials. For materials like aluminum honeycombs that exhibit a nearly constant crushing stress, the work conjugate plateau stress yields a value that is very close to the value that is typically used for the plateau stress. This generalization allows us to compare the energy dissipation performance of materials like H-PXCMs with that of other architectured materials in the Ashby plot of Fig. 9. Since the experiments had values of π1 and π2 that fell outside of the design space for the parametric analysis, these cases were not included in the Ashby plot [8].

The region occupied by the H-PXCM is highlighted in Fig. 9. We observe that the performance of the H-PXCMs has a considerable overlap with that of other comparable materials. It is important to note that this representation is based on a fairly small set of simulated designs, e.g., only one material (steel) was used to model the tape spring ligaments in each of the FE models used for the parametric analysis. Hence, the green region in Fig. 8 should only be treated as a subset of the full extent of this material’s energy dissipation capacity.

7 Summary Remarks and Conclusions

In this study, the mechanics of a family of architectured materials termed chiral PXCMs (C-PXCMs) was explored using quasi-static experiments, nonlinear finite element analyses, and an analytical model. C-PXCMs are periodic cellular materials whose unit cells consist of rigid nodes interconnected by tape spring ligaments. The unit cells have various chiral topologies, viz., hexachiral (H-PXCM), tetra-anti-chiral (TA-PXCM), and the tetra-chiral (T-PXCM). Tape spring ligaments exhibit snap-through instabilities as they transition between a stable (unbent) and metastable (bent) state. These instabilities are associated with the nonequilibrium release of stored strain energy which gives rise to energy dissipation upon loading and unloading at large strains (>5%).

The geometry of a C-PXCM is described using two dimensionless groups: π1 (the slenderness ratio) and π2 (the circular support ratio). Samples of each C-PXCM topology were fabricated with tape spring ligaments and subjected to loading/unloading cycles in an MTS load frame. The measured load–displacement curves showed hysteresis which is indicative of energy dissipation. The energy dissipation capacity of C-PXCMs was observed to be significantly greater than that of comparable chiral materials with flat ligaments. Additionally, several snap-through instabilities as denoted by serrations characterized by sudden drops in the load in the measured load–displacement curves (indicative of phase transformation in the unit cells) were observed.

One of the above C-PXCMs, the hexachiral PXCM (H-PXCM) architecture, was chosen for detailed FE simulation aimed at exploring the effects of tape spring ligament curvature as well as the curvature of hollow-cylinder nodes to which the ligaments are attached. The results from the parametric analyses revealed that the energy dissipated (per unit volume) by the H-PXCM unit cell was primarily controlled by π2, but the peak load of the unit cell was primarily controlled by π1.

Mathematical expressions for an analytical model were derived for the peak load to provide a means for the rapid preliminary design of the H-PXCMs and were compared against the finite element simulation results. The semianalytical model for the peak load of the H-PXCM was observed to follow similar trends to those of the results obtained using finite element simulations. The H-PXCM then was compared with other energy absorbing materials in an Ashby plot relating the energy dissipated per unit volume to the work conjugate plateau stress. We observe that the energy dissipation per unit volume of H-PXCMs is comparable with that exhibited by other traditional energy absorbing materials like foams and honeycombs.

Footnotes

Acknowledgment

The authors gratefully acknowledge the generous financial support of the National Science Foundation (Funder ID: 10.13039/100000001) through the GOALI award CMMI-1538898. P.Z. and C.T. acknowledge the partial financial support from the Undergraduate Research Experience Purdue-Colombia (UREP-C) program. We also acknowledge discussions and collaboration with Gordon F. Jarrold.

Appendix

Tape Spring-Node Connections in the Experimental Models.

The experimental models were fabricated such that the tape spring ligaments were connected to the nodes via multiple screw connections at the point of contact between the ligaments and the nodes. Figures 10(a) and 10(b) are schematic representations of the T-PXCM and TA-PXCM experimental models, respectively. The screw connections between the tape spring ligaments and the nodes are denoted by the small holes in the models, one of which is highlighted by the black arrow in Fig. 10(a).

Experimental Details.

The hexachiral, tetra-anti-chiral, and tetra-chiral honeycomb experimental models with flat ligaments in the unloaded configuration are shown in Figs. 11(a), 11(d), and 11(g). The flat ligaments were cut from sheet steel and the nodes were made of aluminum, the same as was used for the C-PXCM experimental samples described in Sec. 3. The nodes were of the same dimensions as described in the narrative and the flat ligaments had the following dimensions: length L = 80 mm, width w of 19.1 mm, and a thickness t of 0.17 mm. Holes were punched into the ligaments and nodes such that they could be connected together via screws (Fig. 10). It is important to note that a wire (not shown) was used in all experiments to constrain one of the nodes in each experimental sample such that the motion along the x- and z-axes (Fig. 10) was prohibited. For more details on the fabrication process, see the video5.

It is important to note that all strains measured here are engineering strains based upon cross-head displacement. Loading was applied to each of the honeycomb models with flat ligaments along the global y-axis. Figures 11(b) and 11(c) show the hexachiral honeycomb at cross-head displacements of 70 mm (∼25%) and 140 mm (∼50%), respectively. Figures 11(e) and 11(f) show the tetra-anti-chiral honeycomb at cross-head displacements of 75 mm (∼25% engineering strain) and 150 mm (∼50% engineering strain), respectively. Figures 11(h) and 11(i) show the tetra-chiral honeycomb at cross-head displacements of 75 mm (∼24% engineering strain) and 150 mm (∼48% engineering strain), respectively.

The respective load–displacement curves for the chiral honeycomb experimental models with flat ligaments are shown in Figs. 12(a), 12(c), and 12(e), which exhibit no snap-through instabilities. However, these architectures still exhibit hysteresis and energy dissipation. In principle, the load–displacement profiles of the C-PXCMs with flat ligaments should exhibit no hysteresis and no snap-through instabilities. The observed hysteresis is a consequence of friction between the C-PXCM samples and the extruded aluminum alloy AA8020 beams placed on the top and sides of each model during testing. To mitigate the effects of friction, the AA8020 beams were lined with Teflon tape. The efficiency of the experimental sample to dissipate energy was characterized as the average loss factor for each C-PXCM and chiral honeycomb sample, which can be viewed in Figs. 12(b), 12(d), and 12(f). The loss factor is given by η = Wd/(2πWi) where Wd is the energy dissipated by each C-PXCM (tape spring ligaments)/Chiral Honeycomb (flat ligaments) after unloading and Wi is the energy absorbed by each material upon loading [17]. Figures 12(b), 12(d), and 12(f) show that the C-PXCM models equipped with curved tape spring ligaments had a larger loss factor than that of the chiral honeycombs with flat ligaments, meaning that the C-PXCMs dissipate more of the energy that is put into them upon loading.

Table of Parameter Values Used for Parametric Analysis.

Twenty-one FE models for the H-PXCM unit cell were constructed for the parametric analysis conducted in this study, each of which had a different value for π1 and π2. Those values which were controlled by varying the angle of curvature of the tape springs θ and the radius of the nodes r are provided below in Table 1 for reference.

Validation of Imperfections.

The snap-through instabilities in the H-PXCM unit cell FE models caused premature termination of the FE simulations because of an unrealistically small time increment (O(10−11 s)). To address this issue, geometric imperfections were applied to each tape spring in the FE model [39,40]. The first 15 modes of vibration were obtained from an eigenfrequency analysis. Combinations of these modes with (O(L × 10−5)) coefficients were applied to the FE models as imperfections. To ensure that the imperfections did not significantly change the results, an H-PXCM unit cell FE model that was able to finish through a loading and unloading simulation without the application of imperfections was compared with its counterpart with the appropriate frequency models applied as imperfections. Differences in the load–displacement curves and the computed energy dissipated, compared in Fig. 13, were found to be negligible (∼0.5% difference in energy dissipated).

Table of Values Used for the Analytical Equations.

The values of π2, θ′, ξ, ψ, α, β, and γ as well as of F1, F2, F3, Fmax, and the ratios F1/Fmax, F2/Fmax, and F3/Fmax that were used to calculate the analytical peak load as a function of π1 in Fig. 8(a) of the narrative are listed in Table 2. The loads, F1, F2, and F3 were obtained from the FE simulations for model 2.

Visualization of Bending in the H-PXCM Tape Spring Ligaments.

Figure 14 shows the H-PXCM experimental model under quasi-static loading in a load frame. It is important to note that during loading, not all tape spring ligaments that were bending were also exhibiting snap-through instabilities and phase transformation. Figure 14(a) shows the H-PXCM experimental sample in the unloaded configuration, such that none of the ligaments has been bent (tape springs not yet bent were as shown). Figure 14(b) shows the H-PXCM after an applied displacement of approximately 25 mm (∼9% engineering strain). The highlighted tape spring ligaments underwent same-sense bending (see Fig. 1(a)) in which they exhibited no snap-through instability and did not phase transform. The lightly highlighted ligaments underwent opposite-sense bending in which they exhibited a snap-through instability and phase transformation.

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