Energy dissipation in functionally two-dimensional phase transforming cellular materials

Phase Transforming Cellular Materials (PXCMs) are periodic cellular materials whose unit cells exhibit multiple stable or meta-stable configurations. Transitions between the various (meta-) stable configurations at the unit cell level enable these materials to exhibit reusable solid state energy dissipation. This energy dissipation arises from the storage and non-equilibrium release of strain energy accompanying the limit point traversals underlying these transitions. The material deformation is fully recoverable, and thus the material can be reused to absorb and dissipate energy multiple times. In this work, we present two designs for functionally two-dimensional PXCMs: the S-type with four axes of reflectional symmetry based on a square motif and, the T-type with six axes of symmetry based on a triangular motif. We employ experiments and simulations to understand the various mechanisms that are triggered under multiaxial loading conditions. Our numerical and experimental results indicate that these materials exhibit similar solid state energy dissipation for loads applied along the various axes of reflectional symmetry of the material. The specific energy dissipation capacity of the T-type is slightly greater and less sensitive to the loading direction than the S-type under the most of loading directions. However, both types of material are shown to be very effective in dissipating energy.


S1. Design of 2D PXCMs
In this paper, we present a systematic study to design and study the mechanical performance of functional two-dimensional phase transforming cellular materials (PXCMs) that are capable of dissipating energy along various axes of symmetry. We created a series of designs and then we evaluated them through FE simulations ( Fig. S1 and S2). While the extension to 2D PXCMs may seem easy by follow the schematics shown in Fig. 1, the process requires a careful examination of the performance of the different types of designs. As mentioned in section 2.2, 2D PXCMs have three levels of hierarchy structures from level zero to level two. The zeroth level of the hierarchy structure is the elementary building block of the PXCMs which are composed of either single (Fig. S1(a)) or a pair of parallel-connected bent beams ( Fig. S1(b)). To evaluate both designs, the FE simulations of two S-type PXCMs samples with these two types of building blocks under a uniaxial load-unload cycle are created.
We observe that the sample with single bent beams as elementary building blocks shows local wobbling behavior ( Fig. S1(a)), which causes unpredictable and disorderly transformation behavior of materials. This local "wobble" mode is caused by the rotation at the apex of the single bent beam mechanisms. On the other hand, when the elementary building blocks are parallelconnected bent beams, materials transform steadily and progressively ( Fig. S1(b)). This is mainly caused by the fact that rotation at the apex is restricted with parallel beams. Figure.S1(c) shows the result of the simulations (Load-Displacement, F-d curves) of the both cases (single beam and pair of parallel-connected beams). These results indicate that materials with parallel-connected bent beams as zeroth level of the hierarchy structure can have higher better bistability behavior (e.g., larger peak to valley force ratios) which leads to better energy dissipation capacity. Additionally, the thickness of each individual beam in the two parallel-connected bent beams can be tailored independently to influence the performance of the material. To understand the effect of having different values of thickness in the top and the bottom bent beams, we performed FE simulations of a group of S-type 2D PXCMs under the displacement-controlled uniaxial compression. For each building block, the amplitude A and wavelength λ of both bent beams, the thickness t of the bottom bent beam, and supports thickness tstiffer are identical among the simulations. The only variation among different simulations is the thickness t' of the top bent beam ( Fig. S1(d)). We define a parameter Q' =A/t' as an indicator of the value t' (since we kept A constant). The parameter Q' varies from 10 to 15 among our simulations (Fig. S1(e)). The rest of the dimensions follow those indicated in Table S1. The relation between force and normalized displacement (d/A) of all the simulations are displayed in Fig. S1(e). These F-d curves indicate that, when all the other geometry parameters remain constant, the thinner the top bent beam is, the lower the peak and valley forces are, and the mechanism is said to become "more bistable". This means that the bottom part of the force-displacement curve cross the F = 0 line (e.g., the valleys remain negative while the peaks are positive), and the distance between peaks and valleys increases. Another interesting observation is that, during phase transformation, the top bent beam transforms first because the only lateral constraints are provided by the vertical stiffening walls.
However, these top beams push the stiffening walls apart, reducing the constraints on the bottom bent beam. This competing mechanism can be tailored with additional analysis by finding the right combination of Q' and Q. However, to maintain the focus of the paper, most of PXCMs considered in this work have parallel bent beams with the same in thickness (Q=Q'=10, t=t'; see Table S1).
After analyzing these two design options at zeroth level of hierarchy structure, we studied two potential design options in the first level of the hierarchy structure of 2D PXCMs. Triangular and square motifs are natural candidates for building the first hieratical level of 2D PXCMs because they can be tessellated in a 2D plane. The building blocks are assembled by combining the bent beams together into first level motif. The first type of frame connects the bent beams from their ends with spokes that converge at the center of a motif ( Fig. 1(d)). The second type also connects all the bent beams via the ends, but form a polygon frame inside a motif. To evaluate both design choices, we perform FE simulations of S-type 2D PXCMs with these two types of frames as it is shown in Fig. S2(a) and Fig. S2(b). Under a displacement-controlled compressive load-unload cycle, both specimens collapsed steadily. However, the F-d curves (Fig. S2(c)) indicate that the specimen with spokes ( Fig. S2(a)) exhibits almost no hysteresis compared with the specimen with the center frame ( Fig. S2(b)). Therefore, with adopt the center frame as the choice for first level of hierarchy (see Fig. 1(d)). The determination of the second level of hierarchy follows a similar analysis through FEM analysis. The square motifs can be tiled easily as shown in Fig. 1(e). Choosing the support structures for these triangular shape motifs is not straightforward. Two alternative support structure topologies and are shown in Fig. 1(e). To evaluate the support structure topology , a corresponding prototype is fabricated by an Object Connex500 3D printer with a photo-cured polymer (RGD 8530, = 1 and = 19 ). The bent beams are designed to remain elastic during the phase transformation 1,2 . Ten compressive load-unload cycles were applied on the specimen using a universal testing machine (MTS Insight 10 equipped with a 10 kN load cell MTS 661.19F-02) ( Fig. S3(a) and Fig. S3(b)). We observe that the supports and frames of the PXCM deform just enough to eliminate any constrain to the bent beams to obtain bistability. The hysteresis curves indicate that the energy dissipation is mostly plastic deformation rather than phase transformation. Therefore, regular tiling ( ) does not lead to a functionally two-dimensional PXCM. The arrangement shown in ( ) with triangular motifs located at the nodes of a regular hexagon is not a tiling as it includes some empty space at the center of the hexagon. As it is shown in the main paper, this arrangement leads to a functionally 2D PXCM. Optimization could be performed to improve the performance of 2D PXCMs in terms of energy dissipation, strength, or initial stiffness. Figure S3: Example of an unsuccessful design. T-type 2D PXCM with TI arrangement show plastic deformation under ten compressive load-unload cycles.

S2. Characterization 2D PXCMs
Uniaxial, quasi-static, compressive load-unload tests were performed to characterize the response of the S-type and T-type PXCMs along the various axes of symmetry of the materials. These tests were carried out under displacement control. Four specimens corresponding to four tests are fabricated ( Fig.3-6). The volumes of samples are displayed in Table S2. These load cases are tested using nonlinear finite element analysis. This analysis helps us understand whether or not there is phase transformation, acknowledging that the applied displacement in the beams is a function of θ, where θ is the angle between the loading direction and axis of symmetry of a bent beam (see Fig. 4(c)). The applied displacement on each sample follows Eq. s1: where A is the amplitude of a bent beam, is the angle between the axis of symmetry of the ℎ bent beam structure and the loading direction, and n is the total number of such bent beam structures along any column of motifs in the material sample. All the specimens are tested with three compressive load-unload cycles with the loading rate of 1 mm/min. Figure S5 shows the F-d curves of each sample under three compressive load-unload cycles from experiments.  S-type 2D PXCM is loaded at {45°, 135°} loading angles ( Fig. S5(b)), every bent beam has its axis of symmetry 45° rotated from the loading direction (θ=45°). This larger angle θ causes all the bent beams transform though more asymmetric configurations ( Fig. S6(b)) compared with the previous case. As a result, the specimen exhibits mostly metastable behavior and lower energy dissipation capacity. In such cases, when a mechanism transforms via such an asymmetric configuration, we say that those mechanisms go through a secondary pathway. When the T-type 2D PXCM is loaded at {0°, 60°, 120°}, one third of bent beams with their axes of symmetry align with the loading direction. These bent beams go through the primary pathway; similar to half of the bent beams in the S-type 2D PXCM at loading angles {0°, 90°}. The rest of bent beams, which have their axes of symmetry 60° rotated from the loading direction, transform through the asymmetric configurations and exhibit metastable behavior ( Fig. S6(c)). The T-type 2D PXCM at {30°, 90°, 150°} has one third of bent beams parallel to the loading direction. These bent beams clearly do not exhibit phase transformation. The rest of bent beams have their axes of symmetry 30° rotated from the loading direction (θ=30°). These bent beams transform through the noticeable asymmetric configurations, therefore we define the mechanisms go through secondary pathway as well. Since the angle θ is relative small compared with the previous two cases, the sample shows bistability and relatively high energy dissipation ( Figure S6(d)). Figure S6: The simulations (left) and experiments (right) show that the bent beams have phase transformation through more asymmetry configurations when the angle θ between the loading direction and their axes of symmetry increase.

S3.2. Phase Transformation in a Single Mechanism
The experiments and FE simulations discussed in this paper show that the angles between loading directions and axes of symmetry of bent beams affect the performance of materials in terms of bistability and energy dissipation capacity. In order to better understand this influence, we perform ancillary FE simulations of a single mechanism under loads applied at various angles, θ (see Fig.   S7(a)). Figure S7(b) shows the F-d responses as a function of the angle θ. We note that the mechanism response is clearly bistable for =0 with a well defined second stable configuration and a long region with negative stiffness. However, the mechanisms becomes metastable as increases, and the negative stiffness region in the response shrinks rapidly as increases from 0° to 15°. The negative stiffness region disappears completely for > 15° (see Fig. S7(b)). As discussed, energy dissipation in PXCMs arises from the non-equilibrium release of energy accompanying the traversal of the limit points during the loading and unloading of these materials.
Specifically, the energy dissipation in a material with a 'sufficiently' large number of motifs is proportional to the area bounded by the envelope curve shown by a dashed line in Fig S 7(c)-(f) 3,4 . Figure S7(g) shows a plot of this area as a function of θ. We note that the area appears to decrease exponentially with increasing θ and becomes nearly zero for =15°. This shows that the energy dissipation capacity of the single mechanism degrades quickly as the inclination of the applied load with respect to its axis of symmetry increases, and it disappears completely for inclinations as small as 15 degrees. Based on the observations, we conclude that the larger the angle between loading direction and the axes of symmetry of a bent beam, the lower their ability to produce bistable behavior. This eventually adversely affects the capability of the material to dissipate energy. However, the observations from the tests on 2D PXCMs show that when the angle is larger than 15°, the material still exhibit bistable behavior and energy dissipation capacity (Fig. S5). This is due to the collective behavior of all the cells, which produce enough lateral constraint even for those bent beams that have higher values of . This is discussed in the next subsection.

S3.3. Phase Transformation in a 2D PXCMs
The S-type 2D PXCM has four axes of symmetry, but only two of these are aligned with axes of symmetry for its constituent single mechanisms. When a loading direction for these PXCMs is aligned with an axis of symmetry of the material that also happens to be an axis of symmetry for a subset of its constituent mechanisms, this subset of mechanisms contributes the most to the total energy dissipation of the specimen. Half of the mechanisms in the S-type PXCM are aligned with their axis of symmetry lying along the 0 degree direction, and the other half have their axis of symmetry along the 90 degree direction (see Fig. 3). The mechanisms aligned at 0 degrees are not deformed significantly, and hence do not contribute to the total energy dissipation of the sample when the mechanism is loaded along 90 degrees. Similarly, the mechanisms aligned with 90 degrees do not contribute to energy dissipation when the sample is loaded along 0 degrees.
Moreover, as the same number of mechanisms are active contributors to the total energy dissipation when the material sample is loaded along 0 and 90 degrees, we expect the total energy dissipation to be similar in these two cases.
The situation is different when the loading direction for the sample is not aligned with axes of symmetry for a subset of the constituent mechanisms . Based on the prior discussion, we expect any mechanisms oriented in a direction such that their axis of symmetry is inclined 15 degrees or more with respect to the load direction to have a very small contribution to the overall energy dissipation (see Fig. S7). However, an ensemble of mechanisms behaves somewhat differently than the single mechanism due to the internal degrees of freedom possessed by the individual motifs and its interaction with their neighbors. We notice that the individual cells in an ensemble reorient themselves during the deformation of the material such that they reduce the inclination of the force deforming a mechanism with respect to the axis of symmetry of the mechanism undergoing transformation (see Fig. S8-10). This reorientation happens via rotation of the individual motifs based on which subset of its constituent mechanisms is transforming. Thus, we can observe the same motif rotating clockwise and counter-clockwise at different points in the loading history (Fig. S9-10). This behavior results in some energy dissipation contribution even from mechanisms that were oriented such that the direction of their axis of symmetry was inclined by 15 degrees or more with respect to the external force in the undeformed configuration.
Moreover, the percentage of bent beams that undergo transformation in S-type 2D PXCM significantly increases from 50%, when it is loaded at 0°/ 90°, to 100% when a is loaded at 45°.  We observe that the T-type material, which has more axes of reflectional symmetry then the Stype, exhibits slightly lower variation in energy dissipation with changes in the loading direction than the S-type material. This suggests that further increases in the number of axes of reflectional symmetry for the unit cell are likely to reduce the variation in energy dissipation with the changes in loading direction, with the advantage that the relative density of the material does not change significantly while increasing the symmetry of the unit cell. To verify the energy dissipation presented above is still produced by the PXCM when its base material remains in the elastic regime, we carefully analyze other type of sources for potential energy dissipation sources. This will allow us to examine and estimate how much of the total energy dissipated by a PXCM sample can be attributed to the primary and secondary pathways. Figure S5 show respectively the F-d responses of the S-and T-type PXCM samples when they are loaded at 0 degrees, for three back-to-back load-unload cycles. Irreversible (e.g. plastic) deformation across the successive cycles is negligibly small after the first cycle for all four load cases considered in this study (Table S4). This is confirmed not only by the F-d response (Fig. S5), but also by a posteriori examination of the specimens that showed no sign of permanent deformation. This is to be expected, as we designed the mechanisms such that it remained in the elastic strain regime over the complete load-unload cycle.
Estimation of the energy dissipation via the other secondary pathways listed above is not straightforward. The mechanisms are designed to have a bistable mechanical response. They exhibit snap-through behavior under force control, but not under displacement control. If we subject a single mechanism to the same range of deformation under force and displacement controlled conditions, it will dissipate energy via all available dissipation pathways in the former case. However under displacement control, it does not undergo a snap-through and, hence, it does not exhibit dissipation due to the snapping action of the mechanism. We can estimate the energy dissipated by a mechanism via all dissipation pathways except the snapping action by subtracting the energy dissipated by the mechanism under displacement control from that under force control.
Since the S-type PXCM loaded at {0°, 90°} showed the most irreversible deformation in successive load-unload tests, we choose that load case to estimate energy dissipation through pathways other than the snapping action of the beams. The average energy dissipated by the entire sample in cycles 2 -3 is 6594 mJ (see Table S4). We subject a single mechanism from this sample to four successive load-unload cycles under displacement control with a cross head travel rate of 1mm /min (See Fig.   S11). The energy dissipated during a complete load-unload cycle is obtained by measuring the area between the loading and unloading curve (See TableS7-S8). The average energy dissipated 20 by a single mechanism over cycles 2 -4 is 17.5 mJ (see Table S8). Since, 36 such mechanisms undergo a complete load-unload cycle during a complete load-unload cycle on the entire PXCM sample (see Fig.3), we estimate the total energy dissipated by the entire sample due to pathways other than the snapping action of the beams to be 630 mJ. Thus, assuming that all 36 of these mechanisms exhibit identical energy dissipation behavior we estimate that all secondary dissipation pathways other than plastic deformation dissipate approximately 10% of the total energy dissipated by the S-type PXCM sample when it is loaded at 0° and 90°. Figure S11: A single mechanism under displacement control with a crosshead travel rate of 1mm /min.

S.3.4. Auxetic behavior
Under the uniaxial loading conditions along different axes of symmetry, both S-type and T-type samples exhibit auxetic behavior from FE simulations and experiments (See Fig.3-7, and S12). By having these auxetic behaviors, 2D PXCMs gain more benefits such as high indentation resistance, shear modulus, fracture toughness, and synclasticity 5 . These benefits extend the application of 2D PXCMs into medical stent, adaptive clothing, and medical cast [6][7][8] .

S4 Finite Element Simulations
Finite element models are created to quickly capture the essential mechanical response of 2D PXCMs under different loading angles. Element type, element size, contact condition, and base material properties were first studied to determine the most effective simulation set up and assess potential source of error and uncertainties.

S4.1. Element Type
To select a type of computational effective element, we create FE models of a 2D PXCM elementary mechanism (i.e. bent beam) with clamped-clamped boundary conditions and under displacement control with different elements types (Fig. S13(a)). The geometry of the bent beam is identical to the bent beams employed in the T and S-type 2D PXCMs. The elements selected are: two-node linear beam element (B21), four-node bilinear, reduced integration with hourglass control (CPE4R) and eight-node brick element with reduced integration (C3D8R). For 2D and 3D models, 7 elements are assigned throughout the beam thickness (element size = 0.1 mm). Additionally, to check convergence for the 3D model, a model with bent beam with 12 C3D8R elements throughout the bent beam thickness is created. The F-d relations of four models are plotted in Fig. S13(b). As it can be observed in the figure, there is no significant difference between the models with B21, CPE4R, and C3D8R elements. We choose B21 for all the simulations as it is the most computationally efficient and it enables us to create larger models with multiple motifs.

S4.2. Convergence study
Once the element type is chosen, a convergence study is conducted on the bent beam model with B21 elements. The element size varies from 4 mm to 2 mm. The F-d relation curves of five models with these various elements size are shown in Fig. S14(a). Since only minor variation can be observed from the F-d curves, the peak force Fp of each curve is used to quantify the difference.
We plot Fp against element size as shown in Fig. S14(b). The peak load converges when the element size reduced to 2.5 mm. We chose the element size 2.17 mm which is smaller than the threshold of convergence (this is shown as red dot shown in Fig. S14(b)).

S4.3. Friction coefficient
Contact between adjacent beams was modeled using the small sliding formulation in Abaqus 6.14.
Coulomb friction with a friction coefficient of = 0.1 is assumed to be active at all contact interfaces. This value is selected based on the Typical Properties of Generic Acrylonitrile Butadiene Styrene (ABS) 9 . For generic ABS materials, the coefficient of friction varies from 0.1-0.5. To check the sensitivity of the friction coefficient in our models, we build FE models of the S-type PXCM loaded under {0°, 90°} (with  varying from 0.1 to 0.5). The F-d relations of these models are shown in Figure S15. The model with = 0.1 exhibits slightly lower valley force when the 11 th mechanism buckles at the very end of the loading cycle. This is an indication that contact (and friction) only plays a role when most of the motifs are already transformed/collapsed. Overall, as it is evident in Fig. S15, friction seems to play a minor role in full compression and = 0.1 is reasonable assumption to made for FE simulation. Test setup is shown in Figure S16(a). Four increasing load-unload cycles are applied on three samples as shown in Figure S16(b). Sample B1 and B2 have identical cross section areas (i.e., 0.7 mm × 25 mm) compared with the elementary mechanism of 2D PXCMs. Sample S1 also has the same thickness 0.7 mm, but half of the width (i.e. 13 mm) compared with 2D PXCMs (Table S9) Figure S16(b) shows the F-d relation of sample B2 under four increasing displacement load-unload cycles.
During the first two load-unload cycles, the sample exhibits approximately linear elastic behavior.
A plateau is reached when the strain increases to 0.9% and 1.2% at cycle 3 and 4 which indicates that the base material exhibit nonlinear behavior under the large strain. However, for simplicity, we assume a linear elastic material model for the FE simulations, where the modulus is derived from the zero-strain tangent to the F-d curves obtained from a 3-point bending test (Table S 10).
This simplification allows us to capture the essential mechanics of the material behavior in a computationally efficient manner, but it sacrifices the accuracy of the force prediction. This factor could cause the FE models to overestimate the mechanical response of the PXCMs compared with the experiments. To assess the uncertainties introduced by this approach we develop FE models of all four 2D PXCMs samples where we assign a minimum, average and maximum elastic modulus. Such analysis can provide lower and upper bounds on the F-d curves are displayed in Figure S17. The energy dissipation calculated in each sample is summarized in Table S11. Due to the uncertainty of elastic modulus of base material, assuming all the 2D PXCM samples have a constant elastic modulus can cause the energy dissipation capacity of these materials varies from -6% to 7%.

S 5. Energy dissipation rate
Energy dissipation rate is another important factor for engineering applications. To understand the how loading angle influence the energy dissipation rate in 2D PXCMs, we plot the energy dissipation for four samples as a function of the applied displacement. Energy dissipation in PXCMs occurs in a discontinuous way through discrete steps corresponding to snap through transitions in individual building blocks. However, we can define an average energy dissipation rate in the following way: we load the material up to a snap through at a displacement and then unload it completely. As such the average energy dissipation rate, at that given displacement, can be defined as the ratio between the energy dissipated in this complete load-unload cycle and the applied displacement. This is repeated for all snap through events to get the average energy dissipation rate for different displacement values through its loading history. For this we use all the data from Figs. 3-6. These F-d curves can be discretized into cycles. Each cycle starts form the initial state, loads to a snap back point and then unloads back to the initial state ( Figure S18(a)).
The initial stiffness is used to extrapolate the unloading path after snap back happens. Figure   S18 T-type PXCM is not sensitive to the loading direction compared with S-type PXCM. To have explicit comparison, liner interpolation is used to quantify the energy dissipation per unit value and mass varies with applied displacement. Linearized curves are shown in Figure S18 (c)-(d) and the slope of each sample is displayed in  We created FE simulations to investigate the response of a T-type PXCM under biaxial loading condition. The schematic for this load case is shown in Fig. S19(a) where the a1 axis is aligned with the X-axis and a2 axis is aligned with the Y-axis. To ensure all the building blocks achieve phase transformation at the same time, the loading rate in the X direction is 0.39 mm/min and in Y direction is 1 mm/min. Fig. S19(b) shows the undeformed and deformed configurations of the sample at three salient points during its compression as obtained from the finite element simulations. The corresponding points are labelled on the F-d response in Fig. 19(c). The bent beams are color coded according to their status at any point during the deformation process. The 32 beams rendered in gray are still in phase 1 (according to the definition in Design Considerations section Fig. 1 (b) , those shaded green have already transformed to phase 2, and the red ones are undergoing phase transformation.
Unlike the response of T-type PXCM under the uniaxial loading condition, all the building blocks undergo phase transformation under the biaxial loading condition. Expect at loading-unloading transition point, T-type PXCM exhibits higher serrated loading and unloading plateau force in the X direction than in the Y direction. The peak force is reduced to around 50% in both X and Y directions after the first building block transformed (See Fig. S19 (c)). The ratio of energy dissipation capacity of in X to Y direction is 1.3 (Table S13 and video 6).