# Mechanical design of negative stiffness honeycomb materials

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## Abstract

A mechanical system exhibits negative stiffness when it requires a decrease in applied force to generate an increase in displacement. Negative stiffness behavior has been of interest for use in vibro-acoustic damping materials, vibration isolation mechanisms, and mechanical switches. This non-intuitive mechanical response can be elicited by transversely loading a curved beam structure of appropriate geometry, which can be designed to exhibit either one or two stable positions. The current work investigates honeycomb structures whose unit cells are created from curved beam structures that are designed to provide negative stiffness behavior and a single stable position. These characteristics allow the honeycomb to absorb large amounts of mechanical energy at a stable plateau stress, much like traditional honeycombs. Unlike traditional honeycombs, however, the mechanism underlying energy-absorbing behavior is elastic buckling rather than plastic deformation, which allows the negative stiffness honeycombs to recover from large deformations. Accordingly, they are compelling candidates for applications that require dissipation of multiple impacts. A detailed exploration of the unit cell design shows that negative stiffness honeycombs can be designed to dissipate mechanical energy in quantities that are comparable to traditional honeycomb structures at low relative densities. Furthermore, their unique cell geometry allows the designer to perform trade-offs between density, stress thresholds, and energy absorption capabilities. This paper describes these trade-offs and the underlying analysis.

## Keywords

Honeycombs Negative stiffness Bistability Energy absorption Elastic stiffness Stress threshold## Abbreviations

- Δ
_{th} Normalized displacement threshold

*b*Out-of-plane depth for a negative stiffness beam

*d*Transverse displacement

*E*_{0}Specific initial stiffness

*E*_{s}Modulus of elasticity

*F*Normalized force

*f*Transverse force

*F*_{th}Force threshold

*h*Apex height for a negative stiffness beam

*I*Area moment of inertia

*l*Length of a negative stiffness beam

*Q*Ratio of apex height to thickness for a negative stiffness beam

*t*In-plane thickness for a negative stiffness beam

*w*(*x*)Beam-shape coordinate along the vertical axis

*x*Beam-shape coordinate along the horizontal axis

- Δ
Normalized displacement

*ε*_{max}Maximum strain

*ρ*Relative density

*σ*_{pl}Critical stress level

*σ*_{th}Stress threshold

*σ*_{ys}Yield strength

## Background

Honeycombs are ordered cellular materials with prismatic cells. The cells of the honeycomb can assume a variety of cross-sectional shapes, including hexagonal, kagome, square, triangular, and mixed triangular and square [1, 2]. Relative to other low-density materials, such as stochastic foams, honeycombs provide very high levels of compressive strength and energy absorption, and those characteristics are linked directly to cell shape and density [2].

The superior energy absorption capabilities of honeycombs are highly dependent on the relatively flat, extended region of plateau stress in Fig. 1. Once a critical plateau stress is reached, honeycombs absorb very large amounts of energy at the plateau stress level without exposing an underlying structure to additional compressive stress unless the energy imparted to the honeycomb is large enough to cause densification. One disadvantage to utilizing honeycombs for energy absorption applications is that energy absorption in the plateau regime requires plastic buckling, which means that the honeycombs must be replaced after a single use. While it is possible to achieve a plateau stress region with recoverable, elastic buckling for very low density structures (cf. [3]), such cellular structures cannot be fabricated with typical manufacturing methods and also demonstrate very low initial stiffness and plateau stress.

*Q*, the ratio of apex height,

*h*, of the beam to its in-plane thickness,

*t*. As established by Qiu et al. [8], a single curved beam exhibits negative stiffness behavior for a wide range of ratios greater than approximately 1.3 and transitions to bistable behavior at ratios greater than 2.31. The double beams utilized in the honeycomb in Fig. 2 transition to bistable behavior at much higher ratios.

While it has been known for some time that properly constrained curved beams exhibit the behavior illustrated in Fig. 3, the authors of this publication have recently designed a negative stiffness honeycomb structure to leverage this behavior for energy absorption [4, 5]. All of the features in the negative stiffness honeycomb structure shown in Fig. 2 have a specific purpose. The double concentric beams are utilized to constrain the beams to transition from one first-mode-buckled shape to another via the third buckling mode, rather than the second mode, which is known to significantly reduce the force threshold of the beam and the magnitude of its negative stiffness. The flat, horizontal walls constrain the horizontal expansion of the unit cell upon application of in-plane compression, thereby enabling snap-through-like behavior. Chamfers near the intersection of the horizontal and vertical walls help prevent twisting of the cell walls during loading.

The properties of a negative stiffness honeycomb can be tailored by adjusting the dimensions of the unit cells. Assuming that the characteristic cell size is fixed, regular honeycombs offer a single degree of design freedom––density or the thickness of the cell walls––such that adjusting the density results in unique values of plateau stress, initial stiffness, and energy absorption per cell. Similarly, assuming that the characteristic cell length, *l*, is fixed, negative stiffness honeycombs offer two degrees of design freedom––the apex height and in-plane thickness of the beams––such that it is possible to achieve a particular relative density with cells of various geometries, each of which offers different levels of stress threshold, initial stiffness, and energy absorption. This paper describes the analysis and design of negative stiffness honeycombs for energy absorption applications and outlines the types of design trade-offs that can be achieved.

## Methods

*.*[8], the transverse displacement of the center of a curved beam is related to the transverse force applied to the center of the beam as follows:

*F*and ∆ represent normalized force and normalized displacement, respectively. Those terms are related to applied transverse force,

*f*, and transverse displacement,

*d*, as follows:

*E*

_{s}is the modulus of elasticity of the cell wall material and

*I*is the area moment of inertia of the cell wall. The moment of inertia term

*I*accounts for the out-of-plane depth,

*b*, of the beam, assuming a rectangular cross section. Therefore, the relations derived in Equations 1, 2, and 3 are valid for a beam with arbitrary out-of-plane depth, provided it is uniformly loaded throughout its depth. Transverse displacement is measured as illustrated in Fig. 4 for a beam with its undeformed shape defined as follows:

Equation 1 predicts the resulting force-displacement behavior illustrated in Fig. 3 and can be used for any curved beam geometry as long as the beam is constrained to avoid second-mode buckling when it transitions from one first-mode-buckled shape to another.

The force threshold is defined by the peak of the force-displacement curve in Fig. 3. The force threshold can be calculated by taking the partial derivative of Eq. 1 with respect to normalized displacement, setting the partial derivative equal to zero, solving for the normalized displacements, and substituting into Eq. 1. For a beam with a *Q* of 2.31, the force threshold, *F* _{th}, occurs at a normalized displacement, ∆_{th}, of 0.5 and a normalized force, *F*, of 389.

*σ*

_{th}, for comparison with the critical stress level of a regular honeycomb,

*σ*

_{pl}, illustrated in Fig. 1. The stress threshold is found to be

*b*is the out-of-plane depth of the cell. Normalizing by the modulus of elasticity of the cell wall material yields the specific stress threshold:

where *σ* _{ys} and *ρ* represent the yield strength of the cell wall material and the relative density of the hexagonal honeycomb, respectively.

*E*

_{0}/

*E*

_{s}, can be calculated by evaluating the partial derivative of Eq. 1 with respect to normalized displacement and then substituting Eqs. 2 and 3 as follows:

The specific initial stiffness can be compared to the specific elastic stiffness of a regular honeycomb, although the specific initial stiffness of the negative stiffness honeycomb is valid only at the origin and decreases gradually as the stress threshold is approached, as shown in Figs. 3 and 5. In contrast, regular honeycombs typically exhibit nearly linear elastic stiffness prior to the critical buckling stress.

Plastic deformation of the cell wall material occurs when *ε* _{max} exceeds the yield strain or elongation at yield of the material. Thicker, shorter beams with larger apex heights undergo greater strains and risk exceeding the yield strength of the material. Maximum strain serves as a geometric constraint on the relative dimensions of the curved beams.

Equations 1 through 9 are valid for a single curved beam, but the unit cell in Fig. 2 incorporates two concentric curved beams. To represent double curved beams, the specific stress threshold in Eq. 6 and the specific stiffness in Eq. 9 must be doubled. The honeycomb in Fig. 2 is created by combining multiple rows and columns of concentric curved beams. The force threshold of the honeycomb is calculated by multiplying the force threshold of a single set of double beams by the number of columns of double beams, while the overall displacement is calculated by multiplying the displacement of a single set of double beams––equivalent to double the apex height, *h*––by the number of rows of double beams. As with a regular honeycomb, the specific stress threshold and specific stiffness of the honeycomb are equivalent to that of a single set of double beams.

*l*, of 50.8 mm; an apex height,

*h*, of 5.08 mm; thickness,

*t*, of 1.27 mm; and depth,

*b*, of 12.7 mm) was subjected to quasi-static displacement loading in ABAQUS. A finite element mesh of hexahedral elements was created by first creating a two-dimensional mesh of the front face of the honeycomb and then extruding the mesh through its depth,

*b.*The solver used for the simulation was ABAQUS/Explicit with nonlinear geometry enabled. FEA simulations were also performed on a structure consisting of a double beam, a structure with two rows of double beams, and a structure with two columns of double beams. In each simulation, the horizontal displacements of the ends of the structures were constrained to prevent lateral expansion of the structures. The predicted force thresholds obtained from these simulations are summarized in Table 1. The predicted force thresholds match the assumptions in this section precisely. Specifically, the force threshold of a double beam is exactly two times that of a single beam, and the force threshold is proportional to the number of columns of double beams. Additionally, the force threshold of the double beam, two-column structure agrees very closely with the experimentally observed force threshold of approximately 210 N in Fig. 5.

FEA prediction of force thresholds of various beam elements

Element description | Predicted force threshold ( |
---|---|

Single beam | 55 |

Double beam | 110 |

Double beam, two rows | 110 |

Double beam, two columns | 206 |

## Results and discussion

The analytical relationships in Eqs. 1 through 10 can be used to design negative stiffness honeycombs for applications that require combinations of lightweight stiffness and energy absorption. Regular honeycombs are typically designed by adjusting relative density, which defines the ratio of cell wall thickness to characteristic length, and thereby defines the mechanical properties of the honeycomb, including specific stiffness and strength. Negative stiffness honeycombs offer an additional degree of design freedom. In addition to specifying the thickness of the cell wall, *t*, and the characteristic length of the cell, *l*, designers may vary the apex height, *h*, of the curved beams, such that various ratios of cell dimensions provide equivalent relative densities. As a result, honeycombs of equivalent relative densities can be tailored geometrically to provide different combinations of stress threshold and energy absorption. The phenomenon is similar to the functional grading of regular honeycombs to achieve Pareto sets of trade-offs between thermal and structural performance [9, 10], for example, but the focus here is on structural energy absorption.

*d*, of each curved beam equals twice its apex height,

*h*, which is equivalent to a normalized displacement, ∆, of 2. The characteristic length of the unit cells used to generate the curves in Fig. 7 was fixed at 3 cm while the out-of-plane depth was fixed at 1 cm. Clearly, the characteristic length in a negative stiffness cell is a contributing factor to the cell behavior. However, in order to make an unambiguous comparison to regular honeycombs and ensure that they are being compared on the same platform, the length of the beams are fixed in this study. For the purposes of this analysis, the negative stiffness unit cell is defined as two concentric curved beams. The material is specified as a nylon 11 with modulus of elasticity and yield strength of 1400 and 47 MPa, respectively [11]. In Fig. 7, negative stiffness honeycomb contours are generated by maximizing the compaction energy for a fixed relative density and stress threshold, subject to a constraint on the maximum strain,

*ε*

_{max}, in Eq. 10. The maximum strain cannot exceed the elongation at yield of the cell wall material, which is assumed to be 20 % for nylon 11 [12]. The maximum strain constraint determines the boundary of the negative stiffness contours in Fig. 7.

For the range of relative densities plotted in Fig. 7, the compaction energy of regular hexagonal honeycombs increases with relative density. This trend is expected because the hexagonal honeycomb’s plateau stress increases quadratically with the relative density of the material, although this trend is counteracted somewhat by the decrease in distance between opposing cell walls in a denser hexagonal honeycomb. In contrast, for the range of relative densities plotted in Fig. 7, the compaction energy of negative stiffness honeycombs tends to increase with decreasing relative density, and the effect is more pronounced for negative stiffness honeycombs with higher stress thresholds. A particular stress threshold can typically be achieved by relatively thick curved beams with relatively short apex heights or by relatively thin curved beams with relatively tall apex heights. The latter geometry affords more travel and therefore greater compaction energy for beams of equivalent stress threshold with less relative density. These trends illustrate that the mechanism used for energy absorption in negative stiffness honeycombs—buckling in a snap-through-like fashion—leads to a much richer set of trade-offs than those of regular honeycombs. As shown in Fig. 7, for relatively large stress thresholds and low relative densities, the negative stiffness honeycomb provides greater compaction energy than the regular hexagonal honeycomb of equivalent relative density. This advantage diminishes as relative density increases because the hexagonal honeycomb’s increasing plastic buckling strength leads to increasing compaction energy while the negative stiffness honeycomb’s increasing density leads to diminishing apex heights, less travel and a shorter plateau region, and ultimately a lower compaction energy. It is important to note that the characteristic length of the unit cells used to generate the curves in Fig. 7 was fixed at 3 cm while the out-of-plane depth was fixed at 1 cm. Adjusting those values would affect the magnitude of the compaction energy, but the trends observed in Fig. 7 would not change.

Regardless of the relative levels of compaction energy exhibited in Fig. 7, it is important to note that the compaction energy absorbed by the regular hexagonal honeycombs is not recoverable by virtue of the underlying plastic deformation that leads to energy absorption. The negative stiffness honeycombs, in contrast, are designed to return to their initial configurations upon removal of external loading. Indeed, preliminary physical experimental results shown in Fig. 5 indicate that the negative stiffness honeycombs are fully recoverable. Net energy absorbed by a negative stiffness honeycomb depends on the area encompassed by the hysteresis curve in Fig. 5 with the extent of hysteresis influenced by the viscoelastic behavior of the cell wall material among other factors. The presence of hysteresis indicates that the analytical predictions in Fig. 7 most likely overestimate experimentally measured magnitudes of net energy absorbed by a negative stiffness honeycomb. Comparisons are documented in previous research by the authors [4, 5].

## Conclusions

Mechanical energy absorption properties of negative stiffness honeycomb materials have been examined. In contrast to regular honeycombs, which rely on plastic buckling for in-plane mechanical energy absorption, negative stiffness honeycombs rely on tailored elastic buckling phenomena. As a result, they are capable of absorbing large amounts of mechanical energy and returning to their original configuration. Due to their unique energy absorption mechanism, negative stiffness honeycombs offer a multi-dimensional design space for achieving the desired capacity for energy absorption. Specifically, two parameters of the cell geometry, apex height and in-plane thickness, can be altered to achieve the desired performance. This design freedom allows the force threshold to be designed independently of relative density, which is in direct contrast to traditional honeycombs. Negative stiffness honeycombs have been shown to have comparable levels of compaction energy per unit cell as hexagonal honeycombs but only for low relative densities that permit relatively large transverse displacements of curved beams with large apex heights.

This newly introduced honeycomb material offers many opportunities for future work. An initial experimental validation of the behavior of these materials has been conducted by the authors [4, 5], but additional experimentation should be carried out to verify the compaction energy per unit cell predicted in Fig. 7. Additional experimental efforts need to be focused on comparing the predictions in this paper with experimental performance for a wide range of strain rates. Finally, this experimental and theoretical knowledge can be merged to design energy-absorbing materials that achieve new combinations of performance. By adjusting the geometry of the unit cell and the population of unit cells in a negative stiffness honeycomb, for example, it is possible to independently tailor the density, force threshold, and energy absorption capabilities. Potential applications include bumpers, orthotics, and personal protective devices such as helmets.

## Notes

### Acknowledgements

We gratefully acknowledge Professor Desiderio Kovar and Mr. Sergio Cortes for their help in generating the experimental data in Fig. 5. Tim Klatt was instrumental in generating the negative stiffness honeycomb configuration illustrated in Fig. 2 and conducting preliminary proof-of-concept studies to refine the design. We gratefully acknowledge the funding from the Department of Defense Small Business Innovation Research (SBIR) Program under SBIR Topic N142-085 in collaboration with the Maritime Applied Physics Corporation (MAPC).

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