Mechanical design of negative stiffness honeycomb materials
 Dixon M Correa^{1},
 Carolyn Conner Seepersad^{1}Email author and
 Michael R Haberman^{1, 2}
DOI: 10.1186/s4019201500388
© Correa et al. 2015
Received: 30 December 2014
Accepted: 21 April 2015
Published: 2 July 2015
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 vibroacoustic damping materials, vibration isolation mechanisms, and mechanical switches. This nonintuitive 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 energyabsorbing 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 tradeoffs between density, stress thresholds, and energy absorption capabilities. This paper describes these tradeoffs and the underlying analysis.
Keywords
Honeycombs Negative stiffness Bistability Energy absorption Elastic stiffness Stress thresholdBackground
Honeycombs are ordered cellular materials with prismatic cells. The cells of the honeycomb can assume a variety of crosssectional shapes, including hexagonal, kagome, square, triangular, and mixed triangular and square [1, 2]. Relative to other lowdensity 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.
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 firstmodebuckled 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 inplane compression, thereby enabling snapthroughlike 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 inplane 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 tradeoffs that can be achieved.
Methods
Equation 1 predicts the resulting forcedisplacement behavior illustrated in Fig. 3 and can be used for any curved beam geometry as long as the beam is constrained to avoid secondmode buckling when it transitions from one firstmodebuckled shape to another.
The force threshold is defined by the peak of the forcedisplacement 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.
where σ _{ys} and ρ represent the yield strength of the cell wall material and the relative density of the hexagonal honeycomb, respectively.
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.
FEA prediction of force thresholds of various beam elements
Element description  Predicted force threshold (N) 

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 tradeoffs between thermal and structural performance [9, 10], for example, but the focus here is on structural energy absorption.
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 snapthroughlike fashion—leads to a much richer set of tradeoffs 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 outofplane 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 inplane 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 multidimensional design space for achieving the desired capacity for energy absorption. Specifically, two parameters of the cell geometry, apex height and inplane 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 energyabsorbing 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.
Abbreviations
 Δ_{th} :

Normalized displacement threshold
 b :

Outofplane 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 :

Inplane thickness for a negative stiffness beam
 w(x):

Beamshape coordinate along the vertical axis
 x :

Beamshape 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
Declarations
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 proofofconcept studies to refine the design. We gratefully acknowledge the funding from the Department of Defense Small Business Innovation Research (SBIR) Program under SBIR Topic N142085 in collaboration with the Maritime Applied Physics Corporation (MAPC).
Authors’ Affiliations
References
 Gibson L, Ashby M (1999) Cellular solids: structure and properties. Cambridge University Press, Cambridge, UKGoogle Scholar
 Hayes A, Wang A, Dempsey B, McDowell D (2004) Mechanics of linear cellular alloys. Mech Mater 36:691–713View ArticleGoogle Scholar
 Schaedler T, Jacobsen A, Torrents A, Sorensen A, Lian J, Greer J, Valdevit L, Carter W (2011) Ultralight metallic microlattices. Science 334(6058):962–965View ArticleGoogle Scholar
 Correa D, Klatt T, Cortes S, Haberman M, Kovar D, Seepersad C (2014) Negative stiffness honeycombs for recoverable shock isolation. In: Proceedings of the solid freeform fabrication symposium. The University of Texas at Austin, Austin, TXGoogle Scholar
 Correa D, Klatt T, Cortes S, Haberman M, Kovar D, Seepersad C (2015) Negative stiffness honeycombs for recoverable shock isolation. Rapid Prototyp J 21(2):193–200View ArticleGoogle Scholar
 Klatt T, Haberman M, Seepersad C (2013) Selective laser sintering of negative stiffness mesostructures for recoverable, nearlyideal shock isolation. In: Proceedings of the solid freeform fabrication symposium. The University of Texas at Austin, Austin, TXGoogle Scholar
 Fulcher B, Shahan D, Haberman M, Seepersad C, Wilson P (2014) Analytical and experimental investigation of buckled beams as negative stiffness elements for passive vibration and shock isolation. J Vib Acoust 136(3):1–12View ArticleGoogle Scholar
 Qiu J, Lang J, Slocum A (2004) A curvedbeam bistable mechanism. J Microelectromech Syst 13(2):137–146View ArticleGoogle Scholar
 Seepersad C, Dempsey B, Allen J, Mistree F, McDowell D (2004) Design of multifunctional honeycomb materials. AIAA J 42(5):1025–1033View ArticleGoogle Scholar
 Seepersad C, Kumar R, Allen J, Mistree F, McDowell D (2004) Multifunctional design of prismatic cellular materials. J ComputerAided Mater Des 11(2–3):163–181View ArticleGoogle Scholar
 Matbase. Available: www.matbase.com. Accessed 17 Dec 2014
 Leigh D (2012) A comparison of polyamide 11 mechanical properties between laser sintering and traditional molding. In: Proceedings of the solid freeform fabrication symposium. The University of Texas at Austin, Austin, TXGoogle Scholar
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