Compliant composite unit cells were made with negative stiffness constituents. Flexible silicone rubber tubes were incorporated in a post-buckled condition to achieve tunable negative stiffness. Large peaks in mechanical damping tan delta were observed in these systems. Maximum damping was orders of magnitude in excess of the material damping of the silicone rubber. Download pdf from Phil. Mag. Letters, or from here.

Negative Phase delta. Theory diagram: Reuss composite unit cell with negative stiffness phase.

Lakes, R. S.,

Composites with negative stiffness inclusions in a viscoelastic matrix are shown (theoretically) to have higher stiffness and mechanical damping tan delta than that of either constituent and exceeding conventional Hashin-Shtrikman bounds. The causal mechanism is a greater deformation in and near the inclusions than the composite as a whole. Though a block of negative stiffness (negative modulus) is unstable, negative stiffness inclusions in a composite can be stabilized by the surrounding matrix. Such inclusions may be made from single domains of ferroelastic material below its phase transition temperature or from pre-buckled lumped elements.

Download pdf from Physical Review Letters, or from here.

Lakes, R. S., Lee, T., Bersie, A., and Wang, Y. C.,

The force applied to deform an elastic object is in the same direction as the displacement: positive stiffness. The force direction is opposite for negative stiffness, possible if there is stored energy or a power source. Negative stiffness is not illegal; usually only unstable. Negative stiffness (negative modulus) differs from negative Poisson's ratio in which lateral expansion occurs upon stretching. Inclusions of negative stiffness can be stabilized in a composite by the surrounding matrix.

Composites with inclusions of negative stiffness may be called exterlibral since they are on the boundary of balance, or archidynamic since they are based on initial force. They are pertinent to any heterogeneous material in which one constituent undergoes a phase transformation and another does not; also to some materials with a pre-strained constituent. Such composites may be of use as high-performance damping materials, since the figure of merit in the present results exceeds that of commonly used materials (E tan delta less than 0.6 GPa) by a factor of more than twenty. For that purpose, sensitivity to temperature could be reduced via matching inclusion and matrix stiffness.

Bounds on properties of complex heterogeneous materials are generally derived assuming positive phase properties. These bounds can be exceeded if negative stiffness is allowed, permitting extreme properties not previously anticipated. Since in thermoelastic and piezoelectric materials, elasticity is coupled with temperature and electric field respectively, these composites may find use in high performance sensors and actuators.

Experiment diagram. Composite with negative stiffness inclusions. Experimental torsional compliance and mechanical damping tan delta vs. temperature. Open triangles, a composite containing 1 % by volume vanadium dioxide particles in a tin matrix. Points, pure tin for which the damping tan delta = 0.019 over the temperature range considered. Measurements were conducted at 100 Hz, well below resonance, during slow cooling through the ferroelastic phase transition of the inclusions.

Download pdf from Nature or from here.

Wang, Y. C. and Lakes, R. S.,

Particulate composites with negative stiffness inclusions in a viscoelastic matrix are shown to have higher thermal expansion than that of either constituent and exceeding conventional bounds. It is also shown theoretically that other extreme linear coupled field properties including piezoelectricity and pyroelectricity occur in layer- and fiber-type piezoelectric composites, due to negative inclusion stiffness effects. The causal mechanism is a greater deformation in and near the inclusions than the composite as a whole. A block of negative stiffness material is unstable, but negative stiffness inclusions in a composite can be stabilized by the surrounding matrix and can give rise to extreme viscoelastic effects in lumped and distributed composites. In contrast to prior proposed composites with unbounded thermal expansion, neither the assumptions of void spaces nor slip interfaces are required in the present analysis.

link, pdf.

Rosakis, P., Ruina, A., and Lakes, R. S.,

Compressive properties of elastic cellular solids are studied via experiments upon foam and upon single cell models. Open cell foam exhibits a monotonic stress-strain relation with a plateau region; deformation is localized in transverse bands. Single cell models exhibit a force-deformation relation which is not monotonic. In view of recent concepts of the continuum theory of elasticity, the banding instability of the foam in compression is considered to be a consequence of the non-monotonic relation between force and deformation of the single cell.

The negative stiffness represents an instability that can be viewed as a phase transition. Such instability occurs in foams as well as in fiber networks.

Lakes, R. S. and Drugan, W. J.,

Composite materials of extremely high stiffness can be produced by employing one phase of negative stiffness. Negative stiffness entails a reversal of the usual codirectional relationship between force and displacement in deformed objects. Negative stiffness structures and materials are possible, but unstable by themselves. We argue here that composites made with a small volume fraction of negative-stiffness inclusions can be stable and can have overall stiffness far higher than that of either constituent. This high composite stiffness is demonstrated via several exact solutions within linearized and also fully nonlinear elasticity, and via the overall modulus tensor estimate of a variational principle valid in this case. We provide an initial discussion of stability, and adduce experimental results which show extreme composite behavior in selected viscoelastic systems under subresonant sinusoidal load. Viscoelasticity is known to expand the space of stability in some cases. We have not yet proved that purely elastic composite materials of the types proposed and analyzed in this paper will be stable under static load. The concept of negative stiffness inclusions is buttressed by recent experimental studies illustrating related phenomena within the elasticity and viscoelasticity contexts. Download pdf.

Wang, Y. C. and Lakes, R. S.,

When an elastic object is pressed, we expect it to resist by exerting a restoring force. A reversal of this force corresponds to negative stiffness. If we combine elements with positive and negative stiffness in a composite, it is possible to achieve stiffness greater than (or less than) that of any of the constituents. This behavior violates established bounds that tacitly assume that each phase has positive stiffness. Extreme composite behavior has been experimentally demonstrated in a lumped system using a buckled tube to achieve negative stiffness and in a composite material in the vicinity of a phase transformation of one of the constituents. In the context of a composite system, extreme refers to a physical property greater than either constituent. We consider a simple spring model with pre-load to achieve negative stiffness. When suitably tuned to balance positive and negative stiffness, the system shows a critical equilibrium point giving rise to extreme overall stiffness. A stability analysis of a viscous damped system containing negative stiffness springs reveals that the system is stable when tuned for high compliance, but metastable when tuned for high stiffness. The metastability of the extreme system is analogous to that of diamond. The frequency response of the viscous damped system shows that the overall stiffness increases with frequency and goes to infinity when one constituent has a suitable negative stiffness. Download pdf

Wang, Y. C., Ludwigson, M., and Lakes, R. S.,

The figure of merit for structural damping and damping layer applications is the product of stiffness E and damping tan δ. For most materials, even practical polymer damping layers, E tan δ is less than 0.6 GPa. We consider several methods to achieve high values of this figure of merit: high damping metals, metal matrix composites and composites containing constituents of negative stiffness. Download pdf

Wang, Y. C. and Lakes, R. S.,

Systems with negative stiffness constituents can have extreme material properties greatly exceeding those of either constituent. We show that a discrete system with a viscoelastic damping element and a negative stiffness element can be made with overall viscoelastic damping orders of magnitude higher than that of any constituent, or of the system with all elements of positive stiffness. The product of stiffness and damping, important for vibration damping, is also enhanced by orders of magnitude. We show this system is unconditionally stable in the high damping regime. The singularity in damping can be made arbitrarily close to the stability boundary. Download pdf from aip or here

Wang, Y. C. and Lakes, R. S.,

We analytically investigate the stability of a discrete viscoelastic system with negative stiffness elements both in the time and frequency domain. Parametric analysis was performed by tuning both the amount of negative stiffness in a standard linear solid, and driving frequency. Stability conditions were derived from the analytical solutions of the differential governing equations and the Lyapunov stability theorem. High frequency response of the system is studied. Stability of singularities in the dissipation tan delta is discussed. It was found that stable singular tan delta is achievable. The system with extreme high stiffness analyzed here was metastable. We established an explicit link for the divergent rates of the metastable system between the solutions of differential governing equations in the time domain and the Lyapunov theorem. get pdf

Wang, Y. C. and Lakes, R. S.,

Use of negative stiffness inclusions allows one to exceed the classic bounds upon overall mechanical properties of composite materials. We here analyse discrete viscoelastic 'spring' systems with negative stiffness elements to demonstrate the origin of extreme properties, and analyse the stability and dynamics of the systems. Two different models are analysed: one requires geometrical nonlinear analysis with pre-load as a negative stiffness source and the other is a linearised model with a direct application of negative stiffness. Material linearity is assumed for both models. The metastability is controlled by a viscous element. In the stable regime, extreme high mechanical damping tan delta can be obtained at low frequency. In the metastable regime, singular resonance-like responses occur in tan delta. The pre-stressed viscoelastic system is stable at the equilibrium point with maximal overall compliance and is metastable when tuned for maximal overall stiffness. A reversal in the relationship between the magnitude of complex modulus and frequency is also observed. The experimental observability of the singularities in tan delta is discussed in the context of designed composites and polycrystalline solids with metastable grain boundaries. Download get pdf

Jaglinski, T. and Lakes, R. S.,

Wang, Y. C. and Lakes, R. S.,

Jaglinski, T. Stone, D., and Lakes, R. S.,

Wang, Y. C. and Lakes, R. S.,

Jaglinski, T., Frascone , P., Moore, B., Stone, D., and Lakes, R. S.,

Moore, B., Jaglinski, T., Stone, D. S., and Lakes, R. S.,

Jaglinski, T. M., Lakes, R. S.,

Wang, Y. C., Swadener, J. G. and Lakes, R. S.,

Shang, X. and Lakes, R. S.,

go-link. We show that composite materials can exhibit a viscoelastic (Young's) modulus far higher than that of either constituent. The modulus (but not strength) was observed to be substantially greater than that of diamond. These composites contain barium titanate inclusions, which undergo a volume change phase transformation if not constrained. In the composite the inclusions are partially constrained by the surrounding metal matrix. The constraint stabilizes the negative bulk modulus (inverse compressibility) of inclusions. The composites are stiffer than diamond over a narrow temperature range.

Carpick, R., Yap, H, Lakes, R. S.,

Yap, H, Lakes, R. S., Carpick, R.,

Lakes, R., Wojciechowski, K. W.,

Dong, L., Stone, D. S., and Lakes, R. S.,

Substantial softening in the bulk modulus (a factor of five) and a negative Poisson ratio (-0.25) have been observed via broadband viscoelastic spectroscopy in the vicinity of the Curie point of a barium titanate ceramic. These effects were observed under electrical short-circuit conditions at low deformation frequencies. Softening was less in an electric open circuit or at higher frequencies. Softening of individual elastic modulus tensor elements is known to occur near phase transformations, but softening of the bulk modulus has not previously been well reported.

See also piezoelectric materials

Mechanical anomalies damping peaks sharper than Debye peaks, in contrast to a broad relaxation peak were observed in tetragonal barium titanate ceramic via broadband viscoelastic spectroscopy at low frequencies 10 Hz at ambient temperature after aging at 90 degrees C for 15 h. The sharp peaks disappear after aging above the Curie point 150 degrees C for 10 h. Mechanical anomalies are tentatively attributed to negative stiffness heterogeneity.

See also piezoelectric materials

We developed high performance structural dampers based on negative stiffness.

DOI: 10.1177/1045389X15624802 journal link get preprint pdf

doi journal link

We consider stability in Cosserat solids. To obtain restrictions on elastic constants based on positive definite strain energy, energy terms are tacitly assumed to be independent. In finite-size objects, however, the terms are linked in Cosserat materials. Therefore, in contrast to classical solids, the stability of Cosserat solids appears to depend on the size and shape of the specimen, provided strong ellipticity is satisfied. Stability in the presence of stored energy is possible. Solids with microstructure and stored energy offer the potential to facilitate attainment of extreme behavior in the presence of spatial gradients. Snap- through buckling in torsion is envisaged by analogy to the axial buckling concept used for composites with negative stiffness inclusions. It is possible to support compressive load in a stable manner but to dissipate energy in the presence of spatial gradients as in torsion or bending.

journal link

preprint pdf DOI: 10.2140/jomms.2018.13.83

Further lattices exhibiting negative stiffness snap through are in progress.

The negative stiffness dampers based on our original negative stiffness concepts at top (2001) have been built upon, scaled up and used for applications. See this link.

Related papers provide analyses of stability.

W. J. Drugan, "Elastic composite materials having a negative stiffness can be stable"

W. J. Drugan, Wave propagation effects possible in solid composite materials by use of stabilized negative-stiffness components, Journal of the Mechanics and Physics of Solids, Volume 136, March (2020), 103700

doi link

The following article discloses regions of negative stiffness in polymers via molecular modeling.

Yoshimoto, K., Jain, T. S., Van Workum, K., Nealey, P. F., de Pablo, J. J., "Mechanical heterogeneities in model polymer glasses at small length scales", Physical Review Letters 93(17):175501, October 18, (2004).

Related research.

Drugan, W. J.

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase and also many types of multi-phase random linear elastic composite materials. Download pdf.

See also Prof. Walt Drugan's research page.

Snap through. Illustration of instability.

Ring deformation.

Illustration of Reuss model stabilized by displacement control by the hand of the demonstrator. The ring has positive stiffness and the pre-buckled column has negative stiffness. Observe how the ring deforms. First it deforms in compression as it is pressed indicating positive stiffness of the column. As the column is deformed into an S shape, the ring deforms in tension as it is pressed, showing the reversal of force associated with the negative structural stiffness of the column, so negative stiffness in the transverse direction.

See also other publicity.

We thank the National Science Foundation, DARPA, and ARO for sponsorship of some this research.

If you are interested in commercial applications, please contact the Wisconsin Alumni Research Foundation (WARF) for information on US and international patents and licensing.

For current research reports, log in.

This site is copyright, 2002 - 2021, Rod Lakes. All rights reserved.

**A related topic in reversal physics, negative refraction**, References

This was originally suggested by V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of e and m", Sov. Phys. Uspekhi 10, 509-514 (1968).

Nicorovici, N. A. and Milton, G. W., "On the cloaking effects associated with anomalous localized resonance", *Proceedings of the Royal Society of London, A*, 462, 3027 - 3059 (2006).
journal link

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd "Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity" *Science* 12 May 2006: 895-897.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," *Nature* 438, 335-338, 17 Nov. (2005).

N. Fang, H. Lee, C. Sun, X. Zhang, "Sub-diffraction limited optical imaging with a silver superlens" *Science* 308: 534-537, (2005).

J. B. Pendry, "A Chiral Route to Negative Refraction" *Science*, 306: 1353-1355, (2004).

A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental Observations of a Left-Handed Material That Obeys Snell's Law", *Phys. Rev. Lett. * 90, 137401 (2003).

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction" *Science* 292, 77 (2001).

J. B. Pendry, "Negative Refraction Makes a Perfect Lens" Phys. Rev. Lett. 85, 3966 (2000).

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap" *Phys. Rev. B* 62, 10 696 (2000).

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, "Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging" *Science* 291, 849 (2001).

R. J. Blaikie and S. J. McNab, Microelectron. Eng. 61-2, 97 (2002).

V. G. Veselago, *Sov. Phys. Usp.* 10, 509 (1968).

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, "Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial" Appl. Phys. Lett. 78, 489 (2000).

P. M. Valanju, R. M. Walser, and A. P. Valanju, "Wave Refraction in Negative-Index Media: Always Positive and Very Inhomogeneous" Phys. Rev. Lett. 88, 187401 (2002).

N. Garcia and M. Nieto-Verperinas, Opt. Lett. 27, 885 (2002).

N. Garcia and M. Nieto-Verperinas, "Left-Handed Materials Do Not Make a Perfect Lens" Phys. Rev. Lett. 88, 207403 (2002).

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195103 (2002).

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures" Phys. Rev. Lett. 76, 4773 (1996).

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite Medium with Simultaneously Negative Permeability and Permittivity" Phys. Rev. Lett. 84, 4184 (2000).

Nicorovici, N. A. Mcphedran, R. C., Milton, G. W., "Optical and dielectric properties of partially resonant systems." Phys. Rev. B 49: 8479-8482. (1994).

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability" Phys. Rev. E 64, 056625 (2001).

Negative capacitance, as with negative stiffness, is unstable in isolation. Negative capacitance has been studied in the context of phase transitions in ferroelectrics and with reduction of power requirements in microelectronics.