Extreme composite materials with negative stiffness inclusions; negative modulus

Extreme composite materials with negative stiffness inclusions
Rod Lakes, University of Wisconsin

[negative Poisson's ratio main] [Viscoelasticity main]

Articles

Lakes, R. S., "Extreme damping in compliant composites with a negative stiffness phase"
Philosophical Magazine Letters, 81, 95-100 (2001).
Compliant composite unit cells were made with negative stiffness constituents. Flexible silicone rubber tubes were incorporated in a post-buckled condition to achieve 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. lumped negative stiffness

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., "Extreme damping in composite materials with a negative stiffness phase",
Physical Review Letters 86, 2897-2900, 26 March (2001).
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 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., "Extreme damping in composite materials with negative stiffness inclusions",
Nature, 410, 565-567, 29 March (2001).
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. Experimentally, composites were prepared with a dilute concentration of ferroelastic inclusions in a positive stiffness matrix to stabilize them. In a ferroelastic the free energy in the Landau theory has a relative maximum, corresponding to unstable equilibrium, below its transformation temperature T_c . The composites exhibited large peaks in mechanical damping (tan delta), and large anomalies in stiffness, in harmony with composite theory predictions. The inclusions are more effective than diamond in increasing the composite stiffness at selected temperatures. The underlying physical process is interplay between the positive and negative stiffness phases so that the inclusions deform more than the composite as a whole, giving rise to high local strains. Such composites may be useful as high performance damping materials, as stiff structural elements, or in actuators.
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 transition of the inclusions.
Download pdf from Nature or from here.

Wang, Y. C. and Lakes, R. S., "Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase", Journal of Applied Physics, 90, 6458-6465, Dec. (2001).
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., "Microbuckling instability in elastomeric cellular solids",
J. Materials Science, 28, 4667-4672 (1993).
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 non-monotone relation entails negative stiffness of the cell over a range of strain. Download pdf

Lakes, R. S. and Drugan, W. J., "Dramatically stiffer elastic composite materials due to a negative stiffness phase?",
J. Mechanics and Physics of Solids, 50, 979-1009 (2002).
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.

Drugan, W. J. "Two Exact Micromechanics-Based Nonlocal Constitutive Equations for Random Linear Elastic Composite Materials," Journal of the Mechanics and Physics of Solids, 51, 1745-1772, (2003).
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.

Wang, Y. C. and Lakes, R. S., "Extreme stiffness systems due to negative stiffness elements", American J. of Physics, 72, Jan. (2004).
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., "Deformation of extreme viscoelastic metals and composites", Materials Science and Engineering A, 370, 41-49, April (2004).
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., "Stable extremely-high-damping discrete viscoelastic systems due to negative stiffness elements", Applied Physics Letters, 84, 4451-4453 (2004).
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., "Stability of negative stiffness viscoelastic systems" , Quarterly of Applied Math., 63, 34-55, March (2005).
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., "Negative stiffness induced extreme viscoelastic mechanical properties: stability and dynamics", Philosophical Magazine, 35, 3785-3801, Dec. (2004).
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., "Anelastic instability in composites with negative stiffness inclusions", Philosophical Magazine Letters, 84, (12) 803 - 810, Dec. (2004). Download pdf

Wang, Y. C. and Lakes, R. S., "Composites with inclusions of negative bulk modulus: extreme damping and negative Poisson's ratio", J. Composite Materials, 39, 1645-1657, (2005). The bulk modulus is the inverse of the compressibility. Negative compressibility has different stability criteria than negative shear modulus considered earlier. get pdf

bulk-shear map

Jaglinski, T. Stone, D., and Lakes, R. S., "Internal friction study of a composite with a negative stiffness constituent", J. Mater. Research 20 (9), 2523-2533, Sept. (2005). pdf

Wang, Y. C. and Lakes, R. S., "Two-dimensional viscoelastic discrete triangular system with negative-stiffness components", Philosophical Magazine Letters, 86, 99-112, (2006). pdf.

Jaglinski, T., Frascone , P., Moore, B., Stone, D., and Lakes, R. S., "Internal friction due to negative stiffness in the indium-thallium martensitic phase transformation", Philosophical Magazine, 86, (27 / 21) 4285 - 4303, September (2006). Internal friction and dynamic shear modulus in an indium-21 atomic percent thallium alloy were measured as functions of frequency and cooling rate using broadband viscoelastic spectroscopy during the martensitic transformation which in this material occurs around 50oC. Microstructural evolution of martensitic bands was captured using time-lapse optical microscopy. The amplitude of damping peaks due to the temperature induced transformation in the polycrystalline alloy were found to exceed those reported by others for single crystals of similar alloy compositions in contrast to the usual reduction in damping in polycrystals. The high temperature portion of the damping peak occurs before martensitic bands are observed, therefore this portion cannot be due to interfacial motion. Constrained negative stiffness of the grains can account for this damping, as well as for amplification of internal friction peaks in these polycrystals and for sigmoid shaped anomalies in the shear modulus at high cooling rates. Surface features associated with a previously unreported pre-martensitic phenomenon are seen at temperatures above martensite-start. pdf

Moore, B., Jaglinski, T., Stone, D. S., and Lakes, R. S., "Negative incremental bulk modulus in foams", Philosophical Magazine Letters, 86, 651-659, (2006). Negative compressibility is observed in foams under volumetric constraint. Pre-strain is applied. Negative incremental stiffness is known to occur in structures such as post-buckled flexible tubes and single cell models. A single foam cell under uniaxial loading buckles and exhibits a non-monotonic S-shaped deformation curve, which is indicative of negative incremental stiffness. Negative stiffness is not observed in materials, due to instability. For example, individual foam cells display negative stiffness but foams tested in uniaxial compression exhibit a plateau in the stress-strain curve because the buckled cells localise in bands. This behaviour is consistent with the continuum view in which strong ellipticity, hence positive shear modulus G and positive C11 modulus, are required for stability, even for a constrained object. It is hypothesised that a solid with negative bulk modulus can be stabilised by control of surface displacement. Experimentally, foams were hydrostatically compressed under control of volumetric deformation. Negative incremental bulk modulus was observed in foam with 0.4 mm cell size beyond about 20 percent volumetric strain. Foam with large cells 2.5 mm to 4 mm was anisotropic and did not exhibit the cell buckling required for negative modulus. get pdf.

Wang, Y. C., Swadener, J. G. and Lakes, R. S., "Anomalies in stiffness and damping of a 2D discrete viscoelastic system due to negative stiffness components", Thin Solid Films, 515, 3171-3178, (2007). get pdf

cover page

Shang, X. and Lakes, R. S., "Stability of elastic material with negative stiffness and negative Poisson's ratio", Physica Status Solidi (b), 244, 1008-1026 (2007). Cover article. get pdf.

Jaglinski, T., Kochmann, D., Stone, D., Lakes, R. S., Materials with viscoelastic stiffness greater than diamond, Science 315, 620-622, Feb. 2 (2007)
go-link

Carpick, R., Yap, H, Lakes, R. S., "Mechanical instabilities of individual multi-walled carbon nanotubes under cyclic axial compression", Nano Letters, 7(5); 1149-1154 (2007). get pdf

Yap, H, Lakes, R. S., Carpick, R., "Negative stiffness and enhanced damping of individual multiwalled nanotubes", Phys. Rev. B, 77, 045423 (2008). get pdf

Lakes, R., Wojciechowski, K. W., Negative compressibility, negative Poisson's ratio, and stability, Physica Status Solidi, 245, No. 3, 545Ð551, Feb 4 (2008). get pdf

A related paper provides analysis of stability.
W. Drugan, "Elastic composite materials having a negative stiffness can be stable" Physical Review Letters 98, 055502, (2007) get pdf

Related research: see Walt Drugan's research page, and Rob Carpick's research page

Demonstrations

Snap through. Illustration of instability.

Ring deformation.
Illustration of Reuss model stabilized by displacement control by the demonstrator. The ring has positive stiffness and the pre-buckled column has negative stiffness. Observe the compressive, then the tensile deformation of the ring, illustrating the reversal of force.

Blurbs
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Science News, 159, p. 285, May 5 2001. Blurb

We thank the National Science Foundation for sponsorship of this research.

If you are interested in commercial applications, please contact the Wisconsin Alumni Research Foundation (WARF), Patent Agent Marnie Matt, mmatt@warf.ws, tel. (608)-262-7824, FAX (608)-263-1064 for information on US and international patents and licensing.

This site is copyright, 2002 - 2008, Rod Lakes. All rights reserved.

A related topic in reversal physics, negative refraction, Recent references
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