Viscoelasticity Books '98
Composites with negative stiffness inclusions
Negative Poisson's ratio
Internal friction in solids. Viscoelasticity. Anelastic solids.
Viscoelastic materials are those for which the relationship between stress and strain depends on time or, in the frequency domain, on frequency. The slope of a plot of stress vs. strain depends on strain rate. Anelastic solids represent a subset of viscoelastic materials: they have a unique equilibrium configuration and ultimately recover fully after removal of a transient load. After being squeezed, they return to their original shape, given enough time.
Our laboratory is capable of viscoelastic measurements of materials which exhibit anelastic response or internal friction, under isothermal conditions, over as much as eleven decades (a factor of 10 11) of time and frequency, with no appeal to time-temperature superposition. Experimental studies of anelastic, hysteresis, and viscoelastic creep behavior have been conducted with this viscoelastic instrument (broadband viscoelastic spectroscopy or BVS) on solder, alloys, biological materials, biomaterials, high-loss elastomers, cellular solids such as foams, and composites. Many reported experiments are at ambient temperature and cover a wide range of frequency; we can also study materials at temperatures above or below ambient. We also use resonant ultrasound spectroscopy (RUS) in viscoelastic measurements.
To illustrate effects of internal friction, anelastic behavior or material damping, listen to the sound of tuning forks with different damping, in this
If you are interested in commercial applications, please contact the Wisconsin Alumni Research Foundation (WARF) for information on US and international patents and licensing.
Viscoelastic solids, CRC Press, Boca Raton, FL, (1998).
Viscoelastic Materials, Cambridge University Press, (2009).
Rod Lakes and John Quackenbush, "Viscoelastic behaviour in indium tin alloys over a wide range of frequency and time", Philosophical Magazine Letters, 74, 227-232 (1996).
Experimental studies of dynamic and transient viscoelastic response were conducted at 24 deg. C on gamma InSn alloy, and high frequency studies were conducted to extend the frequency range for eutectic InSn. The experiments were conducted in torsion using a viscoelastic instrument capable of determining viscoelastic properties over more than ten decades of time and frequency. The viscoelastic damping, tan delta, depended on frequency to the n'th power, with n ~ -0.2, over many decades of frequency. The high temperature background is observed in these alloys at room temperature. This frequency dependence of internal friction corresponds to a stretched exponential relaxation function, and is attributed to a dislocation-point defect mechanism. It is not consistent with a self-organized criticality dislocation model which predicts n = -2, Dislocation damping in metals is relevant to development of high damping metals, the behaviour of solders and of support wires in Cavendish balances. This plot shows viscoelastic behavior of an indium-tin alloy over more than eleven decades of time and frequency, at 23 degrees C, with no appeal to time-temperature superposition. Also shown is a Debye peak in tan delta, a measure of damping or internal friction. In the frequency domain this corresponds to a single exponential viscoelastic creep or viscoelastic relaxation response in the time domain. Get article pdf
Lakes, R. S., "Thermoelastic damping in materials with a complex coefficient of thermal expansion. ", J. Mechanical Behav. Mts., 8, 201-216, (1997).
Amada, S. and Lakes, R. S., "Viscoelastic properties of bamboo", Journal of Materials Science, 32, 2693-2697, (1997). Get article pdf Viscoelastic properties of bamboo were measured over a range of frequency.
The effect of a complex coefficient of thermal expansion upon the thermoelastic relaxation mechanism is analyzed. A phase angle in the thermal expansion has the effect of generating a very broad band of mechanical damping, in addition to the peak usually observed. Phase angles in the thermal expansion have been observed in several polymers, and they may be generated in composite materials in which one or more phases is viscoelastic. However the resulting enhancement of the overall damping of the composite is modest, unless new materials can be found with very high relaxation strength or with a large intrinsic phase angle in the expansion.
Gibiansky, L. V. and Lakes, R. S., "Bounds on the complex bulk and shear moduli of a two-dimensional two-phase viscoelastic composite", Mechanics of Materials, 25, 79-95, (1997).
The effective complex moduli of an isotropic two-phase, two-dimensional viscoelastic composite material are analyzed in terms of the complex moduli of its phases. The frequency range is assumed to be well below frequencies associated with the inertial terms; the acoustic wavelength is much larger than the inhomogeneities. Bounds are developed for the complex bulk modulus K* = K' + iK'' and complex shear modulus G* = G' + iG'' of the composite with arbitrary phase volume fractions. Shear modulus bounds are obtained subject to one scalar restriction on the phase viscoelastic properties which is valid, in particular, for the phases with real and equal Poisson's ratios. Each of the moduli is shown to be constrained to a lens-shaped region bounded by two circular arcs in the complex bulk or shear modulus planes. The bounds are investigated numerically to explore conditions which give rise to high loss combined with high stiffness. Composite microstructures corresponding to various points on the circular arcs are identified. Influence of anisotropy of the composite on the stiffness-loss map for the bulk and shear type loads are analyzed.
Boyer, D. B., Papadogiannis, Y., Park, H. C., and Lakes, R. S., "Viscoelasticity and aging of dental composites.", Bio-Medical Materials and Engineering, 8, 45-53, (1998).
Effects of aging on several dental composites were studied using a torsion creep apparatus. A constant torque was applied from 1 sec to 3 h, and recovery was observed from 10 sec to two days, for specimens aged from 3 hours to 8 weeks following polymerization. Specimens aged for shorter times exhibited more creep and less complete recovery than specimens aged for longer times. Modulus and viscoelastic creep results were qualitatively consistent with a composite model for particulate inclusions.
Buechner, P.M., Stone, D. and Lakes, R. S., "Viscoelastic behavior of superplastic 37 wt% Pb 63 wt% Sn over a wide range of frequency and time", Scripta Materialia, 41, 561-567, (1999).
Damping of superplastic eutectic Pb-Sn is considerably higher than that of cast Pb-Sn, solder wire, and eutectic Pb-Sn reported by Woirgard et al. Viscoelastic damping tan delta approximately follows a power law dependence over about four decades of frequency. The high temperature background is observed over a broad band of frequency at room temperature in this alloy. There was no appeal to time-temperature superposition. The damping in torsion is higher than in bending; the difference is more marked at high frequency.
The specimen is contained within the brass enclosure to the left in this
Dooris, A., Lakes, R. S., Myers, B. and Stephens, N., "High damping indium-tin alloys", Mechanics of Time Dependent Materials, 3, 305-318 (1999).
Lee, T., Lakes, R. S., Lal, A. , "Resonant ultrasound spectroscopy for measurement of mechanical damping: comparison with broadband viscoelastic spectroscopy", Rev. Sci. Instr. 71 (7) 2855-2861, July (2000). This article compares resonant ultrasound spectroscopy (RUS) and other resonant methods for the determination of viscoelastic properties such as damping. RUS scans from 50 kHz to 500 kHz were conducted on cubical specimens of several materials including brass, aluminum alloys and PMMA, a glassy viscoelastic polymer. Comparison of damping over the frequency ranges for broadband viscoelastic spectroscopy (BVS) and RUS for indium tin alloy in shear modes of deformation discloses a continuation of the tan delta power-law trend for ultrasonic frequencies up to 300 kHz. For PMMA, resonant peaks were sufficiently broad that higher modes in RUS began to overlap. Viscoelastic damping tan delta via RUS and BVS for PMMA agreed well in the frequency range where the methods overlap. RUS is capable of measuring tan delta as high as several percent at the fundamental frequency. Since higher modes are closely spaced, it is impractical to determine tan delta above 0.01 to 0.02 at frequencies other than the fundamental.
Edwards, L. K., Nixon, W. A., and Lakes, R. S., "Viscoelastic behavior of 80In15Pb5Ag and 50Sn50Pb alloys: experiment and modeling", J. Appl. Physics, 87(3), 1135-1140, (2000).
Dynamic and creep behavior of 80In15Pb5Ag and 50Sn50Pb were studied isothermally over nine decades of time and frequency. 80In15Pb5Ag was examined at -6 deg C, 21 deg C, and 50 deg C, and 50Sn50Pb was examined at 21 deg C. Viscoelastic behavior was observed at strains less than 10-5 as a function of temperature and time / frequency. At the small strain used in the study, the alloys exhibit linear viscoelasticity rather than the viscoplasticity observed at larger strain. Viscoelasticity was subject to constitutive modeling. The creep followed a superposition of a stretched exponential and a power law in time, and an Arrhenius form in temperature. Viscoelastic behavior was thermorheologically simple over at least nine decades of true time and frequency.
Lee, T. and Lakes, R. S., "Damping properties of lead metaniobate", IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 48, 48-52, Jan. (2001). Viscoelastic properties of lead metaniobate piezoelectric ceramic were measured over a wide range of frequency at ambient temperature. Tan delta as a measure of internal friction (viscoelastic damping) exhibited a broad minimum at low audio frequency. Damping, tan delta, was high at ultrasonic frequency. There was no appeal to time-temperature superposition.
Lakes, R. S. "High damping composite materials: effect of structural hierarchy", Journal of Composite Materials, 36, 287-298, Feb. (2002).
Lakes, R. S., Kose, S., and Bahia, H., "Analysis of high volume fraction irregular particulate damping composites", ASME Journal of Engineering Materials and Technology, 124, 174-178, April (2002).
Eyre, D. J., Milton, G. W., and Lakes, R. S., "Bounds for interpolating complex effective moduli of viscoelastic materials from measured data", Rheol. Acta, 41, 461-470, August (2002).
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Ludwigson, M. Swan, C. C. and Lakes, R. S. "Damping and stiffness of particulate SiC - InSn composite", Journal of Composite Materials, 36, 2245-2254, (2002).
Kim, H. J., Swan, C. C., and Lakes, R. S., "Computational studies on high stiffness, high damping SiC-InSn particulate reinforced composites", Int. J. Solids, Structures,
39, 5799-5812 Nov. (2002).
Stewart, K. Brand, R., Swan, C. C., and Lakes, R. S., "Micromechanically based poroelastic modeling of fluid flow in Haversian bone" Journal of Biomechanical Engineering 125, 25-37, Feb. (2003). (Get pdf)
Wang, Y. C. and Lakes, R. S., "Resonant ultrasound spectroscopy in shear mode", Review of Scientific Instruments, 74, 1371-1373, Mar. (2003).
American Institute of Physics. Review of Scientific Instruments .
Buechner, P. M., and Lakes, R. S., "Size effects in the elasticity and viscoelasticity of bone", Biomechanics and Modeling in Mechanobiology, 1 (4), 295-301 (2003).
McMillan, K. M., Lakes, R. S., Cooper, R. F., and Lee, T., "The Viscoelastic Behavior of Beta In3Sn and the Nature of the High-Temperature Background", J. Materials Science,
38, 2747-2754, Aug. (2003).
Oza, A., Lakes, R. S. and Vanderby, R., "Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal" Rheologica Acta, 42, 557-568 (2003).
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Lakes, R. S., "Viscoelastic measurement techniques", Review of Scientific Instruments, 75, 797-810, April (2004).
Copyright (2004) American Institute of Physics. Review of Scientific Instruments .
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).
Jaglinski, T. and Lakes, R. S., "Creep Behavior of Al-Si Die-Cast Alloys", Journal of Engineering Materials and Technology, 126, 378-382, Oct. (2004).
Oza, A., Lakes, R. S. and Vanderby, R., "Application of non-linear superposition to creep and relaxation of commercial die-casting aluminum alloys", Mechanics of Time-Dependent Materials 8, 385 - 402, Dec. (2004).
Jaglinski, T. and Lakes, R. S., "Anelastic instability in composites with negative stiffness inclusions", Philosophical Magazine Letters, 84, (12) 803 - 810, Dec. (2004).
Oza, A., Mccabe, R. Vanderby, R. and Lakes, R. S., "Logarithmic pulse generator for long-term creep and relaxation testing ", Rev. Sci. Instr. 76, 056102, 1-3, April (2005).
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).
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
Lakes, R. S. and Wineman, A., "On Poisson's ratio in linearly viscoelastic solids", Journal of Elasticity, 85, 45-63 (2006). Poisson's ratio in viscoelastic solids is in general a time dependent (in the time domain) or a complex frequency dependent quantity (in the frequency domain). We show that the viscoelastic Poisson's ratio has a different time dependence depending on the test modality chosen; interrelations are developed between Poisson's ratios in viscoelastic creep and viscoelastic relaxation. The difference, for a moderate degree of viscoelasticity, is minor. Correspondence principles are derived for the Poisson's ratio in transient and dynamic contexts. The viscoelastic Poisson's ratio need not increase with time, and it need not be monotonic with time. Examples are given of material microstructures which give rise to designed time dependent Poisson's ratios. Some of these microstructures give rise to negative Poisson's ratio over part of the time scale.
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 50 degrees C. 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.
Jaglinski, T., Nimityongskul, A., Schmitz, R., Lakes, R. S, "Study of bolt load loss in bolted aluminum joints", ASME Journal of Engineering Materials and Technology, 129, 48-54, Jan. (2007). 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)
We show that composite materials can exhibit a viscoelastic (Young's) modulus far higher than that of either constituent; indeed, greater than that of diamond.
Calcagno, B., Lopez Garcia, M.d.C., Kuhns, M., Lakes, R. S., "On the nonlinear creep and recovery of open cell earplug foams", Cellular Polymers, 27 (8) 165-178, (2008).
The compressive creep and recovery of foam earplugs was studied at various applied stresses leading to different final creep strains. Creep approximated a power law in time. Nonlinear viscoelasticity was observed: creep compliance depends on stress level. Nevertheless, for strain as high as 24%, recovery follows creep. Recovery does not follow creep at 70% strain. Recovery proceeds to completion given enough time.
Capodagli, J. and Lakes, R. S., "Isothermal viscoelastic properties of PMMA and LDPE over eleven decades of frequency and time: a test of time- temperature superposition", Rheologica Acta, 47, 777-786, September (2008). Many instruments used to measure viscoelastic properties are only capable of subjecting a sample to a limited range of loading frequencies. For thermorheologically simple materials, it is assumed that a change in temperature is equivalent to a shift of the viscoelastic behavior on the log frequency or time axis. For many materials, time - temperature superposition appears to work well for modulus or compliance curves over three decades of time or frequency, but some deviations are known if the window is expanded to five or six decades. To apply a more stringent test of the validity of time - temperature superposition, broadband viscoelastic spectroscopy is used to isothermally study polymethylmethacrylate and low-density polyethylene at several temperatures in the glassy region. Shear modulus and viscoelastic damping (tan delta) are measured isothermally over a wide range (up to 11 decades) of time and frequency. Results indicate that, while modulus curves can be approximately superimposed, the damping (tan delta) curves change in height and shape with temperature. get pdf
Burgers, T. A., Lakes, R. S., Sylvana Garcia-Rodriguez, S., Piller, G. R., Ploeg, H., "Post-yield relaxation behavior of bovine cancellous bone", Journal of Biomechanics, 42, 2728-2733, (2009).
Dandrea, J. and Lakes, R. S., "Creep and creep recovery of cast aluminum alloys", Mechanics of Time-Dependent Materials, 13, 303-315, Nov. (2009). get pdf
Duenwald, S. E., Vanderby, Jr., R., Lakes, R. S., "Constitutive equations for ligament and other soft tissue: evaluation by experiment", Acta Mechanica, 205: 23-33 (2009). get pdf
Dong, L. Stone, D. S., Lakes, R. S., "Softening of bulk modulus and negative Poisson's ratio in barium titanate ceramic near the Curie point", Philosophical Magazine Lett. 90, 23-33, Jan. (2010). Formal reprints are available; if you want one, please request it. Get informal pdf.
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.
Dong, L., Stone, D. S., and Lakes, R. S., "Sharp low frequency dissipative effects in tetragonal BaTiO3 ceramics,"J. Appl. Physics 107, 023514 (2010). get pdf.
Copyright American Institute of Physics AIP link.
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.
Dong, L., Stone, D. S., and Lakes, R. S., "Anlastic anomalies and negative Poisson's ratio in tetragonal BaTiO3 ceramics", Applied Phys. Lett. 96, 141904 (2010). Anelastic anomalies, sharp variations in modulus and damping with temperature, were observed in tetragonal BaTiO3 via broadband viscoelastic spectroscopy after aging at 50 deg C for 15 h. The effect was most pronounced under electrical short circuit condition, at low frequency and under small excitation strain 10-6. Softening in bulk modulus and negative Poisson's ratio were observed near 60 deg C. Effects are attributed to an oxygen vacancy mechanism. A relaxational model cannot account for sharp response at smaller strains. Heterogeneity of negative stiffness is considered as a cause.
APL link. get pdf.
Calcagno, B., Osswald, T., Lakes, R. S., Crone, W. C. "Comparison of viscoplastic properties of polycarbonate, polypropylene, and high density polyethylene", Journal of Plastics Technology, (Zeitschrift Kunststofftechnik) 6(5), 229-254 (2010). pdf
Dong, L., Stone, D. S., and Lakes, R. S., "Viscoelastic and dielectric properties of 3% KNbO3 - BaTiO3 ceramics", Physica Status Solidi, 248 (1), 158-166 (2011).
Viscoelastic behavior (modulus and damping) of polycrystalline 3%KNbO3-BaTiO3 has been studied in torsion and bending via broadband viscoelastic spectroscopy (BVS) over a range of temperature passing through both the orthorhombic-to-tetragonal and tetragonal-to-cubic phase transformations. Softening in bulk modulus and minima in Poisson's ratio as well as peaks in mechanical damping, dielectric constant, and heat capacity vs. temperature were observed in the vicinity of the transformations. Viscoelastic response vs. temperature was sharper than dielectric or thermal response. Transition temperatures of both transformations shifted to lower temperatures in comparison to pure BaTiO3. Damping peaks vs. temperature were wider (5 deg. C) than in pure BaTiO3 (less than 1 deg. C) and of smaller magnitude. Broadening of response is attributed to material heterogeneity on a fine scale.
get pdf. Journal link, online version.
Jaglinski and Lakes, R. S., "Resonant ultrasound spectroscopy of cylinders over the full range of Poisson's ratio" Rev. Sci. Instr. 82, 035105 (2011). get pdf. Journal link, online version.
Dong, L., Stone, D. S., and Lakes, R. S., "Giant anelastic responses in (BaZrO3-ZnO)-BaTiO3 composite materials", EPL, 93 66003 (2011).
get pdf, Journal link, this issue,
Journal link, online version
Composite materials containing particulate BaTiO3 inclusion have been fabricated
with BaZrO3-ZnO as the matrix. Giant anelastic responses (abrupt increase in modulus and damping vs. temperature) have been observed after 75 deg. C aging 24 h in these composites in thetemperature region far below the Curie point of BaTiO3. Anomalies disappeared after 185 deg. C aging 4 h. Such responses are attributed to the constraining negative-stiffness effect of the inclusion. An oxygen vacancy mechanism has been discussed which is considered to be responsible for the negative-stiffness effect. The present work is novel in terms of the idea of manipulation of negative stiffness in ferroelastic inclusions to achieve the desired composite properties.
Dong, L., Stone, D. S., and Lakes, R. S., "Dielectric and viscoelastic properties of KNbO3 doped BaTiO3", J. Appl. Phys., 109, 063531 (2011). Fine grain (1-2 micron) 2%, 3% KNbO3-BaTiO3 ceramics have been synthesized via the solid state reaction method. Fine and coarse grain ceramics sintered at different temperatures exhibit peaks in dielectric constant and internal friction in the vicinity of transition temperatures. Doping lowered the Curie point and raised the temperatures for the structural transformations between orthorhombic and rhombohedral symmetry and broadened the response near the transformations. Dielectric and viscoelastic responses sharpened with increasing sintering cycles. This effect is attributed to a reduced core-shell effect. Doped ceramic exhibited a relaxation peak due to oxygen vacancy with a similar activation energy and relaxation time as pure material. The steep increase in dielectric constant and internal friction in the ferroelectric phase and the reduced dielectric and mechanical anomaly in the vicinity of the transformations in the fine grain ceramics are attributed to the internal stress built up at the grain boundaries. Other possible mechanisms involved such as oxygen vacancy pinning effect, constrained negative stiffness 90 deg. domain bands and the liquid phase effect have also been discussed.
Journal link, this issue,
Journal link, contents.
Dong, L., Stone, D. S., and Lakes, R. S., "Extreme anelastic responses in Zn80Al20 matrix composite materials containing BaTiO3 inclusion", Scripta Materialia, 65, (4), 288-291, August (2011).
Journal link, Scripta Mat., get pdf.
Dong, L., Stone, D. S., and Lakes, R. S., "Viscoelastic sigmoid anomalies in BaZrO3-BaTiO3 near phase transformations due to negative stiffness heterogeneity", Journal of Materials Research, 26,(11), 1446-1452 (2011). Cambridge University Press.
Journal cover pdf.
Journal link, JMR
Delgadillo, R., Bahia, H.., and Lakes, R. S., "A nonlinear constitutive equation for asphalt binders", Materials and Structures, (2011). get pdf.
Dong, L, and Lakes, R. S., "Frequency dependence of Poisson's ratio of viscoelastic elastomer foam", Cellular Polymers, 30, 277-285, (2011). get pdf.
Jakes, J. E., Lakes, R.S., Stone, D., "Broadband nanoindentation of glassy polymers I: Viscoelasticity", Journal of Materials Research,, 27 (2), 463-474 Jan. (2012). get pdf
Jakes, J. E., Lakes, R.S., Stone, D., "Broadband nanoindentation of glassy polymers II: Viscoplasticity", Journal of Materials Research, 27 (2), 475-484 Jan. (2012). get pdf
Dong, L., Stone, D. S., and Lakes, R. S., "Advanced damper with negative structural stiffness elements", Smart materials, 21 07502 (2012). get pdf
Lakes, R. S., "Stable singular or negative stiffness systems in the presence of energy flux", Philosophical Magazine Letters, 92, 226-234, (2012).
Jaglinski, T. M. and Lakes, R. S., "Zn-Al based metal matrix composites with high stiffness and high viscoelastic damping" Journal of Composite Materials, 46: 755-763, April (2012) get pdf
Dong, L. and Lakes, R. S., Advanced damper with high stiffness and high hysteresis damping based on negative structural stiffness, Int. J. Solids Struct., 50, 2413-2423 (2013). get pdf
author in the lab
Viscoelasticity Books '98
Composites with negative stiffness inclusions
Negative Poisson's ratio
Acknowledgment. We thank the National Science Foundation, Darpa, and the U.S. Department of Agriculture for support of some of the work reported here.