Viscoelastic Materials
Internal friction in solids. Viscoelasticity. Anelastic solids.

Demonstrations
Tutorial
Viscoelasticity Books '98
'09
Viscoelasticity Class
Biomechanics
Ultrasonics
Viscoelastic Composites
Composites with negative stiffness inclusions
Negative Poisson's ratio
Piezoelectric materials
Industrial research
QLV
nonQLV
Definitions
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.
Laboratory
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 timetemperature 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, highloss 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
demonstration.
If you are interested in commercial applications, please contact the Wisconsin Alumni Research Foundation (WARF) for information on US and international patents and licensing.
Books
Rod Lakes,
Viscoelastic solids, CRC Press, Boca Raton, FL, (1998).
Rod Lakes,
Viscoelastic Materials, Cambridge University Press, (2009).
Research
More viscoelastic materials prior to 1997
Rod Lakes and John Quackenbush, "Viscoelastic behaviour in indium tin alloys over a wide range of frequency and time", Philosophical Magazine Letters, 74, 227232 (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 dislocationpoint defect mechanism. It is not consistent with a selforganized 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 indiumtin alloy over more than eleven decades of time and frequency, at 23 degrees C, with no appeal to timetemperature 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, 201216, (1997).
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.
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Amada, S. and Lakes, R. S., "Viscoelastic properties of bamboo", Journal of Materials Science, 32, 26932697, (1997). Get article pdf Viscoelastic properties of bamboo were measured over a range of frequency.
Gibiansky, L. V. and Lakes, R. S., "Bounds on the complex bulk and shear moduli of a twodimensional twophase viscoelastic composite", Mechanics of Materials, 25, 7995, (1997).
The effective complex moduli of an isotropic twophase, twodimensional 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 lensshaped 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 stiffnessloss 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.", BioMedical Materials and Engineering, 8, 4553, (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, 561567, (1999).
Damping of superplastic eutectic PbSn is considerably higher than that of cast PbSn, solder wire, and eutectic PbSn 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 timetemperature superposition. The damping in torsion is higher than in bending; the difference is more marked at high frequency. Get pdf
The specimen is contained within the brass enclosure to the left in this lab image.
Dooris, A., Lakes, R. S., Myers, B. and Stephens, N., "High damping indiumtin alloys", Mechanics of Time Dependent Materials, 3, 305318 (1999). This research is directed toward the development of materials of high stiffness and high mechanical damping for the purpose of damping vibrations in structures and machinery. To that end, indiumtin alloys are considered. Cast InSn exhibits substantial damping for a metal. Quenching substantially improved the damping of indiumtin alloy but the effect gradually disappeared due to aging. Cold work of 1.3 % permanent shear strain had the effect of moderately increasing the damping of indiumtin, and 30 % permanent strain substantially increased the damping of tin.
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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) 28552861, 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 powerlaw 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.
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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), 11351140, (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.
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Lee, T. and Lakes, R. S., "Damping properties of lead metaniobate", IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 48, 4852, 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 timetemperature superposition.
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Lakes, R. S. "High damping composite materials: effect of structural hierarchy", Journal of Composite Materials, 36, 287298, Feb. (2002). A combination of stiffness and loss (the product E tan delta) is desirable in damping layer and structural damping applications. Composite materials of structure which gives rise to Reuss or Hashin  Shtrikman lower bound behavior can give rise to such properties. Hierarchical particulate morphologies attain the HashinShtrikman curve. Hierarchical composites give rise to complex Poisson's ratios which, however, have minimal effect on the stiffnessmap. Structural hierarchy is useful in viscoelastic composites in that it enables the attainment of high concentrations of spherical inclusions, and that it facilitates the attainment of both stiffness and damping. A damping layer upon a substrate is considered as the top level of the structural hierarchy. We demonstrate that if the layer itself is a relatively stiff composite, the penalty usually associated with such a geometry for compliant layers is ameliorated.
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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, 174178, April (2002).
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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, 461470, August (2002). The complex effective moduli of viscoelastic materials can be experimentally measured by a number of techniques giving relaxation data for materials at discrete frequencies. In this paper we present a method which allows one to find bounds on the moduli of the material at the unmeasured frequencies. Our bounds can be very tight, and are, therefore, good approximations to the data. Using our bounds, experimentalists can concentrate their efforts on gathering highly accurate data at only a few frequencies and numerically generate data at other frequencies that may be of interest.
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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, 22452254, (2002). Metal matrix composites of silicon carbide particles in indium tin alloy were fabricated with the aim of achieving a high value of the product of stiffness and viscoelastic damping tan delta, without excess density. Stiffness and viscoelastic damping were measured over a wide range of frequency. For monodisperse 40% by volume SiC, and for hierarchical 60% by volume SiC the composite damping increased compared with the matrix at frequencies above 100 Hz. Composite shear modulus was almost a factor two greater than matrix for 40% and a factor of four greater than that of matrix for 60%. The product of stiffness and damping exceeds that of wellknown materials including polymer damping layers. The damping increase at higher frequency cannot be accounted for by a purely mechanical composite model; it is attributed to thermoelastic coupling and an increase in matrix dislocations during fabrication.
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Kim, H. J., Swan, C. C., and Lakes, R. S., "Computational studies on high stiffness, high damping SiCInSn particulate reinforced composites", Int. J. Solids, Structures, 39, 57995812 Nov. (2002).
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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, 2537, Feb. (2003). (Get pdf)
Wang, Y. C. and Lakes, R. S., "Resonant ultrasound spectroscopy in shear mode", Review of Scientific Instruments, 74, 13711373, Mar. (2003). We present an enhancement of the resonant ultrasound spectroscopy method for the determination of elastic and viscoelastic properties. By using shear transducers rather than the usual compressional ones, signal strength for the fundamental is enhanced by one to three orders of magnitude. This enables simplified determination of shear modulus and damping tan delta with off the shelf electronics. Moreover, the polarization of the shear transducers can be used to identify modes of vibration.
American Institute of Physics. Review of Scientific Instruments .
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Buechner, P. M., and Lakes, R. S., "Size effects in the elasticity and viscoelasticity of bone", Biomechanics and Modeling in Mechanobiology, 1 (4), 295301 (2003). Size effects of large magnitude are observed in the torsional shear modulus and damping of bovine plexiform bone. Damping increases and stiffness decreases with specimen size over all sizes studied. Measurements were conducted in torsion using a laser based micromechanics apparatus capable of viscoelastic studies over a range of frequencies up to 100 kHz, upon samples of various size, with no parasitic friction or other errors which could mimic any size effect. Torsional tan delta at 1 Hz varies by about a factor of five over the size range 2.8 to 6.2 mm thick, and is more dependent on specimen thickness at 1 Hz than it is at higher frequency. The size effects are attributed to compliance and viscoelasticity of the interfaces between laminae. These laminae must be substantially stiffer than whole bone. Observed size effects are likely to play a role in understanding scaling laws of bones in living organisms.
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McMillan, K. M., Lakes, R. S., Cooper, R. F., and Lee, T., "The Viscoelastic Behavior of Beta In_{3}Sn and the Nature of the HighTemperature Background", J. Materials Science, 38, 27472754, Aug. (2003). The damping behavior in torsion of single phase beta In_{3}Sn has been evaluated at room temperature over a broad range of frequencies (10^{4} to 10^{3} Hz) and as a function of various mechanical and thermal treatments. The results are consistent with a model for the power law (tan delta proportional to f^{n}), high temperature background absorption being effected by diffusional processes on grain and, particularly, on subgrain boundaries. The results are compared/contrasted with those for damping in single phase gamma InSn_{4} and in the two phase beta  gamma eutectic. Failure of the eutectic material to follow a composite model for damping, combined with a thermal aging effect that lowers damping only for certain frequencies, shows that the boundary based absorption model for the high temperature background applies, too, to phase boundaries.
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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, 557568 (2003).
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Lakes, R. S., "Viscoelastic measurement techniques", Review of Scientific Instruments, 75, 797810, April (2004).
pdf;
Copyright (2004) American Institute of Physics. Review of Scientific Instruments .
RSI link
Wang, Y. C., Ludwigson, M., and Lakes, R. S., "Deformation of extreme viscoelastic metals and composites" Materials Science and Engineering A, 370, 4149, April (2004).
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Jaglinski, T. and Lakes, R. S., "Creep Behavior of AlSi DieCast Alloys", Journal of Engineering Materials and Technology, 126, 378382, Oct. (2004).
pdf
Oza, A., Lakes, R. S. and Vanderby, R., "Application of nonlinear superposition to creep and relaxation of commercial diecasting aluminum alloys", Mechanics of TimeDependent Materials 8, 385  402, Dec. (2004).
pdf
Jaglinski, T. and Lakes, R. S., "Anelastic instability in composites with negative stiffness inclusions", Philosophical Magazine Letters, 84, (12) 803  810, Dec. (2004).
Composites with VO2 particulate inclusions as a negative stiffness phase were fabricated through powder metallurgy. The composites are predicted to exhibit enhanced anelastic damping by virtue of the partially constrained negative stiffness of the inclusions in the vicinity of a ferroelastic phase transformation, and are predicted to become unstable for sufficiently high concentration (5 vol%) of inclusions. Composite specimens with 5 vol% inclusions studied in subresonant dynamic torsion displayed various manifestations of mechanical instability
during cooling in a temperature range including the inclusion transformation temperature. Instability was manifested as macroscopic specimen undulations (slow thrashing) and fluctuation of the damping tan delta Material instability occurs at high inclusion volume fraction in harmony with predictions from composite theory.
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Oza, A., Mccabe, R. Vanderby, R. and Lakes, R. S., "Logarithmic pulse generator for longterm creep and relaxation testing ", Rev. Sci. Instr. 76, 056102, 13, April (2005).
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Jaglinski, T. Stone, D., and Lakes, R. S., "Internal friction study of a composite with a negative stiffness constituent", J. Mater. Research 20 (9), 25232533, Sept. (2005).
Composites with negative stiffness constituents can exhibit material properties that exceed conventional bounds. Composites with VO2 as negative stiffness inclusions and tin as the stabilizing matrix were prepared via powder metallurgy. Specimens were tested over a range of temperature in torsion using broadband viscoelastic spectroscopy. Composites processed via powder metallurgy exhibited internal friction anomalies over a broad range of temperatures, in contrast to the single, sharp anomalies reported previously from cast specimens. The detailed material behavior encompassed a variety of responses, which were also dependent on the number of thermal cycles. Composite theory predictions assuming a distribution of negative shear moduli can account for peak broadening.
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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, 31713178, (2007). get pdf
Lakes, R. S. and Wineman, A., "On Poisson's ratio in linearly viscoelastic solids", Journal of Elasticity, 85, 4563 (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.
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Jaglinski, T., Frascone , P., Moore, B., Stone, D., and Lakes, R. S., "Internal friction due to negative stiffness in the indiumthallium martensitic phase transformation", Philosophical Magazine, 86, (27 / 21) 4285  4303, September (2006).
Internal friction and dynamic shear modulus in an indium21 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 timelapse 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 premartensitic phenomenon are seen at temperatures above martensitestart.
pdf
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, 4854, Jan. (2007). The viscoelastic response of bolted aluminum joints used in the small diecast engine industry at elevated temperatures was studied. Bolt loadloss tests were performed using strain gages in situ. It was found that after a week at temperature, most bolts lost 100% of their initial prestress. Nonlinear constitutive equations utilizing parameters obtained from uniaxial creep and relaxation tests were used in a simple onedimensional model to predict the bolt load loss. The model cannot predict the detailed response and overpredicts retained bolt stress for bolt holes that are not preconditioned. For preconditioned holes, the behavior is intermediate between creep and relaxation. get pdf
Jaglinski, T., Kochmann, D., Stone, D., Lakes, R. S. "Materials with viscoelastic stiffness greater than diamond", Science 315, 620622, 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.
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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) 165178, (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.
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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, 777786, 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 lowdensity 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 GarciaRodriguez, S., Piller, G. R., Ploeg, H., "Postyield relaxation behavior of bovine cancellous bone", Journal of Biomechanics, 42, 27282733, (2009).
Dandrea, J. and Lakes, R. S., "Creep and creep recovery of cast aluminum alloys", Mechanics of TimeDependent Materials, 13, 303315, 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: 2333 (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, 2333, 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 shortcircuit 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 BaTiO_{3} 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 BaTiO_{3} ceramics", Applied Phys. Lett. 96, 141904 (2010). Anelastic anomalies, sharp variations in modulus and damping with temperature, were observed in tetragonal BaTiO_{3} 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), 229254 (2010). pdf
Dong, L., Stone, D. S., and Lakes, R. S., "Viscoelastic and dielectric properties of 3% KNbO_{3}  BaTiO_{3} ceramics", Physica Status Solidi, 248 (1), 158166 (2011).
Viscoelastic behavior (modulus and damping) of polycrystalline 3%KNbO_{3}BaTiO_{3} has been studied in torsion and bending via broadband viscoelastic spectroscopy (BVS) over a range of temperature passing through both the orthorhombictotetragonal and tetragonaltocubic 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 BaTiO_{3}. Damping peaks vs. temperature were wider (5 deg. C) than in pure BaTiO_{3} (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).
Mode structure maps for freely vibrating cylinders over a range of Poisson's ratio are desirable for the design and interpretation of experiments using resonant ultrasound spectroscopy (RUS). The full range of isotropic Poisson's ratio (1 to +0.5) is analyzed here using a finite element method to accommodate materials with a negative Poisson's ratio. The fundamental torsional mode has the lowest frequency provided Poisson's ratio is between about 0.24 and +0.5. For any Poisson's ratio, the torsional mode can be identified utilizing the polarization sensitivity of the shear transducers. RUS experimental results for materials with Poisson's ratio +0.3, +0.16, and 0.3 and a previous numerical study for Poisson's ratio = 0.33 are compared with the present analysis. Interpretation of results is easiest if the length/diameter ratio of the cylinder is close to 1. Slight material anisotropy leads to splitting of the higher modes but not of the fundamental torsion mode.
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Dong, L., Stone, D. S., and Lakes, R. S., "Giant anelastic responses in (BaZrO_{3}ZnO)BaTiO_{3} composite materials", EPL, 93 66003 (2011).
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Journal link, online version
Composite materials containing particulate BaTiO3 inclusion have been fabricated
with BaZrO3ZnO 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 negativestiffness effect of the inclusion. An oxygen vacancy mechanism has been discussed which is considered to be responsible for the negativestiffness 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 KNbO_{3} doped BaTiO_{3}", J. Appl. Phys., 109, 063531 (2011). Fine grain (12 micron) 2%, 3% KNbO3BaTiO3 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 coreshell 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.
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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 BaTiO_{3} inclusion", Scripta Materialia, 65, (4), 288291, August (2011).
Extreme anelastic responses (stiffness greater than diamond and large amplification of damping) due to the presence of a negative stiffness phase have been observed in Zn80Al20BaTiO3 composites not only near the tetragonal orthorhombic transformation temperatures of BaTiO3 anticipated by the classical Landau theory, but far below the Curie point of BaTiO3 which can be manipulated via specific aging processes that operated by an oxygen vacancy mechanism. The idea proposed herein can be utilized to fabricate novel composites for potential engineering applications.
Journal link, Scripta Mat., get pdf.
Dong, L., Stone, D. S., and Lakes, R. S., "Viscoelastic sigmoid anomalies in BaZrO_{3}BaTiO_{3} near phase transformations due to negative stiffness heterogeneity", Journal of Materials Research, 26,(11), 14461452 (2011). Cambridge University Press.
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Journal cover pdf.
Journal link, JMR
Delgadillo, R., Bahia, H.., and Lakes, R. S., "A nonlinear constitutive equation for asphalt binders", Materials and Structures, online (2011); 45, 457473 (2012). get pdf.
Dong, L, and Lakes, R. S., "Frequency dependence of Poisson's ratio of viscoelastic elastomer foam", Cellular Polymers, 30, 277285, (2011). All polymer foams are viscoelastic; for foams used in earplugs, this response is essential to their function. Poisson's ratio was inferred from viscoelastic response of earplug foam in torsion and bending. Poisson's ratio increases with frequency as a result of stress induced air flow. Air within the pores contributes to the overall bulk modulus at high frequency, but not at low frequency at which it easily flows when the foam is deformed. Such behaviour is the opposite of that of solid polymers in which the shear modulus varies with frequency much more than the bulk modulus. The frequency dependent Poisson's ratio is related by Fourier transform to the time dependent Poisson's ratio. get pdf.
Jakes, J. E., Lakes, R.S., Stone, D., "Broadband nanoindentation of glassy polymers I: Viscoelasticity", Journal of Materials Research,, 27 (2), 463474 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), 475484 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).
When a flexible flatends column is aligned in a postbuckled condition, a negative structural stiffness and large hysteresis (i.e., high damping) can be achieved provided the ends of the column undergo tilting from flat to edge contact. Stable axial dampers with initial modulus equivalent to that of the parent material and with enhanced damping were designed and built using constrained negative stiffness effects entailed by postbuckled pressfit flatends columns.
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Lakes, R. S., "Stable singular or negative stiffness systems in the presence of energy flux", Philosophical Magazine Letters, 92, 226234, (2012).
We report stable systems which exhibit quasistatic stiffness that can be negative or tend to infinity without external constraint. They are based on coupled fields in the nonequilibrium presence of energy flux that is modulated by force. They evade thermodynamic restrictions by relaxing a restrictive assumption: equilibrium. Negative values of physical properties, including compressibility and heat capacity, are considered forbidden in classical thermodynamics; such analyses provide bounds on the stiffness and other properties of multiphase materials. Stable negative and singular stiffness is demonstrated experimentally in a piezoelectric system and in a thermoelastic granular material. Coupled fields occur naturally under a wide range of conditions and form the basis for many forms of technology including sensors, actuators, and electric coolers. Because all materials exhibit at least one coupled field effect, the concept is broadly general and is applicable to attaining extreme values of any physical property e.g. stiffness, permittivity, piezoelectricity.
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Jaglinski, T. M. and Lakes, R. S., "ZnAl based metal matrix composites with high stiffness and high viscoelastic damping" Journal of Composite Materials, 46: 755763, April (2012)
A maximal product of stiffness and viscoelastic damping E tan delta, a figure of merit for damping layers, is desirable for structural damping applications. Particulatereinforced metal matrix composites were prepared by ultrasonic agitation of the melt and composed of the zincaluminum (ZnAl) alloy Zn80Al20 (in weight percent) as the lossy matrix and SiC or BaTiO3 as the particulate reinforcements. ZnAlSiC composites were stiffer and exhibited higher damping at acoustic frequencies in comparison to the base alloy. ZnAlSiC composites were superior to SnSiC composites and possessed an E tan delta in excess of 0.6 GPa, the maximum figure of merit provided by commercial polymer damping layers. Furthermore, ZnAlSiC composites displayed a high figure of merit over a broad temperature range. ZnAlBaTiO3composites exhibited anomalies in modulus and damping associated with partial restraint of the phase transformation; one specimen was much stiffer than diamond over a narrow temperature range.
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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, 24132423 (2013). High structural damping combined with high stiffness is achieved by negative stiffness elements. Negative incremental structural stiffness occurs when a column with flat ends is subjected to snapthrough buckling. Large hysteresis (i.e., high damping) can be achieved provided the ends of the column undergo tilting from flat to edge contact. The column configuration provides high structural stiffness. Stable axial dampers with initial modulus similar to that of the parent material and with enhanced damping were designed built and tested. Effective damping of approximately two and stiffnessdamping product of approximately 200 GPa were achieved in such dampers consisting of stainless steel columns. This is a significant improvement for this figure of merit (i.e., the stiffnessdamping product), which generally cannot exceed 0.6 GPa for currently used damping systems. get pdf
Kalathur, H., Lakes, R. S., "Column dampers with negative stiffness: high damping at small amplitude", Smart Materials, 22, 084013 (8pp) (2013). High structural damping combined with high initial stiffness is achieved at small amplitude via negative stiffness elements. These elements consist of columns in the vicinity of the postbuckling transition between contact of flat surfaces and edges of the ends for which negative incremental structural stiffness occurs. The column configuration provides a high initial structural stiffness equal to the intrinsic stiffness of the column material. Columns of the polymers polymethyl methacrylate (PMMA) and polycarbonate were used. By tuning the prestrain, a very high mechanical damping was achieved for small amplitude oscillations. The product of effective stiffness and effective damping as a figure of merit E tan delta effective of about 1.5 GPa was achieved for polymer column dampers in the linear domain and about 1.62 GPa in the small amplitude nonlinear domain. For most materials this value generally never exceeds 0.6 GPa.
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Wang, Z., Lakes, R. S., Golob, M., Eickhoff, J. C., Chesler, N. C., "Changes in large pulmonary arterial viscoelasticity in chronic pulmonary hypertension" PLOS ONE , 8, (11)  e78569, Nov. (2013). Conduit pulmonary artery (PA) stiffening is characteristic of pulmonary arterial hypertension (PAH) and is an excellent predictor of mortality due to right ventricular (RV) overload. To better understand the impact of conduit PA stiffening on RV afterload, it is critical to examine the arterial viscoelastic properties, which require measurements of elasticity (energy storage behavior) and viscoelasticity. Here we hypothesize that PAH leads to frequencydependent changes in arterial stiffness (related to elasticity) and damping ratio in large PAs. Static and sinusoidal pressureinflation tests were performed on conduit PAs isolated in a culture bath at various frequencies (0.0120 Hz) to obtain the mechanical properties in the absence of smooth muscle contraction. Static mechanical tests showed significant stiffening of large PAs with PAH, as expected. In dynamic mechanical tests, structural stiffness increased and damping ratio decreased at a physiologically relevant frequency (10 Hz) in hypertensive PAs.
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Balch, S. and Lakes, R. S., "High vibration damping in in situ InZn composites", Functional Materials Letters, 8, (5) 1550059 (4 pages) (2015).
Indiumzinc in situ composites were fabricated and their viscoelastic properties studied over 8.5 decades of frequency. Material with 5% indium by weight was found to have a stiffness damping product (the figure of merit for damping layers) of 1.9 GPa at 10 Hz; 3 times better than the peak of polymer damping layers and over a wider frequency range. Material with 15% indium had a stiffness damping product of 1.8 GPa. The indium segregated in a platelet morphology, particularly favorable for attaining high damping from a small concentration, as predicted by viscoelastic composite theory.
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Balch, S. and Lakes, R. S., "The effects of composition, temperature, and annealing on the rheological properties of InZn in situ composites", Rheologica Acta, 55, (4), 335341 (2016). doi="10.1007/s003970160915x", High viscoelastic damping is observed in InZn materials over ranges of composition, frequency, temperature, and annealing time. Microscopy reveals InZn when cast segregates into a heterogeneous microstructure resembling an in situ composite consisting of a zinc matrix with soft indium platelet inclusions. This morphology is predicted to be advantageous for maximizing the damping figure of merit E tan delta by viscoelastic composite theory. InZn is found to be linearly viscoelastic, unlike other high damping metals. The damping of InZn varies little over a substantial range of temperature, in contrast with polymers. For 5 % In material, the optimal composition, E tan delta is 2.8 GPa at 10 Hz, compared to a peak of 0.6 GPa for high damping rubbers. After annealing for 13 years, E tan delta was still high at 1.9 GPa. InZn demonstrates high damping under a wide range of conditions.
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Balch, S. and Lakes, R. S., "Amelioration of waves and microvibrations by microbuckling in open celled foam", Cellular Polymers, 36, 111, Feb. (2017). Open cell polymer foam exhibits both an intrinsic microvibration frequency and a cut off frequency associated with resonance of the cell ribs. Compression of the foam causes regions of local buckling of ribs. Incipient buckling gives rise to negative stiffness and enhanced amelioration of waves. Transmitted wave amplitude is reduced by compression below and near the cut off frequency. preprint pdf
Balch, S. and Lakes, R. S., "Lumped negative stiffness damper for absorption of flexural waves in a rod", Smart Materials and Structures, 26 045022 (6pp) (2017) A damper based on negative stiffness from column tilt buckling was used to achieve enhanced mechanical damping of bending vibration of a rod. Damping increased by a factor of three, via a damper column that was only about 0.2 % of the mass of the aluminum alloy rod to be damped.
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Ha, C. S., Plesha, M. E., Lakes, R. S., "Design, Fabrication, and Analysis of Lattice Exhibiting Energy Absorption via Snapthrough Behavior", Materials and Design, 254, 426437 (2018).
In this study, an energy absorption lattice, comprised of multiple tetrabeamplate unit cells with negative stiffness, was designed, fabricated by selective laser sintering method, and analyzed both numerically and experimentally. Snapthrough behavior of the unit cell developed due to negative stiffness caused by geometric nonlinearity from large deflection of the constituent elastic beams, resulting in energy absorption. A criterion for the unit cell to achieve the snapthrough behavior was investigated numerically in terms of the beam slenderness ratio and the inclined angle. This approach was chosen to facilitate control of energy dissipation performance and further design space such as tuning force threshold. The unit cell with the selected geometric parameters was then created and used to construct the energy absorption lattice. Load displacement relationships of the lattices obtained from cyclic loading tests disclosed an area enclosed by two distinct loading and unloading curves, which indicates energy dissipation. This was shown both numerically and experimentally. Drop tests were also performed to investigate energy loss of the lattices due to an impact. An energy absorption phenomenon was revealed by observing a reduced rebound height when the lattice exhibited the snapthrough behavior.
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Acknowledgment. We thank the National Science Foundation, DARPA, ARO, and the U.S. Department of Agriculture for support of some of the work reported here.
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