Viscoelasticity. Anelastic solids which exhibit internal friction.

Abstracts and articles, 19791996
Bone: see below; More recent articles on viscoelasticity
Lakes, R. S., "Shapedependent damping in piezoelectric solids," IEEE Trans. Sonics, Ultrasonics, SU27, 208213, (1980). The piezoelectric contribution to the mechanical loss tangent of a piezoelectric solid is derived from its complex piezoelectric and dielectric coefficients. This loss depends on specimen geometry as a result of differences in effects related to electrical boundary conditions. Inclusion of a positive out of phase piezoelectric modulus results in reduced values of predicted loss. This constitutes an improvement over earlier theories which predict losses exceeding measured losses by a factor greater than two. Get pdf
Rod Lakes, "Logarithmic relaxation spectrum for viscoelastic solids",
Journal of Rheology, 25 (6), 663671, (1981).
A logarithmic spectrum of relaxation times is considered. Analytical forms are presented for the relaxation modulus and for the complex dynamic modulus. Mechanical damping increases at progressively lower frequency. Retardation spectra are prepared numerically for comparison. They increase more rapidly with time than the corresponding relaxation spectra. The logarithmic relaxation spectrum may be of use in describing systems in which the viscoelastic behavior arises from motion along viscous interfaces in an elastic matrix. Zener points out that all observations of such systems indicate a relaxation spectrum whose intensity continually rises or remains stationary with increasing times of
relaxation.
Shipkowitz, A. T., Chen, C. P. and Lakes, R. S., "Characterization of highloss viscoelastic elastomers", Journal of Materials Science, 23, 36603665 (1988).
The behaviour of a variety of high loss viscoelastic elastomers is described. Measurements were conducted using a novel micromechanics apparatus which is capable of creep, constant load rate, subresonant dynamic and resonant dynamic experiments in bending and torsion upon a single specimen. The range of equivalent frequency is from one
microHz to several kilohertz under isothermal conditions.
Chen, C. P. and Lakes, R. S., "Design of viscoelastic impact absorbers: optimal material properties", International Journal of Solids and Structures, 26, 13131328
(1990).
This article describes analytical investigations of impact absorption of linear and isothermal viscoelastic materials. Three methods based on different considerations and approximations are studied, and similarities are shown in their results. For a viscoelastic buffer of given thickness, the optimal loss tangent is determined to be approximately 1. Greater reductions in impact force can be achieved if the high loss is accompanied by stiffness
reduced by a factor of 3 to 4 compared with an elastic buffer. If the impactor is spherical rather than flat, a higher loss tangent, of the order of 10,
is needed to minimize the impact force. Moreover, a more sophisticated interpretation scheme for the ball rebound test for screening the loss tangent of viscoelastic materials is derived.
Chen, C. P. and Lakes, R. S., "Dynamic wave dispersion and loss properties of conventional and negative Poisson's ratio polymeric cellular materials", Cellular Polymers, 8(5), 343359 (1989).
This article describes experimental investigations of the dynamical behaviour of conventional and negative Poisson's ratio foamed materials in torsional vibration. Dispersion of standing waves and cutoff frequencies were observed. Consequently, foamed materials do not obey the classical theory of elasticity or viscoelasticity. The dynamical effects were attributed to microvibrations of the cell ribs in a structural view and were associated with microstructure or micromorphic elasticity in a continuum view. Cutoff frequencies were lower in reentrant foams with negative Poisson's ratios than in the conventional foams from which they were derived. An analytical structural model was developed in which the ribs of the conventional foams were modeled as freefree vibrating beams. The predicted cutoff frequencies were comparable to those observed experimentally.
In more recent parlance, these materials may be referred to as locally resonant metamaterials.
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Lakes, R. S., "The time dependent Poisson's ratio of viscoelastic cellular materials can increase or decrease", Cellular Polymers, 11, 466469, (1992).
In viscoelastic materials, the Poisson's ratio is not a material constant but can depend upon time. For polymeric solids, the shear modulus relaxes much more than the bulk modulus, therefore, the Poisson's ratio is an increasing function of time. In this article we demonstrate that such time dependence is not a necessary consequence of the theory of viscoelasticity. Viscoelastic composite microstructures are presented which result in a time dependent Poisson's ratio which decreases with time.
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Chen, C. P. and Lakes, R. S., "Analysis of high loss viscoelastic composites",
J. Materials Science, 28, 42994304, (1993).
A theoretical study of viscoelastic properties of composites is presented with the aim of identifying structures which give rise to a combination of high stiffness and high loss tangent. Laminates with Voigt and Reuss structure, as well as composite materials attaining the HashinShtrikman bounds on stiffness were evaluated via the correspondence principle. Similarly, viscoelastic properties of composites containing spherical or platelet inclusions were explored. Reuss laminates and platelet filled materials composed of a stiff, low loss phase and a compliant high loss phase were found to exhibit high stiffness combined with high loss tangent. These materials are an example of extremal materials.
A theoretical stiffnessloss map shows the effect of inclusion shape on the properties of a viscoelastic composite.
Chen, C. P. and Lakes, R. S., "Viscoelastic behaviour of composite materials with conventional or negative Poisson's ratio foam as one phase", J. Materials Science, 28, 42884298, (1993).
This article describes experimental investigations of viscoelastic properties (internal friction) of composites consisting of conventional and reentrant negative Poisson's ratio copper foam as a matrix, with high loss filler materials: viscoelastic elastomer, solder, and indium. Viscoelastic properties of gallium and several ferrites were determined as well. The loss tangent of the copper elastomer composite substantially exceeded the (lower) Voigt limit; the loss tangent of the coppersolder and copperindium composites were close to the (upper) Hashin limit for two solid phases and one pore phase.
Gibiansky, L.V. and Lakes, R. S., "Bounds on the complex bulk modulus of a twophase viscoelastic composite with arbitrary volume fractions of the
components", Mechanics of Materials, 16, 317331 (1993).
The dynamic response of isotropic composites of two viscoelastic isotropic phases mixed in arbitrary proportions is considered in the frequency range where the acoustic wavelength is much larger than the inhomogeneities and the properties of the isotropic composite can be described by complex bulk and shear moduli. The effective complex bulkmodulus bounds by Gibiansky and Milton twophase composites with fixed volume fractions of the components are used to obtain the same bounds for the phases mixed in arbitrary proportions. The effective bulk modulus is shown to be constrained to a lensshaped region of the complex plane bounded by the outermost pair of several circular
arcs. The parameters of these arcs depend on the moduli of the original materials. The bounds are investigated numerically in order to find the materials with high loss properties. Microstructures are identified which have bulk moduli that correspond to various points on each of the circular arcs. Get pdf
Papadogiannis, Y., Lakes, R. S., PetrouAmericanos, A., and TheothoridouPahini, S., "Temperature dependence
of the dynamic viscoelastic behavior of chemically and light activated composite resins", Dental Materials,
9, 118122, (1993).
The relationship between temperature and the viscoelastic properties of six composites, three light cured and three chemical cured, was studied, using constant dynamic loading over the narrow range of temperatures (2060 deg C) which
can be encountered in the mouth. The parameters were: storage modulus G', loss modulus G'', tan delta, quality factor Q, coefficient of decay, and dynamic viscosity. It was found that the dynamic viscoelastic properties of the tested materials are temperature dependent, but probably not to a clinically significant degree. Chemical and light cured composites of the same filler loading do not exhibit significantly different viscoelastic dynamic properties.
Brodt, M. and Lakes, R. S., "Composite materials which exhibit high stiffness and high viscoelastic
damping", J. Composite Materials, 29, 18231833, (1995).
Composite microstructures are studied, which give rise to high stiffness combined with high viscoelastic loss. We demonstrate that such properties are most easily achieved if the stiff phase is as stiff as possible. Incorporation of a small amount of damping in the stiff phase has little effect on the composite damping. Experimental results are presented for laminates consisting of cadmium and tungsten and of InSn alloy and tungsten. The combination of stiffness and loss (the product E tan delta) exceeds that of wellknown materials.
A stiffnessloss map shows properties of common materials and also of several high performance viscoelastic composite materials developed and studied in our laboratory. These composites represent a proof of concept. Viscoelastic composites suitable for applications are under development. The product E tan delta is a figure of merit for damping applications. Most common materials, including polymer damping layer materials, have a value less than 0.6 GPa. This is shown by the diagonal line in the diagram. Data for wet bone (right) correspond to ultrasonic frequency and also for less than 0.01 Hz.
Cook, L. S. and Lakes, R. S., "Viscoelastic spectra of Cd0.67Mg0.33 in torsion and bending",
Metallurgical Transactions, 26A, 20372039 (1995).
In both torsion and in bending, the alloy exhibited a viscoelastic relaxation which could be modeled as a Debye peak of internal friction superimposed on a powerlaw lowfrequency background ('high temperature background' at room temperature). In torsion, the relaxed and unrelaxed shear moduli were 9 and 12.2 GPa, respectively; maximum loss tangent was 0.12. In bending, relaxed and unrelaxed Young's moduli were 15 and 35 GPa, respectively; maximum loss tangent was 0.11. Behavior was linear to at least 125 microstrain. These results are significant in that they represent a unique combination of stiffness and loss in a monolithic material. Get pdf
Brodt, M., Cook, L. S., and Lakes, R. S., "Apparatus for measuring viscoelastic properties over ten decades: refinements", Review of Scientific Instruments, 66(11), 52925297 (1995).
This article describes refinements to an instrument for determining the viscoelastic properties of a solid material isothermally, with a single apparatus, over 10 decades of time and frequency. Torque is applied electromagnetically to a specimen fixed at one end. Specimen deformation is determined via a laser beam reflected from the other end upon a split diode detector. Phase resolution is improved by the use of a lock in amplifier at high frequency and by the use of Lissajous figures to measure phase, allowing study of materials of moderate loss (0.008 <= tan delta <= 0.2) in addition to materials with high loss (tan delta ~ 1). The rigidity of the instrument is increased by modifications in the specimen support geometry. The range of equivalent frequency for torsion is from less than one micro Hz to more than ten kHz. Digital methods are incorporated in the creep measurements and in phase measurements.
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Cook, L. S. and Lakes, R. S., "Damping at high homologous temperature in pure Cd, In, Pb, and Sn", Scripta Metall et Mater. , 32, 773777, (1995). get pdf
Quackenbush, J. and Lakes, R. S., "Viscoelastic behavior over a wide range of time and frequency in tin alloys: SnCd and SnSb", Scripta Metall et Mater., 35, 441447, (1996).
Damping of SnCd was lower than that of Cd and at most frequencies lower than that of Sn. Damping of SnSb was lower than that of Sn over most audio frequencies. Both alloys had lower damping than eutectic InSn. Tan delta approximately followed a power law dependence on frequency over a wide range of frequency for these alloys. This is consistent with a high temperature background (dislocation) mechanism. Get pdf
Lakes, R. S. and Quackenbush, J., "Viscoelastic behaviour in indium tin alloys over a wide range of frequency and time", Philosophical Magazine Letters, 74, 227232
(1996).
Experimental studies of dynamic (internal friction) 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 an instrument capable of determining viscoelastic properties over more than ten decades of time and frequency. The damping, tan delta followed a {frequency}^{n} frequency dependence, with n ~ 0.2, over many decades of frequency. This dependence 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 tan delta proportional to {frequency}^{2}. Over much of the frequency range the damping is high temperature background at room temperature. Dislocation damping in metals is relevant to development of high damping metals, the behaviour of solders and of support wires in Cavendish balances.
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Brodt, M. and Lakes, R. S., "Viscoelastic behaviour in indium alloys: InSn, InBi, InCd and
InSnCd",
Journal of Materials Science, 31, 65776581, (1996).
Experimental studies of dynamic (internal friction) and transient viscoelastic response were conducted at 24 deg C on the indium alloys InSn, InBi, InCd and InSnCd. The viscoelastic measurements were conducted in torsion using an instrument capable of determining viscoelastic properties over ten decades of time and frequency. The damping, tan delta, followed a power law dependence at higher frequency and was essentially constant at low frequency. Creep at long times followed a power law dependence upon time. The damping is attributed to a dislocationpoint defect mechanism. High temperature background damping is observed over a broad band of frequency at room temperature.
Lakes, R. S., Katz, J. L., and Sternstein, S. S., "Viscoelastic properties of wet cortical bone: Part I, torsional and biaxial studies." Journal of Biomechanics, 12, 657678, (1979). get pdf
Lakes, R. S., Katz, J. L., "Viscoelastic properties of wet cortical bone:
Part II, relaxation mechanisms," Journal of Biomechanics, 12, 679687, (1979). get pdf
Lakes, R. S. and Katz, J. L., "Viscoelastic properties of wet cortical bone:
Part III, A nonlinear constitutive equation," Journal of Biomechanics, 12, 689698, (1979). get pdf
Lakes, R. S. and Saha, S., "Cement line motion in bone," Science, 204, (1979), 501503.
Compact bovine bone subjected to constant torsional load for long periods of time exhibits large anelastic effects. Displacements occur at the cement lines and are responsible for part or all of the long term deformation. The absence of an asymptotic creep strain is consistent with an interpretation of the cement line as a viscous interface.
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Lakes, R. S., Yoon, H. S. and Katz, J. L., "Slow compressional wave propagation in wet human and bovine cortical bone", Science, 220 513515, (1983).
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Summary on bone viscoelasticity
Bone, a natural viscoelastic composite, exhibits viscoelastic behavior, i.e. the stress depends not only on the strain but also on the time history of the strain. Such behavior can manifest itself as creep, which is a gradual increase in strain under constant stress; stress relaxation, which is a gradual decrease in stress in a specimen held at constant strain; loadrate dependence of the stiffness; attenuation of sonic or ultrasonic
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waves; or energy dissipation in bone loaded dynamically (internal friction). Experimental modalities based on each of the above phenomena have been used in the study of bone. The results have been converted to a common representation via the interrelationships inherent in the linear theory of viscoelasticity, to permit a direct comparison of results. In the case of tension / compression, there is very significant disagreement among the published results. This disagreement may result from nonlinear viscoelastic behavior not accounted for in the transformation process, or from experimental artifacts. In the case of shear deformation, however, there is good agreement between results obtained in different kinds of experiments. The loss tangent, which is proportional to the ratio of energy dissipated to energy stored in a cycle of deformation, achieves a minimum value of about 0.01 at frequencies from 1 to 100 Hz. This is a linear form of hysteresis. At lower and higher frequencies, the loss tangent, hence the magnitude of viscoelastic effects, is greater, e.g. 0.08 at 1 MHz and at one microHz. To compare, the loss tangent of quartz may be less than one part per million, in metals, from one part in ten thousand to 0.01, in hard plastics from 0.01 to 0.1, and in soft polymers, it may attain values greater than 1. It is notable that the minimum energy dissipation in bone occurs in a frequency range characteristic of load histories during normal activities.
