Viscoelastic Properties of Foamed Thermoplastic Vulcanizates and their Dependence on Void Fraction

Transcription

1 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction Junhong Park 1,, Thomas Siegmund * and Luc Mongeau 1, Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN , USA Received: 8 January 003 Accepted: 30 April 003 ABSTRACT Experimental techniques to measure the viscoelastic mechanical properties of foamed elastomers at high frequencies were developed and applied to study foamed thermoplastic vulcaniates. The measurements were performed for foams with a wide range of void fractions such that a transition from a true porous solid to a foam was present. The frequency and time dependence of the dynamic and relaxation moduli, respectively, were measured and were shown to depend on the loss factor of the elastomer. The measured variation of the dynamic and the relaxation moduli with the void fraction did not depend on frequency and time in the frequency and time range of interest. This suggested that the time and frequency dependence of elastic moduli was determined mostly by the void fraction and the corresponding material microstructure. Property measurement results were compared to predictions for porous solids using the Mori-Tanaka method as well as methods for cellular solids. 1. INTRODUCTION Thermoplastic vulcaniates (TPVs) have been proposed as an alternative for thermoset elastomers in many applications. TPVs are a special class of thermoplastic elastomers in which cross-linked elastomer particles are embedded in a thermoplastic polymer matrix (1). Dense TPVs have mechanical properties similar to those of conventional thermoset elastomers. In addition, they are amenable to fabrication into parts using the techniques used for thermoplastic materials such as extrusion, blow molding, and injection molding (1,). The mechanical behavior of TPVs is similar to that of conventional rubber in general, but considerable differences exist. Boyce et al. (3,4) measured the uniaxial and plane strain compressive stress-strain behavior of a class of TPVs and showed that a constitutive model different from those used for rubber material is required to accurately predict the observed behavior of TPVs for various volume fractions of the thermoplastic matrix. The stress- * Corresponding Author Cellular Polymers, Vol., No. 3, /6/03, 8:39 am

2 Junhong Park, Thomas Siegmund and Luc Mongeau strain behavior transited from that of the rubber material to that of the thermoplastic matrix as the volume fraction of the matrix increased, which resulted in higher stiffness and higher yield stress. Conventional thermoset elastomers have been widely used for the suppression of sound and vibration, and the addition of damping to mechanical structures. In many situations, foamed elastomers are advantageous since they require less raw material for a given part volume, and consequently they allow substantial weight savings compared with solid materials. Foams are also useful when very compliant materials are needed. For successful applications of TPVs in such cases, the understanding of the dynamic mechanical properties of foamed TPVs is required. Especially, it is desirable to establish relationships between the void volume fraction and the dynamic moduli of foamed TPVs. Several dynamic material test systems are commercially available for the measurement of the dynamic moduli. Nevertheless, the frequency range of these devices is commonly limited to frequencies of less than 00 H. Then, high frequency material properties are obtained by measurements over a range of temperature, and subsequent extrapolation of measured low frequency data to high frequencies using the WLF equation (5). Alternatively, several methods have been proposed to directly measure the dynamic mechanical properties of materials at high frequencies. Madigosky and Lee (6) used a wave propagation approach to measure Young s modulus and the loss factor for frequencies up to 10 kh. Elongated strips of an elastomer were used as specimens with a phonograph cartridge applied to measure the longitudinal vibrations of the sample induced by an electromagnetic shaker. In this method the length of the specimens is required to be large in order to prevent resonance as it is assumed that there are no reflections from the end of the sample. This requirement limits the applicability of the method to materials for which rather long specimens can be obtained. A measurement method based on standing longitudinal waves was proposed by Prit (7) and Madigosky and Lee (8). Again, a rod-like specimen (length of approximately 15 cm) was excited by a shaker at one end. Longitudinal accelerations were measured at both specimen ends using accelerometers. The elastic modulus and the loss factor of the sample were calculated from the measured transfer function between the two accelerometer signals. This measurement method was eventually adopted as an ANSI standard (9). More recently, a method for the measurement of the frequency-dependent shear and bulk modulus of solids for specimens of arbitrary shapes were 138 Cellular Polymers, Vol., No. 3, /6/03, 8:39 am

3 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction developed by Willis et al. (10,11). Their approach employs laser interferometers for simultaneous measurements of the transverse and longitudinal vibration response of the specimen. The present study reports measurements of the mechanical properties of extruded water foamed TPVs. Extruded, rod-shaped specimens were used in the experiments. High frequency data for longitudinal elastic modulus and loss factor were obtained in the frequency range from 50 H to kh using the transfer function method. Relaxation tests were performed both along the extrusion direction and in the direction perpendicular to the extrusion. The results were compared with predictions from micromechanics models.. VISCOELASTIC PROPERTIES OF FOAMED TPVS.1 Material Santoprene rubber 11-68W8 was extruded into rods of rectangular cross section using water as the sole blowing agent (1). Specimens with six different values of the void volume fraction ranging from 0.0 to 0.7 were considered. All samples were foamed from a dense TPV with a density of 980 kg/m 3. The extrusion process results in a transversely isotropic material. Figure 1 shows the schematic of extruded samples and an appropriate coordinate system. Figures (a) and (b) depict material microstructures captured by microtomography scans for samples of density of 650 kg/m 3 and 50 kg/m 3, respectively. The material microstructures depend strongly on the void volume fraction. Materials with low void volume fraction can be considered as true porous solids, while cellular solids are obtained at large void volume fractions (13). The sie of the voids in the foamed TPV is much larger than the cross-linked rubber particles within the TPV matrix, which are less than µm in diameter (1). Furthermore, it is assumed in the present investigation that the matrix (i.e. the dense TPV) is isotropic. The mechanical properties of the foamed samples are transversely isotropic. In this case, five independent elastic constants - E, E x = E y, the Young s moduli in longitudinal (extrusion) and in the transverse directions, respectively; ν xy, ν x, the Poisson s ratios, and G x = G y, the shear modulus in planes normal to the plane of transverse isotropy describe the elastic properties of the samples (14). In the following section, the complex modulus, Ê, the dynamic, E d, and the relaxation modulus, E r, are defined. The relationships between these quantities are described. Identical procedures apply to both the moduli in longitudinal and transverse directions. Cellular Polymers, Vol., No. 3, /6/03, 8:39 am

4 Junhong Park, Thomas Siegmund and Luc Mongeau Figure 1 Schematic of extruded and foamed specimens together with the appropriate coordinate systems (a) (b) Figure Microtomographic images of microstructure of foamed TPV materials. (a) ρ=630 kg/m 3, and (b) ρ=50 kg/m Cellular Polymers, Vol., No. 3, /6/03, 8:39 am

5 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction. Complex and Relaxation Modulus A commonly accepted method to model the dissipation of vibration energy within a solid is to use complex elastic moduli for the description of the dynamic properties of the solid. For uniaxial vibrations the complex modulus is defined as (15), ( ) ( ) = ( ) + ( ) = ( ) + ( ) ˆ σˆ ω E( ω) = Ed ω iel ω Ed ω [ 1 iη ω ] ˆ ε ω (1) where E d and E l are the dynamic and loss moduli, respectively, i = 1 and η is the loss factor. The Fourier transforms of the stress and the strain are defined iωt iωt asσˆ ( ω) σ ( ) ˆ = te dt and ε( ω) = ε( te ) dt. Under the assumption that the dynamic properties do not exhibit resonance-like frequency dependences the dynamic Young s modulus was approximated by a power law (15) : E d ( ω) = π C ω () d η where C d is a constant determining the frequency dependence of the dynamic moduli. In equation (), the static modulus was assumed to be negligibly smaller than the dynamic modulus. A similar relationship was used by Lagakos et al. (16) to measure the elastic modulus of various polymers as a function of frequency and temperature. The frequency dependence of the material properties can also be inferred from relaxation tests. In a relaxation test, a strain step function with a magnitude ε 0 is imposed onto a specimen, and the resulting time-varying stress is measured. The relaxation modulus, E r (t) = σ(t)/ε 0, is related to the dynamic modulus through (15) Ed ( ω) = ω Er ( t)sinωtdt (3) The time dependent relaxation modulus can be derived from the frequency dependent dynamic modulus of equation (3) and using an inverse Fourier sine transform. The resulting time dependent relaxation modulus is: E t r ( ) = η η C d t = Ct r 1 π π η π η Γ sin (4) π Cellular Polymers, Vol., No. 3,

6 Junhong Park, Thomas Siegmund and Luc Mongeau where Γ is the Gamma function. In equation (4), the relaxation modulus decreases with time and the magnitude of the time dependence in equation (4), η/π, is the same as the frequency dependence of the dynamic modulus shown in equation (). The above time- and frequency- dependence of viscoelastic properties is different from commonly used viscoelastic functions derived from mechanical model analogies such as the Maxwell element and the Voigt element. Despite its simple form, the viscoelastic functions, equations () and (4), well reproduced the trends observed in the experiments as described in the later section, and also were appropriate to be used in determination of the dependence of the viscoelastic properties on the void fraction..3 Prediction of Effective Elastic Moduli Two classes of micromechanical models are being used to predict the elastic properties of the foamed TPV. For low void volume fractions, a true porous solid is present such that its properties can be predicted from methods based on the Eshelby approach (17-0). Voids are spheroidal with aspect ratio l/d. The material anisotropy is taken into account by use of the Eshelby tensor within the Mori-Tanaka method (0). The elastic moduli of foamed TPVs were estimated using the relations: E E 0 = 1+ φ ( 1 0 ) A + ν A A 1 Ex 0A3 1 0 A4 1 0 A5 A 1 E = + ν + ( ν ) + ( + ν ), φ 0 A (5a,b) 1 where E 0 and ν 0 are Young s modulus and Poisson s ratio of the un-foamed TPV, respectively, and φ=1-ρ/ρ 0 is the void fraction. The same notations were used for constants, A and A i, as the one in Tandon and Weng (0). The factors A and A i depend on the aspect ratio. For increased values of the void volume fraction the material under consideration loses the microstructural features of a porous solid. Then the material can better be thought of an interconnected network of a cellular solid consisting of solid struts and plates. For the present material produced by the extrusion processes all wall material is strongly aligned along the extrusion direction. Such a microstructure can be thought of as a honeycomb structure. In the case of a honeycomb model, the longitudinal and transverse elastic moduli are given by (13) : 14 Cellular Polymers, Vol., No. 3,

7 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction E E = Ex φ, E = ( 1 φ) (6a,b) 0 0 If the loss factor is constant, the relations shown in equations (5) and (6) can be used for the prediction of both the dynamic and the relaxation moduli, equations () and (4), independently on the time and the frequency. As observed in the measurement performed in this study (discussed in section 4), the loss factor was not affected by the voids and remained approximately constant in the frequency and time range of interest. 3. TEST PROCEDURES 3.1 The Transfer Function Method Figure 3 shows a sketch of the general configuration used for the transfer function method test. The method is applied to investigations of vibrations in the longitudinal direction of the rod-shaped samples. One end of the specimen is excited at a known frequency; a lumped mass is attached at the other end. The equation of motion governing the one-dimensional longitudinal waves in the sample is w ρ w = Eˆ (7) t where w is the longitudinal displacement, assumed to be small in amplitude, and ρ is the density of the specimen. For a simple harmonic excitation, the longitudinal displacement is expressed as ( )= { } wt we i ω, Re ˆ( ) t After the above separation of the variables, equation (7) is rewritten in the frequency domain as wˆ + ˆ ˆ 0 ( k ) w= (9) where ˆk = β α i is the complex wavenumber, with β and α the real and imaginary parts, respectively. The complex wavenumber is related to the angular frequency through k ˆ = ωρ / E ˆ. From the measured wavenumber (8) Cellular Polymers, Vol., No. 3,

8 Junhong Park, Thomas Siegmund and Luc Mongeau Figure 3 Experimental setup used in measurements of longitudinal dynamic Young s modulus using the resonance method in the solid, the dynamic modulus and the loss factor are calculated as {( )} ωρ β α E d = ( β + α ) αβ, η = (10 a, b) β α ( ) Using equation (10), the viscoelastic properties of the specimen can be calculated from the measured wavenumber. In the transfer function method, a standing wave solution is used to estimate the wavenumber from the measured transfer functions. The measured transfer function between the input and output displacements is related to the wavenumber through (7) wˆ( 0) M Re cosh Lcos L Lsinh Lcos L Lcosh Lsin L wl ˆ( ) = α β + ( α α β β α β ) m (11a) w Im ˆ( 0 ) M sinh Lsin L Lcosh Lsin L Lsinh Lcos L wl ˆ( ) = α β + ( α α β + β α β ) m (11b) 144 Cellular Polymers, Vol., No. 3,

9 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction where M is the mass, L is the length of the specimen, and m is the attached mass. Using equation (11), the complex wavenumber was estimated from the measured transfer function. Note that the transfer function method requires only two measurements to calculate the complex wavenumber. A disadvantage of this method is that small experimental errors may result in large deviations of the measured viscoelastic properties from their actual values. Experimental errors may be caused by the unwanted generation of bending waves, or noise in the acceleration signals (an important factor when the accelerometer attached to the mass is near a node in the longitudinal wave). To calculate the complex wavenumber from equation (11), the Newton- Raphson method was used. There are infinitely many solutions that satisfy equation (11). Among the possible solutions, only one is physically acceptable. The others are artifacts due to the periodicity of trigonometric functions in equation (11) and do not yield the viscoelastic properties with the frequency dependence shown in equation (). Figure 3 shows the experimental apparatus. An electrodynamic shaker (B&K) was used for the generation of longitudinal waves. The excitation was applied to a mass attached to the foam sample using an adhesive. Accelerometers were attached to the mass blocks at both ends of the specimen. Specimen dimensions and mass values are shown in Table 1. Table 1 Mechanical properties of the six TPV materials, and values of the attached mass ρ [ kg/m 3 ] φ S - 10 m ] [ 5 L (transfer function method) [m] m [kg] L (relaxation test) [m] Cellular Polymers, Vol., No. 3,

10 Junhong Park, Thomas Siegmund and Luc Mongeau 3. Relaxation Tests Experimental methods based on one-dimensional wave propagation require specimens with high aspect ratio, and yield results along one specimen direction. For the present case the transfer function method can only be applied for the determination of material properties in the extrusion () direction. To investigate the dynamic material properties for the transverse direction, the relationships between the relaxation and dynamic moduli are explored. Relaxation tests were performed using a low force electro-dynamic test system, ELF 300 by EnduraTEC Systems Corporations. Specimens were tested both along the x- and - directions. Specimen sies in these two directions were identical and are shown in Table 1. The step displacement of 1.0 mm was imposed as initial conditions and the time-dependent relaxation modulus was measured. 4. RESULTS AND DISCUSSION 4.1 Viscoelastic Properties of TPV Figure 4 shows the magnitude and the phase of the transfer function, ŵ(l)/ ŵ(0), measured using the two accelerometers. In the frequency range of interest, three resonances were observed in the measured transfer function for all six Figure 4 Measured phase and magnitude of the transfer function between displacement input, ŵ(o), and resulting displacement, ŵ(l) 146 Cellular Polymers, Vol., No. 3,

11 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction TPV materials investigated. Note that the phase of the transfer function at resonances are π/ or π/. The resonance frequencies of first, second, and third natural modes range from 15 0 H, H, and H, respectively. The resonance frequencies of each mode are larger for the high-density materials, which resulted from the increasing dynamic moduli with decreasing void fractions. Using the measured transfer function, the dynamic modulus and the loss factor of the six samples were obtained through equation (10) and are shown in Figure 5. The frequency dependence of the longitudinal dynamic modulus is linear in logarithmic scales, except at anti-resonances. At anti-resonances, the measured longitudinal dynamic Young s moduli appear to be distorted by the experimental errors related to the limited dynamic range of the accelerometers, and the small actual phase difference between two displacements measured at both ends of the samples. The scattering of acoustic waves from individual voids enhances damping in foams (17). The lowest frequency at which resonant scattering effects associated with the voids affect the mechanical properties of the foam is c L /d, where d is the cavity diameter and c L is the longitudinal wave speed. Since this frequency is lower for the larger cavity diameter, this effect is expected to be most severe in the lowest density material among the samples under study. For the lowest density foam (50 kg/m 3 ) of the present study, the void diameter was approximately 0.4 mm and the wave speed was approximately 00 m/s. The corresponding resonance frequency then is approximately 50 kh, which is much greater than the frequency range of interest. Thus, resonant scattering effects did not contribute significantly in the frequency range of interest, and most of the dissipation of vibration energy was caused by material damping. Consequently, it may be expected that the loss factor, η, does not depend on the relative density. This is confirmed by the experimental results shown in Figure 5(b). The measured loss factors are approximately constant, equal to 0.15, regardless of relative density and frequency. Since the loss factor is not dependent on the void fraction, it also does not depend on the specimen direction, a fact later confirmed in the relaxation tests. From the measured loss factor of the foams, the exponent characteriing the power law dependence of the dynamic material properties, equation (), is thus fixed as x0.15/π = To estimate the remaining constants for equation (), linear regressions of the measured data of longitudinal Young s moduli were performed in the log-log diagram of Figure 5(a). The slope of the regression lines was constraint to η/π. To minimie the effects from the experimental error, the data near anti-resonance was excluded in the interpolation. The estimated value of C d, for the six TPV materials are shown Cellular Polymers, Vol., No. 3,

12 Junhong Park, Thomas Siegmund and Luc Mongeau (a) (b) Figure 5 Viscoelastic properties of foamed TPV materials measured using the resonance method. (a) Dynamic Young s modulus in longitudinal direction and (b) loss factor 148 Cellular Polymers, Vol., No. 3,

13 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction in Table. The results demonstrate that the variation of the longitudinal dynamic Young s moduli as a function of the relative density (or void fraction) did not depend on frequency within the frequency range of interest. The measured relaxation curves for specimens tested along the x- and - directions are given in Figure 6(a) and (b), respectively. Over the time period of the tests, the relaxation moduli followed very closely the power law relaxation behavior of equation (4). A regression of the measured relaxation modulus was performed based on the power law dependence of the relaxation moduli on time, equation (4), and using the already determined loss factor from the transfer function method. The same numerical procedures as for the determination of the dynamic moduli were followed. The resulting values of the constants, C r,x, C r,, for the power law relaxation law for the six TPV materials are shown in Table. Table also shows the measured ratio, C d, /C r,. The ratio obtained from the experimental data is very similar to the ratio given from equation (4), i.e., Γ( 1 ηπ) sin [( π η) ] = 5.. The results show that the power law frequency- and time-dependency shown equations () and (4) accurately characterie the viscoelastic properties of the TPVs within the frequency range and time periods covered. 4. Dependence of Effective Dynamic Moduli on Void Fraction The experimental data for the TPV foams, discussed in previous sections, confirm that the dynamic modulus varies with the void fraction, and does not depend on frequency. Thus, the power law dependence of the dynamic modulus, equation (), can be rewritten as Table Measured material constants, C d,, C r,, and C r,x for six TVP materials φ C d, C r, C r, x C d, / C r, Cellular Polymers, Vol., No. 3,

14 Junhong Park, Thomas Siegmund and Luc Mongeau (a) (b) Figure 6 Relaxation moduli measured for the six TPV materials. (a) x-direction and (b) -direction 150 Cellular Polymers, Vol., No. 3,

15 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction η π E C ω f φ (1) = ( ) d d0 v where C d0 is the corresponding material constant measured for the dense, unfoamed elastomer. A similar expression can be written for the relaxation moduli. The function f v describes the dependence of the modulus on the void fraction. For the present transverse isotropic material, f v has to be established independently for the longitudinal and transverse directions. Predictions for f v can be obtained from micromechanical models, equations (5) and (6). Figure 7 shows the predicted and experimentally determined dependence of f v on the void fraction. To take account of the spheroidal void shapes the predictions of the Mori-Tanaka method, equation (5), were evaluated for aspect ratios l/ d=1.0, 3.0, and For all void fractions the Mori-Tanaka method predicts values of f v that are greater than those determined experimentally. The prediction from the honeycomb model in the -direction is the upper limit of the dynamic moduli of the foam. The predicted value in the x-direction from the honeycomb model agrees well with the measured values when the void volume fraction is large. Figure 7 Measured variations of material constants f v with relative density for TPV and comparison with the predictions from micromechanical models Cellular Polymers, Vol., No. 3,

16 Junhong Park, Thomas Siegmund and Luc Mongeau Figure 8 compares the measured and predicted ratios between the longitudinal and transverse moduli, E r, /E r,x, as a function of the void fraction. The predictions of the anisotropy are considerably different for the Mori-Tanaka model and the honeycomb model. While for the Mori-Tanaka model the anisotropy remains small even for large void aspect ratios, the honeycomb model predicts a rapid increase in the anisotropy as the void fraction reaches 0.6 and approaches 1.0. At low void volume fractions the measured ratios, E r, /E r,x, are well captured by the prediction from the Mori-Tanaka method. However, for void volume fractions larger than 0.6 this is no longer correct. The predictions from the honeycomb material model are in better agreement with the experimentally determined data. Nevertheless, the simple honeycomb model overpredicts the measured degree of material anisotropy mainly due to the over- simplifications made in interpreting the microstructure of Figure (b) in terms of a honeycomb structure. 5. CONCLUSION The viscoelastic properties of foamed TPVs were measured using two different methods. The frequency-dependent dynamic and loss moduli were measured Figure 8 Measured and predicted ratio E r, /E r,x, and its dependence on the void fraction 15 Cellular Polymers, Vol., No. 3,

17 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction using a transfer function method. Time-dependent relaxation moduli were measured using relaxation tests. The measured loss factors were approximately constant in the frequency range of interests. The relationship between relaxation moduli and the dynamic moduli was obtained. It was shown that both quantities depend on the loss factor of the elastomer. This relationship was verified experimentally. The viscoelastic properties were measured for specimens with several void fractions. It was found that the variation of the dynamic and the relaxation moduli with the void fraction does not depend on frequency and time in the frequency and time ranges of interest. The foamed TPVs exhibited strong anisotropic material properties. The degree of anisotropy was evaluated using the relaxation tests. For prediction of the dependence of the material properties, the Mori-Tanaka method and a honeycomb material model were used to take this anisotropy into account. The Mori-Tanaka method captures the material anisotropy well only for low values of the void volume fraction, i.e. as long as a true solid exists. Above a void volume fraction of 0.6, however, a transition to a cellular solid occurs and then models for cellular solids are to be used to capture the material behavior. 6. ACKNOWLEDGEMENT The authors express their thanks to Ford Motor Company and Advanced Elastomer Systems, for financial support and their guidance. The support for testing equipment through the Air Force Office of Scientific Research (grant # ) is acknowledged. The contribution of the Herrick Laboratories Technical staff is also gratefully acknowledged. REFERENCES 1. S. Abdou-Sabet and S. Datta 000 Thermoplastic vulcaniates. In: D. R. Paul and C. B. Bucknall (Eds.) Polymer Blends: Formulation and Performance. New York: John Wiley & Sons (Chapter 35).. A. Y. Coran and R. P. Patel 1996 Thermoplastic elastomers based on dynamically vulcanied elastomer-thermoplastic blends. In: G. Holden, N. R. Legge, R. P. Quirk, and H. E. Schroeder (Eds.) Thermoplastic Elastomers. Cincinatti: Hausen (Chapter 7). 3. M. C. Boyce, K. Kear, S. Socrate, and K. Shaw 001 Journal of the Mechanics and Physics of Solids 49, Deformation of thermoplastic vulcaniates. Cellular Polymers, Vol., No. 3,

18 Junhong Park, Thomas Siegmund and Luc Mongeau 4. M. C. Boyce, S. Socrate, K. Kear, O. Yeh, and K. Shaw 001 Journal of the Mechanics and Physics of Solids 49, Micromechanisms of deformation and recovery in thermoplastic vulcaniates. 5. J. D. Ferry 1980 Viscoelastic Properties of Polymers. New York: John Wiley & Sons. 6. W. M. Madigosky and G. F. Lee 1979 Journal of the Acoustical Society of America 66, Automated dynamic Young s modulus and loss factor measurements. 7. T. Prit 198 Journal of Sound and Vibration 81, Transfer function method for investigating the complex modulus of acoustic materials: Rod-like specimen. 8. W. M. Madigosky and G. F. Lee 1983 Journal of the Acoustical Society of America 73, Improved resonance technique for materials characteriation. 9. ANSI S.-1998, Resonance method for measuring the dynamic mechanical properties of viscoelastic materials. American National Standards Institute, published though the Acoustical Society of America, New York, NY, R. L. Willis, T. S. Stone, and Y. H. Berthelot 1997 Journal of the Acoustical Society of America 10, An experimental-numerical technique for evaluating the bulk and shear dynamic moduli of viscoelastic materials. 11. R. L. Willis, L. Wu, and Y. H. Berthelot 001 Journal of the Acoustical Society of America 109, Determination of the complex Young and shear dynamic moduli of viscoelastic materials. 1. Technical Correspondence 1998 Water Forming of Santroprene Thermoplastic Rubber. Akron, OH: Advanced Elastomer Systems. 13. L. J. Gibson and M. F. Ashby 1999 Cellular Solids: Structure and Properties, Oxford: Cambridge University Press. 14. S. G. Lekhnitskii 1963 Theory of elasticity of an anisotropic elastic body. San Francisco: Holden-Day. 15. T. Prit 1998 Journal of Sound and Vibration 14, Frequency dependences of complex moduli and complex Poisson s ratio of read solid materials. 16. N. Lagakos, J. Jarynski, J. H. Cole, and J. A. Bucaro 1986 Journal of Applied Physics, 59, Frequency and temperature dependence of elastic moduli of polymers. 17. W. Madigosky and K. Scharnhorst 1990 Sound and Vibration damping with Polymers, edited by R. D. Corsaro and L. H. Sperling published by The American Chemical Society, Washington, DC, Acoustic wave propagation in materials with inclusions or voids. 18. R. M. Christensen 1998 Journal of Elasticity 50, Two theoretical elasticity micromechanics models. 154 Cellular Polymers, Vol., No. 3,

19 Viscoelastic Properties of Foamed Thermoplastic Vulcaniates and their Dependence on Void Fraction 19. R. M. Christensen 1990 Journal of the Mechanics and Physics of Solids 38, A critical evaluation for a class of micromechanics models. 0. G. P. Tandon and G. J. Weng 1984 Polymer Composites 5, The effects of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites. APPENDIX A: NOMENCLATURE A, A i parameters in the Mori-Tanaka method C d, C r c L d E x, E E 0 constants determining frequency and time dependence of moduli [Pa] longitudinal wave speed [m/s] radius of the void [m] transverse and longitudinal Young s moduli of specimen [Pa] Young s modulus of dense, un-foamed elastomer [Pa] Ê, E d, E l, E r complex, dynamic, loss and relaxation modulus, respectively [Pa] f v G ˆk =β α i l L m M w constant that determine dependence of modulus on void fraction shear modulus of specimen [Pa] complex wavenumber [rad/m] length of void [m] length of specimen [m] mass of attached mass block [kg] mass of specimen [kg] longitudinal displacement [m] x, y, coordinates Cellular Polymers, Vol., No. 3,

20 Junhong Park, Thomas Siegmund and Luc Mongeau α, β real and imaginary parts of wave number ε ν xy, ν x ν 0 η φ strain Poisson s ratios Poisson s ratios of the dense, un-foamed elastomer loss factor void volume fraction ρ density of specimen [kg/m 3 ] σ ω Stress [Pa] frequency [rad/s] 156 Cellular Polymers, Vol., No. 3,