Abstract
In recent years, several mathematical models have been proposed to describe the quasi-static response of fiber-reinforced materials, consisting of continuous, elastic fibers embedded in a linear viscoelastic matrix. By assuming that geometric dispersion (dispersion resulting from the internal geometry of the material) is small in comparison to viscoelastic dispersion (dispersion resulting from the viscoelastic nature of the material), these proposed constitutive equations can be extended from a quasi-static regime to a dynamic regime. Here, we examine how the extension to the dynamic regime may be accomplished, compare the results with a theoretical model that includes geometric dispersion, and use the results of an experimental program to evaluate the models. In general, the quasi-static constitutive equations predict phase velocities that are larger than that predicted by the model which contains geometric dispersion and attenuation coefficients that are lower; and, the experimental results agree with the theoretical predictions, provided the fibers were spread more or less uniformly over the cross section.
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Abbreviations
- a t :
-
time-temperature shift factor
- C :
-
wave velocity, cm/s
- \(\tilde C\) :
-
complex-wave velocity, cm/s
- E :
-
relaxation modulus, dyne/cm2
- E 1,E 2 :
-
complex moduli, dyne/cm2
- G 1,G 2 :
-
complex shear moduli, dyne/cm2
- h :
-
half the distance between fiber centers, cm
- i :
-
imaginary unit
- M :
-
index for Prony series
- N :
-
momentum transfer term, dyne/cm4
- T :
-
temperature, °C
- t :
-
time, s
- u :
-
displacement, cm
- V :
-
volume fraction
- x :
-
material position, cm
- α:
-
exponential attenuation, Nepers/cm
- γ:
-
wave number, 1/cm
- ξ:
-
fiber radius, cm
- δ:
-
phase angle for complex moduli, rad
- ε:
-
displacement gradient, cm/cm
- ϱ:
-
density, gm/cm3
- σ:
-
stress, dyne/cm2
- τ:
-
reduced time, s
- ω:
-
angular frequency, rad/s
- Ω:
-
reduced angular frequency, rad/s
- E :
-
equilibrium
- f :
-
fiber
- i :
-
index for Prony series
- m :
-
matrix
- o :
-
constant
- R :
-
reference
References
Calvit, H. H., Sutherland, H. J. andWillcox, M. G., Jr., “A Quasi-Static Investigation of Fiber-Reinforced Viscoelastic Materials,”EMRL 1088, The Univ. of Texas, Austin, TX (1970).
Ferry, J. D., Viscoelastic Properties of Polymers, John Wiley & Sons, Inc., New York (1961).
Kolsky, H., “Experimental Studies of the Mechanical Behavior of Linear Viscoelastic Solids,”Pergamon Press, New York (1966);reprinted from “Mechanics and Chemistry of Solid Propellants,” Proc. 4th Symp. Naval Structural Mechanics (1965).
Bedford, A. andStern, M., “Toward a Diffusing Continuum Theory of Composite Materials,”J. App. Mech., Series E,38,8–14 (1971).
Bedford, A. andStern, M., “On Wave Propagation in Fiber-Reinforced Viscoelastic Materials,”J. Appl. Mech., Series E,37,1190–1192 (1970).
Schapery, R. A., “A Simple Collocation Method for Fitting Viscoelastic Models to Experimental Data,” GALCIT, SM61-23A (1961).
Knauss, W. G. andMueller, H. K., “The Mechanical Characterization of Solithane 113 in the Swollen and Unswollen States,”Graduate Aeron. Lab., Polymer Lab. Chemical Engr., Calif. Inst. of Tech., Pasadena, CA, AFRPL-TR-68-125 (1967).
Thompson, D. andSharma, M. G., “Some Aspects of the Deformation of a New-Hookean Material in Compression,”JPL Tech. Rept. No. 5, JPL Contract No. 950875, The Pennsylvania State Univ., University Park, PA (1967).
Arenz, R. J., Ferguson, C. W. andWilliams, M. L., “The Mechanical and Optical Characterization of a Solithane 113 Composition,”Experimental Mechanics,7(4),183–188 (1967).
Williams, M. L., Landel, R. F. andFerry, J. D., “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming,”J. Amer. Chem. Soc.,77,3701–3707 (1955).
Lupton, J., du Pont, personal communication (Feb. 9, 1970).
Williams, M. L. andBender, M. F., “The Extension of Unoriented Nylon 66 Filaments, Part II: Phenomenological Relations,”Textile Research J.,33,1023–1026 (1963).
Kolsky, H., andLee, S. S., “The Propagation and Reflection of Stress Pulses in Linear Viscoelastic Media,”Tech. Rept. No. 5, Office of Naval Research, Brown Univ., Providence, RI (1962).
Ripperger, E. A. andYeakley, L. M., “Measurement of Particle Velocities Associated with Waves Propagation in Bars,”Experimental Mechanics,3(2),47–56 (1963).
Calvit, H. H. andWatson, H., Jr., “An Experimental Investigation of Pulse Propagation in Neoprene Filaments,”EMRL 1050, The Univ. of Texas, Austin, TX (1968).
Leininger, J. R. andNachlinger, R. R., “On the Accuracy of a Dynamic Viscoelastic Experiment,”unpublished note, Univ. of Houston, Houston, TX (1970).
Hunter, S. C., “Viscoelastic Waves,”Progress in Solid Mechanics, I, ed. by I. N. Sneddon andR. Hill, Interscience Publishers, Inc., New York, ch. 1, 3–57 (1960).
Sutherland, H. J., “A Quasi-Static and Dynamic Investigation of Fiber-Reinforced Viscoelastic Materials,”Dissertation, The Univ. of Texas, Austin, TX (1970).
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Sutherland, H.J., Calvit, H.H. A dynamic investigation of fiber-reinforced viscoelastic materials. Experimental Mechanics 14, 304–310 (1974). https://doi.org/10.1007/BF02324952
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DOI: https://doi.org/10.1007/BF02324952