Skip to main content
SpringerLink
Log in

Viscoelastic Waves and Boundary Value Problem

  • Chapter
Mechanical Behaviour of Engineering Materials

  • 432 Accesses

Abstract

When a localized disturbance is applied suddenly in a medium, it will soon propagate to other parts of this medium. This simple fact constitutes a general basis for the interesting subject of “wave propagation”. Well-cited examples of wave propagation in different media include, for instance, the transmission of sound in air, the propagation of seismic disturbances in the earth, the transmission of radio waves, among others. In the particular case when the suddenly applied disturbance is mechanical, e.g., a suddenly applied force, the resulting waves in the medium are due to stress effects and, thus, these waves are referred to as “stress waves” . Our attention in this chapter is focussed on the propagation of stress waves in viscoelastic solid media. In our representation, we consider the solid medium to be a continuum. Hence, the mechanics of wave motion in the medium will be dealt with from a continuum mechanics point of view. The basic concepts of continuum mechanics have been presented in Chapter 2. In such continuum, the solid medium, the disturbance is generally considered to spread outward in a three-dimensional sense (Graff, 1975). A wavefront is considered to be associated with the outward propagating disturbance. Consequently, particles of the medium that are located ahead of the wavefront are assumed to have experienced no motion, meantime, particles that are located behind the front are visualized to have experienced motion and may continue to vibrate for some time.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Achenbach, J.D. and Chao, C.C. (1962) A three-parameter viscoelastic model particularly suited for dynamic problems, J. Mech. Phys. Solids 10, 245–52.

    ADS  MATH  Google Scholar 

  • Achenbach, J.D., Vogel, S.M. and Herrmann, G. (1966) On stress waves in viscoelastic media conducting heat, in: Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids, Eds. H. Parkus and L.I. Sedov, Springer, Berlin, pp. 1–15.

    Google Scholar 

  • Ahrens, T.J. and Duvall, G.E. (1966) Stress relaxation behind shock waves in rocks, J. Geophys. Res. 71, 4349–60.

    ADS  Google Scholar 

  • Alfrey, T., Jr. (1948) Mechanical Behaviour of High Polymers, Interscience, New York.

    Google Scholar 

  • Bailey, P. and Chen, P.J. (1971) On the local and global behaviour of acceleration Waves, Arch. Ration. Mech. Analysis 41, 121. Addendum: Asymptotic Behaviour, ibid 44, 212 (1972).

    MathSciNet  ADS  MATH  Google Scholar 

  • Barberan, J. and Herrara, I. (1966) Uniqueness theorems and speed of propagation ofsignals in viscoelastic materials, Arch. Rat. Mech. Anal. 23, 173–90.

    MATH  Google Scholar 

  • Barker, L.M. (1968) The fine structure of compressive and release wave shapes in aluminium measured by the velocity interferometer technique, in: Behaviour of Dense Media Under High Dynamic Pressures, Gordon and Breach, pp. 483–505.

    Google Scholar 

  • Barker, L.M. and Hollenbach, R.E. (1964) System for measuring the dynamic properties of materials, Rev. Sci. Inst. 35(6), 742–6.

    ADS  Google Scholar 

  • Barker, L.M. and Hollenbach, R.E. (1965) Interferometer technique for measuring the dynamic mechanical properties of materials, Rev. Sci. Inst. 36, 1617–20.

    ADS  Google Scholar 

  • Barker, L.M and Hollenbach, R.E. (1970) Shock wave studies of PMMA, Fused silica and sappire, J. Appl. Phys. 41, 4208–26.

    ADS  Google Scholar 

  • Bland, D.R (1960) The Theory of Linear Viscoelasticity, Pergamon, New York.

    MATH  Google Scholar 

  • Biot, M.A. (1958) Linear thermodynamics and the mechanics of solids, Proceedings 3rd U.S. Natl. Congr. Appl. Mech., ASME, pp.1–18.

    Google Scholar 

  • Boussinesq, M.J. (1885) In Todhunter and Pearson-A History of Theory of Elasticity and of the Strength of Materials, Vol. II, Part 2, Dover reprint (1960), pp. 185–357.

    Google Scholar 

  • Bradfield, G. (1951) Internal friction of solids, Nature 167, 1021–3.

    ADS  Google Scholar 

  • Brull, M.A. (1953) A structural theory incorporating the effect of time-dependent elasticity, in Proceedings 1st Midwestern Conf. On Solid Mechanics, pp. 141–7.

    Google Scholar 

  • Calvit, H.H. (1967) Numerical solution of the problem of impact of a rigid sphere onto a linear viscoelastic half-space and comparison with experiment, Int. J. Solid Structures 3, 951–66.

    Google Scholar 

  • Chao, C. and Achenbach, J.D. (1964) A Simple viscoelastic analogy for stress waves, in: Stress Waves in Anelastic Solids, IUTAM Symposium, Eds. H. Kolsky and W. Prager, Brown University, Providence, RI, Apr. 3-5, 1963, Springer, Berlin, pp. 222–38.

    Google Scholar 

  • Chen, P.J. and Gurtin, M.E. (1970). On the growth of one-dimensional shock waves in materials with memory, Arch. Ration. Mech. Analysis 36, 33–46.

    MathSciNet  ADS  MATH  Google Scholar 

  • Chen, P.J. and Gurtin, M.E. (1972a) On the use of experimental results concerning steady shock waves to predict the acceleration wave response of nonlinear viscoelastic materials, J. Appl. Mech. 39, 295–6.

    ADS  Google Scholar 

  • Chen, P.J. and Gurtin, M.E. (1972b) Thermodynamic influences on the growth of one-dimensional shock waves in materials with memory, Z. Angew, Math. Phys. 23, 69–79.

    MATH  Google Scholar 

  • Chen, P.J., Gurtin, M.E. and Walsh, E.K. (1970) Shock amplitude variation in Polymethyl Methacrylate for fixed values of the strain gradient, J. Appl. Phys. 41, 3557–8.

    ADS  Google Scholar 

  • Christensen, R.M. (1971) Theory of Viscoelasticity, Academic Press, New York.

    Google Scholar 

  • Chu, B.T. (1962) Stress Waves in IsotropiC Viscoelastic Materials, Division of Engineering, Brown University, Providence, March 1962.

    Google Scholar 

  • Coleman, B.D. (1964) Thermodynamics, strain impulses and viscoelasticity, Arch. Ration. Mech. Analysis 17, 230–254.

    ADS  MATH  Google Scholar 

  • Coleman, B.D., Greenberg, J.M. and Gurtin, M.E. (1966) Waves in materials with memory. V. On the amplitude ofacx:eleration waves and mild discontinuities, Arch. Ration. Mech. Analysis 22, 333–54.

    MathSciNet  ADS  MATH  Google Scholar 

  • Coleman, B.D. and Gurtin, M.E. (1965a) Waves in materials with memory. II. On the growth and decay of one-dimensional acceleration waves, Arch. Ration. Mech. Analysis 19, 239–65.

    MathSciNet  ADS  MATH  Google Scholar 

  • Coleman, B.D. and Gurtin, M.E. (1965b) Waves in materials with memory. III. Thermodynamic influences on the growth and decay of acceleration waves, Arch. Ration. Mech. Analysis 19, 266–98

    MathSciNet  ADS  Google Scholar 

  • Coleman, B.D. and Gurtin, M.E. (1965c) Waves in materials with memory. IV. Thermodynamics and the velocity of general acceleration waves, Arch. Ration. Mech. Analysis 19(5), 317–38

    MathSciNet  ADS  MATH  Google Scholar 

  • Coleman, B.D. and Gurtin, M.E. (1966) Thermodynamics and one-dimensional shock waves in materials with memory, Proc. R. Soc. A 292, 562–74.

    MathSciNet  ADS  Google Scholar 

  • Coleman, B.D., Gurun, M.E. and Herrara, R.I. (1965) Waves in materials with memory. I. The velocity of one-dimensional shock and acceleration waves, Arch. Ration. Mech. Analysis 19, 1–19.

    ADS  MATH  Google Scholar 

  • Cornelliussen, A.H. et al (1961) Brown University Report NORD 18594/5.

    Google Scholar 

  • Cunningham, J.R. and Ivey, D.G. (1956) Dynamic properties of various rubbers at high frequencies, J. Appl. Phys. 27, 967–74.

    ADS  Google Scholar 

  • Dally, J. W. and Riley, W.F. (1965) Experimental Stress Analysis, McGraw-Hill, New York.

    Google Scholar 

  • Dove, R.C. and Adams, P.H. (1964) Experimental Stress Analysis and Motion Measurement, Charles Merril Books, Columbus, Ohio.

    Google Scholar 

  • Dunwoody, J. (1972) One-dimensional shock waves in heat conducting materials with memory. I. Thermodynamics, Arch. Ration. Mech. Analysis 47, 117–48.

    MathSciNet  ADS  MATH  Google Scholar 

  • Duvall, G.E. and Alverson, R.C. (1963) Fundamental Research, Tech. Summary Report 4, Stanford Research Inst., Menlo Park.

    Google Scholar 

  • Edelstein, W.S. and Gurtin, M.E. (1964) Uniqueness theorems in the linear dynamic theory of anisotropic viscoelastic solids, Arch. Ration. Mech. Anal. 17, 47–60.

    MathSciNet  MATH  Google Scholar 

  • Ferry, J.D. and Williams, M.L. (1952) II. Approximation methods for determining the relaxation time spectrum of a viscoelastic material, J. Colloid Sci. 7, 347–53.

    Google Scholar 

  • Fung, Y.C. (1965) Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

  • Glauz, R.D. and Lee, E.H. (1954) Transient wave analysis in a linear time-dependent material, J. Appl. Phys. 25, 947–53

    MathSciNet  ADS  MATH  Google Scholar 

  • Golden, J.M. and Graham, G.A.C. (1988) Boundary Value Problems in Linear Viscoelasticity, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Gorsky, W.S. (1936) On the transitions in the CuAu alloy. III. On the influence of strain on the equilibrium in the ordered lattice of CuAI, Phys. Zeit Sowjet 6, 77–81.

    Google Scholar 

  • Graham, G.A.C. (1965) The contact problem in the linear theory of viscoelasticity, Int. J. Eng. Sci. 3, 27–46.

    MATH  Google Scholar 

  • Graham, G.A.C. (1967) The contact problem in the linear theory of viscoelasticity when the time-dependent contact area has any number of maxima and minima, Int. J. Eng. Sci. 5, 495–514.

    MATH  Google Scholar 

  • Graham, G.A.C. (1968) The correspondence principle of linear viscoelasticity for mixed boundary value problems involving time-dependent boundary regions, Quart. Appl. Math. 26, 167–74.

    MATH  Google Scholar 

  • Graff, K.F. (1975) Wave Motion in Elastic Solids, Dover Publications, New York.

    MATH  Google Scholar 

  • Green, W.A. (1964) The growth of plane discontinuities propagating into a homogeneously deformed elastic material, Arch. Ration. Mech. Anal. 16, 79–89.

    MATH  Google Scholar 

  • Greenberg, J.M. (1967) The existence of steady shock waves in nonlinear materials with memory, Arch. Ration. Mech. Analysis 24, 1–21.

    ADS  MATH  Google Scholar 

  • Greenberg, J.M. (1968) Existence of steady waves for a class of nonlinear dissipative materials, Quart. Appl. Math. 26, 27–34.

    MATH  Google Scholar 

  • Gurtin, M.E. and Herrara, I. (1964) A correspondence principle for viscoelastic wave propagation, Quart. Appl. Math. 22, 360–4.

    Google Scholar 

  • Gurtin, M.E. and Herrera, I. (1965) On dissipation inequalities and linear viscoelasticity, Quart. Appl. Math. 23, 235–45.

    MathSciNet  MATH  Google Scholar 

  • Gurtin, M.E. and Sternberg, E. (1962) On the linear theory of viscoelasticity, Arch. Ration. Mech. Anal. 11(4), 291–356.

    MathSciNet  MATH  Google Scholar 

  • Haddad, Y. M. (1995) Viscoelasticity of Engineering Materials, Kluwer, Dordrecht.

    Google Scholar 

  • Hetenyi, M., Ed. (1950) Handbook of Experimental Stress Analysis, John Wiley and Sons, New York.

    Google Scholar 

  • Hill, R. (1962) Acceleration waves in solids, J. Mech. Phys. Solids 10, 1–16.

    MathSciNet  ADS  MATH  Google Scholar 

  • Hillier, K. W. (1949) A method of measuring some dynamic elastic constants and its application to the study of high polymers, Proc. Phys. Soc. LXII, II-B, 701–13.

    ADS  Google Scholar 

  • Hillier, K. W. (1960) A Review of the progress in the measurement of dynamic elastic properties, Int. Symp. on Stress Wave Propagation in Materials, Ed. N. Davids, Inter-science Publishers, London, pp. 183–98.

    Google Scholar 

  • Hillier, K.W. and Kolsky, H. (1949) An investigation of the dynamic elastic properties of some high polymers, Proc. Phys. Soc. B62, 111–21.

    ADS  Google Scholar 

  • Huilgol, R.R. (1973) Growth of plane shock waves in materials with memory, Int. J. Engg. Sci. 11, 75–86.

    MathSciNet  MATH  Google Scholar 

  • Hunter, S.C. (1960a) Viscoelastic waves, in: Progress in Solid Mechanics, Vol. I, Eds. I.N. Sneddon and R. Hill, North-Holland Pub. Co., Amsterdam, pp. 1–57.

    Google Scholar 

  • Hunter, S.C. (196Ob) The Hertz problem for a rigid spherical indenter and a viscoelastic half-space, J. Mech. Phys. Solids 8, 219–34.

    Google Scholar 

  • Hunter, S.C. (1961) The rolling contact of a rigid cylinder with a viscoelastic half-space, J. Appl. Mech. 28, 611–17.

    ADS  MATH  Google Scholar 

  • Hunter, S.C. (1967) The solution of boundary value problems in linear Viscoelasticity, Proc. 4th 1965 Symposium on Naval Structural Mechanics, Eds. A.C. Eringen, H. Liebowitz, S.L. Koh and J.M. Crowley, Pergamon, Oxford, pp. 257–95.

    Google Scholar 

  • Ivey, D.G., Mrowea, B.A. and Guth, E. (1949) Propagation of ultrasonic bulk waves in high polymers, J. Appl. Phys. 20, 486–92.

    ADS  Google Scholar 

  • Keast, D.N. (1967) Measurements in Mechanical Dynamics, McGraw-Hill, New York.

    Google Scholar 

  • Kolsky, H. (1954a) Attenuation of short mechanical pulses by high polymers, Proc. 2nd Int. Congr. on Rheology, Butterworths, London, pp. 79–84.

    Google Scholar 

  • Kolsky, H. (1954b) The propagation oflongitudinal elastic waves along cylindrical bars, Phil. Mag. 45, 712–26.

    Google Scholar 

  • Kolsky, H. (1956) The propagation of stress pulses in viscoelastic solids, Philosophical Magazine 8(1), 693–710.

    ADS  Google Scholar 

  • Kolsky, H. (1958) The propagation of stress waves in viscoelastic solids, Appl. Mech. Rev. 11, 465–8.

    Google Scholar 

  • Kolsky, H. (1960) Viscoelastic waves, Int. Symp. on Stress Wave Propogation in Materials, Ed. N. Davids, Interscience Publishers, London, pp. 59–90.

    Google Scholar 

  • Kolsky, H. (1963) Stress Waves in Solids, Dover Publications, New York.

    Google Scholar 

  • Lee, E.H. (1955) Stress analysis in viscoelastic bodies, Quart. Appl. Math. 13, 183–90.

    MathSciNet  MATH  Google Scholar 

  • Lee, E.H. (1960) Viscoelastic stress analysis, in: First Symposium on Naval Structural Mechanics, Eds. J.N. Goodier and N.J. Hoff, Pergamon Press, New York, pp. 456–82.

    Google Scholar 

  • Lee, E.H., (1966) Some recent developments in linear viscoelastic stress analysis, in: Proceedings, Eleventh Int. Cong. of Applied Mechanics, Ed. H. Gortler, Springer-Verlag, Berlin, pp. 396–402.

    Google Scholar 

  • Lee, E.H. and Kanter, I. (1953) Wave propagation in finite rods of viscoelastic materials, J. Appl. Phys. 24(9), 1115–22.

    ADS  MATH  Google Scholar 

  • Lee, E.H. and Morrison, J.A. (1956) A comparison of the propagation of longitudinal waves in rods of viscoelastic materials, J. Polymer Sci. XIX, 93–110.

    ADS  Google Scholar 

  • Lee, E.H. and Radok, J.R.M. (1960) The contact problem for viscoelastic bodies, J. Appl. Mech., Trans. ASME 27, 438–44.

    MathSciNet  ADS  MATH  Google Scholar 

  • Lee, E.H., Radok, J.R.M. and Woodward, W.B. (1959) Stress analysis for linear viscoelastic materials, Trans. Soc. Rheol. 3, 41–59.

    MathSciNet  Google Scholar 

  • Lockett, F.J. (1962) The reflection and refraction of waves at an interface between viscoelastic materials, J. Mech. Phys. Solids 10, 53–64.

    MathSciNet  ADS  MATH  Google Scholar 

  • Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press.

    Google Scholar 

  • Lubliner, J. and Sackman, J.L. (1967) On uniqueness in general linear viscoelasticity, Q. Appl. Math. 25, 129–38.

    MATH  Google Scholar 

  • Magrab, E.B. and Blomquist, D.S. (1971) The Measurement of Time-Varying Phenomena, Wiley-Interscience, New York.

    Google Scholar 

  • Malvern, L.B. (1951) Plastic wave propagation in a bar of material exhibiting a strain rate effect, Quart. Appl. Math. 8, 405–11.

    MathSciNet  MATH  Google Scholar 

  • Mason, W.P. and McSkimin, H.J. (1947) Attenuation and scattering of high frequency sound waves in metals and glasses, J. Acoust. Soc. Amer. 19, 464–73.

    ADS  Google Scholar 

  • Morland, L.W. (1962) A plane problem of rolling contact in linear viscoelasticity theory, J. Appl. Mech. 29, 345–58.

    ADS  MATH  Google Scholar 

  • Morland, L.W. (1967) Exact solution for rolling contact between viscoelastic cylinders, Quart. J. Appl. Math. 20, 73–106.

    MathSciNet  MATH  Google Scholar 

  • Morland L.W. and Lee, E.H (1960) Stress analysis for linear viscoelastic materials with temperature variation, Trans. Soc. Rheol. 4, 233–63.

    MathSciNet  Google Scholar 

  • Morrison, J.A. (1956) Wave propagation in rods of Voigt material and viscoelastic materials with three-parameter models, Quart. Appl. Math. 14, 153–69.

    MathSciNet  MATH  Google Scholar 

  • Muki, R and Sternberg, E. (1961) On transient thermal stresses in viscoelastic materials with temperature dependent properties, J. Appl. Mech. 28, 193–207.

    MathSciNet  ADS  MATH  Google Scholar 

  • Nashif, A.D., Jones, D.I.G. and Henderson, J.P. (1965) Vibration Damping, John Wiley & Sons, New York.

    Google Scholar 

  • Nunziato, J.W. and Sutherland, H.J. (1973) Acoustical determination of stress relaxation functions in Polymers, J. Appl. Phys. 44(1), 184–7.

    ADS  Google Scholar 

  • Nunziato, J.W. and Walsh, E.K. (1973) Propagation of steady shock waves in nonlinear thermoviscoelastic solids, J. Mech. Phys. Solids 21, 317–35.

    ADS  MATH  Google Scholar 

  • Odell, F. and Tadjbakhsh, I. (1965) Uniqueness in the linear theory of viscoelasticity, Arch. Ration. Mech. Anal. 18, 244–50.

    Google Scholar 

  • Onal, E.T. and Breuer, S. (1963) On uniqueness in linear viscoelasticity, in: Progress in Applied Mechanics, The Prager Anniversary Volume, McMillan, New York, pp. 349–53.

    Google Scholar 

  • Orowan, E. (1934) Zür kristall plastizitåt. III. Ober den mechanismus des gleitvorganges, Zeits f. Phys. 89, 634–59.

    ADS  Google Scholar 

  • Polanyi, M. (1934) Über eine art gitterstörung, die einen kristall plastisch machen könnte, Zeits f. Phys. 89, 660–4.

    ADS  Google Scholar 

  • Predeleanu, M. (1965) Stress analysis in bodies with time-dependent properties, Bull. Math. Soc. Sci. Math., R.S. de Roumanie 9, 115–27.

    Google Scholar 

  • Rayleigh, J.W.S. (1894) Theory of Sound, Dover reprint, Dover Publications Inc., New York.

    MATH  Google Scholar 

  • Read, W.T. (1950) Stress analysis for compressible viscoelastic materials, J. Appl. Phys. 21, 671–4.

    MathSciNet  ADS  MATH  Google Scholar 

  • Rogers, T.G. (1965) Viscoelastic stress analysis, in: Proceedings, Princeton University Conforence on Solid Mechanics, Princeton, New Jersey, pp. 49–74.

    Google Scholar 

  • Rogers, T.G. and Lee, E.H. (1962) Brown Univ. Report NORD 18594/6.

    Google Scholar 

  • Schapery, R.A. (1964) Application of thermodynamics to thermomechanical, fracture, and birefringant phenomena in viscoelastic media, J. Appl. Phys. 35(5), 1451–65.

    MathSciNet  ADS  Google Scholar 

  • Schapery, R.A. (1967) Stress analysis of viscoelastic composite materials, J. Composite Materials 1, 228–66.

    Google Scholar 

  • Schapery, R.A. (1974) Viscoelastic behaviour and analysis of composite materials, Ed. G. Sendeckj, Vol. 2, Academic Press, New York, pp. 86–168.

    Google Scholar 

  • Schuler, K. W. (1970) Propagation of steady shock waves in Polymethyl Methacrylate, J. Mech. Phys. Solids 18, 277–93.

    ADS  Google Scholar 

  • Schuler, K.W., Nunziato, J.W. and Walsh, E.K. (1973) Recent results in nonlinear viscoelastic wave propagation, Int. J. Solid Structures 91, 1237–81.

    Google Scholar 

  • Schuler, K.W. and Walsh, E.K. (1971) Critical-induced acceleration for shock propagation in Polymethyl Methacrylate, J. Appl. Mech. 38, 641–45.

    ADS  Google Scholar 

  • Schwarzl, F. and Staverman, A.J. (1952) Time-temperature dependence of linear viscoelastic behaviour, J. Appl. Phys. 23(8), 838–43.

    ADS  MATH  Google Scholar 

  • Sips, R. (1951) General theory of deformation of viscoelastic substances, J. Polymer Sci. 9, 191–205.

    ADS  Google Scholar 

  • Snoek, J.E. (1941) Effect ofsmall quantities of carbon and nitrogen on the elastic and plastic properties of Iron, Physica 8, 711–33.

    ADS  Google Scholar 

  • Sternberg, E. (1964) On the analysis of thermal stresses in viscoelastic solids, in: High Temperature Structures and Materials, Proc. of the 3rd Symp. on Naval Structural Mechanics, Eds. A.M. Freudenthal, B.A. Boles and H. Liebowitz, Pergamon, Oxford, pp. 348–82.C

    Google Scholar 

  • Sternberg, E. and Gurtin, M.E. (1963) Uniqueness in the theory of thermorheologically simple ablating viscoelastic solids, in: Progress in Applied Mechanics, The Prager Anniversary Volume, Ed. D.C. Drucker, MacMillan, New York, pp. 373–84.

    Google Scholar 

  • Sternberg, E. and Gurtin, M.E. (1964) Further study of thermal stresses in viscoelastic materials with temperature dependent properties, in: Proceedings, IUTAM Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Mechanics, Haifa, pp. 51–76.

    Google Scholar 

  • Thomas, T.Y. (1957) The growth and decay of sonic discontinuities in ideal gases, J. Math. Mech. 6, 455–69.

    MathSciNet  MATH  Google Scholar 

  • Thomas, T.Y. (1961) Plastic Flow and Fracture in Solids, Academic Press, New York.

    MATH  Google Scholar 

  • Ting, T.C.T. (1966) The contact stresses between a rigid indenter and a viscoelastic half-space, J. Appl. Mech. 33, 845–54.

    ADS  MATH  Google Scholar 

  • Ting, T.C.T. (1968) Contact problems in the linear theory of viscoelasticity, J. Appl. Mech. 35, 248–54.

    ADS  MATH  Google Scholar 

  • Tobolsky, A., Powell, R.E. and Eyring, H. (1943) The Chemistry of Large Molecules, Interscience, New York.

    Google Scholar 

  • Truesdell, C. and Toupin, R.A. (1960) The classical field theories, Handbuk der Physik III/I, Ed. S. Flügge, Springer, Berlin.

    Google Scholar 

  • Valanis, K.C. (1965) Propagation and attenuation of waves in linear viscoelastic solids, J. of Mathematics & Physics 44(3), 227–39.

    MathSciNet  Google Scholar 

  • Varley, E. (1965) Acceleration fronts in viscoelastic materials, Arch. Ration. Mech. Analysis 19, 215–25.

    MathSciNet  ADS  MATH  Google Scholar 

  • Varley, E. and Cumberbatch, E. (1965) Nonlinear theory of wave-front propagation, J. Ins. Math. and Appl. 1, June 1965, 1, 101–112.

    MathSciNet  Google Scholar 

  • Volterra, V. (1909) Sulle equazioni integro-differenziali Della teoria dell’ elasticita, R. Accademia dei Lincei, 18(1), 167. Also, Ibid 18(2), 295.

    MathSciNet  MATH  Google Scholar 

  • Volterra, V. (1931) Theory of Functional, Dover Reprint, Dover Publications, New York.

    Google Scholar 

  • Worely, W.J. (Ed.) (1962) Experimental Techniques in Shock and Vibration, ASME, New York.

    Google Scholar 

14.15. Further Reading

  • Abbott, B.W. and Cornish, R.H. (1965) A stress wave technique for determining the tensile strength of brittle materials, Exp. Mech. 22, 148–53.

    Google Scholar 

  • Aboudi, J. (1979) The dynamic indentation and impact of a viscoelastic half-space by an axisymmetric rigid body, Compu. Math. Appl. Mech. Eng. 20, 135–50.

    MATH  Google Scholar 

  • Achenbach, J.D. and Reddy, D.P. (1967) Note on wave propagation in linearly viscoelastic media, Z. Angew. Math. Phys. 18, 141–4.

    MATH  Google Scholar 

  • Alblas, J.B. and Kuipers, M. (1970) The contact problem of a rigid cylinder rolling on a thin viscoelastic layer, Int. J. Eng. Sci. 8, 363–80.

    MATH  Google Scholar 

  • Alfrey, T. (1944) Nonhomogeneous stresses in viscoelastic media, Quart. Appl. Math. 2, 113–19.

    MathSciNet  MATH  Google Scholar 

  • Arenz, RJ. (1964) Uniaxial wave propagation in realistic viscoelastic materials, J. Appl. Mech., Trans. ASME 86(E), Mar. 1964, 17–21.

    Google Scholar 

  • Arenz, R.J. (1965) Two-dimensional wave propagation in realistic viscoelastic materials, J. Appl. Mech., Trans. ASME 32(2), June 1965, 303–14.

    ADS  Google Scholar 

  • Asay, I.R., Lamberson, D.L. and Guenther, A.H. (1969) Pressure and temperature dependence of velocities in Polymethyl Methacrylate, J. Appl. Phys. 40, 1768–83.

    ADS  Google Scholar 

  • Atkinson, C. and Coleman, C.J. (1977) On some steady-state moving boundary problems in the linear theory of viscoelasticity, J. Inst. Maths. Appl. 20, 85–106.

    MATH  Google Scholar 

  • Baker, W.E. and Dove, R.C. (1962) Measurements of internal strains in a bar subjected to longitudinal impact, Exp. Mech. 19, 307–11.

    Google Scholar 

  • Ballon, J.W. and Smith, J.C. (1949) Dynamic measurements of polymer physical properties, J. Appl. Phys. 20, 493–502.

    ADS  Google Scholar 

  • Barton, C.S., Volterra, E.G. and Citron, S.J. (1958) On elastic impacts of spheres on long rods, Proc. 3rd U.S. Nat. Congo Appl. Mech., pp. 89–94.

    Google Scholar 

  • Battiato, G., Ronca, G. and Varga, C. (1977) Moving loads on a viscoelastic double layer: Prediction of recoverable and permanent deformations, in: Proceedings, Fourth Intemational Conf. on Structural Design of Asphalt Pavements, The University of Michigan, Ann Albar, Michigan, USA, pp. 459–60.

    Google Scholar 

  • Becker, E.C.H. and Carl, H. (1962) Transient-loading technique for mechanical impedance measurement, in Experimental Techniques in Shock and Vibration. Ed. W.J. Worley, ASME, New York, pp.1–10.

    Google Scholar 

  • Berry, D.S. (1958) A note on stress pulses in viscoelastic rods, Phil. Mag. 8, 100–2.

    ADS  Google Scholar 

  • Beny, D.S. and Hunter, S.C. (1956) The propagation of dynamic stresses in viscoelastic rods, J. Mech. Phys. Solids 4, 72–95.

    MathSciNet  ADS  Google Scholar 

  • Calvit, H.H. (1967) Experiments on rebound of steel spheres from blocks of polymers, J. Mech. Phys. Solids 15, 141–50.

    ADS  Google Scholar 

  • Chou, P.C. (1968) Introduction to wave propagation in composite materials, in Composite Materials Workshop, Eds. S. W. Tsai, J.C. Halpin and N.J. Pagano, Technomic. Stamford. Conn., pp. 193–216.

    Google Scholar 

  • Chref, C. (1889) The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications, Trans. Camb. Phil. Soc. Math. Phys. Sci. 6, 115–17.

    Google Scholar 

  • Chu, B.T. (1962) Stress waves in isotropic linear viscoelastic materials (part one), J. Mécanique 1(1), 439–62.

    Google Scholar 

  • Chu, B.T. (1965) Response of various material media to high velocity loadings. I. Linear elastic and viscoelastic materials, J. Mech. Phys. Solids 13, 165–87.

    ADS  Google Scholar 

  • Comninou, M. (1976) Contact between viscoelastic bodies, J. Appl. Mech. 43, 630–2.

    ADS  MATH  Google Scholar 

  • Dally, J.W. (1968) A dynamic photoelastic study ofa doubly loaded half-plane, Develop. Mech. 4, 649–64.

    Google Scholar 

  • Dally, J. W., Durelli, A.J. and Riley, W.F. (1960) Photoelastic study of stress wave propagation in large plates, Proc. Soc. Exp. Stress Analysis 17, 33–50.

    Google Scholar 

  • Dally, J. W. and Lewis, D. (1968) A photoelastic analysis of propagation of Rayleigh waves past a step change in elevation, Bull. Seism. Soc. Am. 58, 539–63.

    Google Scholar 

  • Dally, J.W. and Riley, W.F. (1967) Initial studies in three-dimensional dynamic photoelasticity, J. Appl. Mech. 34, 405–10.

    ADS  Google Scholar 

  • Dally, J.W. and Thau, S.A. (1967) Observations of stress wave propagation in a half-plane with boundary Loading, Int. J. Solids Struct. 3, 293–7.

    Google Scholar 

  • Davies, R.M. (1948) A critical study ofthe Hopkinson pressure bar, Phil. Trans. R. Soc. A240, 375–457.

    ADS  Google Scholar 

  • Dohrenwend, C.O., Drucker, D.C. and Moore, P. (1944) Transverse impact transients, Exp. Stress Analysis 1, 1–10.

    Google Scholar 

  • Dunwoody, J. (1966) Longitudinal wave propagation in a rate dependent material, Int. J. of Engineering Sci. 4, 277–87.

    Google Scholar 

  • Dziecielak, R. (1985) The effect oftemperature on the propagation of discontinuity waves in a porous medium with a viscoelastic skeleton, Studia Geotechnica et Mechanica, Vol. VII(2), 17–34.

    Google Scholar 

  • Edelstein, W.S. (1969) The cylinder problem in thermoviscoelasticity, Res. Natl. Bur. Standards 73B, 31–40.

    Google Scholar 

  • Engelbrecht, J. (1979) One-dimensional deformation waves in nonlinear viscoelastic materials, Wave Motion 1, 65–74.

    MATH  Google Scholar 

  • Evans, J.F., Hadley, C.F., Eisler, J.D. and Silverman, D. (1954) A three-dimensional seismic wave model with both electrical and visual observation of waves, Geophysics 19, 120–36.

    Google Scholar 

  • Ferry, J.D. (1961) Viscoelastic Properties of Polymers, Wiley, New York.

    Google Scholar 

  • Fisher, H.C. (1954) Stress pulse in bar with neck or swell, Appl. Scient. Res. A4, 317–28.

    Google Scholar 

  • Fisher, G.M.C. and Gurtin, M.E. (1965) Wave propagation in the linear theory of viscoelasticity, Q. Appl. Math. 23, 257–63.

    MathSciNet  MATH  Google Scholar 

  • Fichera, G. (1972) Boundary value problems of elasticity with unilateral constraints, in: Encyclopedia of Physics, Vol.VIa/2: Mechanics of Solids II, Ed. C. Truesdell, Springer-Verlag, Berlin, pp. 391–423

    Google Scholar 

  • Frederick, J.R (1965) Ultrasonic Engineering, John Wiley and Sons, New York.

    Google Scholar 

  • Frydrychowich, W. and Singh, M.C. (1986) Similarity representation of wave propagation in a nonlinear viscoelastic rod on a group theoretic basis, Appl. Math. Modeling. 10(8), 284–93.

    Google Scholar 

  • Gakhof, F.D. (1966) Boundary Value Problems, Pergamon, Oxford.

    Google Scholar 

  • Gaul, L. (1992) Substructure behaviour of resilient support mounts for single and double stage mounting systems, Computers & Structures, Vol. 44, No. 111, pp. 273–78.

    Google Scholar 

  • Gaul, L., Klein, P. and Kemple, S. (1991) Damping description involving fractional operators, Mechanical Systems and Signal Processing 5(1), 81–82.

    ADS  Google Scholar 

  • Gaul, L., Schanz, M. and Fiedler, C. (1992) Viscoelastic formulations ofBEM in time and frequency domain, Engineering Analysis with Boundary Elements 10, 137–41.

    Google Scholar 

  • Gladwwell, G.M.L. (1980) Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, Alphen aan den Riju.

    Google Scholar 

  • Golden, J.M. and Graham, G.A.C. (1988) Boundary Value Problems in Linear Viscoelasticity, SpringerVerlag, Berlin.

    MATH  Google Scholar 

  • Goldsmith, W., Polivka, M. and Yang, T. (1966) Dynamic behaviour of concrete, Exp. Mech. 13, 65–79.

    Google Scholar 

  • Goodier, J.N., Jahsman, W.E. and Ripperger, E.A. (1959) An experimental surface-wave method for recording force-time curves in elastic impacts, J. Appl. Mech. 16, 3–7.

    Google Scholar 

  • Gopalsamy, K. and Aggarwala, B.D. (1972) Propagation of disturbances from randomly moving sources, ZAMM 51, 31–35.

    Google Scholar 

  • Graffi, D. (1952) Sutla teoria dei materiali elastico-viscosi, Atti Accod. Ligure Sci. Lett. 9, 1–10.

    Google Scholar 

  • Graffi, D. (1982) Mathematical models and waves in linear viscoelasticity, in Wave Propagation in Viscoelastic Media, V52, Ed. F. Mainardi, Pitman, Boston, 1–27.

    Google Scholar 

  • Graham, G.A.C. (1965) On the use of stress functions for solving problems in linear viscoelasticity theory that involve moving boundaries, Proc. R. Soc. (Edin.) A67, 1–8.

    Google Scholar 

  • Graham, G.A.C. (1969) The solution of mixed boundary value problems that involve time-dependent boundary regions, for viscoelastic materials with one relaxation function, Acta Mech. 8, 188–204.

    MATH  Google Scholar 

  • Graham, G.A.C. and Golden, J.M. (1988) The generalized partial correspondence principle in linear viscoelasticity, Q. Appl. Math. 56(3), 527–38.

    MathSciNet  Google Scholar 

  • Graham, G.A.C. and Sabin, G.C.W. (1973) The correspondence principle of linear viscoelasticity for problems that involve time-dependent regions, Int. J. Engg. Sci. 11, 123–40.

    MathSciNet  MATH  Google Scholar 

  • Graham, G.A.C. and Sabin, G.C.W. (1978) The opening and closing of a growing crack in a linear viscoelastic body that is subject to alternating tensile and compressive loads, Int. J. Fracture 14, 639–49.

    MathSciNet  Google Scholar 

  • Graham, G.A.C. and Sabin, G.C.W. (1981) Steady-state solutions for a cracked standard linear viscoelastic body, Mech. Res. Commun. 8, 361–68.

    MATH  Google Scholar 

  • Graham, G.A.C. and Williams, F.M. (1972) Boundary value problems for time-dependent regions in aging viscoelasticity, Utilitas Mathematica 1, 291–03.

    MathSciNet  Google Scholar 

  • Green, W.A. (1960) Dispersion relations for elastic waves in bars, in Progress in Solid Mechanics, Vol. 1, Eds. I.N. Sneddon and R. Hill, Chapter 5, North Holland Publishing Co., Amsterdam.

    Google Scholar 

  • Gurtin, M.E. and Herrara, I. (1964) A Correspondance principle for viscoelastic wave propagation, Quart. Appl. Math. 11, 360–4.

    Google Scholar 

  • Harris, C.M. and Crede, E. (1961) Shock and Vibration Handbook, Vols. I, II and III, McGraw-Hili, New York.

    Google Scholar 

  • Harvey, R.B. (1975) On the deformation of a viscoelastic cylinder rolling without slipping, Q.J. Mech. Appl. Math. 28, 1–24.

    MathSciNet  MATH  Google Scholar 

  • Hatfield, P. (1950) Propagation of low frequency ultrasonic waves in rubbers and rubber-like polymers, Brit. J. Appl. Phys. 1, 252–56.

    ADS  Google Scholar 

  • Hrusa, W.J. and Renardy, M. (1985) On wave propagation in linear viscoelasticity, Quart. of Appl. Math. XLIII(1), July 1985, 237–53.

    MathSciNet  Google Scholar 

  • Hsieh, D. Y. and Kolsky, H. (1958) An experimental study of pulse propagation in elastic cylinders, Proc. Phys. Soc. 71, 608–12.

    ADS  Google Scholar 

  • Hudson, G.E. (1943) Dispersion of elastic waves in solid circular cylinders, Phys. Rev. 63, 46–51.

    ADS  Google Scholar 

  • Hunter, S.C. (1961) Tentative equations for the propagation of stress, strain and temperature fields in viscoelastic solids, J. Mech. Phys. Solids 9, 39–51.

    MathSciNet  ADS  MATH  Google Scholar 

  • Hunter, S.C. (1967) The transient temperature distribution in a semi-infinite viscoelastic rod subject to longitudinal oscillations, Int. J. Eng. Sci. 5, 119–43.

    Google Scholar 

  • Hunter, S.C. (1968) The motion of a rigid sphere embedded in an adhering elastic or viscoelastic medium, in: Proceedings, Edinburgh Mathematical Society 16 (Series II), Part I, pp. 55–69.

    Google Scholar 

  • Jeffrey, A. (1978) Nonlinear wave propagation, ZAMM 58, T38–T56.

    MathSciNet  MATH  Google Scholar 

  • Jeffrey, A and Taniuti, T. (1964) Nonlinear Wave Propagation, Academic Press, New York.

    Google Scholar 

  • Kalker, J.J. (1975) Aspects of contact mechanics, in: The Mechanics of the Contact Between Deformable Media, Ed. A.D. de Pater and J.J. Kalker, Delft University Press, pp. 1–25.

    Google Scholar 

  • Kalker, J.J. (1977) A survey of the mechanics of contact between solid bodies, J. Appl. Math. Phys. (ZAMP) 57, 13–17.

    Google Scholar 

  • Knauss, W.G. (1968) Uniaxial wave propagation in a viscoelastic material using measured material properties, J. Appl. Mech., ASME, Sept. 1968, 449–53.

    Google Scholar 

  • Koeller, R.C. (1984) Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51, 299–307.

    MathSciNet  ADS  MATH  Google Scholar 

  • Kolsky, H. (1960) Experimental wave-propagation in solids, in Structural Mechanics, Eds. J. N. Goodier and N. Hoff, Pergamon Press, Oxford, pp. 233–62.

    Google Scholar 

  • Kolsky, H. (1965) Experimental studies in stress wave propagation, in Proc. Vth U.S. Natn. Congr. Appl. Mech., pp. 21–36.

    Google Scholar 

  • Kolsky, H. and Prager, W., Eds. (1964) Stress waves in anelastic solids, IUTAM Symposium, Brown University, Providence, R.I., April 3-5, 1963, Springer-Verlag, Berlin, pp. 1–341.

    Google Scholar 

  • Lamb, H. (1904) On the propagation of tremors over the surface of an elastic solid, Phil. Trans. R. Soc. A203, 1–42.

    ADS  Google Scholar 

  • Lamb, H. (1917) On waves in an elastic plate, Proc. Roy. Soc. A93, 114–28.

    ADS  Google Scholar 

  • Langhaar, H.L. (1962) Energy Methods in Applied Mechanics, John Wiley and Sons, New York.

    Google Scholar 

  • Lee, E.H. (1966) Some recent developments in linear viscoelastic stress analysis, in Proceedings of the 11th Congress of Applied Mechanics, Ed. H. Gonler, Springer-Verlag, Berlin, pp. 396–402.

    Google Scholar 

  • Lifshitz, J.M. and Kolsky, H. (1964) Some experiments on anelastic rebound, J. Mech. Phys. Solids 12, 35–43.

    ADS  Google Scholar 

  • Lifshitz, J.M. and Kolsky, H. (1965) The propagation of spherical divergent stress pulses in linear viscoelastic solids, J. Mech. Phys. Solids 13, 361–76.

    ADS  Google Scholar 

  • Lindholm, U.S. (1964) Some experiments with the split Hopkinson pressure bar, J. Mech. Phys. Solids 12, 317–35.

    ADS  Google Scholar 

  • Lindsay, R.B. (1960) Mechanical Radiation, McGraw-Hill, New York.

    Google Scholar 

  • Lockett., F.J. (1961) Interpretation of mathematical solutions in viscoelasticity theory illustrated by a dynamic spherical cavity problem, J. Mech. Phys. Solids 9, 215–29.

    MathSciNet  ADS  MATH  Google Scholar 

  • Lockett, F.J. (1962) The reflection and refraction of waves at an interface between viscoelastic materials. J. Mech. Phys. Solids 10, 53–64.

    MathSciNet  ADS  MATH  Google Scholar 

  • Lockett, F.J. and Morland, L.W. (1967) Thermal stresses in a viscoelastic thin-walled tube with temperaturedependent properties, Int. J. Engg. Sci. 5, 879–98.

    Google Scholar 

  • Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticlty, Dover Publications, New York.

    Google Scholar 

  • MahaJanabis, R.K. and Mandal, B. (1986) Propagation of thermomagneto-viscoelastic waves in a half-space of Voigt-type material, Indian Journal of Technology 24, Sept. 1986, 365–567.

    Google Scholar 

  • Mainardi, F. and Nervosi, R. (1980) Transient-waves in finite viscoelastic rods, Lett. Nuovo Cim 29, 443–7.

    Google Scholar 

  • Mainardi, F. and Turchetti, G. (1975) Wave front expansions for transient viscoelastic waves, Mech. Res. Comm. 2, 107–12.

    Google Scholar 

  • Mainardi, F. and Turchetti, G. (1979) Positive constraints and approximation methods in linear viscoelasticity, Lett. Nuovo Cim. 26, 38–40.

    Google Scholar 

  • Margeston, J. (1971) Rolling contact ofa smooth viscoelastic strip between rotating rigid cylinders, Int. J. Mech. Sci. 13, 207–15.

    Google Scholar 

  • Margeston, J. (1972) Rolling contact of a rigid cylinder over a smooth elastic or viscoelastic layer, Acta Mech. 13, 1–9.

    Google Scholar 

  • McCartney, L.N. (1978) Crack propagation in linear viscoelastic solids: Some new results, Int. J. Fract. 14, 547–54.

    MathSciNet  Google Scholar 

  • Medick, M.A. (1961) On classical plate theory and wave propagation, J. Appl. Mech. 28, 223–8.

    MathSciNet  ADS  MATH  Google Scholar 

  • Meyer, M.L. (1964) On spherical near fields and far fields in elastic and viscoelastic solids, J. Mech. Phys. Solids 12, 77–111.

    MathSciNet  ADS  MATH  Google Scholar 

  • Miklowjiz, J. (1964) Pulse propagation in a viscoelastic solid with geometric dispersion, in Stress Waves in Anelastic Solids, Springer-Verlag, Berlin, pp. 255–76.

    Google Scholar 

  • Morland, L. W. (1963) Dynamic stress analysis for a viscoelastic half-plane subject to moving surface traction, Proc. London Math. Soc. 13, 471–92.

    MathSciNet  MATH  Google Scholar 

  • Morland, L.W. (1968) Rolling contact between dissimilar viscoelastic cylinders, Q. Appl. Math. 25, 363–76.

    MATH  Google Scholar 

  • Morse, P. and Feshbach, H. (1953) Methods of Theoretical Physics, Vols. I and II, McGraw-Hill, New York.

    MATH  Google Scholar 

  • Nachman A. and Walton, J.R (1978) The sliding ofa rigid indenter over a power law viscoelastic layer, J. Appl. Mech. 45, 111–13.

    ADS  MATH  Google Scholar 

  • Norris, J.M. (1967) Propagation of a stress pulse in a viscoelastic rod, Experimental Mechanics 7(7), 297–301.

    Google Scholar 

  • Nunziato, J.W. and Walsh, E.K. (1973) Amplitude behavior of shock waves in a therrnoviscoelastic solid, Int. J. Solids Structures 9, pp. 1373–83.

    MATH  Google Scholar 

  • Oliver, J. (1957) Elastic wave dispersion in a cylindrical rod by a wide-band short duration pulse technique, J. Acoustic Soc. Am. 29, 189–94.

    ADS  Google Scholar 

  • Pao, Y.H. (1955) Extension of the Hertz theory of impact to the viscoelastic case, J. Appl. Phys. 26, 1083–8.

    ADS  MATH  Google Scholar 

  • Petrof, R.C. and Gratch, S. (1964) Wave propagation in a viscoelastic material with temperaturedependent properties and therrnomechanical coupling, J. of Applied Mechanics, ASME, Sept. 1964, 423–9.

    Google Scholar 

  • Pindera, J.T. (1986) New research perspectives opened by isodyne and strain gradient photoelasticity, in Proceedings of the International Symposium on Photoelasticity, Tokyo, pp. 193–202.

    Google Scholar 

  • Reinhardt, H.W. and Dally, J.W. (1970) Some characteristics of Rayleigh wave interaction with surface flaws, Mater. Eval. 28, 213–20.

    Google Scholar 

  • Renardy, M. (1982) Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids, Rheologica Acta 21, 251–4.

    MathSciNet  MATH  Google Scholar 

  • Ricker, N.H. (1977) Transient Waves in Viscoelastic Media, Elsevier, Amsterdam.

    Google Scholar 

  • Riley, W.F. and Dally, J.W. (1966) A photoelastic analysis of stress wave propagation in a layered model, Geophysics 31, 881–89.

    ADS  Google Scholar 

  • Ripperger, E.A. (1953) The propagation of pulses in cylindrical bars. An experimental study, Proc. 1st Midwest Conf. Solid Mech., 29–39.

    Google Scholar 

  • Rogers, T.G. (1965) Viscoelastic stress analysis, in: Proceedings of the Princeton Univ. Conf. on Solid Mechanics, Princeton, N.J., USA, pp. 49–74.

    Google Scholar 

  • Rogers, C.A., Barker, D.K. and Jaeger, L.A. (1988) Introduction to smart materials and structures, in Proceedings, Smart Materials, Structures and Mathematical Issues Workshop, Virginia Polytech. Inst. & State University, Blacksburg, VA, Sept. 15-16, 1992, pp. 17–28.

    Google Scholar 

  • Rubin, J.R (1954) Propagation oflongitudinal deformation waves in a prestressed rod of material exhibiting a strain-rate effect, J. Appl. Phys. 25, 528–36.

    MathSciNet  ADS  MATH  Google Scholar 

  • Sabin, G.C.W. (1975) Some Dynamic Mixed Boundary Value Problems in Linear Viscoelasticity, Ph.D. Thesis, Univ. of Windsor, Windsor, Ont., Canada.

    Google Scholar 

  • Sabin, G.C.W. (1987) The impact ofa rigid axisymmetric indenter on a viscoelastic half-space, Int. J. Eng. Sci. 25, 235–51.

    MATH  Google Scholar 

  • Sackman, J.L. and Kaya, I. (1968) On the propagation of transient pulses in linearly viscoelastic media, J. Mech. Phys. Solids 16, 349–56.

    ADS  MATH  Google Scholar 

  • Schapery, R.A. (1955) A method of viscoelastic stress analysis using elastic solutions, J. Franklin Institute 279(4), 268–89.

    MathSciNet  Google Scholar 

  • Schapery, R.A. (1962) Approximate method oftransforrn inversion for viscoelastic stress analysis, Proc. 4th U.S. National Cong. Appl. Mech., ASME, New York, pp. 1075–85.

    Google Scholar 

  • Schapery, R.A. (1974) Viscoelastic behaviour and analysis of composite materials, Mechanics of Composite Materials, Vol.2, Ed. G.P. Sendeckj, Academic Press, New York, pp. 85–168.

    Google Scholar 

  • Schapery, R.A. (1978) A method for predicting crack growth in nonhomogeneous viscoelastic media, Int. J. Fract. 14, 293–309.

    Google Scholar 

  • Schapery, R.A. (1979) On the analysis of crack initiation and growth in nonhomogenous viscoelastic media, in: Fracture Mechanics, Proceedings of the Symposium in Applied Mathematics of the A.MS. and S.I.A.M, Vol. XII, Ed. R. Burridge, American Math. Soc., Providence, pp. 137–52.

    Google Scholar 

  • Sips, R. (1951) Propagation phenomena in elastic viscous media, J. Polymer Sci. 6, 285–93.

    ADS  Google Scholar 

  • Skalak, R. (1957) Longitudinal impact ofa semi-infinite circular elastic bar, J. Appl. Mech. 34, 59–64.

    MathSciNet  Google Scholar 

  • Sokolnikoff, I.S. (1956) Mathematical Theory of Elasticity, 2nd ed., McGraw Hill, New York.

    MATH  Google Scholar 

  • Stackgold, I. (1967) Boundary Value Problems of Mathematical Physics, Vol. 1, McMillan. New York.

    Google Scholar 

  • Stoneley, R. (1924) Elastic waves at the surface of separation of two Solids, Proc. R. Soc. A106, 416–28.

    ADS  Google Scholar 

  • Sultanov, K.S. (1984) Longitudinal wave propagation in a viscoelastic semi-space including an absorbing layer, J. Appl. Mech. Tech. Phys. 25(5), Sept.-Oct. 1984, 790–5.

    MathSciNet  ADS  Google Scholar 

  • Sutherland, H.J. and Lingle, R. (1972) An acoustic characterization ofPolymethyl Methacrylate and three epoxy formulations, J. Appl. Phys. 43(10), 4022–6.

    ADS  Google Scholar 

  • Tanagi, T. (1990) A Concept of intelligent material, U.S. Japan Workshop on Smart/Intelligent Materials and Systems, Eds. C.A. Rogers, C. Andrew and A. Masuo, March 19-23, 1990, Honolulu, Hawaii, pp. 3–10.

    Google Scholar 

  • Tanary, S. and Haddad, Y.M. (1988) Characterization of Adhesively Bonded Joints Using Acousto-Ultrasonics, final Report prepared under contract serial number 31946-6-0012/01-ST, Department of Mechanical Engineering, University of Ottawa, Ottawa, Canada.

    Google Scholar 

  • Tatel, R.E. (19.54) Note on the nature ofa seismogram II., J. Geophys. Res. 59, 289–94.

    Google Scholar 

  • Thau, S.A. and Dally, J.W. (1969). Subsurface characteristics of the Rayleigh wave, Int. J. Eng. Sci. 7, 37–52.

    MATH  Google Scholar 

  • Timoshenko, S.P. (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Phil. Mag. 6(41), 744–6.

    Google Scholar 

  • Timoshenko, S.P. (1928) Vibration Problems in Engineering, Van Nostrand, New Jersey.

    MATH  Google Scholar 

  • Ting, T.C.T. (1969) A mixed boundary value problem in viscoelasticity with time-dependent boundary regions, in: Proceedings of the 11th Midwestern Mechanics Conf., Eds. H.J. Weiss, D.F. Young, W.F. Riley and T.R. Rogge, Iowa Univ. Press, pp. 591–8.

    Google Scholar 

  • Ting, E.C. (1970) Stress analysis for a nonlinear viscoelastic cylinder with ablating inner surface, Trans. ASME, J. Appl. Mech. 37E, 44–7.

    ADS  Google Scholar 

  • Tsai, Y.M. and Kolsky, H. (1968) Surface wave propagation for linear viscoelastic solids, J. Mech. Phys. Solids 16, 99–109.

    ADS  Google Scholar 

  • Tschoegl, N. W. (1989) The Phenomenological Theory of Linear Viscoelastic Behaviour, Springer, Berlin.

    Google Scholar 

  • Viktorov, L.A. (1967) Rayleigh and Lamb Waves: Physical Theory and Applications, Plenum Press, New York.

    Google Scholar 

  • Walsh, E.K. (1971) The decay of stress waves in one-dimensional polymer rods, Trans. Soc. Rheology 15:2, 345–53.

    ADS  Google Scholar 

  • Watson, G.N. (1960). A Treatise on the Theory of Bessel Functions,Cambridge University Press, New York.

    Google Scholar 

  • Whitham, G.B. (1974) Linear and Nonlinear Waves, J. Wiley & Sons, New York.

    MATH  Google Scholar 

  • Willis, J.R. (1967) Crack propagation in viscoelastic media, J. Mech. Phys. Solids 15, 229–40.

    ADS  MATH  Google Scholar 

  • Zemanek, J. (Jr.) and Rudnick, L. (1961) Attenuation and dispersion of elastic waves in a cylindrical bar, J. Acoust. Soc. Am. 33, 1283–8.

    ADS  Google Scholar 

  • Zener, C. (1948) Elasticity and Anelasticity of Metals, Univ. Press, Chicago.

    Google Scholar 

  • Zukas, J.A. (1982) Stress Waves in Solids, in Impact Dynamics, Eds. J.A. Zukas et al., John Wiley & Sons, New York, Chapter I, pp. 1–27.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, Mechanical Engineering, University of Ottawa, Canada

    Yehia M. Haddad

Authors
  1. Yehia M. Haddad
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Haddad, Y.M. (2000). Viscoelastic Waves and Boundary Value Problem. In: Mechanical Behaviour of Engineering Materials. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0436-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-0436-7_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0350-9

  • Online ISBN: 978-94-010-0436-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation