Rheological Response and Validity of Viscoelastic Model Through Propagation of Harmonic Wave in Non-Homogeneous Viscoelastic Rods

Document Type : Research Paper


1 Principal, DIPS Polytechnic College, Hoshiarpur

2 Faculty of Applied Sciences, BMSCE, Muktsar-152026, India


This study is concerned to check the validity and applicability of a five parameter viscoelastic model for harmonic wave propagating in the non-homogeneous viscoelastic rods of varying density. The constitutive relation for five parameter model is first developed and validity of these relations is checked. The non-homogeneous viscoelastic rods are assumed to be initially unstressed and at rest. In this study, it is assumed that density, rigidity and viscosity of the specimen i.e. rod are space dependent. The method of non-linear partial differential equation (Eikonal equation) has been used for finding the dispersion equation of harmonic waves in the rods. A method for treating reflection at the free end of the finite non-homogeneous viscoelastic rod is also presented. All the cases taken in this study are discussed numerically and graphically with MATLAB.


[1] Alfrey T., 1944, Non-homogeneous stress in viscoelastic media, Quarterly of Applied Mathematics 2: 113-119.
[2] Barberan J., Herrera J., 1966, Uniqueness theorems and speed of propagation of signals in viscoelastic materials, Archive for Rational Mechanics and Analysis 23(3): 173-190.
[3] Achenbach J.D., Reddy D. P., 1967, Note on the wave-propagation in linear viscoelastic media, Zeitschrift für angewandte Mathematik und Physik (ZAMP), 18(1):141-144.
[4] Bhattacharya S., Sengupta P.R., 1978, Disturbances in a general viscoelastic medium due to impulsive forces on a spherical cavity, Gerlands Beitr Geophysik Leipzig 87(8): 57-62.
[5] Acharya D. P., Roy I., Biswas P. K., 2008, Vibration of an infinite inhomogeneous transversely isotropic viscoelastic medium with a cylindrical hole, Applied Mathematics and Mechanics 29(3): 1-12.
[6] Bert C. W., Egle D. M., 1969, Wave propagation in a finite length bar with variable area of cross-section, Journal of Applied Mechanics 36: 908-909.
[7] Biot M.A., 1940, Influence of initial stress on elastic waves, Journal of Applied Physics 11(8):522-530.
[8] Batra R. C., 1998, Linear constitutive relations in isotropic finite elasticity, Journal of Elasticity 51: 243-245.
[9] White J.E., Tongtaow C., 1981, Cylindrical waves in transversely isotropic media, The Journal of the Acoustical Society of America 70(4):1147-1155.
[10] Mirsky I., 1965, Wave propagation in transversely isotropic circular cylinders, part I: Theory, Part II: Numerical results, The Journal of the Acoustical Society of America 37:1016-1026.
[11] Tsai Y.M., 1991, Longitudinal motion of a thick transversely isotropic hollow cylinder, Journal of Pressure Vessel Technology 113:585-589.
[12] Murayama S., Shibata T., 1961, Rheological properties of clays, 5th International Conference of Soil Mechanics and Foundation Engineering, Paris, France 1:269 – 273.
[13] Schiffman R.L., Ladd C.C., Chen A.T.F., 1964, The secondary consolidation of clay, rheology and soil mechanics, Proceedings of the International Union of Theoretical and Applied Mechanics Symposium, Grenoble, Berlin 273 – 303.
[14] Gurdarshan S., Avtar S., 1980, Propagation, reflection and transmission of longitudinal waves in non-homogeneous five parameter viscoelastic rods, Indian Journal of Pure and Applied Mathematics 11(9): 1249-1257.
[15] Kakar R., Kaur K., Gupta K.C., 2012, Analysis of five-parameter viscoelastic model under dynamic loading, Journal of Solid Mechanics 4(4): 426-440.
[16] Kaur K., Kakar R., Gupta K.C., 2012, A dynamic non-linear viscoelastic model, International Journal of Engineering Science and Technology 4(12): 4780-4787.
[17] Kakar R., Kaur K., 2013, Mathematical analysis of five parameter model on the propagation of cylindrical shear waves in non-homogeneous viscoelastic media, International Journal of Physical and Mathematical Sciences 4(1): 45-52.
[18] Kaur K., Kakar R., Kakar S., Gupta K.C., 2013, Applicability of four parameter viscoelastic model for longitudinal wave propagation in non-homogeneous rods, International Journal of Engineering Science and Technology 5(1): 75-90.
[19] Friedlander F.G., 1947, Simple progressive solutions of the wave equation, Mathematical Proceedings of the Cambridge Philosophical Society 43: 360-73.
[20] Karl F. C., Keller J. B., 1959, Elastic waves propagation in homogeneous and inhomogeneous media, Journal of Acoustical Society America 31: 694-705.
[21] Moodie T.B., 1973, On the propagation, reflection and transmission of transient cylindrical shear waves in non-homogeneous four-parameter viscoelastic media, Bulletin of the Australian Mathematical Society 8: 397-411.
[22] Carslaw H. S., Jaeger, J. C., 1963, Operational Methods in Applied Math, Second Ed., Dover Pub, New York.
[23] Bland D. R., 1960, Theory of Linear Viscoelasticity, Pergamon Press, Oxford.
[24] Christensen R. M., 1971, Theory of Viscoelasticity, Academic Press.