Abstract
This chapter examines the propagation of harmonic and shock waves in viscoelastic materials of integral and differential type. For simplicity, the different topics are introduce in one dimension, presenting the balance of linear momentum across the shock front, and the jump equations in stress, strain and velocity without obscuring the subject matter. As must be expected, harmonic waves in viscoelastic media are always damped. Also shown is that shock waves travel at the glassy sonic speed of the viscoelastic material in which they occur; that is, at the speed of sound in an elastic solid with modulus of elasticity equal to the glassy modulus of the viscoelastic material at hand.
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Notes
- 1.
The root of index n of a complex number, \( z = re^{j(\theta + 2k\pi )} \), is given by: \( z^{1/n} = r^{1/n} e^{j(\theta + 2k\pi )/n} \), as shown in Appendix A.
- 2.
In mathematical terms, this has to be so because every Laplace transform must vanish at s→∞, per the limit theorems [c.f. Appendix A].
- 3.
In this case, the solution in physical space can be established from a table of Laplace transforms and it is [3] \( \sigma (x,t) = \rho cVe^{ - t/(2\tau )} I_{o} \left( {\frac{1}{2\tau }(t^{2} - x^{2} /c^{2} )} \right)H(t - x/c) \), where I o (·) is the “modified Bessel function of order zero”.
References
H. Kolsky, Stress Waves in Solids (Dover Publications, NY, 1963), pp. 41–53
W. Flügge, Viscoelasticity, 2nd edn. (Springer, Berlin, 1975), pp. 121–140
R.M. Christensen, Theory of Viscoelasticity, 2nd edn. (Dover, NY, 1982), pp. 110–116
N. Cristescu, I. Suliciu, Viscoplasticity (Martinus Nijhoff, The Hague, 1982)
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Gutierrez-Lemini, D. (2014). Wave Propagation. In: Engineering Viscoelasticity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8139-3_10
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DOI: https://doi.org/10.1007/978-1-4614-8139-3_10
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