Eur. J. Mech. ASolids 18 1999 539–553
Elsevier, Paris
Wave splitting in linear viscoelasticity
M. Romeo
D.I.B.E., Università, via Opera Pia, 11a, 16145 Genova, Italy Received 12 November 1997; revised and accepted 30 June 1998
Abstract
– The problem of transient propagation in a homogeneous linear viscoelastic solid is investigated by means of the wave splitting technique. Two mechanical viscoelastic models are considered in particular, corresponding respectively to a single Maxwell element and to the series of a Maxwell
element plus a Kelvin element. Analytical solutions for the split field are evaluated in terms of the wave propagator via the Laplace transforms. The reflectivity of a viscoelastic half-space is explicitly for the two mechanical models and the corresponding inverse problem is approached. Finally, the
reflection and transmission problem for a transient wave is solved for a viscoelastic layer.
Elsevier, Paris
wave propagation in solids linear viscoelasticity
1. Introduction
The aim of the present paper is to obtain some definite results for the one-dimensional problem of transient propagation in a viscoelastic medium. This problem is of common interest for various investigations ranging
from seismic phenomena in geophysics to the mechanical behaviour of viscoelastic polymers. Within the context of a linear theory of viscoelasticity, the direct problem is based on the knowledge of a relaxation
function Gτ which characterizes the stress state of the material for any given history of the applied strain. In some cases the available mechanical model is not sufficiently refined to give a relaxation function which
works effectively for practical purposes. This fact is also due to the difficulty of measuring specific parameters such as the characteristic relaxation times. In those cases in which a stress–strain law holds in a differential
form, a definite expression can be worked out for Gτ as a sum of exponentials of τ . The most simple example of such relaxation functions arises from the Maxwell model which require the knowledge of only
one relaxation time. A more realistic behaviour of viscoelastic media can be achieved by the Maxwell–Kelvin model, which involves two relaxation times. It has been remarked that this last model generally agrees with
experimental results on creep tests Jaunzemis, 1967. Moreover, as previously shown Caviglia and Morro, 1990, the analysis of creep tests for such materials allows us to obtain reliable values of the relaxation times
and, in turn, a satisfactory expression of Gτ . Applications of the Maxwell–Kelvin model or, more generally, of exponential-like relaxation functions are customary in numerical computations for wave scattering problems
Ammicht et al., 1987; Caviglia and Morro, 1992, but no exact or approximate analytical solutions have been derived for the propagation and the reflection problems.
In this paper we consider a transient wave which propagates in a viscoelastic homogeneous isotropic medium described by the Maxwell model or by the Maxwell–Kelvin model and obtain analytical solutions for the
propagation and reflection problems. Our analysis parallels the approach by He and Ström 1996, to an analogous problem in a time-dispersive electromagnetic medium. In practicular we adopt a wave-splitting
technique to decompose the mechanical velocity field into two independent backward and forward propagating modes. Similar techniques have been widely used in electromagnetism to treat direct and inverse problems,
540 M. Romeo
especially in connection with transmission lines He, 1993; Åberg et al., 1995; Kristensson, 1995. The viscoelastic model is outlined in Section 2 where a differential form of the stress–strain law is used to obtain the
relaxation function Gτ . The wave splitting formulation is applied in Section 3 to both pure compressive and pure shear motions. Then, in Section 4, we analyze the split field in terms of a wave propagator which, via a time
convolution, relates the wave field at a fixed point of the space to the wave field at any other point. The integro- differential equation for the wave propagator is analytically solved for the Maxwell model and for a suitable
approximation of the Maxwell–Kelvin model. An exact solution for the reflectivity is derived in Section 5 at an elastic–viscoelastic interface for each viscoelastic model and it is shown how the reflectivity data at t = 0 can
be exploited to derive the characteristic relaxation times. Finally, the problem of transient propagation through a viscoelastic layer is solved in Section 6 arriving at a suitable expression for the reflection and the transmission
functions in terms of a series expansion.
2. Linear viscoelastic model
