Energy dissipation during wave propagation in fragmented geomaterials can be caused by independent movement of fragments leading to energy loss on their impact. By considering a pair of impacting fragments at times much greater than the period of their oscillations, we show that at a large timescale, the dynamics of the pair can be described by a linear viscous model with damping coefficients expressed through the restitution coefficient representing energy loss on impact. Wave propagation in fragmented geomaterials is also considered at the large timescale assuming that the wavelengths are much larger than the fragment sizes such that the attenuation associated with wave scattering on the fragment interfaces can be neglected. These assumptions lead to the Kelvin–Voigt model of damping during wave propagation, which allows the determination of a dispersion relationship. As the attenuation and dispersion are not related to the rate dependence of rock deformation, but rather to the interaction of fragments, the increased energy dispersion at low frequencies can be seen as an indication of the fragmented nature of the geomaterial and the capacity of the fragments for independent movement.

Geomaterials are often fragmented, with the fragments covering different
scales. This makes it important to understand the properties of wave
propagation in such geomaterials. Fragmented materials are characterized by
three major features. First is the bilinear nature of contacts when stiffness
in compression is considerably higher than stiffness in tension. Bilinear
oscillators feature multiple resonances, both multi-harmonic and sub-harmonic

Second is the possibility of block rotations. The bending between fragments
leads to elbowing of the neighbouring fragments in the course of their mutual
rotations

Third is the energy dissipation associated with the impact of blocks
characterized by low restitution.
The main feature of this type of dissipation is that it acts only at the
neutral position of the oscillators formed by pairs of adjacent blocks. In
this paper we consider only this special type of energy dissipation and its
influence on

Mathematically, the basic dynamic element involved in the process of wave
propagation through fragmented geomaterials is a pair of neighbouring
fragments, which can be modelled as a free undamped oscillator consisting of
a single mass on an undamped spring complimented by a condition that velocity
decreases each time the system passes through the neutral point and a
second-order differential – Eq. (

In this system, each next impact starting from the second occurs after time

Vibrations for fragmented media (red solid line) and
impact oscillator (blue dashed line) for absolute values of

Although the analysed model is relatively simple to describe, its implementation for energy dissipation during wave propagation is a challenging problem, the reason being that the boundaries of the time intervals where the system behaves linearly are not known a priori as they are influenced by incomplete restitution and hence need to be determined step by step. Therefore, an equivalent continuous damping model with effective coefficients should be selected and the relationship between it and the original model should be established.

Here, viscous damping governed by the Kelvin–Voigt model is chosen due to
its simplicity and popularity. This model represents a free oscillation of a
mass with damping that can be characterized in a dimensionless form by the
damping coefficient

Among those three solution types, only the underdamped solution is physically
admissible for a comparison with the proposed model because the other two
types do not intersect the axis

Comparing the expressions inside the

Relationship Eq. (

It is seen from Eq. (

When both types of energy dissipation take place, the restitution coefficient
should be replaced by a damping coefficient; therefore, a reverse
relationship is also important to define. It can be found relatively easily
from Eq. (

It is seen that between the impacts, the functions can be quite different
even for high values of

Vibrations for fragmented media (solid lines) and
the equivalent Kelvin–Voigt model (dashed lines) for

The dissipated energy of vibrations with the same parameters is given for
both cases by the following equations, where

Dissipated energy of vibrations for fragmented media
(solid line) and the equivalent Kelvin–Voigt model (dashed line) for

Now, after establishing the large timescale equivalence of the discrete and
continuous dynamics of a pair of fragments, the energy loss during wave
propagation in fragmented geomaterials can be modelled by replacing the
fragmented geomaterial with a visco-elastic continuum described by a
Kelvin–Voigt model. The

Velocity characteristics of a

It is seen that both the wave velocity and coefficient of absorption increase
with frequency; however, the increase in the wave velocity becomes weaker as
the restitution coefficient increases. Subsequently, the dispersion vanishes
as the restitution coefficient tends to 1, i.e. the impacts are not
accompanied by energy loss. It is also noteworthy that these formulae can be
implemented for the

A possible mechanism of wave attenuation in fragmented geomaterials with fragment sizes much smaller than the wavelengths is the energy loss on impact of the contacting fragments with each other. The energy loss is characterized by the restitution coefficient. It is shown that the energy loss during wave propagation in such a discrete material can be modelled by an equivalent visco-elastic continuum if the characteristic times involved are considerably greater than the periods of oscillations of all neighbouring pairs of fragments. The attenuation is modelled by the Kelvin–Voigt model, its equivalent damping being expressed through the restitution coefficient and the period of oscillations of contacting fragments averaged over all pairs. For all restitution coefficients smaller than 1, the wave velocity shows a dispersion relationship, which is stronger the smaller the restitution is. The attenuation and dispersion are not related to rate-dependent rock deformation, but rather to the interaction of fragments. For that reason the effect is long-wave. Therefore, increasing damping and dispersion at low frequencies can be seen as an indication of the fragmented nature of the geomaterial and the capacity of the fragments for independent movement.

No competing interests are present.

This article is part of the special issue “Waves in media with pre-existing or emerging inhomogeneities and dissipation”. It is a result of the EGU General Assembly 2016, Vienna, Austria, 17–22 April 2016.

The first author gratefully acknowledges the scholarship support from the University of Western Australia through the Research Training Program (RTP) Scholarship.Edited by: Sergey Turuntaev Reviewed by: two anonymous referees