VOLUME 85, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JULY 2000
Universal Slow Dynamics in Granular Solids
James A. TenCate, Eric Smith, and Robert A. Guyer
Los Alamos National Laboratory, Earth and Environmental Sciences Division, Los Alamos, New Mexico 87545
(Received 27 September 1999; revised manuscript received 22 December 1999)
Experimental properties of a new form of creep dynamics are reported, as manifest in a variety of
sandstones, limestone, and concrete. The creep is a recovery behavior, following the sharp drop in
elastic modulus induced either by nonlinear acoustic straining or rapid temperature change. The extent
of modulus recovery is universally proportional to the logarithm of the time after source discontinuation
in all samples studied, over a scaling regime covering at least 10
3
s. Comparison of acoustically and
thermally induced creep suggests a single origin based on internal strain, which breaks the symmetry of
the inducing source.
PACS numbers: 62.40. + i, 62.65. + k, 91.60. – x
Relaxation processes in which some stress is relieved as
the logarithm of time after a step function is applied are
remarkably common: in mechanical response of rocks [1]
and metals [2], thermoremnant magnetization relaxation
in spin glasses [3], dc susceptibility in granular magnetic
media [4], and even fluid invasion percolation in soils [5].
Though these systems have completely unrelated micro-
scopic dynamics, they have in common that these forms of
creep all monotonically minimize a free energy in response
to a constant external stimulus, respecting the symmetry of
the source.
In studies of nonlinear elasticity of many rocks, it was
found that, at strains 10
26
, retarded effects resembling
creep appeared, which could not be explained with equi-
librium elastic theory, either classical [6] or hysteretic [7].
Their universal feature was a persistent drop in elastic
modulus, and increase in material damping, which could
be induced by harmonic acoustic stressing, and we have
since found also by thermal shocking. After the stress or
shock is removed, the material properties recover toward
their original values as the logarithm of elapsed time, over
a featureless scaling regime lasting hours to days. An im-
portant feature of this effect, which we call slow dynamics,
is that the elastic modulus decreases in response to sym-
metric stress cycling or temperature change of either sign,
thus violating the symmetry of the inducing source.
This Letter reports properties of slow dynamic response
in a variety of sandstones, limestone, and concrete. A reso-
nance method is used to enhance sensitivity to small shifts
in material properties: reduced modulus causes resonant
frequency to drop, and increased damping lowers the res-
onant quality factor Q (defined so that successive cycles
of an undriven oscillator decrease in magnitude by e
2pQ
[8].) The materials studied are sufficiently different that
there appears to be no universally shared chemistry, char-
acteristic scale, or microstructure, suggesting that slow dy-
namics may be an emergent form of creep not seen before.
The symmetry breaking of slow dynamics resembles the
quick loss of microscopic contact area, and its subsequent
restoration as log (time), in the slipstick of a static friction
bond [9]. Depending on geometry, however, slip may lead
to irreversible change (damage accumulation) while slow
dynamic recovery is (at least macroscopically) perfect,
even over hundreds of cycles spanning more than a year.
The combined resemblance to stress-relieving creep and
bond rupture raises the question of whether slow dynam-
ics may arise from a glasslike state, somehow intermediate
between equilibrium elasticity and damage formation.
Creep in all familiar systems [1–5,9] is understood as
a result of thermal activation, and even in the absence of
a full microscopic understanding of slow dynamics, this
leads to a generic prediction of temperature dependence of
the susceptibility. Therefore, slow dynamic recovery was
measured as a function of temperature as well as sample
type and, in one case, humidity. The suspected connec-
tion to bond rupture suggests that modulus and damping
changes are induced by internal straining. Therefore, the
shifts from acoustic driving and thermal shocking were
also compared. From the known thermal expansion an-
isotropy of the underlying grain materials, the compari-
son shows all slow dynamic effects to be consistent with a
single strain-dependent origin.
The experimental method, described in detail in
Refs. [10,11], is to drive a suspended cylindrical sample
of rock or concrete in the fundamental longitudinal
elastic mode (Young’s mode), with a piezoelectric force
transducer cemented between one end of the sample and
a massive backload. Acceleration of the opposite end of
the sample is measured with a lightweight accelerometer
and processed with a lock-in amplifier referenced to the
driving signal. The driving force is a harmonic acoustic
wave, incremented through the fundamental resonance
frequency of the bar, to produce the frequency-dependent
lumped-parameter response function of the resonant
barbackload system. These response functions can cur-
rently be processed to resolve shifts in Young’s modulus
of one part per million of the static value.
Figure 1 shows the amplitude of a typical response func-
tion, for frequency incremented several times per second,
first upward and then downward through resonance. An
elastic system described by an equilibrium equation of
state, either single-valued (classical) [6] or multivalued
1020 0031-9007 00 85(5) 1020(4)$15.00 © 2000 The American Physical Society