Granular Dynamics in Compaction and Stress Relaxation

J. Brujic1, D. Johnson1, H. A. Makse2, O. Sindt1,

1 Schlumberger Doll Research, Ridgefield, Connecticut, U.S.A.2 Levich Institute and Physics Department, City College of New York, New York, NY 10031, US

Abstract

The nonlinear elastic and viscoelastic properties of cohesionless granular assemblies are stud-ied experimentally, under uniaxial compression. As a prelude to granular dynamics measure-ments, we investigate the system’s exploration of all possible static configurations through anovel compaction procedure at varying confining pressures. Once the system is fully compacti-fied we assume no further changes in volume and study the slow relaxation dynamics purely interms of grain-grain interactions and possible rearrangements through grains sliding and rolling.

1

fixed

walls

Figure 1: A model system

1 Introduction

The mechanical properties of granular materials are under investigation in many areas of research

due to their intriguing rheological properties, which govern many of their diverse applications.

Unconsolidated grains exhibit fluid behavior under flow and build up a yield stress provided their

packing density is sufficiently high. The transition between the fluid and solid-like mechanics of

the system reveals several specific features. The transmission of stress through the highly packed

granular medium is by no means homogenous, but suggests there is a more complex mechanism

withstanding the applied load: namely via force chains and arching. In order to gain a better

understanding of the material’s mechanical properties it is important to investigate the static shear

stress of the system and its response to an applied perturbation. Since this is a many body problem,

the relaxation of an applied stress is a very slow process involving collective particle motion, on

much longer timescales than that of the microscopic constitutive particles. The mechanism of stress

relaxation and the macroscopic response of the granular system to a confining pressure give insight

into the physics underlying the observed rheological behavior.

2 The Problem

When hard, rough particles (e.g. glass beads) form a granular pack such that all the particles are

touching their neighbors they experience a structural arrest and are referred to as ‘jammed’. Static

granular materials are characterized by two basic features: their packing density and the details of

its geometry, and stress propagation through the medium once an external force has been applied.

The density of a granular pack is dependent on the method of its creation, and in turn affects the

mechanism of stress relaxation by determining the likelihood of grain rearrangements. Since the

energy barrier to rearrange the jammed particles is several orders of magnitude higher than their

kinetic energy at room temperature, an external energy input is required to overcome the barriers

of friction in order to move the grains.

Chicago experiments [4] have shown that an adequate way of providing motion to the grains is

to introduce a vertical tapping mechanism of varying amplitude (and therefore acceleration) which

2

result in history dependent packing densities. They find that an increasing amplitude of the tapping

initially eliminates unstable loose voids, leading to a low-density, close-packed configuration. This

first compaction is irreversible. Subsequent taps of decreasing amplitude are able to reach denser

configurations and therefore result in a maximum packing density (e.g. φ = 0.64 for perfect

spheres). This result is reversible and confirms that there is an intrinsic property governing the

relation between the external energy input (i.e. the amplitude, frequency and the number of taps)

and the most favorable packing density of the material. Furthermore, they have shown that the

mechanism of the compaction process leading to a steady-state density is extremely slow, namely,

it is logarithmic in the number of taps. Many further studies have been done on the mechanisms

of compaction and simple models to describe the slow dynamics of the system [1].

These experiments support the ideas proposed by Edwards [5] that granular materials indeed

have entropy and their behavior can therefore be understood in terms of concepts of classical

statistical mechanics. This thermodynamical analogy arises from the assumption that there is a

large number of metastable states available to the system at a given energy and volume, which

have equal statistical probability. The dynamical properties of the system’s exploration of the

energy landscape lead to phenomenological analogies with other glassy materials. There are several

simple models that account for slow compaction available in the literature and can be meaningfully

adapted to granular materials.

It is important to distinguish between processes related to granular compaction and those related

to supporting stress once the volumetric conditions have been satisfied. According to the number

and magnitude of the taps, the grains search for the ‘equilibrium’ configuration, i.e. to occupy the

least volume. Once this criterion has been achieved, an application of an external pressure will

result in dissipation mechanisms quite different to the compaction process because of the volume

constraint.

2.1 Theoretical Background

It is perhaps useful to differentiate between the two regimes by considering the statistical mechanics

equations applicable to granular assemblies [2].

The granular analogue of the Gibbs distribution, δ(E − H)e−S/kT , can only depend on the

volume and the number of particles, and therefore leads to an entropy as the logarithm of the

number of jammed configurations:

S = log

∫δ(V − W )Θd all (1)

where V is the volume, W is the volume function and Θ is the function which ensures all particles

are jammed.

During compaction, the corresponding configurational analogue of temperature is defined as the

compactivity X:

X−1 = ∂Sconf/∂V (2)

and is a measure of how far the system is from its equilibrium packing density. However, at

constant volume, where X is zero, the relevant quantity becomes the effective temperature, arising

when the system has an internal energy:

3

T−1

conf=

∂Sconf

∂E. (3)

This quantity is also related to the number of jammed states available to the system, but this

time it is determined by the constraint on the energy. There have been interesting simulations and

theoretical work to show that both of those parameters coincide for a given system in glasses [7, 9, 8]

and in granular matter [6]. However, the macroscopic rheological behavior of the system in response

to external perturbations needs a clear distinction between these two measures of ‘temperature’.

2.2 Pressure-Dependence of Granular Assemblies

Previous studies of granular systems provided with an internal energy by the application of external

pressure have shown there is a strong dependency of the material properties (i.e. the bulk and the

shear modulus) on the confining pressure, which raised many interesting theoretical questions [3].

There have been mechanical rheological and sound propagation experiments probing the pressure

dependent behavior of granular matter. There is a lack of experimental data in the low pressure

regime, thus making the determination of a governing law for the pressure dependency all the

more difficult. The mechanistic features describing the grain-grain interactions under pressure and

the microscopic motion and grain dynamics experienced under pressure have also as yet not been

studied experimentally. The problem is complicated by the increased number of grain contacts as

pressure is increased, as well as the non-affine part of the motion including grain rearrangements

which are difficult to account for theoretically. In this work, we aim to test the dependence of

the bulk and shear modulus on confining pressure at low pressures and the mechanism of stress

relaxation in response to an external perturbation.

3 Experimental Set-Up

The experiments were performed using an INSTRON(spec.) press to measure the mechanical

properties of granular assemblies confined in a cylindrical cup and piston. The machine was strain

controlled and enabled oscillatory, step and ramp shear tests up to a maximum limit of 300kN load.

At such high pressures it was important to choose a very stiff material (in our case steel) for the

cup and piston, in order not to affect the material’s viscoelastic response. The cylinder had the

following dimensions: diameter d = 2inchs, height h = 1inch and wall thickness l = 1/3inch. These

dimensions were chosen to achieve a good statistical ensemble for monodisperse (to within 6%)

beads of average diameter 355µm for soda-lime glass (MO-SCI) and 400µm for acrylic. Moreover,

it was thought that the Jansen effect is avoided at heights less than “10” bead diameters and we

therefore filled only 1/2 inch of the cup’s height. By applying the strain in any of the available

shearing methods and measuring the resulting stress we were able to extract the compressional

moduli of the sample and therefore the macroscopic mechanical response of the system.

4

0 2000 4000 6000 8000 10000

0

−1000

−3000

−4000

−2000

Load

(N)

Load

000000time (s)

−0.8

−0.6

−0.4

−0.2

0

Dis

plac

emen

t (m

m)

Displacement

Figure 2: Compaction procedure: Sinusoidal displacement of the grain pile in response to anoscillatory stress around the mean value of 10MPa. The height (i.e. the packing density) at zeroload changes after the procedure according to the compaction amplitude.

4 Compaction Experiments

In this paper we present a novel way to compactify a granular material by a uniaxial oscillatory

compression technique. Since the grains were situated in a heavy, steel cup appropriate for use in

the INSTRON press it was not possible to use the mini-shaker for compaction due to its weight,

the ultrasonic vibrations could not provide a high enough amplitude for the beads to move and

the mechanical vibrator could not achieve full compaction. The material is first compressed with a

constant, slow velocity to a mean value of pressure, then oscillated between zero and double that

mean value with a frequency of 1Hz. Each compaction procedure consists of 10,000 cycles after

which the material is again slowly released to its uncompressed state, as shown in Fig. 2. The

sequence is then repeated for increasing values of the mean pressure and the correspondingly larger

amplitudes of oscillation. The final volume fraction of the material after each compaction is plotted

against the mean pressure to show the same trend as the tapping experiments discussed in Section

2 (See Fig. 3). The fact that the reversible packing density curve can be achieved in a system

which has internal energy and where the grains are confined under a pressure suggests that the

mechanism of supplying energy to the particles is irrelevant. This result could give validity to the

hypothesis that quasi-static shearing would have equivalent effects. More importantly, if it were

to be true that the same granular material could be compactified by two independent methods to

achieve the same reversible curve of the packing density, then the thermodynamic ideas of Edwards

would need no further justification.

This experiment provides the preliminary data for such a discovery, as well as a novel method to

achieve the necessary exact comparison. In this case, however, the system sizes and bead diameters

5

0 2 4 6 8

Compaction Mean Pressure (MPa)

61.5

62

62.5

63

63.5

Vol

ume

fract

ion

(%)

irreversiblereversible

Figure 3: Density fluctuations induced by oscillatory compression experiments at varying meanpressures and their corresponding amplitudes.

are different from the Chicago experiments, leading to different density values. The dynamics of

compaction, involving a slow evolution from one static state to the next, is well described by the

function y = A + B ln t (shown as the white line of best fit in Fig. 2), but one could argue that

the steady-state density has not been achieved on the timescale of the experiment and therefore

prohibit further speculation as to the exact form of the relaxation. To obtain good statistics

for the calculated relaxation time of 1000s, one would require a compaction procedure of the

order of one month, which time restrictions would not permit. This relaxation mechanism is quite

different in principle to the step-strain response experienced by the fully compactified material

under compression.

5 Stress Relaxation Experiments

The mechanical properties of the material are probed by stress relaxation experiments of both glass

and acrylic bead-packs under uniaxial compression. In particular, we investigate the viscoelastic

properties under low confining pressures, ranging from 0.2 to 10MPa. Prior to rheological measure-

ments we perform the above-mentioned stress-controlled procedure to ensure that the beads are

fully compactified, i.e. at the maximum random close packing configuration. The system is then

perturbed by an infinitesimal, constant strain amplitude which is ‘instantaneously’ applied, and the

relaxation properties are observed by recording the stress over time. The y-axis is labelled ‘Mod-

ulus’ throughout this report to represent the normalized G(t)/G(0), facilitating the comparison of

the relaxations alone.

6

Compaction

Procedure

reversiblereversible

<p>

58%

64%=

irreversible

RCP

Confining

Pressure

Φmax no pressure

ΦΦΦΦ

p

Step StrainFrequency

Sweep

t

eqmeqm

stabilisestabilise

Dissipation

Mechanisms

t

σσσσ

γγγγ

Figure 4: Sequence of the experimental procedure for compaction and relaxation measurements.

7

0 1 10 100 1000t (s)

760

780

800

820

840

860

Mod

ulus

(MP

a)

Figure 5: Stress relaxation after a step strain γ = 0.04% under a confining pressure of 8MPa.

We find two characteristic relaxation times at low confining pressures, independent of pressure

and amplitude of the applied strain. The characteristic relaxation curve is presented in Fig. 5, and

is fitted exactly with the following equation:

M = 778.62 + 79.95 × e−t/1.42− 1.682 ln(x/472.41) (4)

This experimental fit carries much significance in terms of interpreting the underlying mech-

anisms of energy dissipation within the system. It is closely related to some of the free volume

arguments presented in glassy dynamics models and will be discussed further in the following sec-

tion. The simplest relaxation model is a single exponential, if it is independent of the free volume

itself. It is intuitively clear that the dynamics is self inhibitory, in a sense that it is the free volume

itself that allows further relaxation. Therefore, it is expected that the relaxation vanishes when all

the grains have relaxed. It thus follows that the number of grains remaining to relax is proportional

to the change in the stress. The second hypothesis is that the grains relax only when they have

room to; a condition which is more difficult to satisfy as fewer grains remain mobile (unrelaxed).

A void has to be created by the neighboring grains, which determines the barrier height of the

attempt rate for each grain to move. This kind of self-retardation mechanism is best understood by

considering the probability of grain motion in terms of the number of voids available to it. At short

times, many grains remain unrelaxed and the relaxation is therefore exponential (fast), whereas at

long times the retardation mechanism dominates, crossing over to a logarithmic, slow relaxation.

This result has been derived by many simple models for collective relaxation mechanisms which

arrive at similar functional forms[].

8

5.1 Dependence of Stress Relaxation on Confining Pressure

The stress relaxation experiments were performed at different initial confining pressures to measure

the viscoelastic properties of the system. At each pressure, a sinusoidal shear was applied to ensure

a stable packing with no change in volume at the given constant pressure. The subsequent step

strain applications provided infinitesimally small perturbations to the system, thus probing the

macroscopic mechanical properties.

As the confining pressure on the glass beads increases their elastic modulus and therefore

stiffness increase, much beyond that of the steel container. Therefore, it gets progressively more

difficult to separate the deformation due to glass beads and that of steel. To make matters more

complicated, we used two different load cells within the INSTRON machine, namely the 10kN cell

for low confining pressures and the 300kN cell beyond the limits of the small cell in order to achieve

the highest possible accuracy. Naturally, the two set-ups have different mechanical responses, which

must be taken into account when considering the deformation of the beads themselves. It turns out

that since the small load cell is very soft relative to the beads, it takes up all the deformation, thus

prohibiting the calculation of the bead modulus. In the case of the 300kN load cell, at confining

pressures of up to 5MPa it is still possible to separate the two responses and isolate the elastic

modulus of the glass beads, but above that value, yet again the deformation is purely attributable

to the cell set-up. Since the applied stress is transmitted through both the sample and the set-up

and there are no long relaxation processes in the set-up alone, it is reasonable to assume that it is

the beads which provide the observed slow collective relaxation mechanism. It is therefore useful

to consider the relaxation processes alone, and disregard the calculated modulus comparisons.

5.1.1 10kN Load-cell Results

In Fig. (6), there is a definite trend which should be noted. It is that the profundity of the

relaxation is dependent on either the amplitude of the step strain, or the order number of the

experiment in the sequence. If the amplitude were the determinant factor, one would expect to

see less relaxation for smaller steps, which is in fact the contrary to the observed trend. It later

becomes obvious that all the applied strains are well within the linear viscoelastic limit and should

therefore not affect the relaxation, however, the sample is history dependent and memory effects on

the relaxation become obvious. Each time a step relaxation is performed, there is subsequently a

small compaction of the sample as a whole and a smaller relaxation follows. Since the difference in

the density is minuscule between experiments (on the order of 0.01%) it is not always obvious how

to differentiate between temperature effects and true compaction. However, this small compaction

effectively means that each run is performed on a ‘different’ sample. Once there is no change in the

response of the system we can assume a behavior which reflects on the true pressure dependency

of the system dissipation.

Furthermore, it seems that the key feature of the mechanism of relaxation is the amount of

motion available to the grains, which is closely tied to their initial packing density, or in other

words, their proximity to the ‘equilibrium’, optimal packing at a given pressure. Despite enormous

efforts to fully compactify the system as described in Fig 4 prior to these measurements, it seems

that the system still manages to compactify further (by very small amounts on the order of a few

9

0.1 1 10 100 1000

time (s)

0.8

0.85

0.9

0.95

1

Mod

ulus

I _ 7µmII _ 12µmIII _ 17µmIV _ 21µmV _ 31µmVI _ 41µmVII _ 54_µmVIII _ 117µm

Figure 6: Stress relaxation after infinitesimal step strain applications at a constant confining pres-sure of 0.2MPa.

0.1 1 10 100 1000

time (s)

0.96

0.97

0.98

0.99

1

Mod

ulus

I_12µmII_23µmIII_24µmIV_45µmV_96µm

Figure 7: Stress relaxation after infinitesimal step strain applications at a constant confining pres-sure of 0.5MPa.

10

0.1 1 10 100 1000

time (s)

0.96

0.97

0.98

0.99

1

Mod

ulus

I_7µmII_32µmIII_51µmI_101µm

Figure 8: Stress relaxation after infinitesimal step strain applications at a constant confining pres-sure of 1MPa.

mum) in search of a lower energy minimum. The chances are that it can never find the lowest

energy state (or packing density) available to it in the available phase space using experimental

techniques on feasible time-scales. The lower the confining pressure, the harder it is to obtain

reproducible relaxation curves, which would indicate a system-specific mechanism of relaxation.

This observation can be rationalized in terms of the reduced degree of freedom available to each

grain as the pressure is increased. Hence, grain rearrangement becomes more difficult to achieve

and the system achieves the equilibrium configuration with fewer relaxation attempts.

In support of this argument, the data at 0.5MPa pressure shown in Fig. (7) shows that all

the relaxations exactly overlap, independent of the step amplitude. It appears that the initial, fast

relaxation is more pronounced for the highest step amplitude, which could be interpreted as an

increased dissipation due to grain motion within its ‘cage’ of neighbors. The relaxation at long

times remains unchanged. It marks the onset of non-linear viscoelastic behavior.

The results at 1MPa pressure shown in Fig. (8) show further evidence of a linear behavior with

no signs of compaction and more profound dissipation at the highest strain amplitude, confirming

that the trend is a true characteristic of the system, rather than an artifact of the experimental

technique. Note that in both of the mentioned cases, the dissipation increases above a yield strain,

contrary to the observations in the first set of experiments shown in Fig. (6), explained by com-

paction. The data obtained at 2MPa (see Fig. (9))and 3MPa pressure repeat the results of lower

pressures, although there is an additional pattern evolving, making the behavior almost predictable:

If the sample is fully unloaded prior to the application of pressure with a constant velocity and

11

0.1 1 10 100 1000

time (s)

0.95

0.96

0.97

0.98

0.99

1

Mod

ulus

I_15µmII_42µmIII_49µmIV_98µm

Figure 9: Stress relaxation after infinitesimal step strain applications at a constant confining pres-sure of 2MPa.

subsequent step strains, it necessarily experiences a sequence of profound, compaction relaxation

curves until it reaches some equilibrium configuration. It seems that unloading and loading the

sample with a constant, slow velocity builds up a frustration within the system in some way (even

though it is supposedly at the random close packed limit to begin with) and it therefore searches to

accommodate that frustration, reflected in the sequence of relaxation curves which typically look

like those presented in Fig. (6).

5.1.2 300kN Load-cell Results

The results for the 5MPa pressure are again a representation of the system’s search for equilibrium,

which is only achieved after 10 repeated step experiments. It is additionally the only pressure at

which the deformation of the beads can be separated from that of the load cell set-up, and therefore

gives an idea of the elastic modulus obtained at that pressure (see Fig. (10). Since the error bars

in the elastic modulus (i.e. at t = 0)are largest for small steps (+ − 1µm), it is reasonable to take

the value obtained from the largest step as the most accurate (assuming, justifiably, that we are

in the linear regime). This would indicate a modulus of 5GPa at a pressure of 5MPa, which is in

reasonably good agreement with simulation predicted data (K = 2GPa). It is, of course, futile to

draw conclusions from a single data point, but nonetheless it shows that the experimental technique

could probe the relevant quantity.

The reproducible curves at the higher pressure of 8MPa shows consistent results with previous

12

0.1 1 10 100 1000

time (s)

1000

1500

2000

2500

3000

3500

4000

4500

5000

Mod

ulus

(MP

a)

0.01%0.03%0.06%0.05%0.034%

Figure 10: Stress relaxation after step strain applications at a constant confining pressure of 5MPa.The modulus corresponds to the displacement of the beads only.

data (See Fig. (12)), although the 10MPa results consistently show an increased slope of relaxation

after 100 seconds. Perhaps this is indicative of further relaxation processes enabled by overcoming

the frictional thresholds between individual beads which expose them to exploring configurational

phase space previously inaccessible to them (see Fig. (13)).

So far we have determined the characteristic nature of the relaxation of stress for each confining

pressure and its dependence on a.) the compaction processes (related to the sample history) and b.)

the more profound relaxation above the yield point. When comparing the reproducible relaxation

within the linear viscoelastic regime for different pressures, shown in Fig. (14), the first obvious

remark is that they are all very similar, perhaps even independent of pressure if we consider the

error bars involved. In addition, the extremes at 0.2MPa pressure can be explained by the fact

that the sample was not yet fully ‘equilibrated’, as discussed, and at 10MPa pressure exhibits quite

different relaxation behavior speculated to arise from a new mechanistic origin.

6 Effect of Velocity of Loading on Stress Relaxation

The velocity of approach of the crosshead to apply the step amplitude was found to drastically

affect the observed stress relaxation. The faster the speed of approach, the more pronounced

the fast relaxation (during the first 10 seconds) appeared. It confirmed the existence of a second

characteristic relaxation time of the system and showed the larger extent of dissipation at short

times. A set of step relaxation data was taken at 8MPa constant pressure and constant step

amplitude to investigate the effect of increasing approach velocity, but the rest of the measurements

were taken at 1mm/min. The reason for this moderate speed was to ensure a carefully controlled

13

0.01 0.1 1 10 100 1000 10000

time (s)

0.85

0.9

0.95

1

Mod

ulus

0.01%0.03%0.06%0.05%0.034%

Figure 11: Normalized stress relaxation after infinitesimal step strain applications at a constantconfining pressure of 5MPa.

Figure 12: Stress relaxation after infinitesimal step strain applications at a constant confiningpressure of 8MPa.

14

0.1 1 10 100 1000

time (s)

0.975

0.98

0.985

0.99

0.995

1

Mod

ulus

21µm35µm fast35µm

Figure 13: Stress relaxation after infinitesimal step strain applications at a constant confiningpressure of 10MPa.

0.1 1 10 100 1000

time (s)

0.96

0.97

0.98

0.99

1

Mod

ulus

0.2MPa0.5MPa1MPa2MPa3MPa5MPa8MPa10MPa

Figure 14: Characteristic stress relaxations at various confining pressures.

15

0.01 0.1 1 10 100 1000

time (s)

0.9

0.95

1

Nor

mal

ised

σ (t

)

44µm 0.1mm/min44µm 1mm/min44µm 5mm/min48µm 10mm/min45µm 15mm/min47µm 15mm/min

Figure 15: Effect of velocity of strain application on the observed relaxation curves. The legendstates the strain amplitude followed by the velocity of approach.

experiment without overshooting the targeted step amplitude. The results are shown in Fig. (15).

Note that the input strain amplitude was the same for all velocities, but the faster ones were not

able to stop at the appropriate displacement. The danger is that if the crosshead misses the required

step amplitude it must be stopped manually, or else it can crush the sample.

6.1 Hysteretic Behavior

Another way to probe the mechanisms of energy dissipation of the fully compactified system is by

performing oscillatory compression experiments and observing any hysteretic behavior. Attempts

were made to quantify the phase difference between the applied stress and the resulting strain,

however the data were not well approximated by a simple sinusoidal fit, and therefore need further

analysis including the higher harmonics to obtain the loss modulus.

As a preliminary result, hysteretic behavior was examined as a function of frequency, as shown

in Fig. 16, proving that the system is indeed dissipative. It is evident from the graph that the

higher the frequency, the more dissipation is observed. Moreover, the graph shows the perfect

superposition of 1000 cycles at a given frequency, thus showing that there is no further evolution

of the system with time.

In principal, it should be possible to extract the loss modulus from the area within the hysteresis

loop, but since the displacement could not be attributed to the beads, this kind of quantitative

analysis was not possible. The apparatus was limited to the low frequency range (0.001 to 2Hz)

due to the slow acquisition time, but we were able to investigate the full frequency range by

Fourier transforming the stress relaxation data. The frequency-dependence of the loss modulus

was extracted using the standard equations of viscoelasticity theory from the measured σ(t).

16

-0.02 -0.01 0 0.01 0.02

Displacement (mm)

3000

4000

5000

6000

7000

p (k

Pa)

0.1Hz0.5Hz1Hz2Hz

Figure 16: Hysteresis during an oscillatory strain of amplitude 20µm and frequency 1Hz repeatedover 1000 cycles.

Fig. (??) shows the loss modulus as a function of frequency as obtained from one of the typical

relaxation curves. It is in good agreement with the observed trend in the hysteretic behavior in

that the loss modulus increases up to the maximum dissipation at a frequency of 4Hz, which is

higher than the INSTRON instrument limitations. Therefore, within the overlapping range, the

trends were consistent between the two types of experiments. The result revealing a broad peak in

the loss modulus at a ‘resonant’ frequency is theoretically expected for a viscoelastic material.

7 Ambient Temperature Effects on Viscoelastic Measurements

Another important observation is that the stress relaxation results are greatly affected by ambient

temperature fluctuations on the time scale of the experiments. This effect is attributed to the

thermal expansion of the beads, which is of the same order of magnitude as the step strain amplitude

applied to the system as a whole. Consequently, the duration of the relaxation which could be

trusted was limited to 1000 seconds, illustrated in Fig. (19).

Attempts were made to control the temperature externally by placing the sample cell in an

oven at a temperature with an accuracy of one degree. Rather than improving the experimental

technique, this control confirmed the previous predictions of the importance of temperature effects

on the viscoelastic measurements. It became evident that the fluctuations in the relaxation were

on the timescale of the order of 100s, directly correlated with those of the temperature. Moreover,

there was clearly an additional drift of the temperature on longer timescales, making it impossible

to correct for the problem to obtain a clean relaxation. This effect is shown in Fig. (18), illustrating

the relaxation of stress.

17

1×10-2 1×10-1 1×100 1×101

ω [Hz]

0

0.1

0.2

0.3

0.4

0.5

G"(

ω)

Figure 17: Frequency dependence of the loss modulus (G′′(ω) extracted from stress relaxation data.

1 10 100 1000

time (s)

2400

2420

2440

2460

2480

Forc

e (N

)

step (139µm) _ 0.25MPa

Figure 18: Temperature fluctuations within the oven govern the relaxation process.

18

1 10 100 1000

Time (s)

60

80

100

Mod

ulus

(MP

a)

Figure 19: Characteristic stress relaxation affected by temperature fluctuations at long times.

8 Validity of the Hertz Law in Granular Packings

With the goal of comparing the response of a single bead and that of an assembly, we performed

additional sensitive measurements of the force/displacement relation for a single bead. The DMA

result showed a force law with the power law exponent of 1.29, shown in Fig. (21), slightly underes-

timating the exponent of the theoretical Hertz law for elastic particles which predicts F ∝ x3/2. The

collective response, however, obtained from compressing the whole sample, overestimates the same

law by measuring a power low exponent of 1.72. The discrepancy between the two measurements

shows that collective particle behavior and particle interactions do indeed change the property of

the material itself. The mechanism by which they do so is yet to be deciphered.

9 Experimental Difficulties

During the course of experimentation, many different systems were tried without success. It is

useful to keep a log book of these, so as not to repeat them. Firstly, it was important to find

beads of the appropriate size to suit the set-up. Beads larger than 500µm were very difficult to

compactify in the small cell used and did not give a very good statistical ensemble to probe the

collective relaxation. On the other hand, beads smaller than 100µm were small enough to fit in the

gap between the piston and the cylinder and hence cause it to jam. Another observation for the

small beads was that they aggregate under vibrations due to possible hydration and therefore form

clusters. The sample purity and narrow size distributions were also important control parameters,

19

0.001 0.01 0.1

Displacement (mm)

100

1000

10000

F (N

)

F=112928 * x1.72

Figure 20: Loading and unloading of the sample with velocity v = 1mm/min to test the intergrainforce law.

1 10 100

x (microns)

0.1

1

10

100

F (N

)

F = 0.236 * x1.287

Figure 21: DMA experiment to measure the force law for a single glass bead (d = 355µm).

20

so as to eliminate relaxations attributable to impurities and for comparison of different bead sizes,

respectively.

During sinusoidal measurements, the higher the frequency, the lower the actual amplitude the

INSTRON can achieve. Therefore, for frequency sweeps with constant amplitude, it is necessary

to compensate for this error by falsely increasing the input amplitude.

It is important to state the experimental elastic limit of the glass beads, which is lower than the

elastic moduli quoted in the literature. The 300µm glass beads were found to break at pressures

above 20MPa. The grinding of beads causes fine powder formation which can be detected at the

surface of the sample. During the compaction experiment, an unusually high packing density was

detected which did not fit with the compaction trend, and indeed corresponded to damage being

done to the beads.

10 Conclusions

During the course of this internship, there were several important results which emerged once the

data was carefully analyzed, and could be very useful in future INSTRON experimentation as well

as granular materials research as a whole. The most important conclusions are summarized below:

1) It was shown that granular compaction can be induced by a novel method under compression,

instead of the well known vertical tapping mechanisms. This discovery is useful both in terms of

industrial processing where tapping the sample is inaccessible yet a close packed system is required,

as well as further support to the far-reaching theoretical argument which claims that entropic

considerations will determine the density of a granular pack independent of the type of energy

supplied for grain motion.

2) It is imperative to distinguish between stress relaxation processes related to reaching a

granular ‘packing equilibrium’ and the remaining relaxation characteristics of the system without

further changes in volume. The sample is history dependent and therefore must be presheared each

time a new pressure is introduced.

3) Once the system is in equilibrium at a given pressure, the relaxation seems to be independent

of the confining pressure and is characterized by a fast, exponential relaxation followed by a slow,

logarithmic decay. The fast relaxation indicates the grain-grain interaction within its cage. The

slow decay can be interpreted in terms of previously described collective relaxation mechanisms

also used in compaction dynamics.

4) Several important experimental parameters were identified and need further consideration for

future experimentation, such as the stiffness of the apparatus in the INSTRON, the effects of room

temperature fluctuations, the use of appropriate bead characteristics and the importance of sample

preparation in gaining insight into the properties of this rather unpredictable class of materials.

As a final remark, I would add that despite the lack of conclusive evidence of pressure depen-

dency of the material, these results have brought substantial insight into the relevant questions

which need to be asked and further investigated.

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