Permanent Deformation of Granular Materials: Modeling & Analysis

Construction and Building Materials, Vol. 12, No. 1, pp. 9]18, 1998
Q 1998 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0950]0618r98 $19.00 q 0.00
PII:S0950–0618(97)00078–0
Modelling permanent deformation behaviour of
unbound granular materials
F. LekarpU † and A. Dawson‡
U
Division of Highway Engineering, Royal Institute of Technology, S-100 44
Stockholm, Sweden
‡Department of Civil Engineering, University of Nottingham, Nottingham NG7 2RD,
UK
Received 15 April 1997; revised 13 November 1997; accepted 19 November 1997
A state of the art is presented on modelling permanent deformation behaviour of unbound granular
materials. The existing numerical models are verified by a series of laboratory repeated load tests. A
new model is introduced expressing the accumulated permanent axial strain at any given number of
cycles as a function of applied stresses, taking into account the maximum shear stress ratio and the
length of the stress path in p-q space. The results show similarities with the concept of the shakedown
theory. At low levels of stress ratios the accumulation of permanent strain ceases, resulting in an
equilibrium state. However, there are indications of the existence of a threshold stress ratio, known as
the shakedown limit, beyond which the material experiences incremental collapse or gradual failure.
Q 1998 Elsevier Science Ltd. All rights reserved.
Keywords: permanent strain; granular materials; plastic behaviour
Introduction
these small plastic deformation increments could lead
to an eventual failure of the pavement due to excessive
rutting. For performance prediction, it is of great importance to know whether a given pavement will experience progressive accumulation of permanent deformation, or whether the increase of permanent deformation will cease, resulting in a stable and fully resilient response.
The objective of this paper is to increase the understanding of the permanent deformation behaviour of
unbound granular materials. The results of previous
research from various sources have been collected
through a literature survey, which is presented in the
first part of this paper. The second part discusses the
findings from a series of repeated load laboratory tests
conducted at the University of Nottingham to study the
behaviour of a number of unbound aggregates. The
laboratory tests have been used to study the effect of
number of load repetitions and of stress level on plastic
strains.
A mechanistic pavement design procedure requires that
the constituent materials are efficiently used in regard
to their strength and stiffness characteristics. The current knowledge concerning the granular materials employed in pavement structures is limited. To date, no
overall framework has been established to explain satisfactorily the behaviour of unbound granular materials
under the complex repeated loading which they experience. During its service life, a pavement experiences a
large number of stress pulses each consisting of vertical, horizontal and shear stress components. These
stresses are transient and change with time as shown in
Figure 1. The shear stress is reversed as the load passes
and there is thus a rotation of the principal stress axes.
Granular materials exhibit two types of deformation
when subjected to repeated loading: resilient deformation which could lead to fatigue cracking of the overlying bound Že.g. asphalt. surface, and permanent Žplastic. deformation. Although the permanent deformation
during one cycle of loading is normally just a fraction
of the total deformation produced by each load repetition, the gradual accumulation of a large number of
Literature review
In modelling the long-term behaviour of pavements, it
is essential to take into account the gradual accumula-
†Corresponding author. Tel.: q46 8 7908707; fax: q46 8 108124;
e-mail: [email protected].
9
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
10
Figure 1 Stresses beneath a rolling wheel load
tion of permanent strain with number of load applications and the important role played by stress condition.
Perhaps the main objective of research into long-term
behaviour should be to establish a constitutive relationship which allows an accurate prediction of the amount
of permanent strain at any number of cycles at any
stress level. This section aims to present a brief state of
the art on modelling the permanent deformation behaviour based on the literature available to date.
Correlation between static and cyclic loading tests
Several research workers have attempted to correlate
repetitive loading and simple static loading test
results 1 ] 5. This approach has met mixed reaction since
the behaviour of granular materials is generally regarded to be very complex and cyclic and static loadings do not necessarily induce the same structural
response. Lentz and Baladi 3 used the static stress]
strain results of samples of sand to predict the cumulative permanent strain of identical samples tested under
cyclic loading condition. They suggested that, provided
the samples used in the static and cyclic tests are
identical in every aspect, the permanent strain under
cyclic loading can be expressed by a constitutive expression of the form
« 1, p s « 0.95S ? ln Ž 1 y qrS .
q
½
y0 .15
n ? Ž qrS .
? ln Ž N .
w 1 y m ? Ž qrS .x
5
Ž1.
where
« 1, p
s accumulated permanent strain after N load
repetitions
« 0.95S s static strain at 95% of static strength
q
s deviator stress s s 1 y s 3 Ž s 1 and s 3 are
principal stresses .
S
s static strength
N
s number of load cycles
n, m s regression parameters, which vary with confining pressure.
Lentz and Baladi obtained a good correlation
between measured and calculated values in their investigation. However, they pointed out that the equation
was based on the results from a single sand subgrade
material and additional research was needed. The validity of Equation Ž1. was investigated by Sweere 6 , for
both sands and granular base course materials, and
rejected.
Gerrard et al.5 suggested a new approach by relating
the applied stress level to the static shear strength
given by the Mohr]Coulomb static failure envelope. In
this case the static failure envelope is plotted on a
normal Mohr]Coulomb diagram as shown in Figure
2 Ž a.. The Mohr circles of the stresses applied in each
repetitive loading test are then drawn and the permanent strains marked on the circles at the point nearest
to the static failure envelope. The applied stress level
during the cyclic triaxial test is referred to by the ratio
of the cyclic shear stress to the corresponding static
shear strength, i.e. the ratio CArCB in Figure 2 Ž a..
The permanent strain contours are then drawn after a
certain number of load cycles as shown in Figure 2 Ž b ..
Whilst this approach allows the relationship of permanent strain to stress to be estimated, it makes no
prediction of development with number of load appli-
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
11
Figure 2 Mohr]Coulomb representation of permanent strains 5
cations. Barrett and Smith7 used this approach to express the permanent strain response of a crushed dolerite with a clay binder and reported good correlation
with observations. No other verification was found in
the literature.
q
N
s deviator stress
s number of load repetitions.
The experimental results were then used to express
the permanent deformation moduli as functions of
number of cycles by
Correlation between resilient and plastic beha¨iour
Veverka8 studied both resilient and plastic behaviour
of granular materials and found a correlation between
the two. He proposed a simple relationship between
permanent and resilient strains given by
« 1, p s a? « r ? N b
Ž2.
in which « r is resilient strain, and a and b are material
parameters. This equation is very simplistic, requiring
considerable testing as the stress dependency is not
explicit, and has not been confirmed by other researchers.
Permanent deformation moduli
Most of the work on modelling the permanent deformation behaviour deals with axial and horizontal
stresses and strains. Jouve et al.9 introduced a new
approach by decomposing both stress and strain fields
into volumetric and shear components. In a similar
manner to the elasticity theories, plastic deformation
‘moduli’ were defined by
p
q
K pŽ N . s
, Gp Ž N . s
Ž
.
Ǭ , p N
3 « s, p Ž N .
Ž3.
where
K p Ž N . s bulk modulus with respect to permanent
deformation
Gp Ž N . s shear modulus with respect to permanent
deformation
« ¨ , p Ž N . s permanent volumetric strain for N ) 100
« s, p Ž N . s permanent shear strain for N ) 100
p
s mean normal stress s Ž s 1 q s 2 q s 3 .r3
Ž s 1 , s 2 , and s 3 are principal stresses .
Gp s
A 2'N
'N q D 2
,
Gp
A 3'N
s
Kp
'N q D
Ž4.
3
in which the parameters A 2 , A 3 , D 2 and D 3 are
functions of the stress ratio qrp. The parameter A 2 is
dependent on the stress ratio by a relationship of the
type arx b while A 3 , D 2 and D 3 are connected linearly
with this ratio.
Permanent strain and number of cycles
Barksdale1 performed a comprehensive study of the
behaviour of several different base course materials
using cyclic load triaxial tests with 10 5 load applications. He found that the accumulated permanent axial
strain was proportional to the logarithm of the number
of load cycles and expressed the results by log]normal
expression of the form
« 1, p s aq b? log Ž N .
Ž5.
in which a and b are constants for a given level of
s 1 ] s 3 and s 3 . Once again, the definition of stress
dependency is less than ideal.
Sweere 6 applied 10 6 load applications in a series of
repeated load triaxial tests and observed that the lognormal approach did not fit his test results. He then
suggested that for a large number of load repetitions a
log-log approach should be employed, and expressed
the results Žalso without an explicit stress dependency
definition. by
« 1, p s a? N b
in which a and b are regression parameters.
Ž6.
12
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
The applicability of the log-log model presented by
Sweere was later challenged by Wolff and Visser 10 who
performed a full-scale Heavy Vehicle Simulator ŽHVS.
testing with several million load applications. They
described the permanent deformation behaviour as being composed of two phases. In the HVS testing results, an initial phase, up to 1.2 million load repetitions,
was observed with a rapid development of permanent
deformation but a constantly diminishing rate of increase. During the second phase, permanent strain
development seemed much slower and the rate of
increase approached a constant value. The log]log
model failed to give reliable estimates of permanent
strain at large numbers of load cycles. Wolff and Visser,
therefore, suggested an improved stress]strain model
given by
« 1, p s Ž m ? N q a. ? Ž 1 y eyb N .
Ž7.
where a, b and m are regression parameters.
Equations Ž5. ] Ž7. all imply that accumulation of
permanent strain with number of load cycles proceeds
indefinitely. Yet several other researchers 11 ] 21 have
reported that at least at certain levels of applied stresses
the induced permanent strain eventually levels off,
resulting in an equilibrium condition.
Khedr 13 used repeated load triaxial tests to study the
permanent strain behaviour of a crushed limestone and
came to the conclusion that the rate of permanent
strain accumulation decreases logarithmically with the
number of load repetitions according to an expression
of the form
«p
s A ? Nym
N
Ž8.
in which m is a material parameter and A a material
and stress-strain parameter given as a function of shear
stress ratio and resilient modulus. While Khedr reported correlation values quite close to unity for all
tested samples, no other verification of this model was
found in the literature.
Paute et al.19 suggested that permanent strain increases gradually towards an asymptotic value. Using a
similar approach to that of Equation Ž4., they expressed the relationship between permanent axial strain
and number of cycles by
U
« 1,
ps
A ? 'N
'N q D
Ž9.
in which « 1,U p is additional permanent axial strain after
the first 100 load cycles, and A and D are regression
parameters.
In a separate, recent study, Paute et al.20 proposed a
new approach to express the influence of number of
load applications on development of permanent deformation in granular materials. In this case, the accumulated permanent strain after an initial period of 100
cycles is given by
« 1,) p s A ? 1 y
N
100
yB
ž ž / /
Ž 10 .
in which A and B are regression parameters. According to this equation, « 1,U p goes towards a limit value
Žequal to A. as N increases towards infinity. The
parameter A is, therefore, considered as the limit value
for total permanent axial strain.
Permanent strains and their relationship to stress
Pervious research has shown that stress level has a
significant influence on the development of permanent
deformation in pavement structures. Several researchers who carried out repeated load triaxial tests
on unbound granular materials have found that permanent deformation behaviour is principally governed by
some form of stress ratio. Lashine et al.22 conducted
repeated load triaxial tests on a crushed stone and
found that the measured permanent axial strain settled
down to a constant value related to the ratio
qmaxrs 3max , where qmax and s 3max are maximum deviator stress and maximum confining pressure, respectively. Similar results have been reported by Brown and
Hyde 23 who studied the response of crushed stone
under cyclic triaxial condition with constant confining
pressure. Brown and Hyde further stated that the same
results are obtained from tests with variable confining
pressure if the mean value of the applied confining
stress is used in the analysis.
Barksdale1 used repeated load triaxial tests to relate
the permanent axial strain to the ratio of repeated
deviator stress and constant confining pressure. He
employed the general hyperbolic expression given by
Duncan and Chang 24 for static triaxial tests, and found
a close fit to plastic stress-strain curves obtained from
repeated load test results. Barksdale suggested that for
a given number of load applications the variation of
permanent axial strain with stresses can be expressed
by
« 1, p s
qrK? s 3n
Ž R f ? q . r2 Ž C ? cos f q s 3 ? sin f .
1y
Ž 1 y sin f .
Ž 11 .
where
K? s 3n s relationship defining the initial tangent modulus as a function of confining pressure, s 3
Ž K and n are constants .
C
s apparent cohesion
f
s angle of internal friction
Rf
s a constant relating compressive strength to
an asymptotic stress difference.
Pappin12 performed triaxial tests with variable confining pressure on specimens of a well-graded crushed
limestone. He reported that permanent shear strain
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
rate can be expressed as a function of the length of the
stress path in p]q space and the applied shear stress
ratio. He also calculated a shape factor for the variation of permanent strain with number of cycles, and
expressed the total permanent shear strain by
« s, p s Ž fnN . ? L ?
q8
p8
ž /
Ž 12 .
max
accumulated permanent shear strain
shape factor
stress path length
modified deviator stress s 2r3 ? q
modified mean normal stress s '3 ? p.
'
Pappin stated that unless the material was stressed
close to the static failure limit, large permanent strains
did not occur. However, the mathematical expression
suggested by Pappin is not asymptotic to failure and
predicts finite permanent strain even at or beyond the
static failure stress.
Other researchers have reported that the amount of
permanent strain is determined by how close the applied stresses are to the static failure stress. Barret and
Smith7 and Raymond and Williams 25 used the stress
ratio qmaxrqfailure to characterise the results of permanent deformation tests. Thom26 , on the other hand,
suggested that the permanent shear strain is better
related to the stress ratio Ž qfailure y qmax .rqmax .
In a recent study, Paute et al.20 defined a limit value
for the maximum permanent axial strain, A-value as
described in Equation Ž10., and suggested that it varies
with maximum shear stress ratio, qmaxrŽ pmax q pU .,
according to a hyperbolic expression given by Equation
Ž13.. This hyperbolic relationship indicates that A increases when the maximum shear stress ratio increases,
and that there is a limit value to the maximum shear
stress ratio Žequal to m. for which A becomes infinite.
This approach suggests that the static failure line,
defined by parameters m and pU , could be estimated
using the results from cyclic triaxial tests. On the other
hand, if the failure parameters are known, a single
triaxial test is needed to define the expression. Lekarp
et al.21 discussed this approach and showed that the
model generally results in quite unreasonable values of
failure parameters if these are estimated, or very low
correlation in cases where the failure line is determined by static failure tests.
qmax
Ž pmax q pU .
As
qmax
b? m y
Ž pmax q pU .
ž
pmaxs maximum mean normal stress
pU s stress parameter defined by intersection of the
static failure line and the p-axis in p]q space
m s slope of the static failure line
b s regression parameter.
2.8
where
« s, p s
fnN s
L s
q8 s
p8 s
13
Shakedown theory
The literature review revealed that several researchers1,14,17,18,27 ] 30 , having related the magnitude of permanent strain to shear stress level, concluded that at
low levels of stress ratio the resulting permanent strain
would eventually reach an equilibrium condition. At
high levels of stress ratio, however, they stated that
permanent strains are likely to increase rapidly resulting in an eventual failure. This has raised the possibility of the existence of a critical stress level separating
the stable and failure condition.
A certain theory of plasticity known as the ‘shakedown’ theory has been used for the response of structures subjected to repeated cyclic loading. This theory
which was first introduced by Melan 31 has been widely
applied to structures such as trusses, frames and plates.
Sharp and Booker 30 and Sharp 32 suggested that shakedown principles can also be employed for pavement
design. Applied in this manner, the shakedown theory
predicts that a pavement is liable to show progressive
accumulation of plastic strains under repeated loading
if the magnitude of the applied loads exceeds a limiting
value called the shakedown load. The pavement is then
said to exhibit an incremental collapse or incremental
failure. On the other hand, if the applied loads are less
than this shakedown load, the growth of plastic strains
will eventually level off and the pavement is said to
have attained a state of shakedown by means of adaptation to the applied loads. The pavement’s response
will then be elastic under additional load applications.
It is, therefore, postulated that the life of a pavement
under traffic is directly related to its resistance to
incremental failure, and a satisfactory pavement is one
that shakes down17 .
Although the possible application of the shakedown
theory as a convenient pavement design tool has been
recognised by several researchers, very limited effort
has been devoted to developing simple numerical models based on the theory. The few proposed numerical
methods found in the literature 15,17,33 deal with the
response of the whole pavement as a single, unified
structure. No work was found on adapting the shakedown theory to explain the behaviour of granular materials.
Ž 13.
/
where
A s limit value for maximum permanent axial strain
qmaxs maximum deviator stress
Experimental
A series of repeated load laboratory tests was performed at the University of Nottingham to study the
permanent deformation behaviour of unbound granular materials. Five different aggregates, mainly used as
subbase material in pavement structures, were tested.
14
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
Figure 3 Particle size distributions for the tested materials
The materials consisted of: crushed dolomitic limestone ŽLS., crushed granidiorite ŽGr., crushed slate
waste ŽSW., sand and gravel ŽS& G. and Leighton
Buzzard sand ŽS.. The particle size distribution curves
are given in Figure 3. In this investigation the Repeated
Load Triaxial Apparatus ŽRLTA. and Hollow Cylinder
Apparatus ŽHCA. available at the University of Nottingham were employed. Detailed description of the
testing equipment and the specimen preparation are
given elsewhere 34 .
A test programme was planned with the objective of
studying the development of cumulative permanent
strain with number of load applications and its variation with stresses. The test programme was chosen with
great consideration to the newly proposed European
standard 35 regarding repeated load triaxial testing. Details of the testing programme are given in Table 1. The
sand ŽS. was tested in the HCA using constant confining pressure since the existing equipment does not
allow application of variable confining pressure. In the
case of the other materials, however, the RLTA was
employed and both deviatoric and confining stresses
were cycled. All the tests were performed in a drained
condition to avoid development of excess pore water
pressures.
Results and discussion
Accumulation of permanent axial strain with number of
cycles
Development of permanent axial strain with number of
load applications is shown for each material in Figure 4
Žsee abbreviations at top of paper.. The data were
compared with some of the existing models discussed
previously ŽEquations Ž5. ] Ž10... The Paute model of
Equation Ž10. showed the least error in most cases and,
subjectively, seemed to fit the data best 34 . The predicted results by the Paute model are also given in
Figure 4. Table 2 shows the actual stresses applied in
each test and the results of the analysis for the Paute
model.
In most cases a very high correlation was found with
the Paute model, and the rate of increase of permanent
strain reduced gradually to such an extent that towards
the end of the test the specimen seemed to be in an
equilibrium condition. However, in a few cases ŽP1 for
Gr, P2 and P4 for LS, and P2 and P3 for S& G. the
model did not seem to fit the data well and no stabilisa-
Table 1 Details of the intended test programme
Material
Stress path
s3
N
q
Min
Max
Min
Max
Gr
DDs 2.09
MCs 1.9
P1
P2
P3
P4
P5
80 000
80 000
80 000
80 000
80 000
0
0
0
60
45
20
50
120
165
150
0
0
0
120
20
300
600
600
700
420
LS
DDs 2.26
MCs 5.0
P1
P2
P3
P4
P5
P6
80 000
80 000
80 000
80 000
80 000
80 000
0
0
0
0
60
45
20
50
75
120
165
150
0
0
0
0
120
20
300
600
300
600
700
420
SW
DDs 2.16
MCs 4.6
P1
P2
P3
P4
80 000
80 000
80 000
80 000
0
0
0
0
20
100
200
75
0
0
0
0
300
600
600
300
S& G
DDs 2.05
MCs 4.0
P1
P2
P3
80 000
80 000
80 000
100
100
100
135
285
220
0
0
0
200
500
400
S
DDs 1.52
MCs 0.0
P1
P2
P3
40 000
40 000
40 000
70
70
70
70
70
70
0
0
0
80
90
105
DD, dry density Žgrcm3 .; MC, moisture content Žgrcm3 .; N, number
of load applications; s 3 , confining pressure ŽkPa.; q, deviator stress
ŽkPa..
In the case of Gr and SW the test P3 had to be stopped at 20 000
cycles due to puncturing of the membrane.
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
15
Table 2 Results of permanent deformation tests
Stress path
qmin ŽkPa.
qmax ŽkPa.
pmin ŽkPa.
pmax ŽkPa.
Ž qrp.max
« 1, p Ž100. Ž10y4 .
A Ž10y4 .
B Ž10y4 .
R2 Ž10y4 .
Gr
P1
P2
P3
P4
P5
0.0
0.0
0.0
122.6
23.6
292.7
595.4
585.6
719.5
441.2
0.0
0.0
0.0
103.8
57.4
112.1
245.5
310.2
417.5
311.5
2.61
2.42
1.89
1.72
1.42
9.37
21.96
17.64
19.40
9.57
342.49
25.99
6.25
25.20
3.66
0.004
0.089
0.416
0.040
0.155
0.990
0.991
0.993
0.995
0.997
LS
P1
P2
P3
P4
P5
P6
0.0
0.0
0.0
0.0
131.7
23.1
296.1
593.0
298.1
594.4
726.2
440.6
0.0
0.0
0.0
0.0
107.2
55.8
117.5
245.9
170.8
312.6
419.4
310.6
2.52
2.41
1.74
1.90
1.73
1.42
7.10
20.57
4.72
16.96
9.77
7.42
14.41
608.68
4.16
1520.32
6.97
1.36
0.049
0.018
0.104
0.002
0.165
0.102
0.993
0.845
0.992
0.981
0.996
0.991
SW
P1
P2
P3
P4
0.0
0.0
0.0
0.0
295.4
589.4
591.5
295.4
0.0
0.0
0.0
0.0
117.6
292.5
388.1
170.0
2.51
2.01
1.52
1.74
25.89
211.66
71.60
94.78
1092.59
1226.00
4994.08
276.48
0.004
0.037
0.005
0.054
0.967
0.992
0.997
0.985
S&G
P1
P2
P3
0.0
0.0
0.0
198.0
498.1
395.7
100.0
100.0
100.0
199.3
429.0
346.5
0.99
1.16
1.14
3.73
29.26
14.45
177.58
6093.86
1218.27
0.004
0.002
0.003
0.816
0.980
0.864
S
P1
P2
P3
0.0
0.0
0.0
78.8
91.0
103.4
69.9
68.6
66.6
96.2
98.9
101.1
0.82
0.92
1.02
39.52
30.72
121.01
45.47
666.41
178.10
0.205
0.015
0.197
0.999
0.995
0.998
Material
tion was achieved. A possible explanation for how well
each test fits the model is given in the following section
where the effect of stresses is discussed.
Effect of stresses on permanent deformation beha¨iour
None of the models found in the literature seemed to
give a satisfactory explanation of the influence of
stresses on accumulation of permanent strain. However, further analysis of the test results indicated a
possible relationship between accumulated permanent
axial strain, the maximum shear stress ratio and the
length of the stress path. It was clearly observed that
accumulated permanent strain increased with maxi-
mum shear stress ratio. It was then argued that at a
constant shear stress ratio the amount of permanent
strain should increase with the length of the applied
stress path to reach this maximum point, as a result of
the greater energy dissipated in hysteresis during the
load-unload cycle. A consistent pattern was observed
when the ratio of permanent strain at a given number
of cycles and stress path length was plotted against the
maximum shear stress ratio. The relationship can be
given by a simple expression of the form
« 1, p Ž Nr e f .
q
s a?
L
p
Figure 4 Cumulative permanent axial strain versus number of load repetitions
ž /
b
Ž 14 .
max
16
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
where
« 1, p Ž Nr e f . s accumulated permanent axial
strain at a given number of
cycles Nr e f , Nr e f ) 100
L
s length of stress path
q
s deviator stress
p
s mean normal stress
Ž qrp. max s maximum stress ratio
a, b
s regression parameters
w}x
wkPax
wkPax
wkPax
w}x
w}x
The equation above is not dimensionally correct.
This can be avoided by dividing the stress path length
by a reference stress of, for instance, 1 kPa. By introducing p 0 as the reference stress Equation Ž14. would
become
« 1, p Ž Nr e f .
q
s a?
p
Ž Lrp 0 .
ž /
b
Ž 15.
max
It can be shown that Equation Ž15. is applicable for
any given number of load cycles greater than 100.
Figure 5 shows the results for Nr e f s 20 000 cycles. It
should be mentioned that for LS and SW the stress
path P1 Žwhich has a high stress ratio. shows a very
distinct deviation from the general pattern. These extreme cases were regarded as outliers and excluded
from the statistical analysis.
For the type of analysis explained here, it is very
important that a wide range of shear stress ratios are
applied. However, the limited test programme used in
this investigation does not provide a sufficient range of
shear stress ratios for all the materials tested. The
three stress paths used for S& G did not provide enough
information for the analysis and no correlation was
found.
Figure 5 Variation of permanent axial strain with stresses for N s 20 000
Although the Paute model of Equation Ž10. was
shown above to fit the strain-number of cycles data in
most cases, in some cases the correlation was not quite
satisfactory and the observed values indicated signs of
progressive accumulation of permanent strain. The
analysis of the effect of stresses shows that at low levels
of shear stress ratio a very high correlation is found
with the Paute model, the growth of permanent strain
levels off and the material behaviour approaches an
equilibrium state. At high shear stress ratios, however,
the correlation with the Paute model is relatively poor
and the material exhibits a form of gradual collapse
rather than stabilisation. In the case of Gr, for instance, the least correlation with the Paute model is
found for the stress path P1 followed by P2, both
showing signs of gradual collapse. Figure 5 shows that
for Gr the applied shear stress ratio is highest for the
stress path P1 followed by P2. For the remaining stress
paths, with lower shear stress ratios, the high correlation with the Paute model indicates that the permanent
strain increases asymptotically and the material
stabilises. This behaviour is even more clear for LS, for
which incremental collapse is apparent for the stress
paths P2 and P4, with the least correlation obtained for
P2. According to Figure 5 the applied stress ratio for
LS is highest for the stress path P2 followed by P4. The
same argument can be used for the other materials
tested.
If low levels of stress ratio ultimately result in an
equilibrium state and high stress ratios cause gradual
collapse of the material, there must be a certain level
of stress ratio at which this change of behaviour occurs.
This stress ratio can be considered as the boundary
which separates the final stabilised behaviour from the
incremental failure of the material. This hypothesis
shows close similarities with the concept of the shake-
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
down theory in pavement structures, with the existing
boundary being the shakedown limit. However, no attempt is made here to determine this shakedown limit,
as the number of tests performed and the range of
stress ratios applied do not seem to be sufficient and
further research is needed in this regard.
Previously, several research workers have attempted
to relate gradual accumulation of permanent strain
under cyclic loading to the ultimate shear strength of
the material under static loading. The use by Paute et
al.20 of an asymptotic strain value based on failure
stress conditions ŽEquation Ž13.. is an example of this.
The results of this investigation do not support the
existence of such a relationship and it is argued that
the failure in granular layers in a pavement under
repeated loading is perhaps a gradual process rather
than a sudden collapse as in static failure tests. On this
basis, ultimate shear strength and stress levels that
cause sudden failure are of no great interest for the
analysis of the material behaviour when the increase of
permanent strain is incremental. The new model of
Equation Ž15. explains the material behaviour while
the accumulated permanent strain is finite and the
applied stresses do not cause sudden failure.
Conclusions
The main conclusions from this study can be summarised as follows.
1. Unbound granular materials show a complex behaviour under cyclic loads, with a gradual accumulation of permanent strain with each load application.
2. A new model has been introduced which describes
the relationship between accumulated permanent
axial strain and stresses at any given number of
cycles greater than 100. The model takes into account the maximum shear stress ratio and the length
of the stress path in p-q space. The results of the
laboratory repeated load tests show very good correlation between the proposed model and the observed
values.
3. At low levels of shear stress ratios, the growth of
permanent strain seems to level off indicating that
the material will eventually reach an equilibrium
condition with almost total resilient behaviour. For
these cases the Paute model of Equation Ž10. was
found to be quite successful in predicting the relationship between permanent axial strain and number of load applications.
4. At high shear stress ratios, the accumulation of
permanent strain is more progressive with signs of
gradual deterioration of the material. The Paute
model, which is based on the assumption of eventual
stabilisation of the behaviour, clearly loses its accuracy at high stress ratios.
5. If low shear stress ratios result in an ultimate equilibrium condition and high shear stress ratios cause
gradual failure, then a threshold stress ratio must
17
exist at which this change of behaviour occurs. This
threshold stress ratio could be a form of the, socalled, shakedown limit.
6. More research is needed to determine the shakedown limit, and to investigate the possibility of using
a modified shakedown theory to study the behaviour
of granular materials under cyclic loading.
Acknowledgements
This study was made possible by a 1-year joint research
programme between the University of Nottingham,
Nottingham, England and the Royal Institute of Technology ŽKTH., Stockholm, Sweden. The Swedish part
of the work was financially supported through the ‘Ballast Research Programme’ at the KTH, by SBUF ŽDevelopment Fund of the Swedish Construction Industry.,
GMF ŽNational Swedish Sand, Gravel and Crushed
Stone Association., KFB ŽSwedish Transport and Communications Research Board., and NUTEK ŽSwedish
National Board for Industrial and Technical Development.. Thanks are due to Mr. T. Gleitz, formerly of the
Technishe Universitat
¨ Dresden, some of whose triaxial
test data was made available to the authors, and to Mr.
I.R. Richardson, in the Civil Engineering Department
of the University of Nottingham, who carried out the
hollow cylinder tests on sands.
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