statistical evaluation of resilient models characterizing coarse granular materials

7/28/2019 Statistical Evaluation of Resilient Models Characterizing Coarse Granular Materials

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O R I G I N A L A R T I C L E

Statistical evaluation of resilient models characterizing

coarse granular materials

Jonas Ekblad

Received: 23 January 2007 / Accepted: 8 May 2007 / Published online: 7 June 2007

RILEM 2007

Abstract Consistent material modeling is a pre-

requisite for a mechanistic approach to pavement

design. The scope of this investigation was to

statistically evaluate the efficiency of various resilient

models commonly encountered in highway engineer-

ing. These models were categorized as describing

either resilient modulus or shear and volumetric

strains. Triaxial tests using constant and cyclic

confining pressure were performed on coarse granular

materials of various gradings (maximum particle size

90 mm). Two statistical methods, the extra sum ofsquares F-test and the Akaike information criterion,

were used for model comparison. Concerning resil-

ient modulus, the Uzan model provided, in general, a

statistically significant improvement compared to the

kh model. However, this improvement is lost if a

constant Poisson ratio is used to predict shear and

volumetric strains. In case of the shearvolumetric

approach, no single model was most likely to be the

best model for all gradings studied.

Keywords Unbound granular materials Resilientmodulus Shearvolumetric models Akaikeinformation criterion Highway engineering

1 Introduction

Reliable material models are an integral part of a

mechanistic framework for road pavement design.

Traditional use of empirical design methods is an

obstacle if new materials or construction types are

introduced. Design criteria are limited to the context

under which they were derived. This could be

remedied, at least partly, by development of design

procedures accommodating for more adequate and

fundamental material models. Unbound granularmaterials are extensively used in pavement construc-

tions and play an important role in the overall

performance of the structure. The resilient response

to loading for this type of materials is complex,

essentially being nonlinear inelastic. Furthermore,

granular materials are influenced by environmen-

tal factors, especially water content. The stresses

induced in granular layers of a pavement by a

passing wheel are indeed complex. Compared to in-

situ stresses, the triaxial test setup used in this

study represents a simplified stress state, yet it isdifficult to properly model measured stressstrain

behavior.

The scope of the work described in this paper

was to evaluate common relationships describing

resilient behavior encountered in the field of high-

way engineering. Triaxial testing was performed

using constant and cyclic confining pressures fol-

lowing different stress paths. The study was limited

to one type of aggregate and four different gradings

J. Ekblad (&)

Division of Highway Engineering, Royal Institute of

Technology, Stockholm 100 44, Sweden

e-mail: [email protected]

Materials and Structures (2008) 41:509525

DOI 10.1617/s11527-007-9262-9


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with maximum particle size 90 mm. Before submitted

to the test schedule used in this work, all samples

were saturated by water and freely drained. If it is

assumed that the granular layers in pavements

generally remain unsaturated, this state will corre-

spond to the state of maximum water content.

Modeling of results was performed in terms ofresilient modulus and in terms of volumetric and

shear strain. The models were statistically compared

using two different statistical techniques: the extra

sum of squares F-test and the Akaike information

criterion, respectively.

2 Experimental

2.1 Equipment

In Fig. 1, the main features of the cylindrically

confined triaxial setup used in this investigation are

shown. The diameter of the samples is 500 mm and

the height 1,000 mm. Axial and confining stresses are

applied by two independent hydraulic actuators.

During this work, testing was carried out using air

to provide constant confining pressures and silicone

oil for cyclic confining pressures. In the right part of

Fig. 1, transducers used to measure deformation are

shown. To measure axial deformation of the central60 cm part of the specimen, three LVDTs, spaced

1208 apart, were attached to anchoring devices buried

during compaction. Circumferential measurement is

provided by a strain-gaged extensometer attached

between the ends of a roller chain, wrapped around

the specimen at mid height. The rollers on the chain

extend outside the link plates. Steel springs are used

to connect the chain ends. In addition to transducers

shown in Fig. 1, a load cell, a confining pressure

sensor and a LVDT measuring top-plate movements

are connected. The left part of Fig. 1 shows in-sampleinstrumentation. Although results obtained using

these transducers were not specifically used in this

investigation, they are shown for reference.

Fig. 1 Specimen

instrumentation

(dimensions in mm)

510 Materials and Structures (2008) 41:509525


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2.2 Materials and samples

The origin of the material used was Skarlunda

(Ostergotland, Sweden). The material is characterized

as a foliated medium grained granite with (based on

point counting) quarts, K-feldspars and plagioclase as

main constituents. Phyllosilicates (muscovite, biotiteand chlorite) are also present, comprising about 10%

by point counts. The target particle size distributions

(gradings) were derived using the equation:

P d

Dmax

n1

where P is the percent smaller than d, Dmax is the

maximum particle size and n is the grading coeffi-

cient, describing the shape of the curve [1]. In this

investigation, a maximum particle size of 90 mm andfour different distributions corresponding to grading

coefficients: 0.3, 0.4, 0.5 and 0.8, respectively, were

used. This gives a range spanning from a fairly coarse

material to a material containing a substantial

amounts of fines. The nominal gradings were com-

posed using eight fractions of more narrow distribu-

tion to reach a close concurrence to target grading. In

Fig. 2, the target and nominal gradings are shown

denoted by grading coefficient.

Densities for the different gradings are given in

Table 1. Maximum bulk density was determinedaccording to ASTM D4253 [2], using a vibratory

table on which a sample is contained in a cylinder and

heavily vibrated under a static pressure and a surplus

of water. The specified validity of this method is

restricted to cohesionless, free-draining soils contain-

ing less then 15% by weight particles smaller than

74 mm and with maximum particle size of 75 mm.

Even though the gradings conflict with the specifica-

tions in this method, no attempt to adapt to these

criteria, e.g. by removing larger particles, was made.

Density ratio is calculated as the ratio of actual

sample bulk density to maximum bulk density.While there is only a small variance (


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the specimens were allowed to drain freely, after

which they were tested according to the schedule

used in this investigation in which the samples were

tested using constant and cyclic confining stresses

(c.f. Sect. 2.3.). That means that all tests were

performed on drained samples. This stage represents

water retentive capacity at drying. At this stage, thewater content is higher compared to the correspond-

ing wetting sequence, due to hysteresis of the so-

called soilwater characteristic curve. In other words,

this stage represents maximum water content, given

the geometry of the sample. The matric suction at the

bottom of the sample was 0 kPa and at the top 10 kPa.

Because water content is a function of matric suction,

which can be described by the soilwater character-

istic curve, there is a gradient in terms of water

content in the vertical direction. Predictions of this

distribution are given in Fig. 3 for the variousgradings. The predicted values are based on the

estimated soilwater characteristic curve of each

grading and were compared to measured values

obtained using the buried TDR-probes (at 20 cm and

80 cm from the bottom). The differences between

estimates from the soilwater characteristic curve and

measurements using time domain reflectometry were

all less than 0.5% mass fraction.

From Fig. 3 it can be seen that the water content

varies within the sample, but also that this variation

with height is most pronounced below the lower level

of axial deformation measurement. Within the part of

the sample where deformation was measured, thewater content change is commonly smaller. Predicted

sample average water contents (% mass fraction)

between upper and lower level were: 1.2% (grading

0.8), 2.9% (grading 0.5), 4.1% (grading 0.4) and

6.6% (grading 0.3), respectively. In this connection, it

should be noted that the draining curve of grading 0.8

could not be measured, why model parameters were

estimated from the wetting curve and parameter

relationships based on the other gradings. However, as

already mentioned this measure provided reasonable

predictions when compared to contents measuredusing time domain reflectometry.

2.3 Triaxial test procedure

In the cylindrically confined triaxial test, where only

principal stresses can be applied and two principal

stresses are equal, usually r2 and r3, the following

stress invariants commonly are utilized:

p r1 2r33

2

q r1 r3 3

h r1 2r3 4

where p is the mean normal stress, q deviatoric stress

and h is the sum of principal stresses. The mean

normal stress, p, is equal to the mean stress in the

volumetric and deviatoric stress tensors, h is the firstinvariant of the principal stress tensor and q is a

function of the second invariant of the deviatoric

tensor. Stresses are expressed in terms of total

stresses, which mean that any influence of matric

suction, or elevated pore pressure, on mean stress is

disregarded. In a prior study [3], it was found that the

influence of internally acting stresses was not easily

interpreted; it was difficult to explain the resilient

behavior for these materials, at different levels of

Fig. 3 Water content distribution for the triaxial samples of

various gradings. Upper and lower levels of on-sample axial

strain measurements are also indicated



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matric suction, in terms of a single stress invariant

e.g. effective stress. Furthermore, all the measure-

ments in this investigation were performed at equal

levels of matric suction, 5 kPa at midheight of the

sample, which is far below the levels of applied

stresses.

The corresponding strain invariants are definedas:

ev e1 2e3 5

eq 2

3e1 e3 6

where ev is volumetric strain, eq shear strain, e1 major

principal strain (axial) and e3 minor principal strain

(radial). Details of the resilient test schedule, using

constant and cyclic confining pressures, are shown in

Fig. 4 as total stresses. Each stress path was repeated

for 50 cycles as sinusoidally (haversine) oscillating

loads at a frequency of 1 Hz. For the finer gradings,

some of the most severe stress conditions were

omitted to avoid failure or excess permanent defor-

mation. Measured resilient behavior might be influ-

enced by the test protocol used, as indicated by e.g.

Andrei et al. [4]. Results presented by e.g. Raad and

Figueroa [5] indicate that granular soil samples

maintain their resilient behavior even after the

occurrence of large permanent deformations.

In addition to the state variables defined by Eqs. 2

6, two resilient parameters, resilient modulus, Mr, and

strain ratio, Re, were calculated under constant

confining pressure stress paths, using the following

equations:

Mc r1 r3

e1

q

e17

and

Re e3

e18

where e3 and e1 are the resilient radial and axial

strains, respectively. Resilient modulus and strainratio are determined as secant values in terms of

deviator stress. The definitions of resilient modulus

and strain ratio are similar to elastic or Youngs

modulus and Poisson ratio, respectively, which are

applicable to linear elastic response. It should be

noted that, at stress levels encountered during this

investigation, granular materials are commonly

inelastic. However, the term Poisson ratio will be

used interchangeably in following sections, where

reference is given to previously published results or

procedures.

2.4 Resilient models

A common approach in pavement design is to

consider the response of granular materials as linear

isotropic elastic, i.e. the materials can be character-

ized by two elastic parameters e.g. Youngs modulus

and Poisson ratio. However, the resilient response of

granular materials is unmistakably nonlinear and

depends on stress (or strain) level. To recognize this

circumstance, Youngs modulus is replaced by resil-ient modulus to indicate typical inelastic resilient

behavior. The resilient modulus is expressed as a

function of stress level. This practice represents one

category of resilient models. Researchers have also

extended the resilient modulus framework by intro-

ducing degrees of anisotropy in the derivation of

elastic parameters required to describe the resilient

behavior [68]. In the case of cross-anisotropy

(transverse isotropy), five elastic parameters are

Fig. 4 Stress paths used in

the triaxial tests (total

stress)

Materials and Structures (2008) 41:509525 513
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required to characterize the behavior: resilient mod-

ulus and Poisson ratio in the vertical and horizontal

direction, respectively, and shear modulus. However,

it is not possible to uniquely derive the resilient

parameters from cylindrically confined triaxial tests;

either some of the parameters have to be assumed

constant or they are derived using numerical optimi-zation. Furthermore, the techniques described by [6

8] would preferably require different testing sched-

ules and/or triaxial equipment than was used in this

investigation. Another category is based on decom-

posing the stresses and strains into shear and

volumetric components. In this work, the stress

dependent resilient modulus is used to model results

obtained from triaxial tests using constant confining

pressure while the shear and volumetric approach is

used mainly for measurements using cyclic confining

pressure. It should be noted that neither of thesemodel categories are restricted to one type of test

mode (constant or cyclic confining pressure).

2.4.1 Resilient modulus models

Probably the most well-known and commonly used

model is generally called the kh model. It assumes

that resilient modulus can be related to the sum of

principal stresses h according to:

Mc k1hk2 9

where k1 and k2 are regression parameters. The origin

of the model dates back to at least 1960s. Brown and

Pell [9] plotted the modulus versus the first stress

invariant (I1 or h) in a loglog diagram. Seed et al.

[10] suggested a power law relationship relating

resilient modulus to sum of principal stresses, while

Hicks and Monismith [11] concluded that Eq. 9 was

verified in most of their test results.

From analyzing field data, May and Witczak [12]

concluded that the resilient modulus is not solelydependent on sum of principal stresses but also on

shear strain level. This shear effect was later recog-

nized by Uzan [13], who suggested the following

relationship:

Mc k1hk2 rk4d 10

where k1, k2 and k4 are regression parameters and rdis the deviatoric stress invariant q.

A disadvantage with these models is that they do

not cater for strains in the transversal direction.

Commonly, a constant Poisson ratio is used in this

connection. However, the models can also be

extended by use of stress dependent Poisson ratio

relationships to improve shear and volumetric strain

predictions [14]. Furthermore, In 1992 Uzan [15]extended the resilient description expressed by Eq. 10

with a relationship expressing a nonlinear stress

dependent Poisson ratio as a function of sum of

principal stress (I1 or h) and the second deviatoric

stress invariant (J2), together with the exponents of

Eq. 10, and additionally two regression constants. To

estimate the parameters, Uzan suggests that the

resilient modulus and Poisson ratio are simultaneously

fitted using nonlinear regression. Basically, this inves-

tigation was not extended to include relationships

describing stress dependent Poisson ratio. However, asan example of this category of models, results are also

fitted using the extended Uzan model [15].

2.4.2 Shearvolumetric models

The second category of resilient models is based on

decomposing the principal stress and strain tensors,

respectively, into two tensors: a deviatoric (or shear)

and a volumetric. In this section, five different

models evaluated in this investigation are briefly

introduced.In 1980, Boyce [16] presented general equations

for characterizing nonlinear behavior in terms of bulk

(K) and shear modulus (G), which are expressed as

stress dependent. Basic assumptions are elastic and

isotropic material behavior. In the case of granular

materials, bulk and shear modulus are expressed as

power functions of mean normal stress (p). By

imposing a constraint of no net loss of energy,

volumetric (ev) and shear (eq) strains are expressed as:

ev pn

K11 b q

2

p2

11

eq pn

3G1

q

p12

b 1 n K1

6G113

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where n, K1, G1 are regression parameters. The b

parameter is an effect of the thermodynamic con-

straint of no loss of energy and couples the volumet-

ric and shear behavior.

Brown and Pappin [17] developed a model com-

monly referred to as the contour model. The basic

idea is to mathematically describe observed strain

contours in terms of shear and normal stress state. For

this purpose, they derived the following equations:

ev p

A

m1 B

q

p

n 14

eq Cq

p D

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 q2

pp=2

c15

where A, m, B, n, C, D and r are regression

parameters. The term within parentheses in Eq. 15

depends on the length of the stress path. In contrast to

the Boyce model, the contour model does not couple

shear and volumetric behavior.

Based on the Boyce model, Jouve and Elhannani

[18] investigated the efficiency of a number of

modifications to the original model. By assuming

orthotropy and introducing an orthotropic parameter

(n), they proposed to express strains as:

ev p1na p

n 1

Ka nn

q

p

b

Ka

q2

p2

16

eq p1na p

n 1

3Ga

q

p n

17

where n, Ka, n, b and Ga are regression parameters,

and pa is a constant used to normalize the dimension

of the mean normal stress term. In the most general

formulation, as used in this investigation, b is anindependent parameter but could also be determined

by Eq. 13.

Another way of introducing anisotropy (orthotropy

or cross-anisotropy) to the Boyce model was

described by Hornych et al. [19]. In their approach,

the major principal strain is adjusted by an empirical

parameter, coefficient of anisotropy c, which leads to

the following equations:

ev pn

pn1a

c 2

3Ka

n 1

18Gac 2

q

p

2

c 1

3Ga

q

p

18

eq 2

3

pn

pn1a

c 13Ka

n 118Ga

c 1 q

p

2

2c 1

6Ga

q

p

19

where n, c, Ka and Ga are regression parameters and

pa is a constant, mainly aimed at normalizing the

dimensions. The corresponding modified stress in-

variants become:

p cr1 2r3

320

q cr1 r3 21

In contrast to the models presented so far, Hoff

et al. [20] presented a different approach although

still using the concept of dividing the model into

volumetric and shear components. The derivation of

their relationship is based on hyperelasticity. For

general stress conditions, Hoff et al. define the strain

energy (U) function as:

U K Ie1 2

2 DIe1J

e2 2GJ

e2 22

where K is bulk modulus, G is shear modulus, D is a

dilatancy constant used in the coupling of the first

strain invariant Ie1

and the second deviatoric strain

invariant Je2

. The stressstrain relationship is

obtained by partial differentiation and for the triaxial

test stress state it becomes:

q

p

!

3G DeqDeq=3 K

!eqev

!23

To comply with the previously described model, the

compliance matrix is required instead of the stiffness

matrix in Eq. 23. Unfortunately, due to the coupling of

strains in the strain energy function, the stiffness

matrix cannot be inverted. In this investigation, a



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different regression model was used to determine the

parameters of the Hoff model. In contrast to Eqs. 11

19, where the parameters were estimated by nonlinear

regression minimizing the squared error between

measured and estimated strain, the nonlinear regres-

sion of the Hoff model is based on minimizing the

squared error between actually applied stresses andestimated stress levels required to reach the measured

strains. In other words, for a given strain state,

determined by measured volumetric and shear strain,

the nonlinear regression estimate the stresses required

to reach this strain state and minimize the difference to

actually applied stresses. Consequently, the goodness-

of-fit statistics of the Hoff model cannot easily be

compared to the other models used. Although, results

from fitting of the Hoff model are presented for

reference and visual assessment.

An important model feature, influencing the sta-tistical evaluation, is number of parameters to be

estimated. Table 2 summarizes the models and

number of parameters used in each model.

2.5 Statistical analysis

To compare the ability of the different models to

explain measured mechanical behavior, two different

statistical procedures were used: the extra sum of

squares F-test and the Akaike information criterion

(AICc). The extra sum of squares is a measure of themarginal increase in the regression sum of squares

gained by adding another parameter to the model,

after which hypothesis testing (significance testing) is

performed to determine whether this increase could

be regarded as statistically significant. This approach

is commonly used in multiple regression to determine

which variables should be used in a statistical model.

The basis of the Akaike information criterion is

different. In this case, the procedure is based on

maximum likelihood and information entropy and is

not readily comparable to hypothesis testing used in

the extra sum of squares procedure. The Akaike

information criterion provides an estimate of the

relative distance between the fitted model and the true

mechanism. This distance can be used for comparison

of the different models. More detailed information onthese statistical techniques can be found in textbooks

(e.g. [21, 22]) although a short introduction of the

calculated statistics is given here.

The extra sum of squares test compares sum of

squares obtained from the full model (maximum

number of parameters) and reduced models (exclud-

ing parameters). The F-ratio is determined by:

F SSER SSEF

dfR dfF 0SSEF

dfF24

where SSER and SSEF is the sum of squared errors for

the reduced and full model, respectively, and dfR and

dfF are the corresponding degrees of freedom. The

extra sum of squares can be used when the compared

models are nested, i.e. the full model contains only an

addition to the reduced model. The null hypothesis is

that the additional parameter equals zero. In this

investigation, results obtained using the kh and Uzan

model fitting of resilient modulus are tested using the

extra sum of squares test. The criterion for statistical

significance is based on the P-value; if obtained

values are lower than 0.05, the null hypothesis is

rejected.

When comparing the shearvolumetric models, the

Akaike information criterion is used. This criterion is

defined as:

AICc n ln SSE n ln n 2K 2K K 1

n K 125

where n is the number of data points, SSE is the

sum of squared error and K is the number ofparameters plus one. The model with the lowest

AICc-value is estimated to be most close to the true

mechanism generating the data. From Eq. 25, it can

be seen that the first term depends on sum of

squared error, which commonly decreases with

increased number of model parameters. The last

two terms depends on number of parameters and

these will increase AICc as the number of estimated

parameters increase, i.e. these terms balance the

Table 2 Summary of resilient shearvolumetric models andthe number of regression parameters

Model Number of

parameters

Boyce [16] 3

Brown and Pappin [17] 7

Jouve and Elhannani [18] 5

Hornych et al. [19] 4

Hoff et al. [20] 3



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reduction of SSE reached by introducing additional

parameters. In a sense, there is a penalty in the

criterion of adding too many parameters, to avoid

overfitting or overparameterized models. Using more

parameters in a model will commonly decrease the

bias of the model while the variance is increased.

Furthermore, the precision of estimated parametersmight be poor. If the model is too complex, compared

to empirical data available, it might include spurious

effects or anomalies in measured data; even if a

model is physically incorrect, it might provide a

reasonable fit to measured data if the number of fitted

parameters is high. When comparing models using

AICc, the absolute magnitude is not important but

rather differences between model candidates. A

model with a lower value of AICc is considered to

be a better model. The absolute value of AICc

depends on units used and number of observations. Inthis investigation, results are given as DAICc i.e. the

increase of AICc for each model compared to the

model with the lowest value.

3 Results

3.1 Constant confining pressure

From measurements using constant confining pres-

sure, resilient modulus and strain ratio were calcu-lated and are summarized in Fig. 5. For resilient

modulus, results are fitted to a power law relationship

in similarity to the kh model. The strain ratio is

expressed in terms of the stress ratio q/r3. The scatter

indicated in both diagrams of Fig. 5 is mainly caused

by the fact that the stress dependence of neither of

these two resilient parameters can be related to a

single stress invariant.

Figure 5 gives an impression of the differences in

mechanical behavior between the various gradings

tested at maximum water content. The different

gradings range from 0.8 showing the stiffest response

to 0.3, which is significantly softer. There is also a

difference in stress dependency. Grading 0.8 shows

the largest increase in modulus with increased mean

normal stress, whereas 0.3 is influenced to a lesser

degree. Concerning strain ratio, all gradings except

0.8 undergoes dilation at higher deviator stress levels,

grading 0.3 showing the highest levels.

The efficiency of the kh and Uzan models in

fitting measured results are visualized in Fig. 6, wherefitted resilient modulus is shown versus measured

modulus of all the gradings used. Also shown is the

line of equality and, consequently, a successful

regression of results should adhere closely to this

line. Fitted parameters are given in Table 3.

In Fig. 6, it can be seen that the Uzan model

regression yields a closer fit to measured results, even

though the kh model also performs reasonably well.

In this context, it should be remembered that the

addition of another parameter in the model is

expected to reduce the error between measured andfitted results.

Table 4 summarizes goodness-of-fit statistics and

the result obtained using the extra sum of squares F-

test for the various gradings. The extra sum of square

procedure tests whether the added parameter cause a

significant increase in sum of squares explained by

the regression.

Overall, for both models, the degree of explained

variance (R2) is high; as expected, it is slightly higher

Fig. 5 Resilient modulus and strain ratio of the four different

gradings at maximum water content



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for the Uzan model. It can also be seen that for allgradings, except 0.3, the improvement caused by

adding the deviator stress in the model is statistically

significant. From Fig. 5 it seems that for grading 0.3,

the influence of stress level on resilient modulus is

smaller compared to the other gradings and in

particular grading 0.8 (i.e. the slope of the curve is

smaller).

Even though the kh and Uzan models are mainly

intended for description of resilient modulus, shear

and volumetric strains can be predicted using an

estimated (constant) Poisson ratio. In this investiga-

tion, the average strain ratio value of the entire stress

range for each grading was used (cf. Fig. 5). It was

found that this measure caused the smallest predictive

errors. Used values ranged from 0.53 for the 0.3

grading to 0.24 for the 0.8 grading. For comparison,measured shear and volumetric strains were also

fitted using the Boyce model. Since the ratio q/p is

constant (3/1) for all stress paths using constant

confining pressure, q/p was calculated as Dq/pmax,

where pmax includes the confining pressure. Acquired

results for all gradings are shown in Fig. 7 using the

Uzan and Boyce models, respectively. The results

obtained using the kh model are omitted since

predicted strains are essentially showing the same

pattern as those obtained using the Uzan model. In

the lower part of Fig. 7 fitted results using theextended Uzan model is also shown for comparison.

From Fig. 7 it can be seen that, using the Uzan

model and a constant Poisson ratio, shear strain

predictions are closer to measured results compared

to volumetric strains. The prediction of volumetric

strain is poor. At higher strain levels, it seems as the

Uzan model underpredicts the shear strains. The

prediction of negative volumetric strains is an effect

of using the average measured Poisson ratio, which

for grading 0.3 was 0.53. It should be noted that this

value is inconsistent with an elastic framework andisotropic behavior. Compared to the Uzan model, the

Boyce model yields an improved fit for both shear

and volumetric strains. The Boyce model tends to

underestimate the volumetric strains during dilation.

By extending the resilient description with a stress

dependent Poisson ratio, as in the lower part of Fig. 7

(extended Uzan), the prediction of shear and volu-

metric strains is greatly improved. When comparing

the outcomes shown in Fig. 7 (the two upper

diagrams) and in Table 5, it should be remembered

that the parameters in the Boyce model are estimatedbased on the shear and volumetric strains, while the

strains for the Uzan model are predictions obtained

by fitting of resilient modulus and using a constant

Poisson ratio. However, the results and inferences

would be basically the same if the parameters of the

kh and Uzan models are derived based on regression

of shear and volumetric strains. In Table 5, statistical

comparisons between the applied models are given.

In terms of shear and volumetric strains, the kh and

Fig. 6 Fitted versus measured values of resilient modulus for

all gradings and corresponding line of equality

Table 3 Parameters obtained for the different models and

gradings using constant confining pressure

Grading kh

k1 (kPa) k2

08 16.6 0.69

05 27.2 0.53

04 27.2 0.49

03 33.2 0.41

Uzan

k1 (kPa) k2 k4

08 21.0 0.87 0.22

05 32.4 0.66 0.16

04 31.3 0.58 0.11

03 39.6 0.48 0.10

Boyce, GK model

K1 (kPa) G1 (kPa) n

08 23.8 12.0 0.52

05 26.2 7.99 0.50

04 24.8 5.21 0.46

03 53.4 4.83 0.50

Stress input was in kPa



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Uzan models are compared using the extra sum of

squares F-test and the comparisons with the Boyce

model are made using the Akaike information

criterion. In all cases, the Boyce model gave the

lowest AICc-value.

As indicated in Table 5, there is no statisticallysignificant improvement (P > 0.05) of predicted

results by using the Uzan model compared to the

more simple kh model. Furthermore, DAICc-values

are only marginally lower for the Uzan model

compared to the kh model. For all gradings, the

AICc-value was at least 10 units lower for the Boyce

model compared to the other models, which is a

strong indication of higher likelihood that this is a

better description of measured strain data. By using

the extended Uzan model, the goodness-of-fit statis-

tics becomes more comparable to the Boyce model;in two cases the Akaike information criterion indi-

cates similar likelihood of the two models and in the

remaining two cases the Boyce models is more likely

to yield an improved description of empirical data.

Another way of comparison of the models is to use

the Boyce fitting of volumetric and shear strains for

prediction of resilient modulus. Overall, the predicted

values from this calculation are comparable to results

obtained using the kh model, i.e. slightly less precise

than the Uzan fit.

3.2 Cyclic confining pressure

Results obtained at cyclic confining pressure were

analyzed using the shearvolumetric approach. Fig-

ure 8 illustrates typical results of such tests for

grading 0.5; shear and volumetric strain are shown as

function of mean normal stress at various ratios of

deviator to normal stress (q/p). In Fig. 8, the

nonlinear regressions of the Boyce and Brown

Pappin models, respectively, are also shown. The

Boyce model is the one (among the described in this

paper) containing the lowest number of estimated

parameters and the BrownPappin model the one

containing the highest number.

Visually from Fig. 8 the pattern of stress depen-dency is clearly seen. Both volumetric and shear

strain depend not solely on mean normal stress but

also on stress ratio. At lower levels of q/p, the

volumetric strain mainly depends on mean normal

stress, but as the stress ratio increases, this pattern is

changed. At the highest stress ratio used (2.5), the

volumetric strain is almost constant in the normal

stress range tested. The shear strain is also strongly

influenced by stress ratio; the higher the strain ratio

the higher the strain for a given mean normal stress.

As can be expected the lowest levels of shear strainwas measured under isotropic loading. Considering

the nonlinear regressions shown, it seems as the

BrownPappin model provides a closer adherence to

measured data compared to the Boyce model. How-

ever, this is what would be expected considering the

higher number of parameters used.

The outcome of the nonlinear regressions is

summarized in Table 6, in which all estimated

parameters for the different models and gradings

are given. It should be noted that the dimension of the

models and parameters may be inconsistent. In theJouveElhannani and Hornych relationships, this

inconsistence is remedied by introducing a constant

to render the stress term nondimensional. This

measure is more a technical solution and does not

change the bases or rationale of the models.

The nonlinear regression algorithm provides a

numerical optimization to determine the best fit. It

should be noted that there might exist several local

minima of the squared error, of which the results in

Table 4 Statistical comparison of kh and Uzan models. Coefficient of determination, sum of squared errors and test of the extra

sum of squares (F-ratio and probability) are shown

Grading kh Uzan Extra sum of squares test

R2 SSE R2 SSE F P-value

08 0.96 0.032 0.99 0.0047 133 0.00*

05 0.96 0.019 0.99 0.0047 70 0.00*

04 0.97 0.012 0.99 0.0055 27 0.00*

03 0.93 0.015 0.94 0.012 4.1 0.06

*Statistically significant (P < 0.05)



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Table 6 represent one minimum. Furthermore, esti-

mated parameters might depend on the initial values

chosen for the regression. Presented results were

obtained from nonlinear regressions using the GRG2-

algorithm (generalized reduced gradient) imple-

mented in Excel (Microsoft Corp.), by minimizing

the squared error between measured and fitted strains.

The goodness-of-fit for the nonlinear regression

fitted using the different models are illustrated in

Fig. 9. The diagrams show fitted volumetric and shear

strains versus measured strains as well as line of

equality.

The BrownPappin model shows the closestadherence to the line of equality. However, in most

cases all models seem to generate reasonable esti-

mates of measured strains. Except for the Brown

Pappin model, negative volumetric strains seems to

be underestimated, while no obvious deviations of

shear strain can be distinguished. The goodness-of-fit,

for the various models was also analyzed using the

Akaike information criterion statistic. Table 7 sum-

marizes the coefficient of determination (R2) and

differences in AICc for the different models and

gradings, used.The degree of explained variance (R2) is typically

fairly high. In all cases, the BrownPappin model

shows the highest R2-values. When comparing results

using AICc, the model with lowest DAICc is most

likely to be closest to the true mechanism i.e., the

larger the difference in AICc the less empirical

support of a particular model compared to the best

performing model. Based on results presented in

Table 7, no distinct conclusions can be drawn; no

single model performs best for all materials studied.

To further explore the performance, the models areranked in order of increasing AICc as shown in

Table 8.

To evaluate whether there is a statistically signif-

icant difference in the ranking, a Friedman test by

ranks was performed. In this case no statistically

significant differences were indicated (P = 0.21).

As mentioned previously, results acquired using

the Hoff model could not be statistically compared to

the other models since the basis of the regression was

Fig. 7 Predicted strains using the Uzan (top), Boyce (mid) and

extended Uzan (bottom) models versus measured strains. Line

of equality is also shown

Table 5 Statistical comparison of estimation of shear and

volumetric strains using kh, Uzan and Boyce models. Coef-ficient of determination (R

2), DAICc (compared to the Boyce

model) and P-value of the extra sum of squares F-test between

kh and Uzan models

Grading kh DAICc Uzan DAICc F-test (P-

value)

Boyce R2

08 10 10 0.83

05 48 47 0.40 0.94

04 33 31 0.34 0.90

03 11 10 0.39 0.91

No reduction of SSE using the Uzan model



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different compared to the other models used. Never-

theless, the performance of the Hoff model can be

qualitatively assessed by visually comparing fitted

results in Fig. 8, with the nonlinear regression

obtained using Eq. 23 as shown in Fig. 10.

It should be noted that the axes are reversed in

Fig. 10 compared to Fig. 8. This is because the

nonlinear regression using the Hoff model estimated

the stress levels required for a given strain level, in

contrast to the other models.In similarity to the previous visual summaries of

model efficiency (cf. Fig. 9), Fig. 11 shows fitted

stresses using the Hoff model versus actually applied

stresses.

The individual R2-values for the different gradings

obtained by fitting to the Hoff model were: 0.98

(grading 0.8), 0.92 (grading 0.5), 0.97 (grading 0.4)

and 0.73 (grading 0.3). It should be noted that the

nonlinear regression was performed using conditions

different to those used in for the other models thus

making a direct comparison less meaningful. How-ever, the overall performance of the Hoff model is

considered fairly accurate. The scatter around the line

of equality is slightly smaller for the shear stress

compared to the mean normal stress.

Since equipment able to apply cyclic confining

pressure is considerably more complicated than

equipment using constant confining pressure, it would

be advantageous to be able to predict shear and

volumetric behavior when both stresses are cycled,

from triaxial tests using constant confining pressures.

The Boyce model parameters were estimated by

fitting results obtained using constant confining

pressure. These parameters were subsequently used

to predict the behavior using cyclic confining

pressure. In Fig. 12, a comparison between predicted

and measured values is shown.

Overall, predicted values show poor agreement

with measured values, especially for the volumetric

strain. The result seems to be in agreement withKarasahin et al. [23] who concluded, from perform-

ing a similar prediction, that model parameters

derived from constant confining pressure can only

approximately predict behavior when both stresses

are cycled.

4 Discussion and conclusions

Even for the comparably simple stress conditions

applied at cylindrically confined triaxial tests, it isdifficult to describe the resilient stressstrain rela-

tionship in a more general way. The comparably great

number of, more or less empirical, models found in

the literature indicates this complexity. In this

context, it should be remembered that the resilient

framework itself is a simplification of the complex

inelastic response of granular materials, including

e.g., nonlinearity and stress history dependence.

Furthermore, laboratory testing of coarse granular

Fig. 8 Comparison of

measured and fitted strains

for different stress paths

(grading 05)

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materials is inherently difficult in respect to mini-

mizing the influence of the highly particulate nature

of the larger aggregates. There is a practical conflict

between minimizing the sample size and maintainingapproximate continuum conditions. Generally, on-

sample instrumentation is preferred to avoid end

effects close to the platens but buried anchors might

be more susceptible to particle effects compared to

the averaging effect plate-to-plate measurements

show. By determining radial strains from measure-

ments of circumferential length change, an averaging

effect is achieved. In addition, particle effects are

probably also influenced by stress path, especially

when using cyclic confining pressure as the aggregate

matrix may deform in different ways under different

stress paths. Altogether, this poses difficulties when

analyzing triaxial testing of granular materials, bothof a theoretical and technical nature.

An important feature of the models, based on shear

and volumetric components is that nonlinear regres-

sion is required to estimate the parameters. This

makes the outcome dependent on the algorithms used

and the initial parameter values needed for optimi-

zation. The final solution might represent a local

maximum of the goodness-of-fit. Sometimes, the

regression might even fail to provide a solution. In

Table 6 Parameters obtained for the different models and gradings

Grading Boyce, GK model

K1 (kPa) G1 (kPa) n

08 14.7 15.2 0.516

05 9.13 9.06 0.443

04 14.1 12.9 0.547

03 3.84 2.62 0.263

Brown and Pappin, contour model

A (MPa) B m n C D (kPa) r

08 3.70 0.0615 0.688 2.81 1.70 104

0.296 1.08

05 7.45 0.128 0.635 1.95 2.61 104

0.105 0.539

04 5.78 0.0290 0.646 3.72 5.56 104

0.187 0.330

03 90.0 0.0527 0.522 3.68 3.89 104

0.097 0.445

JouveElhannani

Ka (kPa) Ga (kPa) n b n pa (kPa)

08 130 155 0.667 0.108 2.09 107

10005 116 122 0.538 0.111 3.52 10

7100

04 109 120 0.648 0.121 1.23 107 100

03 110 91.7 0.468 0.213 1.88 107 100

Hornych et al.

Ka (kPa) Ga (kPa) n c pa (kPa)

08 118 103 0.622 0.805 100

05 108 94.1 0.538 0.845 100

04 101 82.4 0.618 0.849 100

03 104 63.1 0.378 0.828 100

Hoff et al.

K (MPa) G (MPa) D (GPa)

08 180 97.8 475

05 176 73.9 476

04 157 82.0 238

03 156 100.0 210

Stress input was in kPa



15/17


16/17

When modeling shear and volumetric strains from

the cyclic confining pressure triaxial tests, no

single model was most likely to be correct, for all

the gradings tested. Although not statistically

significant, the model proposed by Hornych et al.

showed the highest mean ranking.

A nonlinear regression method for estimating the

parameters of the model presented by Hoff et al.[20] was proposed. However, this measure inval-

idates a direct statistical comparison of the

efficiency of this model to the others.

When using parameters estimated by triaxial tests

at constant confining pressure for prediction of

mechanical behavior under cyclic confining

pressure, predicted and measured results showed

poor agreement.

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statistical evaluation of resilient models characterizing coarse granular materials

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