Cyclic and Dynamic Mechanical Behaviour of Soils

8/17/2019 Cyclic and Dynamic Mechanical Behaviour of Soils

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RFGC. – 7/2003. Geodynamics and Cycling Modelling, pages 881 to 910

Cyclic and dynamic mechanical behaviour

of granular soils: experimental evidence

and constitutive modelling

Claudio di Prisco* - Clara Zambelli*

*Milan University of Technology (Politecnico di Milano)

Department of Structural Engineering

P.za L. da Vinci, 3220133, Milan, Italy

[email protected] [email protected]

ABSTRACT: The mechanical behaviour of granular soils under transient loading conditions is

quite complex because of the number of factors influencing it. In this presentation, to make

order in the large number of experimental observations and theoretical approaches

introduced in the last decades, a systematic approach has been chosen.

A basic description of standard laboratory experimental devices conceived to describe the

cyclic and dynamic mechanical behaviour of soils will be firstly reported. The main features

of the mechanical response, in drained and undrained conditions, of granular soils are

described and critically compared. Finally, within the framework of elasticity and elasto-

plasticity theories, a brief discussion about some constitutive modelling aspects is introduced.

KEY WORDS: cyclic and dynamic mechanical behaviour, constitutive modelling,

experimental tests, sand, densification, liquefaction


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882 RFGC. – 7/2003. Geodynamics and Cycling Modelling

1. Introduction

The analysis and the understanding of the mechanical behaviour of soils

cyclically and/or dynamically loaded can be considered up till now among the most

stimulating subjects of soil mechanics, in particular, when large strains as well as a

large spectrum of frequencies are taken into consideration and the problem of the

coupling between volumetric and shear irreversible strains is analysed.

Usually, this subject is studied by starting from two opposite points of view. The

cyclic mechanical behaviour is highlighted by means of sophisticated constitutive

models capable of reproducing the volumetric-deviatoric coupling and the

irreversibility of the constitutive relationship, and by disregarding at all the time

factor. On the contrary, the dynamic mechanical response is tackled by means of

elasto-viscous approaches that are linear and allow us to solve boundary value

problems in the frequency domain. Solely in the last decade non-linear numerical

analyses of dynamic problems have been performed, but, as far as constitutive

modelling is concerned, according to the authors, a great effort of synthesis of the

different experiences must still be done.

These few following pages will be devoted to enumerate the experimental

devices that are usually employed, to describe the experimental results and finally to

critically analyse some constitutive approaches conceived to highlight some aspects

of the problem and suggest new research items.

2. Experimental evidence

In the last thirty years many experimental test series were performed with the

aim of describing and highlighting the mechanical behaviour of soils under cyclic

and dynamic loads both in drained and undrained conditions. When cyclic tests are

considered, a quasi-static evolution of the material microstructure is assumed,

inertial forces are negligible but the time factor can play a role because of the time

dependency of the material mechanical behaviour. When dynamic and impulsive

tests are taken into consideration, the interpretation of the mechanical problem

becomes more complex because the time dependency of the mechanical behaviour is

superimposed to the inertial effects.

Finally, even the number of cycles as well as the amplitude that characterise the

loading disturbance are important peculiarities of the problem. When seismic actions

are taken into consideration, the number of cycles is quite small but the cyclic loadamplitude can be considerable. On the contrary, when wind actions, travelling loads

or vibrating machine foundations are considered, the loading cycle amplitude is

smaller but the representative number of cycles is enormous.


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Cyclic and dynamic mechanical behaviour 883

2.1. A brief introduction to e xperimental laboratory devices

The most widespread experimental test apparatus to study the cyclic mechanical

behaviour of soils are the triaxial cell, the simple shear and the torsional shear

device. Whereas, the resonant column and more recently the torsional shear device

are employed to study the dynamic mechanical response.

1- Triaxial cyclic experimental test series are usually performed to analyse the

problem of cyclic liquefaction of saturated granular materials. Both compression and

extension cycles in the effective triaxial plane (q-p’, where a r q σ σ = − ,

' ( 2 ) / 3a r p σ σ ′ ′= + , ij ij ijuσ σ δ ′ = − , u is the pore pressure and ijδ is the Kroneker

symbol) are usually performed. The number of cycles necessary to reach in

undrained conditions the material liquefaction is evaluated and the effective stress path is recorded. Usually, the total stress path is imposed, the axial load is cyclically

varied and both pore pressure and axial strain are recorded. The type of

consolidation (oedometric or isotropic), the consolidation pressure, the relative

density of the specimen, the initial stress level, the strain history are usually varied.

When soil specimens are tested in drained conditions, a viscoelastic approach is

usually assumed to interpret the material mechanical response (§.2.3). The single

loop in the aq ε − plane is taken into consideration and the pseudo-elastic stiffness E

as well as the damping ratio D are directly evaluated from the stress-strain curve. A

more detailed definition of these parameters is introduced here below.

2- Conversely to the triaxial cell, the simple shear test is conceived to reproduce

the stress paths followed by soil layers during seismic ideal events. The initial state

of stress is imposed to be oedometric and the cyclic perturbation is characterised by

the change in the value of the shear stress τ. Analogously to the triaxial apparatus, as

is synthetically illustrated in Fig. 2.1. with reference to the torsional shear device,

these tests allow the direct evaluation both of the shear stiffness G and of the shear

damping D.

3- The torsional shear test puts together some peculiarities of the two previously

cited laboratory experimental apparatus. Cyclic shear stresses (it is worth noting that

in this case the cylinder is hollow) imposed by means of a torque moment applied

along the vertical axis (Fig. 2.1.) are superimposed to a general triaxial state of

stress. By recording the applied torque moment and the relative rotation, it is

possible to evaluate the corresponding stress paths within the soil specimen. If the

internal and external pressures are independently controlled [SYM 88], [MIU 86]the principal axis rotation of the state of stress can be uncoupled of the change in the

stress level.

In all the three test apparatus cited above the loading history is assumed to be quasi-

static and inertial forces are disregarded. Sometimes, to evaluate the stiffness of a

soil at very small strain levels, impulse tests are performed. The perturbation

frequencies are largely higher than those characterising the natural frequency of the

soil sample. The state of stress is a priori imposed by means of the test apparatus


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employed and the propagation rates of P and S waves within the soil specimen are

recorded.

Figure 2.1. Stress state and schematic interpretation of results in a cyclic torsional

shear test [VIN 96].

4- The resonant column is a triaxial apparatus equipped with a cyclic torsional

loading system (Fig. 2.2.). It is based on the theory of wave propagation in prismatic

rods. The axial and radial stresses are usually kept constant during the test and they

define the initial state of stress. On the contrary, the torsional loading frequency ischanged continuously to obtain the soil specimen resonance. Sometimes,

compression waves instead of shear waves are propagated through the soil

specimen. The resonant frequency is a function of the soil specimen mass and of the

material shear stiffness G [WOO 78]. Damping is determined by switching off the

driving power at resonance and recording the amplitude of the decaying vibrations.

Figure 2.2. Stress state in a resonant column apparatus [VIN 96].


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2.2. Cyclic mechanical behaviour: experimental results

2.2.1. Sand liquefaction

During quasi-static cyclic standard triaxial compression and/or extension tests

performed in undrained conditions on saturated loose sand specimens, the inhibited

tendency of the granular assembly to suffer a volumetric compaction may cause a

continuous increase in pore water pressure. When the pore water pressure equalises

the total main pressure, i.e. when the effective mean pressure becomes about zero,

the material behaves like a viscous fluid and its shear strength disappears. This

phenomenon is evident in nature when severe seismic actions are applied to loose

sand layers and it causes many dramatic structural collapses and enormous damagesall around the world every year. An example of such a behaviour is illustrated in

Fig. 2.3., from [ISH 75]. Both the effective stress path and the stress-strain curve are

illustrated. From point A to point B of Fig. 2.3.a the mechanical response of the Fuji

River sand specimen is vergin, from point B to point C both the pore pressure and

the effective stress level (q/p’) continuously increase. Nevertheless the increase in

the deviatoric strain ( 2 / 3( )a r ε ε ε = − , where a stands for axial and r for radial)

(Fig. 2.3.b) is quite small. When point C is reached, the mechanical response

changes abruptly: large variations of the pore water pressure are recorded and a

dramatic increase in the deviatoric strain is observed. It is evident from Fig. 2.3. that

the effective mean pressure does not become completely zero and a sort of strength

recover is observed every cycle. In this case, the term used in literature to describe

such a phenomenon is cyclic mobility and this is associated to the undrained

mechanical behaviour of medium loose sands. This latter is due to the dilatant

tendency of the medium dense or medium loose sand when large strains take place.

During the occurring of such a phenomenon, the strain distribution is no more

uniform within the soil specimen and some zones of localized strains can be

observed. The companion mechanical response, which is obtained by performing an

undrained strain controlled test on the same material, at the same relative density is

illustrated in Fig. 2.4. When the pore pressure increases the amplitude of the cycles

in the effective triaxial plane decreases according to a precise condition: the Δq

varies in order to keep approximately constant the ratio Δq/p’. As Δq continuously

decreases, while Δεa is kept constant, the ratio Δq/Δεa decreases, too. This implies a

proportional decrease in the value of Gsec and even a cyclic degradation of the

material stiffness, however easily foreseeable because of the decrease in the current

effective mean pressure.

The factors which influence the onset of the phenomenon can therefore be

summarised as follows:

the current relative density. As is evident from Fig. 2.5. [CAS 69], when a very

loose sand specimen is subject to a similar total stress path (in this case the cycle

is asymmetric with respect to the hydrostatic axis) liquefaction takes place very

rapidly and no strength recover is observed. On the contrary, when dense sand


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specimens are tested, neither liquefaction nor cyclic mobility occur. In fact, in

this case, the increase in pore pressure is not continuous. At a certain instant of

time, the increase in the number of cycles is not associated to any decrease in the

effective mean pressure and a sort mechanical stabilisation takes place (shake

down). As is implicitly illustrated in Fig. 2.6., such a phenomenon takes place

when the amplitude of symmetric cycles does not reach a certain threshold.

Conversely, when such a threshold is exceeded, the anisotropic hardening can

induce cyclic mobility as in Fig. 2.3. The same observations are not valid when

cycles are asymmetric and the initial state of stress is characterised by a large

value of the initial effective mean pressure (see also the following points).

Figure 2.3. Stress path of loose Fuji River sand: a) effective stress path, b) stress-

strain curve [ISH 75].

The cycle amplitude. By increasing this latter the number of cycles necessary for

the mechanical instability to occur decreases very rapidly.

The deviatoric medium value. In some cases, when the cycles are asymmetric,

the progressive accumulation in pore pressure can become less rapid but the

proximity to the unstable condition in the effective triaxial stress plane generally

dominates and reduces the number of cycles N necessary for the liquefaction to

occur.

The initial effective mean pressure. Such a dependency, jointly to the two factors

cited above, infers that the most important parameters are the initial stress level

and the ratio 0/ 'q pΔ , that, when cyclic torsional tests on oedometrically

consolidated specimens are considered, becomes 0/ 'vτ σ (Fig. 2.6).


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a) b)

Figure 2.4. Effective stress path (a) and stress-strain curve (b) of an undrained

strain controlled test performed on a loose sand specimen [ISH 75].

The type of perturbation. Liquefaction can be obtained even by performing

cycles at constant stress level and continuously changing the Lode angle by

means of a true triaxial apparatus [LAN 91], by rotating the principal axes

and by keeping constant both the mean pressure and the second invariant of

stresses (thanks to the hollow cylinder), by increasing simultaneously the

principal axes and the second invariant of the stress deviator.

Figure 2.5. Liquefaction of loose sand under cyclic loads: a) effective stress path,

b)stress-strain curve [CAS 69].


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Figure 2.6. Number of cycles necessary for the liquefaction to occur during simple shear tests in double amplitude according to varying relative densities [SEE 76].

The strain history of the sample. In accordance with the experimental evidence

concerning the static liquefaction of loose sand [CAN 89], [DIP 95], isotropic or

anisotropic overconsolidations and previous strain histories can induce marked

modifications of the liquefaction potential value [ISH 93].

2.2.2 Cyclic compaction

When drained tests on saturated sand specimens are performed, the volumetric

strain is recorded while the pore pressure is kept constant. Analogously to what was

observed at point 2.2.1., with reference to undrained tests, the mechanical responseduring drained cycles on sand specimens is mainly a function of the current relative

density and of cycle amplitude. In this case, however, it is worth noting that the

relative density changes with the progressive increase in the cycle number. This

implies that, during the evolution of the test, the material presumably changes its

mechanical properties. Such an observation is confirmed by the experimental results

illustrated in Fig. 2.7. [MOH 83]. In this case, a standard drained compression-

extension test performed in strain controlled conditions on a medium loose Hostun

RF sand specimen, characterised by large values of strains reached during each

cycle both in compression and extension, is taken into consideration. As is evident,

the overall result of the transient load is a material compaction. Correspondingly to

the increase in the cycle number, the compaction tendency decreases and a marked

increase in the compression stiffness is recorded.The mechanical response of the granular assembly is completely different if

cycles of small amplitude are performed (Fig. 2.8., after [LUO 80]) and the medium

effective state of stress is changed. This figure shows clearly that the same dense

sand specimen can cyclically densify or dilate according to the medium effective

state of stress, or better to the medium stress level (q/p’) imposed. These results

confirm implicitly the undrained experimental evidence shown above. By following

the simplifying theoretical approach introduced by Luong, we can affirm that, when


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drained tests are taken into consideration, cycling above the characteristic state leads

to dilation of the sand and hence build up of strains; whereas below the

characteristic state, compaction occurs with progressive stiffening. The

characteristic state marks the boundary between incremental collapse and

shakedown. A similar threshold is provided for undrained loading: for stress levels

lower than the characteristic state, positive pore pressure and eventual liquefaction

can develop; at higher stress levels the prevented dilation leads to a tendency to

negative pressures which rapidly stabilize the deformation.

Different peculiarities characterise the mechanical response of granular materials

subject to large numbers (N) of loading cycles. In primis, we can cite the experience

of Suiker [SUI 02], who performed drained standard compression triaxial cyclic

tests on subballast specimens (N>106

). Besides observing that the permanentdeviatoric strains generated under asymmetric cyclic loading conditions are severely

dependent on the cyclic stress level ( η cyc ) and much less on the hydrostatic pressure,

whereas the permanent volumetric deformation are governed both by the cyclic

stress level and the main pressure p, he underlined that:

1. a large number of drained cyclic tests may cause grain degradation and change

the granulometric curve. Such a phenomenon can become not negligible when

calcareous sands, expanded clays or angular grain sands are considered and

tested at high pressures.

2. The deformation rate measured during cyclic tests generally decreases by

increasing the number of load cycles. In fact, as is shown in Fig. 2.9, that

concerns standard drained compression triaxial tests in load controlled conditions

(n=η/ηF =0.98), both deviatoric and volumetric strains generated between one

million and five millions is quite negligible with respect to the strains generated

during the overall loading history. The strain rate progressively decreases, even

though the cyclic stress level is very close to the static failure level of the virgin

material. Such a phenomenon is usually known as shakedown.

Figure 2.7. Drained triaxial tests on a medium loose Hostun RF sand specimen: a)

Stress-strain curve, b) volumetric behaviour [MOH 83].

v o l .

s t r a i n ( % )


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3. The process of cyclic densification influences the strength and the stiffness

properties of the granular materials. By comparing the curves illustrated in Fig.

2.10.a (the three curves were obtained by testing a virgin subballast specimen

and three specimens previously cyclically compacted -one million of cycles- at

three different stress levels n), we can note that an increasing cyclic stress level

causes a significant increase in the post-cyclic peak strength of the material.

Figure 2.8. Cyclic loads at constant confining pressure performed on a dense

Fointanbleu sand specimen [LUO 80].

Many methods for estimating cumulative plastic strains associated to large

numbers of loading cycles for fine-grained soils have been published in the past. For

instance, Dingqing and Selig [DIN 96] have proposed such a simple formula:

m b

pε =Aβ N [1]

where pε stands for the cumulative irreversible strain, d sβ= σ σ represents the

amplitude of cycles normalised with respect to the peak value of stress sσ , while A

can be considered conceptually as the produced soil plastic strain when the soil is

loaded to failure under a monotonic loading, i.e., when β=N=1, ε p=A. Exponent b

can be considered as a constant introduced to quantify the accumulation rate of

plastic strain under repeated loading. When the load cycle amplitude is changed, the

total amount is calculated as is shown in Fig. 2.11.


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a) b)

Figure 2.9. Cyclic response of subballast material (load frequency = 5 Hz): a)

evolution of total deviatoric strain (solid line) and permanent deviatoric strain

(dashed line) with load cycles; b) evolution of total volumetric strain (solid line)

and permanent volumetric strain(dashed line) with load cycles [SUI 02].

a) b)

Figure 2.10. Static response of subballast material after pre-loading at various

cyclic stress levels n=η/η F : a)stress ratio –q/p versus the total deviatoric strain; b)

total volumetric strain versus the total deviatoric strain [SUI 02].

2.3. Dynamic behaviour

The most simple way of interpreting the dynamic response of soils consists in

defining an elasto-viscous model in which the elastic stresses are added to the

viscous ones and both contributes are obtained by assuming the linearity of the

constitutive relationship. By following such an approach, the coupling between the


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shear and the volumetric components is missing, and, consequently, these models

are not capable to capture all the phenomena associated to the coupling between

water and soil skeleton, in which soil liquefaction is included.

Figure 2.11. Model for total cumulative plastic strain [DIN 96].

According to such an interpretation of the material mechanical behaviour, as was

partially already observed in §.2.1, the only three constitutive parameters to be

defined are the shear stiffness G, the bulk modulus K and the damping ratio D which

is usually calibrated on cyclic shear test results and solely associated to this latter

component of stress. In order to appropriately discuss such an approach, we must in

primis remind that the deformation characteristics of soil are non-linear and, as aconsequence, both the shear modulus and the damping ratio vary significantly with

a) the amplitude of shear strains under cyclic loading, b) the confining pressure, c)

the stress level and d) the strain history.

In Fig. 2.12. [IWA 78] some experimental results relative to the dependency on

the amplitude of shear strain under cyclic loading are reported. These results are

obtained by means of a torsional shear test device, in drained conditions on

isotropically consolidated Toyoura sand specimens. The specimen was sheared in

quasi static conditions. By increasing the shear stress or better the shear strain

amplitude the shear stiffness G decreases whereas the damping ratio D increases. An

analogous trend is recorded when the dependency of G and D on the number of

cycles is discussed: by increasing it the shear modulus increases whereas the

damping ratio decreases. The importance of the factors previously cited is wellknown and accepted. On the contrary, the dependency of the shear modulus and

damping ratio on the loading frequency is more complex and less investigated, in

particular when granular soils are concerned. As was recently shown by [MEN 03]

with reference to cohesive materials, the shear modulus depends linearly and not

dramatically on loading frequency, while the damping ratio dependency is severe

and highly non linear (Fig. 2.13.). It is worth noting that both these dependencies are

also markedly influenced by the strain amplitude experimentally analysed.


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a) b)

Figure 2.12. Torsional shear tests on Toyoura sand: a) stress-strain records at 10th

cycle of I to VII stages, b) shear moduli and damping ratios [IWA 78].

From Fig. 2.13. we derive that the range taken into consideration is relative to

low frequencies and that the minimum value of the damping ratio corresponds with

values of frequency smaller than 1 Hz. As was observed by Shibuya [SHI 95], thethree zones of the curves illustrated in Fig. 2.13. are due to two antagonistic factors:

the time dependency of the material mechanical behaviour and the viscous effects

associated to high frequencies.

Analogously, as far as granular soils are concerned, Lin et al [LIN 96] performed

cyclic torsional shear tests on Ottawa and I-lan river sand specimens and recorded

the mechanical response during each cycle. They analysed very small strain

amplitudes 5 410 10γ − −≤ ≤ and took into consideration the frequencies ranging

from 0,1 to 20 Hz, i.e. they diregarded smaller values of frequency where the time

dependency of the material mechanical behaviour is dominant. Therefore, they

observed that the shear modulus is approximately constant while the damping ratio

linearly increases with frequency f . Such a dependency induced the authors to

suggest the following expressione for D:

f f D A B f = + [2]

where the two parameters Af and Bf describe the hysteretic and the viscous

damping, respectively. The values of both terms seem to increase with the increase

in shear strain and to decrease with the increase in the confining pressure.

D


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Contrary to what observed by Lin et al, Bolton and Wilson [BOL 89], by

performing resonant column tests, explored the ranges of high frequencies. By

interpreting the experimental data by means of the equation of a one dimensional

viscoelastic system, they observed that the mechanical response of the material in

terms of stiffness (G) and damping (D) (Fig .2.14.) are independent of load

frequency. In this figure, it is illustrated an almost satisfactory comparison between

experimental data and numerical simulations, at different shear strain amplitudes,

where constitutive parameters were calibrated on quasi-static cyclic torsional

experimental test results.

Figure 2.13. Frequency effect on material damping ratio [MEN 03].

Naturally, the linear interpretation based on the viscoelastic model works solely

when high frequencies are considered, in fact, during quasi static loading cycles (i.e.

when both strain acceleration and strain rate are negligible) the numerical loop

disappears. This derives from the fact that the viscosity parameter is confused with

the irreversibility of the mechanical response of the material.

To confirm the conceptual framework outlined in Fig. 2.13., we report the

experimental data of Santucci et al [SAN 98] (Fig. 2.15.) obtained by testing in

undrained conditions Metrano silty sand specimens at small frequencies. It is

evident that by increasing the strain rate the stiffness increases and the damping ratio

decreases abruptly. By decreasing the strain rate, the deformation properties become

less linear and less reversible (i.e. G decreases and D increases).


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Figure 2.14. Dynamic response [BOL 89].

Figure 2.15. Effects of strain rate on Young’s modulus E 0 and damping ratio D0 at

strains less than 0.001 % from cyclic undrained triaxial tests on Metramo silty sand

[SAN 98].

Analogously, here below (Fig. 2.16.) some experimental results obtained by

performing undrained triaxial tests on Banckok clays [TEA 02] are reported. In

particular, the influence of the rate of loading on the value of the measured secantYoung’s modulus is illustrated: samples tested at a faster loading rate exhibit higher

values of E, expecially at moderate strains (εa ≅ 0.02-0.2 %). Moreover, an increase

in the excess pore pressure accumulation corresponds with slower loading rates (Fig.

2.16.b). This confirms that by increasing the loading rate, the mechanical response

becomes more reversible.

Finally, in Fig. 2.17. the dependency on the loading frequency of the time

necessary for the liquefaction to occur is illustrated. Two identical samples of quick


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Tiller clay have been tested at frequencies of 0.1 and 0.4 Hz: in the former case the

number of cycles at liquefaction was 11, while in the latter one was 180.

a)

b)

Figure 2.16. Undrained triaxial tests (compression, extension and cyclic loading)

on samples of soft Bangkok clays: a) variation of secant Young’s modulus; b)

development of excess pore water pressure for compression tests [TEA 02].

3. Mathematical modelling: the elastoplastic approach

The theoretical approaches usually followed to simulate the mechanical

behaviour of soils subject to cyclic loading at low frequencies can be roughly

separated into two distinct classes: a) incrementally non-linear models [DAR 78],

[DAR 84] and b) plastic models. Within the framework of the former class, the

hypoplastic approach, as was defined by [KOL 91], is particular successful. The

irreversibility of the material mechanical behaviour is associated both to loading and


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unloading stress paths. Such a theory by avoiding the classical notions and

definitions of elastoplasticity (yield locus, plastic potential, flow rule and

consistency rule) defines a unique equation for loading and unloading.

Figure 2.17. Cyclic standard triaxial compression tests on quick Tiller clay

specimens at different frequencies of loading [VOZ 99].

Within the framework of plasticity, instead, we distinguish three different classes:

(a) the bounding surface models, (b) the generalised plasticity and (c) the multiple

mechanism plastic models. Class (a) is characterised by two surfaces, an outer or

consolidation surface, and an inner loading locus, and provides a rule for thedefinition of the hardening modulus in the space between them [DAF 86]. In order

to obtain a satisfactory simulation of the material mechanical behaviour an

anisotropic hardening for both the inner loading locus and the outer boundary

surface must be provided [GAJ 99]. In the generalized plasticity (class b), instead,

the concepts of loading and unloading, as well as of flow rule, are extended. As is

exhaustively explained in [PAS 85], [ZIE 85], [DIP 00] the elastic domain

disappears as well as plastic potential and consistency rule.

Finally, the natural extension of single-mechanism plasticity is provided by

multiple-mechanism plasticity, in which two or more yield surfaces and plastic

potentials are defined. An example of such a class is the Milan Model 2002 [ZAM

02], an elasto-plastic-cyclic model characterised by two uncoupled plastic

mechanisms: the first one which is associated to a generalised and global evolution

of the material internal fabric, the second one associated to small strains and small

loops taking into account the fabric rearrangements due to small size cyclic load

disturbances. Such an approach allows to separate, perhaps in a unphysical manner,

the two distinct contributes but allows a very simple calibration of the constitutive

parameters. As was previously observed with reference to bounding surface models,

in order to achieve a satisfactory simulation of cyclic mechanical behaviour of soils,


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both yield surfaces and plastic potential must be characterised by an anisotropic

hardening.

As is usually done when shear and volumetric contributes are dealt with

separately, such an approach consists in adding three distinct terms as follows:

el ir cir

ij ij ij ijε =ε +ε ε+ , [3]

whereel

ijε stands for the elastic strain rate tensor,ir

ijε for the standard irreversible

strain rate tensor andcir

ijε for the irreversible strain rate tensor associated to the

second cyclic irreversible mechanism. In the constitutive model previously cited,

that will be here in the following very briefly outlined, the elastic strain rate tensor isdefined by starting from the definition of an elastic potential [LAD 87] [MOL 88],

from which the elastic incremental stiffness matrixel

ijhk C can be easily derived. Both

the two irreversible strain rate tensors are calculated by assuming the existence of a

yield locus f, a plastic potential g, a flow rule and a hardening rule. As far as the first

plastic mechanism is concerned, a viscoplastic Perzjna approach is followed ([PER

63], [PER 66], [DIP 96]), i.e.:

ir 1ij 1 '

ij

gε ( )

σ f

∂= Φ

∂ [4]

this choice derives from the aim to achieve three distinct goals:

The simulation of the time dependent unstable mechanical behaviour of

granular materials during drained and undrained creep tests ([LIN 81], [DIP 96]).

To reproduce the dependency of the mechanical behaviour on the time factor

when they are subject to dynamic actions of large intensity (§.3.2.).

To describe the dependency of the mechanical behaviour on current relative

density. In fact, when Eq.(4) is introduced and the consistency rule is avoided,

the updating of constitutive parameters on the relative density becomes very

simple [DIP 02].

As was described in [DIP 93], both f 1 and g1 are characterised by an anisotropic

hardening, that is governed by a tensor χij (Fig. 3.1.). The yield locus f 1 is a function

of two additional hidden variables: r c, which describes its size, and βf , its shape,respectively.

As is schematically illustrated in Fig. 3.1., even the cyclic mechanism is

characterised by a very elongated cone shaped yield locus f 2, rotating in the effective

stress space in accordance with the evolution of its axis χij2 [ZAM 02] and keeping

constant its size. The definition of the yield locus derives from the assumption that


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cyclic irreversible strains take place only when the stress level (η in the triaxial

plane) changes: i.e. radial cyclic effective stress paths are approximately reversible.

Figure 3.1. Two mechanisms constitutive model: a schematical representation of the

two yield loci.

In this case, by assuming a time independent mechanical response, a standard

elastoplastic approach has been chosen:

cir 2ij cir '

ij

gε λ

σ

∂=

∂ [5]

where cir λ

is the relative plastic multiplier calculated by imposing the consistency

rule and a hardening rule defining a Prager type dependency of2ij

χ on cir ijε [PRA

52]. Even this second plastic mechanism has been assumed to be non associate: the

plastic potential g2 rotates with f 2 but its size evolves as a function of'

ijσ and 2ijχ .

The two mechanisms allow us to simulate independently and simultaneously two

distinct aspects of the mechanical behaviour of granular materials: the first one

which is associated to large strains, to material collapse and unstable phenomena,

the second one to irreversible strains associated to small loading cycles and to

volumetric compaction or dilation associated to shear cyclic perturbations. In other

worths, such an approach makes possible the interpretation and the simulation of the

previously defined (Eq.2) hysteretic damping.

3.1. Numerical examples

In order to illustrate the capability of the constitutive model briefly presented in the

previous paragraph, to simulate the cyclic mechanical behaviour of granular soils,

here in the following some cyclic experimental test results will be compared with

relative numerical simulations [ZAM 02]. In Fig. 3.2. standard drained strain


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controlled triaxial compression/extension tests are considered. The strain amplitude

of cycles is large enough to make active the anisotropic hardening, i.e. the rotation

of the yield locus f 1. It is interesting to note that, during the compression unloading

paths, both experimental data (Fig. 2.7.) and numerical simulations show a marked

volumetric loop. This is due to the initial pseudo-elastic dilation associated to the

unloading that is cancelled by the successive plastic compaction. The global

volumetric response is compactant but such a behaviour is progressively inhibited

by the evolution of the current relative density.

a) b)

Figure 3.2. Numerical simulation of the experimental data shown in Fig. 2.7.

Drained cyclic compression/extension test on medium loose Hostun RF sand: a)

Stress-strain curve, b) Volumetric behaviour.

A completely different mechanical response is observed when smaller cycles are

performed (Fig. 3.3. and Fig. 3.4.). In this case the evolution of the yield locus

associated to the macro-fabric is negligible and only micro-sliding phenomena take

place. It is evident the marked change in the mechanical response when the

maximum deviatoric load is exceeded (Fig. 3.3.) and it is important to underline the

absence of the phenomenon of ratcheting: i.e. the stress-strain curves show,

whatever the stress cycle amplitude or the medium stress level are, a well definedloop (Fig. 3.3.).

As was previously observed, when cyclic loads are considered, the importance of the

medium stress level is crucial, in particular when the volumetric response is

concerned. Obviously, in undrained conditions this implies a completely different

mechanical response and varying effective stress paths (Fig. 3.4.). In this case the

increase in the pore pressure is inhibited by the increase in the effective stress level


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and a sort of phenomenon of stabilisation takes place. On the contrary, when a loose

sand specimen is cyclically loaded, liquefaction takes place (Fig. 3.5.). The

difference between the two mechanical responses is essentially due to the capability

of the material of reaching the yield locus . In fact, both in Fig. 3.4. and 3.5., the

effective stress path is within the yield locus f 1 and irreversible strains are due to the

second micro-plastic mechanism. In Fig. 3.4. f 1 is not reached, while in Fig. 3.5., this

latter is reached and instability takes place. It is essential to underline, that the

mechanical response, when the effective stress image point is within the yield locus

f 1, is always stable, while when the effective stress point belongs to such a yield

locus, volumetric instability becomes possible [NOV 91].

a) b)

c) d)

Figure 3.3. Drained triaxial test on Hostun medium dense sand: a) and b)

numerical simulation, c) and d) experimental data [MOH 83].


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a) b)

c) d)

Figure 3.4. Undrained cyclic compression triaxial test on a Toyoura dense sand

specimen: a) and b) numerical simulation, c) and d) experimental data [HYO 91].

Figure 3.5. Numerical simulation of a cyclic, load controlled, undrained

compression triaxial test on a loose sand specimen: a) stress-strain curve, b)

effective stress path [ZAM 02].


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3.2. D ynamic modelling

When granular media under dynamic actions are considered, an important

question naturally arises: is the constitutive relationship time independent? That is:

during cyclic tests is the frequency value an important factor?, what kind of

constitutive modelling is more suitable for describing the material mechanical

response during cyclic or impulsive loading conditions? Definitive answers to these

questions are not yet be introduced and in literature experimental observations lead

often to contradictory conclusions. For the sake of simplicity, it is at any rate

important to distinguish between the mechanical response recorded during small

strain cyclic load tests and that one we obtain when large strain loading cycles are

performed or impulsive load increments are imposed on virgin soils.

If an elastoplastic approach is taken into consideration, the dependency of the

mechanical response on the load frequency, during cyclic tests, or on the load period

when impulsive tests are analysed, is completely provided by the inertial term; in

fact the incremental elastoplastic constitutive relationship is assumed to be time-

independent. Conversely, the elasto-viscoplastic constitutive models take actually

into consideration the dependency of the mechanical behaviour of soils on the time

factor, but such a dependency is very well known as far as cohesive soils are

concerned but less when soft rocks and granular soils [DIP 96] are considered.

In accordance with Tatsuoka et al [TAT 98], the time effects that can be

observed during laboratory experimental tests on sand specimens can be summarised

as follows:

a)

creep deformation at constant effective stress state,

b) strain relaxation at constant axial strain,

c) temporary overshooting and undershooting in stress immediately after

change in constant strain rate,

d) volumetric instabilities, i.e. sudden collapses, of loose sand specimens

tested in load controlled conditions [DIP 97], [DIP 00a].

Whereas phenomena a and b can be satisfactorily simulated by means of a standard

elasto-viscoplastic approach, the phenomena recalled at points c and d can be

simulated only by introducing in the constitutive relationship the dependency on

strain acceleration [DIP 00b], [DIP 03], or non linearity in the hardening rules.

An example of the time dependency of the mechanical behaviour of granular soils is

illustrated in Fig. 3.6. [DIP 96]. The experimental data are relative to a drained

triaxial creep test characterised by an instantaneous increase in the axial load. The

relative numerical simulations are obtained by means of the previously introduced

MM 2002 constitutive model, in which an exponential definition for the viscous

nucleus was assumed.


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Figure 3.6. Creep test experimental data and numerical simulation obtained by

means of the Milan 2002 constitutive model previously cited corresponding to an

instantaneous axial load increment of 5 kPa (cell pressure of 100 kPa, mobilised

friction angle of 16) [DIP 96].

It is also worth noting that elasto-viscoplastic constitutive models are capable of

reproducing both the dependency of the number of cycles necessary for the

liquefaction to occur on the stress rate imposed ( in Fig. 2.17.), and the dependency

of the damping ratio on frequency when this later varies between 0 and 0,1 Hz

(Fig.2.15.) (where such a range has not to be interpreted in a strict manner because it

depends on the amplitude of the loading cycles).

As far as the former aspect is concerned, in Fig.3.7. some numerical simulations

of a standard undrained triaxial cyclic compression test performed at varying stress

rates are shown. Unfortunately, as is evident from the comparison of the three

numerical curves with those shown in Fig. 2.17. [VOZ 99], such a dependency is a

consequence only of the undrained mechanical response calculated during the first

cycle. In fact, in the model numerically implemented only the first plastic

mechanism is viscoplastic.

In the definition of the viscoplastic relationship a crucial role is played by the the

function describing the viscous nucleus. In fact, this latter dramatically affects the

numerical system response during the evolution of time. A definition which can be

acceptable when long term mechanical behaviour is taken into consideration can

become unsatisfactory when the time range is changed. For instance, by choosing anexponential function (Fig. 3.9.a), the material mechanical response, for high

frequencies or impulsive loads, becomes almost instantaneous, whereas the time

dependency remains evident when creep tests are considered. Such a high non

linearity is quite important to capture some important aspects of the mechanical

response (i.e. this latter justifies the evident creeping behaviour during load

controlled tests and the negligible effect of strain rate during strain controlled tests)

but must be dealt with extreme caution. By changing the definition of Φ (in this case


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a maximum threshold Φ

has been introduced (Fig.3.8.a.), that becomes active only

when dynamic and/or unstable phenomena take place) and by reducing the value of

Φ

, it is possible to observe a continuous change in the system mechanical response,

that passes from plastic to elastic. In Fig. 3.8 [ZAM 02], three numerical simulations

of an horizontal dense sand infinitely long stratum, subject to a Ricker wave action

imposed along the x direction at the bottom boundary (Fig. 3.8.b), at increasing

viscous nucleus thresholds Φ

, are compared. In Fig. 3.8.c the imposed action,

whereas in Fig. 3.8.d, e and f, the horizontal displacements along the sand stratum

during the evolution of time are illustrated.

a) σa

=0.059 kPa/s b) σa

=0.59 kPa/s

c) σa

=5.9 kPa/s

Figure 3.7. Numerical simulation of a standard undrained triaxial cyclic test on

Fuji River loose sand specimen at varying loading rate [ZAM 02].

These observations become very important when large strain cycles are triggered

by seismic actions. In this case plastic waves occur, but the choice in the constitutive

model and in the viscous nucleus definition can become dramatically important

[FAC 95].


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a) b)

c) d)

e) f)

Figure 3.8. Numerical analysis of an ideal infinitely long stratum dynamically

loaded [ZAM 02]: dynamic excitation (c), horizontal displacements numerically

simulated by increasing the viscous nucleus thresholds (d-f) [ZAM 02].

4. Concluding remarks

Even if the experimental study and the constitutive modelling of the cyclic and

dynamic mechanical behaviour of soils is quite complex and, as has been shown in


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this brief written discussion, not yet completely highlighted, it is important to

summarise the following experimental and theoretical observations:

a) within the framework of the most sophisticated constitutive modelling

approaches previously cited (bounding surface, generalized plasticity,

hypoplasticity and so on) constitutive relationships capable of reproducing the

quasi-static cyclic mechanical behaviour of soils can be conceived.

b) When the number of cycles is quite large, and/or the considered granular

material is loosely compacted, the evolution of relative density becomes an

essential factor influencing the mechanical response.

c) When rapid load increments or high frequencies in cyclic loading conditions are

taken into consideration, the time factor must be introduced in the constitutiveapproach.

d) Standard viscoplasticity seems to be capable of reproducing the dependency of

the mechanical behaviour of soils on the time factor, in particular when low

frequencies and hysteretic damping are considered.

e) Conversely, the dependency of the numerical response on instantaneous

changes in strain rate, the unstable collapses of loose sand specimens taking

place during triaxial load controlled tests are phenomena that can be reproduced

only by modifying the standard elasto-viscoplastic approach and by adding non

linearity or strain acceleration dependency.

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Cyclic and Dynamic Mechanical Behaviour of Soils