Triaxial Loading lab

8/12/2019 Triaxial Loading lab

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Recently, similar experimental results were published by Sasitharanet al.

,

3,4

Skopeket al.5

In the first papers, the authors interpreted the collapses as the consequence of an instantaneous and

undrained mechanical response of the material. Instead, in the second one, because the sand speci-

mens are dry, the authors are not able to theoretically justify the phenomenon experimentally

observed.

If a drained standard triaxial strain-controlled test is performed, the same material, up to the steady

state, manifests a stable, continuously hardening, mechanical behaviour. Consequently, it seems

evident that the same loose sand specimen may show two different mechanical responses, according

to the type of test performed: strain or load-controlled.

In the second part of this paper, a theoretical interpretation is proposed, which analyses the

unstable phenomenon from a microdynamic point of view. In order to investigate the dynamic aspect

of this unstable phenomenon, it is necessary to take into account the effects of the time factor on the

mechanical response of the granular assemblies. The concept of time dependency on the mechanical

behaviour of the material has already been experimentally demonstrated and theoretically consideredin di Prisco and Imposimato.6 In that paper, the authors presented the experimental results obtained

by performing drained standard triaxial tests, characterized by finite load increments and by different

time periods between two successive load steps.

As regards low stress levels, the authors showed that the asymptotical trend is not reached

immediately and that during a finite time period the strain rate is not nil, even though the load is kept

constant. Experimental evidence suggests that the microstructural evolution, in a single load incre-

ment, might be interpreted as a microdynamic, statistically determined phenomenon, which passing

through a transient condition, reaches a steady condition.

According to the authors, the collapses are caused by an unstable transient condition.

In the following section, the phenomenon considered and its theoretical interpretation will be

proposed as a useful tool in highlighting the static liquefaction phenomenon. The increase in pore

pressure will be considered as a consequence of a previous collapse, and not as the triggering cause.By accepting this mechanical explanation, in boundary value problems, the pore pressure wave

becomes only a vehicle of instability, by making the instability propagation more rapid and prob-

able.79

2. SOME REMARKS ON STATIC LIQUEFACTION

2.1. Traditional laboratory test results

Liquefaction is an important phenomenon, causing dramatic effects. The phenomenon of sand

changing its behaviour from solid to liquid was recognized in the early stage of soil mechanics

development. The term spontaneous liquefaction was coined by Terzaghi10 to indicate the sudden

change of loose deposits of sand into flows, much like those of viscous fluid, triggered by a slight

disturbance.Further studies on the liquefaction phenomenon taking place under static loadings1115 concluded

that the material must be loose for liquefaction to occur, i.e., contractive during shearing at large

strain, and assumed that the loadings must not allow the water to flow prior to the collapse.

In order to clarify the liquefaction phenomenon, both static and cyclic, undrained, strain-con-

trolled, triaxial loading tests, on loose granular specimens, were performed by Konrad,16 Ishihara,17

and di Prisco, Matiotti and Nova.18 Over more than 20 years, a large number of monotonic undrained

test results on loose sand specimens have shown that the effective stress paths are characterized by a

peak in the effective triaxial plane, and by a peak of the deviatoric-axial strain curve, see Figure 1

(from Castro11).

94 C. DI PRISCO AND S. IMPOSIMATO


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On the other hand, the undrained triaxial cyclic tests on loose saturated sand specimens 17 are

characterized by a continuous increase in pore pressure, by an associated increase in stress level, and

lastly, if the test is stress-controlled, by a sudden collapse. All these test results are traditionally used

to clarify the liquefaction phenomenon, because undrained conditions are commonly assumed to be

physically realistic, even if boundary value problems and a material like sand are considered.

In order to justify the link between undrained laboratory experimental results and the physical

reality of the in situ conditions, the triggering collapse disturbances and the associated mechanical

response of the material must be assumed to be rapid.

2.2. Comments on undrained conditions

By considering various case histories of spontaneous liquefaction, it appears difficult to recognize

rapid triggering causes. Liquefaction commonly concerns local steepenings owing to erosion pro-

cesses and to seepage pressures during falling tide or rapidly accumulating sediments.19

Moreover, also the rapidity of the mechanical soil response needs to be discussed. In fact, the

authors have shown experimentally that the irreversible strains are delayed. With reference to loose

sand triaxial specimens, the time period required to reach the final strain is a few minutes. This effect

is irrespective of the presence of water; in fact, even if dry specimens are considered, the delay

phenomenon occurs.

In order to closely examine this particular aspect, a triaxial compression test on a loose iso-

tropically consolidated saturated sand specimen was performed. At the beginning, after the firstdrained load controlled triaxial phase, the drainage valve was closed (point A in Figure 2(a)) and a

load increment was imposed.

In Figure 2(b) the trend of the pore pressure versus time is shown. A time dependency of the

increase in pore pressure is evident. This trend is also shown in Figure 2(a): point B corresponds to

the pore pressure value one minute after the load increment; point C to that recorded half an hour

later and D one hour later.

In undrained conditions, even though the load increment is applied almost immediately, the final

increase in pore pressure is reached only after a considerable time lag. Consequently, when the

drainage valve is open, the complete undrained increase in pore pressure (point D ) will never be

Figure 1. Standard undrained triaxial compression on a very loose sand specimen (from Castro 11): (a) effective stress path; (b)stressstrain behaviour

LOOSE SAND SPECIMEN COLLAPSES 95


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reached: the value of the pore pressure increase will depend on the material permeability coefficient

and the strain rate. The main factor, in reality, is the ratio between these two variables.

With reference to particular boundary conditions and to particular hydromechanical characteristics

of the material, in time the pore pressure values may induce dangerous collapses. Nevertheless,

according to the authors, in some other cases, the instabilities may be caused by a completely

different mechanical mechanism, in which the pore pressure (i.e., the water presence) does not play

the role of the triggering factor.

Figure 2. Undrained axial load increment after a drained standard triaxial compression phase on a very loose Hostun sandspecimen: (a) total and effective stress path; (b) pore pressure versus time curve



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The aim of this paper is, precisely, to introduce this new interpretation, by analysing the experi-mental results briefly described in the following paragraph. The doubts about the mechanism, which

in situ triggers of liquefaction phenomena, as already clearly expressed by Eckersley, 2 derives from

laboratory experimental observations. By analysing the collapses involving the laboratory model

slopes he monitored, Eckersley concluded that the flow slides can initiate under essentially static,

drained conditions. The liquefaction occurs subsequently to failure initiation with excess pore

pressures being generated in relatively thin shear zones.

Moreover, the collapses described below by the authors, although they concern drained tests, are

qualitatively very similar to those observed when undrained conditions are imposed. The collapses

are unexpected and may occur very rapidly. When the specimens are saturated, it is possible to

observe a sudden increase in pore pressure following the specimen failure.

The material, from a macroscopic point of view, is always uniformly deformed, even after the

collapse takes place. Consequently, according to the authors, the experimental results described

below, together with those recently presented by other researchers,1,4,6,2023 lead directly to a re-examination of the liquefaction problem.

3. EXPERIMENTAL OBSERVATIONS

3.1. Experimental programme and procedure

A series of triaxial load controlled tests was performed. All the tests were carried out on specimens

(140 mm high, 70 mm wide) of loose Hostun RF sand (Dr 20%). A more detailed laboratory

device and test procedure description may be found in di Prisco and Imposimato. 6

Two different types of triaxial tests were carried out, and are drawn in Figure 3. The first is a

drained standard triaxial compression test, obtained by increasing only the axial load and by keeping

the cell pressure constant. The second is characterized initially by a standard drained triaxial com-

pression phase, followed by a q constant effective stress path. This second phase is obtained by

Figure 3. Drained load controlled tests: standard compression and q constant stress paths.



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keeping the axial load constant and either increasing pore pressure when the specimen is saturated, orby directly decreasing the cell pressure, when the specimen is dry.

All the stress paths were obtained by imposing finite incremental stress steps ( qor p). The time

elapses between two consecutive stress increments and the magnitude of stress increments were set as

variable: every test has a particular time history.

3.2. Strain controlled and load controlled tests

In order to underline the difference between the mechanical response obtained by imposing strain-

control and load-control in drained standard triaxial tests, it is interesting to compare the experimental

results of two different 100 kPa consolidated saturated specimens.

The strain-controlled test is performed by imposing a constant axial strain rate ( 1 mm min);

whereas the load-controlled test is carried out by imposing finite axial load-increments. The axialload time history is illustrated in Figure 4.

In Figure 5 the experimental results are shown. The strain-controlled test is characterized by a

continuous hardening regime up to the steady state ( 32 ). On the contrary, the load-controlled

test at the stress level characterized by a mobilized friction angle of 25 suffers a sudden collapse.

Up to this point, the mechanical responses of both tests are roughly the same, but the sudden

instability is peculiar only to the load-controlled test.

In Figure 6, the collapse points of the various load-controlled tests24 are collected, and compared

with the ultimate state points obtained by means of strain-controlled loadings. In Figure 6, not only

the standard triaxial test, but also the q constant test results are collected.

In order to explain the large dispersion of the points corresponding to the sudden collapses, it is

possible to draw the same locus in a Drqp space or more simply in a meplane (Figure 7). But, as

will be clarified below, the unstable phenomenon considered cannot be completely explained by onlytaking into account the relative density and the effective stress state: the problem is more complex.

The phenomenon is caused by a structural collapse and is a result of progressive destabilization of

the grain structure. In order to explain the difference in the materials mechanical behaviour, obtained

Figure 4. Drained standard triaxial tests [ c 100 kPa (T100d)]: axial loadtime history



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by performing strain and load-controlled tests, it is necessary to outline the main aspects of the two

test procedures.

When a strain-controlled test at constant strain rate is carried out, the system is kinematically

controlled: no global axial acceleration is allowed. On the contrary, when a load-controlled (i.e.

creep) test is performed, the system is free to accelerate and evolve without any kinematical con-

straint.

In order to highlight the causes of instability, in the following section the load controlled test will

be described more accurately.

Figure 5. Comparison between two drained standard triaxial compression tests performed by controlling the axial strain or theaxial load: [ c 100kPa] (T100d): (a) deviatoric stress versus axialstrain curves; (b) volumetric behaviour



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3.3. Load-controlled tests: stable and unstable mechanical responses

The load-controlled tests were carried out by imposing finite load increments. The load increments

are followed by time periods at constant stress level.

Among the 20 tests performed, the test T100a will be analysed. The relative load time history is

illustrated in Figure 8(a), and in Figure 8(b) the corresponding mechanical response is drawn. In order

to highlight the nature of the phenomenon, it may be interesting to compare the material mechanicalresponse to the load increments at different stress levels.

Figure 6. Comparison between collapse points in load-controlled tests and steady-state locus

Figure 7. Mobilized friction angles versus void ratio values relative to collapse points



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In Figure 9(a) the first seven increments are considered. The axial strain versus time curves show

stable and clearly time dependent mechanical behaviour.6 The static disturbance induces a delayed

deformative response, which has been defined as delayed plasticity. The transient regime takes many

minutes, and the internal structure continues to evolve subsequently, too. The micro-structural

rearrangement, when it takes place, is not immediate but delayed.

When the mobilized friction angle of 22.3 is reached, the type of response illustrated in Figure

9(b) is observed for the first time.

Figure 8. (Test T100a), standard triaxial test, [ c 100 kPa]: (a) loadtime history; (b) stressstrain behaviour



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Initially, the axial strain rate increases, even if the effective axial stress remains constant. However,

after a while, the asymptotical value of strain is reached. At a stress level, characterized by a

mobilized friction angle of 25 , the collapse takes place (Figure 9(c)).

Each experimental load controlled test, both standard triaxial and q constant, is characterized by

the same mechanical trend (Figure 10). At a low stress level, the axial strain rate continuously

decreases (curve a). By increasing the stress level, the system initially accelerates, but subsequently a

strain asymptotical stabilization takes place (curve b), and lastly collapse occurs (curve c).

Figure 9. (Test T100a), standard triaxial test, [ c 100 kPa]: (a) dq 2 kPa, stable mechanical behaviour; (b) dq 2 kPa,unstable mechanical behaviour; (c) dq 2 kPa, collapse increment



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By assuming the traditional mechanical definition of stability,25 it is possible to define the material

mechanical behaviour, described by curve c, as unstable. In fact, the continuity of the material

response is lost. Small stress disturbances cause large strain increments.

If no instability occurred, the steady state would be reached and, by increasing the stress level,

strain increments would continuously increase. On the contrary, when such an instability occurs, a

gap between the previous load step and the unstable one (Figure 9(c)) may be observed.

If curve B of Figure 9(b) is considered, together with the previous and the subsequent load steps (A

and C, respectively (Figure 11)), it appears evident that curve C is more rigid than curves A and B.

This means that the incipient instability (seen in curve B) induces a more stable micro-structure and

consequently a more rigid macroresponse in the subsequent load steps. Moreover, it is important to

observe that the incipient material instability recorded is associated with compactive volumetric

behaviour (see section 4.5).

In order to demonstrate that the instability phenomenon previously shown is not linked to the

rapidity of the load test considered, many different load-controlled tests were performed. Among

these the collapse obtained by means of a q constant test performed very slowly will be illustrated.

Moreover, this experimental result clearly shows that the onset of instability is not due to a rapid

increase in pore pressure.

Figure 9. (continued)

Figure 10. Schematical mechanical responses to a generic axial load increment



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The collapse takes place very slowly (Figure 12); acceleration is continuously positive, though

initially almost nil. The collapse occurs some hours after the load increment has been applied. The

pore pressure increases when the collapse has already taken place. Before that point, no increase in

pore pressure can be observed. Therefore the phenomenon may be assumed to be unrelated to the

presence of water.

3.4. Axial strain rate and axial strain acceleration analysis

In order to highlight the actual nature of the phenomenon, it would be very interesting to describe,

during the single time periods between two consecutive load increments, the internal fabric.

Figure 11. Comparison between the unstable mechanical behaviour and the mechanical behaviour corresponding to theprevious and the successive load increments

Figure 12. Collapse occurrence after a long time period



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Unfortunately, using ordinary geotechnical laboratory devices, this cannot be done. Consequently,the following analyses will be made only by interpreting the external measures (the phenomenon

consequences) and not by considering the microstructural causes.

As previously observed, when a load-controlled or creep test is performed, the problem must be

dynamically analysed: the trend of the kinetical variables such as axial-strain rate and axial-strain

acceleration must be described.

In order to compare the stable and unstable material responses, the first seven load steps are shown

in Figure 13. While in Figure 14, the unstable response is illustrated together with the previous load

step (Figure 9(a)).

In Figure 13, it is evident that the load-increment causes a strain rate increase (points Ai). This is

followed by a continuous decrease up to the subsequent load increment.

The peak velocities, which are recorded a minute after increasing the axial load, will continue to

grow gradually. With reference to increasing stress levels and by keeping the time period elapsing

between two succeeding load increments constant, the material strain response becomes more rapid.Nevertheless, the single material response to the load disturbance remains stable, in fact the loose

sand specimen mechanical behaviour is continuous and characterized by typical exponential decay.

In the same manner, if we draw the acceleration trend relative to the load steps of Figures 9(a) and

11, the curves of Figures 15 and 16 are obtained. In Figure 16 the unstable load step and the previous

one are compared.

From Figure 16, it appears evident that when instability takes place, the system accelerates over a

significant period of time.

3.5. Dry sand specimens: further experimental observations

Recently, by conducting a series ofq-constant, load-controlled, triaxial tests, on dry loose Ottawa

sand specimens, Skopeket al.5 concluded that the structural collapse of very loose dry sand is a result

of progressive destabilization of the grain structure.The same conclusions on dry loose sand specimens may be obtained, by analysing the experi-

mental results of Figure 17(c). The relative stress path is illustrated in Figure 17(a), while the p

Figure 13. Axial strain-rate versus time: load increments corresponding to those illustrated in Figure 9(a)



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versus time curve is shown in Figure 17(b). Similar results were obtained by performing load-

controlled, standard compression, triaxial tests on dry loose sand specimens. The collapse point of

Figure 17(a) corresponds with a mobilized friction angle of 23 . The axial strain versus time curve,

illustrated in Figure 17(c), is relative to the phase of the test characterized by a constant value ofq

(from point B to C of Figure 17(a)). The points of the curve are relative to the instant of time

corresponding to the load increment application.

The particular strain versus time behaviour shows an initial rigid mechanical response. This is

followed by more considerable strain increments. This may be clarified by interpreting the material

mechanical behaviour in the light of elastoplasticity. In fact, initially the response may be assumed to

be elastic and, subsequently, to be elastoplastic.

Figure 14. Axial strain-rate versus time: load increments corresponding to the unstable response and the previous one

Figure 15. Axial strain acceleration corresponding to the load increments of Figure 9(a)



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The loose sand specimens designated as dry, are not actually completely dry, because of the

preparation method. In fact, to obtain the desired relative density, the traditional method introduced

by Bjerrum, Kringstad and Kummeneje26 was used. Before the specimen preparation, a small per-

centage of water (2% in weight) is added. Consequently, the specimens are not completely dry and

obviously, because of suction forces, this may cause a different mechanical response in comparison

with the saturated specimens.

By taking into account Figure 17, it is evident that the material behaves qualitatively as if it weresaturated. However, the stress level at which the collapses take place is quantitatively different. In

particular, if we compare the collapse points of the saturated specimens and the dry ones, it is

possible to conclude that suction makes the microstructure more stable, and this effect is highlighted

by the increase in the mobilized friction angle at which sudden collapses occur.

4. THEORETICAL INTERPRETATION

The phenomenon shown experimentally in section 3 may be defined as delayed instability and,

consequently, may be interpreted as an unstable creep. Generally, delayed instability is classified as

tertiary creep and considered as peculiar to cohesive materials; in fact, they are studied within the

framework of fracture mechanics and are assumed to be the ultimate result of unstable fracturepropagation.

On the contrary, as regards granular assemblies, a theoretical framework, capable of justifying the

instability considered, does not exist. Therefore, in order to conceive constitutive models which are

able to reproduce such an unstable phenomenon, the problem of microstructure evolution from a

probabilistic point of view has been tackled.

The aim of the following paragraphs is to extend the applicability of the traditional elastovisco-

plastic theory to dynamic problems. In fact, according to the authors, when load-controlled tests are

performed, the microstructure evolution cannot be considered as quasi-static, but must be interpreted

dynamically. Within the framework of strain-hardening elastoplasticity, this implies the redefinition

Figure 16. Axial strain-acceleration versus time: load increments corresponding to the unstable response and the previous one



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of the hardening rules which govern state variable evolution. In the following, a new constitutive

model is not presented, but only a logical path is outlined.

In this perspective, the theoretical attempt presented below lays no claim to being exhaustive, but

rather seeks to give rise to scientific debate.

The interpretation outlined in Sections 4.24.4 makes use principally of two variables to describe

system evolution: the configurational entropy Sc and the kinetic energyEc.Scand Ecare assumed to

determine, respectively, how and whether the granular system evolves.

Figure 17. Load controlled triaxial q constant test on dry very loose Hostun sand specimen: (a) stress path; (b) mean pressure

time history; (c) axial strain versus time curve relative to the q constant phase



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Thanks to some simplifying assumptions, in Sections 4.4.3 the approach proposed is shown to be

capable of justifying the kind of instability under consideration.

Naturally, according to the authors, the validity of these theoretical observations will have to be

confirmed by their application to constitutive relationships; nevertheless, in this paper such appli-

cations are not presented so that the approach may not lose its general applicability.

4.1. Stability definition

According to Liapunovs definition of stability, if small consequences correspond to small dis-

turbances, the system is defined as stable. This definition may be mathematically converted into a

continuous dependence of solution on data. Therefore, in this sense, the collapses previously illu-

strated are the ultimate consequences of an unstable mechanical response. For instance, if we

compare the experimental mechanical responses corresponding to steps A and B of Figure 11, it

appears evident that a discontinuity takes place. The inputs are continuous, but the response is

discontinuous. On the contrary, when the material reaches the steady state, by means of strain-

controlled loadings, the mechanical response appears to be very different. In this case, the mechanical

behaviour is stable, because the dependency of the mechanical response on disturbances remains

continuous.

As anticipated in the introduction, the unstable phenomena considered will be theoreticallyinterpreted by taking into account the time dependency of the granular materials mechanical

behaviour. This time dependency may be observed whether the mechanical response be stable or

unstable. The difference seems to consist in the evolution trend. From the analysis of the experi-

mental results shown above (Figure 9), it seems that, as the axial load is kept constant and if the

response is stable (Figure 9(a)), the axial strain rate continuously decreases whereas, if the response is

unstable (Figure 9(b)), the strain rate increases. Nevertheless, this difference is only apparent.

In reality, when the time period following a single instantaneous load increment is taken into

account, the axial velocity initially increases and subsequently continuously decreases. The experi-

mental results illustrated in Figure 9(a) do not show such behaviour. The initial acceleration takes

Figure 17. (continued)



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place during the first minute and consequently is not recorded. The linear interpolation of experi-mental data does not allow us to describe the actual initial trend (Figure 9(a)).

The initial system acceleration may take a few seconds when the mechanical response is stable, or

several minutes when instability occurs. The experimental results, obtained by Delage et al..,21

characterized by more accurate time recordings, confirm the initial axial acceleration, with reference

to all load increments. Therefore, with reference to the single axial-strain versus time trends, the

difference between stable and unstable mechanical responses is not qualitative but only quantitative.

The correct way to evaluate the instability occurrence is based on the analysis of the overall

mechanical behaviour during the load test.

In order to adapt Liapunovs definition of stability to granular assemblies, it may be useful to

interpret the different grain microconfigurations as points of a set, and to relate them to the points

defined in a space in which the co-ordinates are the state variables. This link is meaningful if, and

only if, a representative volume continues to exist. This is true if no strain localization takes place.

Consequently, drained strain localization is excluded a priori, because the considered phenomenonappears to be globally diffused and not localized.

With reference to granular continua, it may be useful to define as state variables, for instance, the

relative density (a scalar variable), and tensors ( ij) describing the directional characteristics of the

material. With regard to these, many authors27,28 have recently introduced different tensors, which

describe directional properties of the microstructure, which may be used as state variables.

Between the points defining the single microconfigurations, and those to which they correspond,

defined in the space of state variables, there is no one to one correspondence (Figure 18). A single

point in the state variable space corresponds to an enormous number of different microconfigurations.

Having theoretically introduced a one to one relationship between the starting set and the points

belonging to the state variable space, and since it is possible to make this space topological, using any

mathematical definition of distance, it now becomes possible to apply Liapunovs definition of

stability thermodynamically interpreted to granular assemblies:A granular system is defined as stable, when the distance between the starting and the finishing

point, defined in the state variable space, continuously decreases in direct proportion to the decrease

in the size of the load disturbance, of whatever kind it may be.

4.2. Configurational entropy

In order to describe how the system evolves, the configurational entropy 29 Scmay be introduced.

The configurational entropy is assumed to be a function of the probability Pof occurrence of a certain

Figure 18. Schematic correspondence between the set of microconfigurations and the hyper-space of the state variables



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state described by a pointA

i in the state variable space (section 4.1):Sci Sc Pi 1

The value ofPiis obtained by summing thepikover the set ofkmicroconfigurations corresponding to

the same point Ai, where pik are the single probabilities associated with the microconfigurations k:

Pi

Ni

k 1

pik 2

where Niis the number of microconfigurations belonging to the i set.

In the following section, a simplifying definition ofp and P is proposed. This is necessary because

of the complexity of the system. The aim consists of highlighting the variables upon which Scdepends, by disregarding a description of such a dependency.

4.2.1. Density of probability p. If we consider a fixed representative volume of sand, which is filled

with a great and variable number of grains, it is possible to affirm that the number of

microconfigurations that may be generated, depends only on the dimensions of the elemental

volume and on the geometrical characteristics of grains.

We can choose to define these different microstates geometrically, without any hypothesis as to

how the specimen is filled. In this case, the density of probability p, associated with each micro-

configuration, is the same. This ceases to be true when the gravity force is considered or the material

is assumed to be subjected to a certain state of stress. In fact, should this be the case the equilibrium

acts as a constraint and the density of probability p must be redefined.

Some microconfigurations miss any possibility of existing, because the microstructure must sustain

its own weight and the state of stress applied. Moreover, among the possible microconfigurations

some are characterized by high values ofp, others by lower ones.In order to define the value of p associated to a certain microconfiguration and to a particular

effective state or stress, i.e., to define the following dependency:

p p1

ij 3

it is possible to introduce the method described below.

We may assume the density of probability p associated with a certain microconfiguation (Figure

19), defined by the measure of the surrounding domain of i j*(which is defined in the effective state

of stress), held by the maximum hyper-spherical domain within which the microconfiguration

Figure 19. Hyperspace of the effective state of stress: domain containing the states of effective stress which may be staticallysustained by the microconfiguration considered



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considered may exist (that is where the equilibrium condition is statically satisfied). The system mayor may not be balanced, depending on the geometry and on the mechanical properties of grains.

4.2.2. Probability P. Corresponding to a certain sand and to a given effective state of stress ( i j*),

in section 4.2 a definition ofp has been introduced.

Phas been defined as the sum of the pvalues associated with all the microconfigurations belonging

to the same set. Once the set of micro-configuration is defined, from section 4.2.1 the value ofP may

be derived.

As we have introduced a relationship between the microconfigurations and the points belonging to

the state variable space, the sum previously defined is meaningful and a value ofP may be associated

with each point defined in the state variable space.

As observed in section 4.2.1, if the granular assembly is considered only from a geometrical point

of view ( i.e., disregarding the mechanical aspects) the density of probability p is the same for each

microconfiguration. However, it is important to observe that the number of elements which form eachset of microconfigurations, is variable. In particular, among the sets associated to the points char-

acterized by the same value of relative density Dr, the sets associated with random micro-

configurational distributions of grains (isotropic microconfigurations) are larger.

Consequently, if we consider the points characterized by the same value ofDr, we observe that the

probability P connected to the points characterized by isotropic microstates is greater. This is no

longer true when the material is loaded and an effective state of stress is applied.

4.3. Micro-structural evolution

4.3.1. Kinetic Energy. If we consider load-controlled tests and, in particular, one load-incrementand the subsequent time period during which the state of stress is kept constant, we may observe the

following:

If the current microstructure cannot statically exist under an increased state of stress (i.e., the

current microstructure cannot sustain the increased state of stress), the grains accelerate and a

system evolution takes place. As time passes during this process, each grain develops its own

momentum.

The introduction of the internal kinetic energy Ecallows us to produce a rough description of the

kinetic aspects of the dynamic evolution of the system. Ec is defined as the integral over the

representative volume of the overall kinetic energy of grains. This is strictly related to the specific

load disturbance and to the dissipated energy along contacts. In fact, it develops during microrear-

rangements and is dissipated by the interparticle frictional slidings. The internal kinetic energy

depends on the disturbances size and type but, in particular, on the vulnerability of the currentgranular fabric.

It may be interesting to clarify the relationship between Ecand the strain rates, which are recorded

during experimental tests by interpreting the loose sand triaxial specimen as a unique macroelement.

This relationship is difficult to determine because, a priori, the two quantities cannot be assumed to

coincide but, if no turbulence takes place within the specimen, the link becomes explicit. In fact, if we

assume the displacement field to be homogeneous within the specimen, from the boundary condition

it is possible to derive an internal kinematic description of the continuum.

On the other hand, if any turbulence takes place, homogeneity within the specimen is lost and the

boundary conditions are not sufficient to derive the internal displacement field.



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In the following, E

cwill be interpreted as a useful tool in describing the onset of turbulence.According to the authors, the instability takes place when the turbulence is not dumped, but itself

feeds.

4.3.2. State Variable evolution. In section 4.2 we have introduced the configurational entropy Scby statically describing the granular system. In fact, the definitions ofp and P introduced above were

related only to static quantities and the system, as previously described, was assumed to be

motionless. On the other hand, in the present section the microstructural evolution is taken into

consideration and the configurational entropy itself will allow us to describe such an evolution. This

is possible because the variable Sc has been defined as a monotonic increasing function of the

probability P. Consequently, the coincidence between the most probable state and the state which

most frequently happens is translated in the following condition on Sc:

In the state variable space, the granular system evolves towards the point characterized by thecurrent relative maximum value ofScwithin the infinitesimal surroundings of the initial state. Screpresents the variable which determines the way the granular system evolves, in the same way

as Ecdetermines the evolution itself. In fact, the system stops evolving when Ecgoes down to

zero.

4.3.3. Further remarks. The evolution of a granular system is necessarily associated with a strain

increment. On the other hand, a strain increment is not necessarily associated with a state variable

evolution. For instance, we may obtain considerable strain increments, even though the image point

of the internal state of the material does not change. This is possible if the strain increments are

caused by the evolution of the fabric within the same set of microstructures. This happens when the

material reaches the so-called steady state.

This means that the system, defined in the state variable space, does not recognize any path which

allows an increase in Scin the infinitesimal surroundings of the current state. On the contrary, up tothis condition, a state variable evolution takes place, but by increasing the stress level this evolution

takes place more slowly.

It is interesting to observe that, even if the strains associated with the steady state are considerable,

according to the thermodynamical definition introduced in section 4.1, the system remains stable. In

fact, according to the definition of steady state, the distance between the starting and the finishing

point, defined in the state variable space, associated with load disturbance, goes down to zero. Only

by analysing the phenomenon in the interior of the state variable space, is it possible to highlight a

clear separation between stable and unstable behaviour. In fact, by considering only the strain

variables, the mechanical behaviour of the material may be wrongly interpreted.

4.4. Instability analysisIn section 4.1 the occurrence of the instability has been described. In this paragraph the authors

propose a new approach that, thanks to the definition of the configurational entropy Sc previously

introduced, highlights the mechanical instabilities experimentally shown in section 3.

The new theoretical approach is based on the assumption that the density of probability p (defined

in section 4.2.1) depends on the kinetic variables describing the motion of each grain. In fact, during

the granular system evolution, the definition ofpcannot remain static, but must become dynamic and

thus the evolution of the system becomes dependent on the time factor.

In section 4.3 the evolution of the micro-structure has been taken into account, nevertheless this

has been assumed to be quasi-static. This means that the kinetic energy is not nil, but is small enough



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not to influence the mechanical behaviour of the material at all (i.e., to be negligible in determiningSc). In fact, in the definition ofp introduced in section 4.2.1, no kinematic variable appears.

The extension of this theoretical approach to real phenomena, implies disregarding any dynamic

effect on the mechanical behaviour of the material. Such an assumption is implicit in all the elas-

toplastic constitutive models but also in the elastoviscoplastic ones. In the latter the evolution of the

state variables takes place with time, but is not dependent on kinetic variables.

4.4.1. A simplified dynamic definition of p. When a load-controlled test is considered, as the load-

increment may be assumed to be impulsive, an accurate dynamic description of the system response

must be outlined. From a probabilistic point of view, this entails a dynamic description of the density

of probabilityp. In the following, the authors will assume that the dynamic description of the system

may be simplified and that the value ofp is a function only of the effective state of stress (section

4.2.1), of the micro-inertia and of the kinetic energyEc(section 4.3.1). This implies that the momenta

of grains are disregarded.The direct dependency of p on Ec may be physically interpreted by means of a very simple

hydraulic analogy. As happens in fluid flows, the increase in Ecmay cause turbulent motions in the

continuum and these are assumed to influence the mechanical behaviour of the material.

Then, we may write:

p p1

i j micro-inertiae p2 Ec 4

where p1is the extension ofp defined in equation (3) (section 4.2.1) to the dynamic case, while p2will be described below.

By means of equation (4), we assume that the density of probability p is a function of two

independent factors:

the equilibrium condition (dynamically adapted),

the possible onset of turbulence.

In the first term, the micro-inertiae allow the granular system to sustain dynamically some effective

states of stress excluded in static conditions. More simply, thanks to micro-inertiae in dynamic

conditions, the number of possible micro-configurations changes and the value of p1 (defined in

section 4.2.1) changes, too.

Instead, the second term p2 of equation (4) may be interpreted as a modulating function, which

takes into account the disorder induced by turbulence.p2 may be assumed to be a constant function of

unit value, when Ec is sufficiently small, in all other cases p2 is a weight function. This is char-

acterized by larger values corresponding to disordered microstructures and smaller values corre-

sponding to ordered microstructures.

4.4.2. Drained triaxial strain controlled tests. As the authors in the previous sections underlined

the possible importance of the Ecvariable in the description of the granular system evolution, it maybe interesting to discuss the possible strain-rate dependence of the materials mechanical behaviour.

Some experimental results, obtained by performing strain controlled standard triaxial compression

tests, are illustrated in Figure 20. The three curves are relative to the same type of test on specimens

characterized by the same initial relative density but these tests were performed by imposing different

strain rates.

The experimental curves show a clear dependency of the mechanical response on the strain rate.

Naturally, this dependency can neither be reproduced by means of a traditional elasto-plastic con-

stitutive model, nor by means of an elasto-viscoplastic one (for instance Perjna30). By using an elasto-

viscoplastic constitutive model and by increasing the strain rate, an initial more rigid behaviour



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would be obtained. Moreover, the value of the q limit would not be influenced by the value of strain

rate.

Such experimental results may be theoretically interpreted only by a global discussion of the

influence of the strain rate on the microstructural evolution.

First of all, we must assume that the kinetic energy Ecis constant, during constant strain rate tests.

This is possible because the granular system capability of dissipating energy allows us to exclude that

a quantity of energy is stored as grains vibrational energy. Therefore, we may assume that, also with

Figure 20. Strain controlled standard triaxial tests on saturated very loose Hostun sand specimen, performed at different strain

rates: (a) stressstrain behviour; (b) olumetric curve.



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reference to high values of strain rate, a monotonic dependence of the external value of E

con theinternal one exists. Moreover during strain controlled loadings, characterized by a constant axial

strain rate, we can disregard the values of the micro-inertial terms.

In Section 4.4.1 we have observed that, by changing Ec, p2 and p change too. In particular, by

increasing Ec, the probability ofp associated with more disordered microstructures increases. Con-

sequently, by increasing the strain rate (i.e., by increasing the Ec), it derives a different path in the

state variable space and a steady state characterized by a less ordered microstructure.

Generally, the system has two different possibilities to satisfy the equilibrium condition: to

improve the contact distribution along the direction of maximum stress (i.e., by inducing the

directional rearrangement of the microstructure-induced anisotropy) or by increasing the global

number of contacts (by increasing the relative density). The increase in Ec is assumed to partially

inhibit the induced anisotropy. Consequently, a denser microstructure is obtained.

In fact, in Figure 20(b), the experimental curves show a compaction influenced by the strain rate. In

particular, by increasing the strain rate, the compaction increases. Instead, by analysing the axialstress-strain curves of Figure 20(a) and Figure 21, we may observe that, by increasing the strain rate,

the maximum q value decreases. This seems to suggest us the following observation:

the increase in relative density is not large enough to balance the decrease in contact surfaces

along the direction of maximum stress.

These experimental results seem to imply a curious softening behaviour of the material linked to

the strain rate and not envisaged, by performing a constant strain rate test.

Such a strain rate dependency is analogous to that analysed by Rice and Ruina31 with reference to

the stability of fault slip.

From equation (4) of section 4.4.1. we may conclude that the increase in Eccauses a change in the

path followed by the granular system. This causes two different effects:

Figure 21. Straincontrolled tests: asymptotic values of q corresponding to different axial strain-rates



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the ultimate state of the system is denser the maximum q value is lower.

Both these effects may be justified, by assuming an ultimate state, described in the state variable

space which is less ordered and a slightly denser.

4.4.3. Instability occurrence. By simplifying the approach outlined in section 4.4.2, the system

evolution may be described schematically by means of two different variables:

Dr

where Dris the relative density (a state variable) and is not a state variable, but a scalar quantity

introduced for simplicity which may be explained as follows.

By introducing in the effective stress space a state variable tensor ij(section 4.1) which describes

the anisotropy of the material, we may define the scalar quantity as follows:

hk ij 5

may be interpreted as the eccentricity of ijwith reference to the current ij.

Therefore, each microstructure evolution may be schematically classified as in Figure 22. Point I

represents the current state of the material and each arrow defines the direction of possible evolution.

Type 1 is characterized by an increase in and a decrease inDr; type 2 by an increase in andDr;

and type 3 by a decrease in and an increase inDr.

During a standard triaxial compression test on loose sand specimens, when the mechanical

behaviour is stable, the evolution type 2 is followed by the system. On the other hand, according to

the authors, when the instability occurs, evolution type 3 takes place.

In order to highlight when and why this kind of evolution occurs and why it is unstable, it isnecessary to describe the system evolution during the time period following the load increment which

causes the unstable mechanical response.

We assume that the system initially is motionless. Subsequently, at the time instant t0 successive

to the load-increment, the micro-inertial terms may be not negligible, but Ecmay be assumed to be

nil. From equation (4) it derives that the current path is characterized by an increase in Drand in

(Path number 2). Then, Ec increases and the values of p associated with more disordered micro-

structures increase too. After a certain period of time, because of the disturbance caused by the Ecdeveloping, among the values of the density of probability p associated with the microconfigurations

belonging to the infinitesimal surroundings of the current microconfiguration, the maximum corre-

sponds to a less ordered internal fabric (path number 3).

Figure 22. System evolution: stable and unstable paths.



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If the micro-inertial terms are nil, within the quadrant 3, a line may be drawn separating thepossible and the impossible states (section 4.4.1). When these terms are not negligible, this limit

rotates clockwise. According to the authors, during the time period following the load increment

mentioned previously, point J, possibly thanks to the micro-inertial terms, becomes the most prob-

able.

When evolution type 3 is chosen by the system, even if the density increases, the increase in

density may be not enough to replace the contacts lost along the direction of the maximum stress

caused by the negative anisotropic evolution.

In this case, because of the decrease in the number of contacts along the direction of maximum

stress, both the micro-inertial term and the Ecmay increase. The system continues to accelerate and

the point image in the state variable space continuously changes.

According to the definition given in section 4.1, this behaviour may be defined as unstable, because

corresponding to a small load disturbance, a great distance between the starting and the finishing

points (defined in the state variable space) is obtained. The phenomenon is self-feeding, because thepath causes an acceleration and an increase in Ec automatically. The collapses may occur slowly or

quickly, but the mechanism does not change.

During the collapse, when the microstructure becomes isotropic enough, the increase in relative

density (i.e., the increase in number of contacts) allows the system to decelerate. Ecdecreases and,

consequently, the phenomenon stops. In this manner, we may assume that the instabilities experi-

mentally shown are caused by the dependency of the mechanical behaviour of loose sand on the

kinetic energy.

4.4.4. Further remarks. In section 4.4.2 the increase in the relative densityDrassociated with the

instability phenomenon was analysed. As observed experimentally (section 3.3), if a loose sand

specimen is saturated, a rapid increase in pore pressure takes place.

The presence of water during collapse causes a decrease in the effective mean pressure andconsequently an increase in the effective stress level. This combines with the dynamic factor

described above in determining the unstable mechanical behaviour of the granular system.

When the liquefaction phenomenon is considered with reference to boundary value problems, the

propagation of the instability must be taken into account. Therefore, when the propagation of this

type of instability in saturated soils is analysed, it is important to consider both the mechanical factors

previously mentioned: the dynamic effect and the subsequent rapid increase in pore pressure.

5. CONCLUSIONS

Sudden and unexpected collapses, taking place during experimental drained load controlled triaxial

tests on loose Hostun sand specimens, were shown.

From an experimental point of view, it was underlined that these instabilities are not necessarilylinked to the presence of water within the sand specimens. In fact, these collapses may occur whether

saturated or dry loose sand specimens are tested.

The instability considered appears to be strictly linked to the load-control, i.e., it may be inter-

preted as an unstable creep phenomenon.

In order to justify the experimental results shown above from a theoretical point of view, the

authors tried to highlight the factors influencing the mechanical behaviour of the material.

The mechanical response was associated with the microstructural evolution of the granular system

and, according to the authors, such an evolution may be described theoretically by means of the

configurational entropy Scand the kinetic energy Ec.



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Scwas interpreted as the variable determining the way in which the granular system evolves at themacrolevel, while Ec determines whether this evolution takes place. The configurational entropy

associated with a certain point in the state variable space was assumed to be dependent on the

effective state of stress and of the current kinetic energy Ec.

In fact, the density of probability p at the microlevel, introduced in order to define the config-

urational entropy Scat macrolevel, was assumed to be a function of two independent probabilistic

constraints:

the equilibrium condition (dynamically adapted)

the energetic content.

The simplifying assumptions introduced allowed the authors to outline the dynamic aspects of the

unstable phenomenon, by paying particular attention to the influence of the energetic content of the

system on its own evolution.

In the light of these considerations, the incapability of reproducing the instability considered,shown by the elastoplastic and elastoviscoplastic constitutive models commonly implemented, may

be justified.

In fact, these constitutive relationships assume the process of fabric evolution to be quasi-static.

This appears to be unrealistic with reference to the phenomenon considered and, in this paper, the

authors have tried to introduce a new way of defining the rules which govern such an evolution. The

whole theoretical interpretation has been developed by analysing laboratory experimental test results.

Any extension of this approach, in order to interpret boundary value problems (spontaneous lique-

factions, flow slides), should be coupled to a more accurate analysis of the phenomenon propagation,

both in dry and saturated continua.

ACKNOWLEDGEMENTS

This research was conducted within the framework of Project 2, localization phenomena in geo-

mechanics, of the A.L.E.R.T. Geomaterials Programme, funded by the E.U. (Human Capital and

Mobility). Financial support from Italian C.N.R. and M.U.R.S.T. is also gratefully acknowledged.

Moreover, the authors would like to acknowledge Prof. R. Nova for his helpful and precious

support, Prof. I. Vardoulakis for his hints, Ing. F. Calvetti for pleasant discussions and Dr. S. Losasso

for his careful linguistic observations.

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