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Relationship between dilatancy, stresses and plastic dissipation in a granular material with rigid grains
Pierre Evesque, Christian Stefani
To cite this version:
Pierre Evesque, Christian Stefani. Relationship between dilatancy, stresses and plastic dissipation in a granular material with rigid grains. Journal de Physique II, EDP Sciences, 1991, 1 (11), pp.1337-1347.
�10.1051/jp2:1991143�. �jpa-00247595�
J Phys. II France 1(1991) 1337-1347 NOVEMBER 1991, PAGE1337
Classification
Physics Absttacti 46.10 47.20
Shom Communication
Relationship between dilatancy, stresses and plastic dissipation
in a granular material with rigid grains
Pierre Evesque(~) and Christian Stefani(2)
(~) I-aboratoire de Mdcanique: Sols-Structures-Mat£riaux, Ecole Centrale Paris, 92295 Chatenay- Malabry Cedex, France
(~ I-aboratoire Central des Ponts et Chaussdes, Antenne d'orly, 58 boulevard Lefebvre, 75732 Paris Cedex 15, France
(Received 19 November 199fl revbed 20 June 1991, accepted5 Septe>nber1991)
R4sum4. En utilisant un postulat de reproductibilit£ des essais triaxiaux et une relation liant
l'4nergie dissipde, la dilatance fi et [es contraintes impos£es a un £chantillon, on d4montre qu'un
mat6riau granulaire ne peut que se contracter apr~s avoir subi une contrainte de confinement isotrope (I.e. q
= 0), que la rupture spontande a lieu aprds un maximum de dilatance, qu'il existe un dtat caract6ristique (au sons de Luong et Habib) et qu'il existe un stat "critique" (au sens de Schofield et Wroth). Nous donnons deplus une mdthode pour estimer la dissipation .plastique durant un essai
triaxial h partir des r6sultats exp6rimentaux.
Abstract. By considering a drained cohesionless granular sample made up of rigid grains and
submitted to a triaxial test, we derive an equation relating the dilatancy K, the deviatoric stress q and the confining pressure p to the energy losses Dp~~i;c due to plastic yielding. We demonstrate that the system is contracting (h' < 0) at q = 0, when q is increasing and that spontaneous uncontrolled
yielding begins occurring when dilatancy fi is maximum. We also demonstrate the existence of the characteristic state introduced byLuong and Habib and the existence of the critical state of Schofield and Wroth. Finally, we give a method to determine the plastic losses during a triaxial cell test using
the experimental data.
Introduction.
It seems that the physics of "sandpiles" has been attracting a great amount of work recently.
Most of these theoretical [1-3] or experimental [4-fl papers concern the problem of grain
avalanches. Thin is due to the pioneering model of Bak, lbng and Wiesenfeld [1, 2] (BTW) which relates the avalanche process to a self-organized critical state and to the well-known problem of I/f noise generation. Unfortunately, most experimental studies [4-5] on sandpile avalanches have
proved the existence of a typical avalanche size. This is in contradiction with the BTW predictions
1338 JOURNAL DE PHYSIOUE II N° II
(only when the pile is small enough, one may observe avalanches obeying scaling laws and a I/f noise [6j; the latter point proves the existence of a finite size effect).
The reason for the discrepancy between the BTW model and the experimental results has been
described by one of us recently[7-9], and the sketch ofa coherent view wliich predicts both a quasi- periodic regime of intermittences, with quasi-periodic avalanches, and finite size effects which
exhibit self-organized criticality has been given [7-9]. The trick to obtain this unified scheme is to introduce the effect of dilatancy in the BT~V model of sandpile (dilatancy and its mechanical
properties will be defined later, see Eq. (2)).
It turns out that a classical procedure in soil mechanics to understand and quantity the effect of dilatancy is to study the soil behaviour through a triaxial test, as explained in [7-9]. It remains
then that the main reason for the misfit between the BTW prediction and the experiments has
come from the ignorance of the real and wellknown properties of sand.
The first purpose of this paper b to introduce to the non-specialist reader the general me-
chanical properties of sand, (or non-cohesive granular material), by describing typical triaxial cell results. The main purpose here is to give a better understanding of the deformation and dis-
sipation processes occuring in a pile; this will be done by deriving some relationships between diffement quantities measured with this set-up. This will allow a tentative theoretical explanation
of the main features exhibited by the triaxial data within very simple arguments and assumptions;
the main assumption is the form of the energy dissipation (Eq. (6)). We hope this will make soil behaviour more understandable.
It is worth noting that most of these connections have been already observed experimentally and are basic experimental evidences in soil mechanics, although we have not found these equations
derived in literature. We also derive new relationships which may be used as a test for the validity
of our approach.
AS a final remark, we want to warn the non-specialist reader that calculations based on triaxial cell data are not the only technique used by soil mechanics specialists to predict soil behaviour;
for instance one may also use the yield design theory [10] which analyses the limit of stability and which determines the most probable way of failure. However, the great advantage of a method based on the triaxial cell technique is to assume a continuous evolution of the media so that the small macroscopic changes which are recorded define eventually yieldings. In other words, this
"triaxial-test" method allows and even assumes intrinsically that finite element calculations are
possible; more details about this point can be found in [11], and recent references on these two kinds of approaches.
Synopsis of the triaxial test.
Soil-mechanics specialists are commonly using triaxial tests [12-17] in order to characterize the mechanical behaviour of granular materials, of clays or even of rocks. A triaxial apparatus is composed of a cylindrical flexible membrane of radius R and height h closed at its bottom and top by two pistons (volume v = wR~h). This membrane contains the material to be studied (I.e.
the granular medium here) which is immersed in a liquid at a given pressure p and is submitted to a vertical overload q, called the deviatoric stress (see Fig. I).
The basic principle of the experiment is to submit the sample to a given set of successive stress- strain transformations and to record the material response. For instance, one may submit the
sample to cyclic deviatoric loadings or to cyclic compressions, but the simplest tests consist in
applying a monotonous one-way path such as a single compression (p = constant, q increases), a single oedometric compression(R = constant h decreases byincreasing continuously the overload
q, which in turn implies that p increases), and so on. The mechanical state at a given stage of
q
Liquid
..i~~
v~
Fig. 1. Triaxial test: a granular material is contained in a flexible cylindrical membrane closed at its bot- tom and top by txo pistons. The membrane is immersed in a liquid at pressure p and a vertical additionnal stress q is applied to ihe top piston. q is called the deviatoric stress. The test consists in measuring simul- taneously the variations of height 6h
= ho h, of volume iv
= uo u, of q and of p for a given series of mechanical transformations.
transformations is defined by a set of four experimental data: the liquid pressure p, the deviatoric
stress q, the volume change bu
= vo ii and the height decrease bh
= ho h. (Note that volume- and height-decreases are positive as is the use in soil mechanics).
In this paper, we will be exclusively concemed by tests which consist of a single continuous decrease of the sample height, (d(bh)/dt > 0 where t is time). Furthermore, the rate d(bh)/dt
will be sufficiently small to keep the sample in a quasi-static equilibrium, at the limit of plasticity,
so that time will not directly enter the test response. Such h decreases may be performed under many different conditions as already described in the previous para graph. We will not consider any oedemetric test (I.e. implying a constant sample radius) in the following. We will exclusively
consider tests which keep a given relationship r between q and p constant and we will impose
to r to have the dimension of stress for sake of simplicity; for instance
r may be the confining
pressure p, (r = p); but soil mechanics specialists also use an other classical compression test which consists in keeping the mean stress
r = p + q/3 applied to the material constant during the
compression. Our theoretical approach may apply to this last test.
When one wants to investigate the mechanical properties of a consolidated soil, such as drained
clay or rocks, one puts directly in the triaxial cell a boring of the natural soil so that the initial conditions of the sample are the natural in-sint conditions. But when studying an unconsolidated soil, such as sand or any non-cohesive granular material, this cannot be done and the way of
settling the sample in the cell defines the initial conditions. We have then to discuss briefly the set of initial conditions which may be studied and the way of making a sample obeying such initial
conditions.
1340 JOURNAL DE PHYSIQUE II N°11
In the case of non-cohesive materails, one knows that piles may be built at different densities by dropping the granular medium from a hooper: the higher the fall from the hooper and/or the
smaller the hooper aperture, the denser the pile. Furthermore, a good sample homogeneity is obtained when spraying the sand all over the surface and building the pile in successive horizontal slices. One may also submit the sample to vibrations in order to bet very large densities. In sum, there are many different procedures of sample preparation which depend on the soil-mechanics
specialist. Nevertheless, it is well accepted by these specialists that homogeneous and isotropic piles may be built at different densities. They also know that an isotropic homogeneous medium
submitted to a uniforrn compression stays isotropic and homogeneous during the compression,
but it reaches a density slightly larger than its initial density; (however, we must point out that due to solid friction, it is only in the case of very loose packings that one observes a real increase of the
density of the packings during a uniform compression, when grains are rigid). So, we may admit that one can build an isotropic homogeneous sample at a given initial density (within a certain
range) and submitted to a uniform compression pot this will be the set of initial conditions we will consider in the following. On the other hand, it is accepted that triax%I tests on homogeneous granular samples are reproducible and that the mechanical responses depend only on the initial
density of the material and on its isotropy.
Typical experimental results on sand.
We consider here the triaxial test on non-cohesive granular media made of rigid grains, such as sand. The initial pile is assumed to be isotropic, homogeneous and at a given density do. So,
the initial conditions of the triaxial test are the pressure po, the deviatoric stress qo
= 0, the initial volume vo, the initial height ho and the initial density do.
The granular sample is then submitted to >an imposed decrease bh of the vertical height h ob- tained byvarying the overload stress q applied to the top piston and by keeping a stress function
r constant. bh (and then q) are varied slowly enough to keep the material in a quasi-static equi-
librium so that the system is kept in its limit of plasticity. The test consits in keeping the sample
as homogeneous as possible and in recording at the same time the volume change bv and the de- viatoric stress q as functions of the height variations bh. The results are commonly summed up by giving the two plots of q/r and bv/vo as functions of bh/ho. Typical results are shown in figure
2. They have been obtained when keeping the rate dh/dt of decrease of the height small and constant, and keeping p constant, (r = p here); the three sets of two curves of figure 2a and 2b
correspond to the same sand at different initial densities. We recall that positive bh and bv cor-
respond to height and volume decrease (according to soil-mechanics) so that the sample volumes
larger than the initial volume are located above the bh/ho axis as usually.
the variation of q/p versus bh/ho (Fig. 2) indicate that the deviatoric stress ratio q/p tends towards the same limit value M at large bh/ho (I.e. bh/ho > 0.3), independent of the initial density
of the packing. Furthermore, a systematic investigation has demonstrated that M is independent
of p [12-16j; it only depends on the nature of the material and may be considered as an intrinsic parameter which is related to the friction angle 4l [12-lfl; (sin ~a
= q/(2p + q) according to a
simple Mohr-Coulomb approach without cohesion, so sin 4l = Ml(2 + M). A typical value of the friction angle is 30°, and a typical value of M is 2 (# ranges between 25° -40° depending on the material). Soil-mechanics specialists say that the material h in its "critical" state when the material has yielded and when its q/p value has reached M. It also appears that the density dc of the pile in its critical state depends only on the pressure p [12-16]; dc is called the "critical" state
density.
However the way the material reaches the "critical" state depends on the initial density. Fig-
q/p la)
'- . M =
6hn~~'1'
o
16)
6hn~
_---
_-~
,
', ~,
$V/Vo
Fig. ~ Three typical results obtained with a triaxial cell on the same sand compacted at three different initial densities. The sand is continuously kept at the limit of plastic yielding. The pressure p is kept constant in this test. (-) dense sand; (- -) intermediate density; (.-.-.-.-.) loose packing. a) variations of the ratio q/p between the deviatoric stress and the horizontal pressure as functions of the relative variation of height 6h/ho. b) relative variations of 6u/uo as a function of the relative height variations 6h/ho. The curves
corresponding to the two dense media exhibit both some dilatancy effect (Ji > 0). The "characteristic state"
of Luong and Habib [13-14] is the state for which the volume is minimum (fi
= 0) and for which q/p
crosses the M horizontal line; the so-called "critical" state is obtained at large plastic yielding, which is also characterized bya h' = 0 dilatancy. The maximum value of 6h/ho in a contraction test is obviously 1.
ure 2 shows that when the initial sample is very losse, the ratio q/p increases monotonically and the material h always contracting. on the contrary, for an initial sufficiently dense packing, the deviatoric stress ratio q/p begin to increase, crosses the q/p = M line, reaches a maximum qm/p
which depends on the initial density and then decreases before reaching the M value asymptoti- cally. If one looks at the same time at variations of the volumetric stain iv/vo, one sees that the
material is first contracting until the q/p ratio crosses the q/p
= M line. It h then dilatant until it reaches and asymptotic value. Now looking at the density dc of the material at this final state,
one finds experimentally that it is independent of the initial densitydo [12-16]. The point at which
q/p crosses the M horizontal line has been called the "characteristic state" by Luong and Habib
[12-14].
ljpical experimental values are:
M = 2; # = 30°; M < qm/p < lit for Hostun sand: 1.4 < do < 1.7;
typical investigated pressure range: 30 kPa-10 MPa;
typical investigated range for bh/hoi 0.3;
typical investigated range for bv/voi 0.2;
bh/ho = 0.01 0.05 for the characteristic state;
At this stage, we want to emphasize two points. First, we have considered a material with rigid grains, such as sand. The elastic strength of this material is negligible so that most of the strain
imz JOURNAL DE PHYSIOUE II N°11
in Figure 2 is due to plastic irreversible yielding. It might occur, (and it does) that, under some
special test procedure, a finite stress ratio qo/Pis required to get a small yielding. Thus the curves in Figure 2 may contain a segment of the vertical axis from 0 to the ordinate qoIF- For instance,
this is observed when the pile is not isotropic initially or when the test is carried out after a first
loading-unloading cycle. This is not observed when the pile isisotropic initially.
The second point concerns the instability of a packing which has reached its qm/p maximum value and has been submitted to adq/dt
= constant triaxial test: a slight increase of q is no longer possible at this maximum of q and a microscopic stable d(bh)/ho response is impossible, so that the packing is unstable and a macroscopic motion, which is often a failure, occurs. This h the
rather large difference between triaxial tests carried out under strain- and stress-rate control.
Tentative theoretical description of triaxial test results on sand.
Let us now try to undersand these experimental features within a simple scheme. In order to do so, we calculate the mechanical work supplied to the material during an infinitely small transformation and compare it to the plastic dissipation. The two quantities should be equal, since the grains are rigid, so that the material does not gain any elastic energy during the transformation.
the supplied mechanical work is completely dissipated through friction. In a second stage, we
assume agiven form for the dissipation which allows us to derive the exhtence of the characteristic
and "critical" states.
For a sample submitted to a set (p, q) of stresses, the mechanical work dW which h supplied
to the sample h dW = IIR~(p + q)dh + 2IIRhpdR when the sample height is decreased from h to h dh and the sample volume from v to u du. AS the volume variation du is such that
dv/v = dh/h + 2dR/R, dW may be rewritten dW
= flR~h(p duIv + q dh/h) or as:
dW = (q I(p)flR~ dh (1)
if we define the dilatancy K byi
Ii= -(dv/u)/(dh/h) (2)
lst result: The ~ysteni is contracting iii < 0) at q = 0. We now remark that plastic dissipation
must be positive, (since it is an energy dissipation), that p is positive and that dh ispositive, (since
it is a height decrease). Equation (I) implies that Ii h negative when q = 0 or that dh
= 0 at
q = 0, Indeed, this is observed experimentally since the system isalways contracting at q = 0 for
isotropic pile, (seeFig. 2 and Refs. [12-14,17j), or dh is 0 at q = 0 for anisotropic ones, (the curve starting at qo in this case).
Let us write the plastic dissipation dwr in the following way:
dwr = DpiasiflR~ dh (3)
Equation (3) is only a
different way of writing dWr and defines Dplag. Dpjag is a coefficient which governs the energy losses due to plastic yielding, (typically solid friction). We will discuss possible expressions for this
coellicent Dplag later.
Let us now write the energy balance: dW = dwr since the grains are rigid and the medium is not elastic. Combining Equations (I) and (3) leads to:
q ii p = Dp~g j4)
