IOANNIS VARDOULAKISO and IOANNIS-ORESTIS …

SOILS AND FOUNDATIONS Vol. 45, No. 2, 61-76, Apr. 2005

Japanese Geotechnical Society

THE "STRESS-DILATANCY" HYPOTHESIS REVISITED:

SHEAR-BANDING RELATED INSTABILITIES

IOANNIS VARDOULAKISO and IOANNIS-ORESTIS GEORGOPOULOS11)

ABSTRACT

In this paper we will firstly consider the range of validity of Taylor's "stress-dilatancy" hypothesis. From this study we conclude that Taylor's rule is strictly valid only for coaxial deformations. For smooth non-coaxial deformations Taylor's rule must be modified according to the rule proposed by Gutierrez and Ishihara (2000), which includes a "non -coaxiality" correction factor. Strictly speaking the stress-dilatancy rule breaks down if in a process an abrupt

rotation of principal axes is imposed, as this is the case at the onset of localization. During this phase the corre-sponding stress function decreases monotonously, whereas the dilatancy oscillates between large negative and large

positive values. Finally it is shown that in globally undrained, displacement controlled tests this incipient post-failure contractancy to dilatancy oscillation should result in a dynamic instability.

Key words: dynamic instability, non-coaxiality, stress-dilatancy (IGC: D6/F6)

INTRODUCTION

In the "Fundamentals of Soil Mechanics", Taylor

(1948) on pages 345 to 347 discusses the basic idea of his famous "stress-dilatancy" theory (cf. Schofield, 1999), by using a few data from direct-shear tests on Ottawa standard sand. Taylor's intuitive assumption is based on the interpretation of the expression for the incremental work done by the stresses during a direct shear test,

(1)

In this expression a' and T are the normal (effective compressive) and shear stresses, acting on planes parallel to the shear-plane, and, uh, and uv are the relative horizontal and vertical displacements of the shear-band

boundaries (Fig. 1).Equation (1) can be written in the form of the rate of

work done by the stresses:

(2)

Equation (2), involves two functions of the horizontal displacement uh, which in turn acts as a history parameter for the considered loading process. These are the so-called stress-ratio or mobilized friction function

(3)

and the dilatancy function

i) Professor NTU Athens, Faculty of Applied Mathematics and Physics, Section of Mechanics, Laboratory of Geomaterials, Greece

([email protected]).ii)

PhD Student, Civil Engineer, ditto (I.Georgopoulos(a)mechan.ntua.gr).

The manuscript for this paper was received for review on April 14, 2004; approved on October 27, 2004.

Written discussions on this paper should be submitted before November 1, 2005 to the Japanese Geotechnical Society, 4-38-2, Sengoku,

Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

Fig. 1. Idealized kinematics and statics of the dilatant shear-band

61

62 VARDOULAKIS AND GEORGOPOULOS

(4)

In these definitions Os and vs are the mobilized Coulomb friction angle and the dilatancy angle in the shearing

plane, respectively.As far as the sand is concerned, we assume that all

elastic stored energy is negligible and that practically all the work done by the internal forces is dissipated,

(5)

On the basis of the 2nd Law of Thermodynamics this assumption yields that the first order incremental work is

positive, o W> 0, which in turn results to the statement that: for a granular material at any state of shearing , the mobilized dilatancy angle cannot be greater than the

corresponding mobilized friction angle,

(6)

Taylor's stress-dilatancy theory is based on the assump-

tion that the rate energy dissipation is constant , and independent of the state of hardening, i .e.

(7)

This hypothesis has led several authors to assume that , for consistency reasons, the angle c/) in Eq . (7) must be identified with the friction angle of the material at the critical state, Oes, where the sand presumably deforms isochorically and the mobilized friction angle is constant . While some authors found it tempting to investigate , whether or not 0, can be identified with the mobilized friction q5c, at the state, where the deformation turns from contractant to dilatant.

Following Taylor's constant dissipation hypothesis , Eq. (7), we deduce from Eqs. (2) and (5) that frictional and dilatant sand behaves from the standpoint of energy dissipation, like a purely frictional Coulomb material , that deforms isochorically and possesses a constant fric-tion coefficient (Fig. 2),

(8)

In writing Eq. (8) in the form

(9)

we retrieve Taylor's decomposition of shearing strength of sands into pure "frictional resistance among grains" and into a resistance due to "interlocking" of grains. This interpretation is an attempt for a plausible micro-mechanical interpretation of the strength of sand. It should be taken into note that Rowe (1962, 1971) and de Joselin de Jong (1976) have used the same concept as above and they have called the parameter 0, a "true" angle of friction and identified it to the inter-particle friction coefficient (/),. We should notice however that this hypothesis is not verified by distinct element simulation with idealized cylindrical or spherical particles (cf . Calvetti et al., 2003).

By using the definitions for the stress and dilatancy functions: from Eqs. (3) and (4) we may write Eq . (9) in the following form

(10)

where we have put

(11)

Equation (10) yields to a plausible phenomenological interpretation of Taylor's hypothesis: Eq . (11) is the simplest generalization of the normality condition of the classical flow-theory of plasticity , which is included in it, for the degenerate case of a zero effective friction coefficient,

(12)

In the literature of Plasticity Theory it is customary to call the flow-rule, Eq. (12), an associated flow-rule , whereas its simple generalization , Eq. (10), a non-associ-ated flow-rule. At any rate we observe that Taylor's rule

provides a simple way to relate the normal vectors to the plastic potential- and the yield-surface by assuming that they produce in stress space always the same angle , which, due to the inequality Eq. (6) , cannot be zero (Fig. 3). At this point we should highlight that the adop-tion of the non-associated flow rule for the constitutive law causes non-symmetry of stress-strain stiffness matrix . This non-symmetry becomes a serious issue when bound-ary value problems are actually solved. However as can be seen from the above demonstrations , non-associativi-ty, as expressed by Eq. (10), is the result of simplifying the assumption of a straight yield locus (Fig . 3).

Several decades after Taylor's partially unnoticed remark, the same idea can be found in a number of

publications (Rowe, 1962, 1971; King and Duickin, 1970;

Fig. 2. Stresses and velocities in ideal direct shear of: a) a frictional-dilatant granular material like dry sand and b) the equivalent in energy-dissipa-

tion incompressible solid according to Taylor's theory

STRESS-DILATANCY 63

de Josselin de Jong, 1976; Dietrich, 1976; Bolton, 1986; Schanz and Vermeer, 1996; Wan and Guo, 1999; Li and Defalias, 2000). The majority of these and other similar studies refer mainly to the behavior of sand tested under triaxial conditions, where the principal axes of the stress-and of the rate of deformation tensor are fixed, coincid-ing with both the geometric axes of the tested specimens. Under these conditions Taylor's hypothesis is found to be basically valid, and most of this type of work focuses in addressing the issue of critical state and the realistic determination of the critical frictional parameter as a function of the voids ratio of the sand and the isotropic effective stress.

As it can be seen from Fig. 4, the evaluation of Taylor's original data does not support the simple rule, Eq. (10), but rather a modification of it in the form of:

(13)

with a slope c =0.5.As we will observe in this paper this modification of

Taylor's original stress-dilatancy equation, Eq. (13), is explained theoretically by the fact that, unlike triaxial

experiments, direct shear tests involve rotations of the

principal axes of the stress and of the strain-rate. The importance of considering correctly the effect of principal axes rotations has been recently demonstrated in a paper by Gutierrez and Ishihara (2000), who proposed a modification of Taylor's rule of the form of Eq. (13). In the next sections we re-derive and we investigate the range of validity of the modified Taylor-Gutierrez-Ishihara stress dilatancy rule, Eq. (13).

NON-COAXIAL SOIL PLASTICITY

The "Coaxial" Flow-ruleIn classical flow theory of Plasticity, the plastic rate of

deformation tensor is given by a flow-rule, which is usually defined through a set of isotropic tensor functions of the stress tensor and of a suitably chosen non-decreas-ing plastic hardening parameter, W= I W,

(14)

where

(15)

and

(16)

I1σ,J2s,J3s in the argument list of the functions Qv( •E) are

the 1st invariant of the stress tensor and the 2nd and 3rd invariant of the stress deviator, respectively.

In view of the flow-rule, Eq. (14), we recall first that the

principal directions of a tensor are determined by its devi-ator. The stress tensor aij, its deviator si, as well as its 2nd

power have the same principal axes. Thus from the flow-rule, Eq. (14) follows that the stress tensors and the

plastic rate of deformation tensor have the same principal axes. Assuming that the background elastic behavior is isotropic, we conclude that both the elastic rate of defor-mation /-); and the total rate of deformation Di; = + Dr; have the same principal axes as the stress tensor au. We may call such a set of tensors faii, sii, Di;, Diej,Dpij,...}acoaxial set and the corresponding flow-rule a

coaxial flow-rule. In order to further illustrate the proper-

Fig. 3. Visualization of the Taylor's flow-rule in direct shear by using the terminology of plasticity theory

Fig. 4. Evaluation of the stress-dilatancy relationship for the direct

shear test on the Ottawa-standard sand shown in Fig. 1 (data repro-

duced from Taylor, 1948)


ties of the coaxial flow-rule Eq. (14), we evaluate this flow-rule in the coordinate system 0(xli) of the common

principal axes of the coaxial set,

(17)

where

(18)

Thus for such a coaxial flow-rule the shear components of

the rate of deformation tensor in the coordinate system

of the common principal axes are purely elastic,

(19)

To our knowledge Professor Andrew Palmer (1973) was the first point out clearly to the Soil Mechanics communi-ty the deficiencies and shortcomings of classical flow theories in describing the behavior of real solids. Within the realm of Soil Mechanics and based on the experimen-tal evidence, the validity of this theoretical result, Eq. (19), has been questioned also by the authors in a study concerning the role of the so-called out-of-axes shear moduli on shear banding (Vardoulakis , 1980; Desrues and Chambon, 2002). At this point we should comment that presently a variety of constitutive models for soils have been proposed, which aim at a more realistic description of soil behavior. A thorough review of the corresponding literature and in particular the extensions of the classical plasticity concepts can be found in a recent paper by Hashiguchi et al . (2002). In this paper the idea of reverse plasticity (i .e. plastic deformation inside the yield surface) is considered in detail.

Plane Deviatoric Stress PathsIn order to investigate directly the validity of the co-

axial flow-rule we must perform experiments that allow for controllable stress principal axes rotations . For this reason let us consider an ideal test , where we realize the following deformation history of a material element: Starting from an isotropic, compressive state of stressC(0),

(20)

the soil element is subjected to a plane deviatoric stress-

path C"•¨C(t). The stress in C(t) is defined as follows

(Fig. 5):

[σ(t)ij]=[σ(0)ij)]+[Sij(t)]

(21)

where

(22)

This state of stress can sufficiently be well realized in the hollow-cylinder, torsional apparatus (Pradel et al., 1990) .

We remark that for the considered plane-deviatoric stress paths, the 2nd deviatoric stress invariant is given as

(23)

whereas the 3rd deviatoric stress invariant vanishes

identically,

(24)

We consider now, as shown in Fig. 6, the stress-path of

the transition C(0)•¨ C(t) in the particular deviatoric planel

O(X,Y)with the axes

(25)

For proportional loading the stress-path is represented in this plane by the straight line (OA). For this reason we call the ray (OA), a radial stress path. The length of the radial stress-path (OA) coincides with the shearing stress intensity,

(26)

In the considered Ishihara deviatoric plane , the inclina-tion of the radial stress path (OA) with respect to the X-

1We may call this plane after Professor K. Ishihara, who has introduced it in Soil Mechanics through his studies on soil behavior for non -coaxial

stress- and strain-rate sets.

Fig. 5. Plane-deviatoric state of stress

STRESS-DILATANCY 65

axis is denoted here by 20,

(27)

As it can be seen in the Mohr plane (Fig. 7), the angle 0 coincides with the angle of the major principal stressσi(σ2≦ σ3≦ σ1<0),with respect to the x1-axis of the

underlying Cartesian coordinate system in the physical

space

(28)

According to Budiansky (1959), for an infinitesimal

continuation C(t)-> C(t+•¢t) the deviation of direction of

the stress probe vector, d si, = zl t, from the direction of

the stress deviator si, is conveniently measured by the

following mixed-invariant function' (Fig. 6),

(29)

Let mi, be the unit tensor in direction of the stress

deviator,

(30)

In the vector sub-space of the pair {mij, sij } we define the tensor which is normal to si, as:

(31)

We notice that in the pertinent literature we encounter in

particular the following loading programs:a) Radial loading, with

(32)

i.e. a continuation of the previous radial stress path.b) Tangential loading, with

(33)

i.e. a stress path that is perpendicular to the radial stress vector.

c) A mixed-mode or "oblique" loading program, which contains both radial and tangential components.

For any such a loading program, the angle between the stress deviator vector and the incremental plastic deviatoric strain vector, ZI t, in the deviatoric stress-space is measured by the following mixed-invariant function;

(34)

The relation between the Budiansky angles fis and fie, depends on the material, the stress-path and its continua-tion and is empirically restricted on the basis of suitable experimental programs.

Experimental EvidenceAs far as the mechanical behavior of sand is concerned,

Ishihara and co-workers have clearly falsified experimen-tally the co-axial flow-rule of ordinary Plasticity Theory. This evidence stems from hollow cylinder torsional experiments on Toyoura sand, which included radial loading experiments, pure stress-rotation experiments as

2In an updated Lagrangian formulation all stress rates appearing hereafter should be replaced by corresponding objective stress-rates like the

Jaumann-rate.

Fig. 6. Ishihara's stress diagram

Fig. 7. Mohr's circle of stress with pole


well as mixed-mode tests. These tests have shown that:

a) For radial stress paths (fis= 00) the plastic strain-

increment is practically co-axial with the stress (fisp=0)3 .

b) For pure stress rotation experiments (fls= 90•‹) the

plastic strain-increments are indeed generated and their

direction angle fis, is found to be in general a weakly

decreasing function of both the angle 0 and of the

shearing stress intensity T; Figs. 8 and 9.

These experimental results contradict clearly the

classical concept of a yield surface as a function of the

stress invariants of the form,

(35)

This is because, if we assume such a yield function, then

for the considered hollow cylinder torsional test

experiments we have that,

(36)

From Eqs. (35) and (36) we obtain that

(37)

In that case the trace of any possible yield surface in the considered (X, Y)-deviatoric sub-space will always be a circle, Eq. (37). Thus, according to the classical flow-theory of plasticity, pure stress-rotation experiments should correspond to "neutral loading" and should not

produce any plastic strains; c.f. Eq. (19). However the aforementioned experiments yield that in pure stress rotation plastic strains are generated along the circular stress path, T= const., and with that the contradiction between experiment and the classical flow-theory of

plasticity is evident.

The Modified Stress-dilatancy Rule Following these experimental observations one is

compelled to abandon the concept of a unique yield surface and of a coaxial flow-rule and to assume two mechanisms of plastic deformation. For example, according to Papamichos and Vardoulakis (1994) and Hashiguchi and Tsutsumi (2001) the deviatoric part of the plastic rate of deformation may be decomposed formally into two parts4:

(38)

In the decomposition Eq. (38), the tensor D( has the direction of mi; (i.e. it is coaxial with the stress deviator8ij), whereas-D'igpn is normal to it,

(39)

The magnitude of the coaxial part of the rate of deforma-

tion is computed as,

(40)

The non-coaxial part is then written as;

(41)

and with that we obtain the following expression for the magnitude of the non-coaxial part of the rate of deforma-

tion,

(42)

For the considered plane deviatoric stress paths in Fig. 10, the corresponding components of the plastic rate

of deformation in radial and circumferential direction are

depicted,

(43)

With the above notations the first order rate of dissipa-

tion becomes,

3The small deviations of the plastic strain increment vectors from the radial direction are attributed to the inherent anisotropy of the specimen ,

caused by the method used for its preparation (e.g. by pluviation).4In Eq . (38) we are using the notation of Papamichos and Vardoulakis, where the superscripts (pc) and (pn) to denote plastic "coaxial" and plastic

"normal", respectively. Notice that Hashiguchi and Tsutsumi are using the term "tangential" instead of "normal" .

Fig. 8. The evolution of cos le, in a pure stress rotation test (fis = 90•‹):

T= 35.4 kPa (Data taken from Gutierrez and Ishihara, 1991)

Fig. 9. The evolution of cos Pe, in a pure stress rotation test (fis = 90•‹):

20= 45•‹ (Data taken from Gutierez and Ishihara, 1991)

STRESS-DILATANCY 67

(44)

This is because from Eq. (41) we obtain that the first order dissipation of the normal component of the deviatoric plastic rate is vanishing.

(45)

We notice that due to Eq. (45) the contribution of the normal component of the deviatoric plastic rate of deformation to the dissipation rate is of second order,

(46)

According to Eq. (44) this 2nd order dissipation vanishes either if the stress-path is radial (fis = 0) or if the plastic deviatoric rate of deformation tensor is coaxial with the stress deviator (fie, 0). Thus in coaxial plasticity formu-lations this contribution to the second order plastic work is disregarded, leading to false predictions concerning loss of uniqueness and instability in several occasions.

Let

(47)

(48)

Then Eq. (44) yields the following expression for the first order rate of dissipation,

(49)

We underline that the first term in Eq. (49) corresponds to the dissipation due to the coaxial part of the plastic rate of deformation deviator (cf. Eq. (45)),

(50)

We define further the invariant stress-ratio and the

dilatancy functions as follows

(51)

(52)

We notice that in pure stress rotation tests the corre-

sponding invariant stress ratio function is constant

(53)

These definitions lead finally to the following simple

general expression for the rate of first order dissipation

(54)

where

(55)

Gutierrez and Ishihara (2000) arrived to a similar expres-sion and they called the factor c the "non-coaxiality" factor. In case of torsional tests with axes rotation these authors found that, within a good approximation

(56)

In particular for the tested Toyoura sand they found that

fc=o.48,where by using their notation,fc=ηc//3.

As mentioned in the introduction, Eqs. (55) and (56)

constitute a generalization of Taylor's original "stress-dilatancy" equation. This modification was proposed by Gutierrez and Ishihara (2000) and was found to hold for non-coaxial deformations of sand specimens, as these deformations were realized in the hollow-cylinder tor-sional apparatus.

THE COLLAPSING MICROSTRUCTURE

Goldscheider (1975) in true triaxial experiments with dry sand, reported that principal axes rotation is always accompanied with the tendency of the granular medium to contract. This result was indirectly validated later by Ishihara and Towhata (1983) in undrained torsional experiments, where they found that principal axes rotation is always accompanied with pore-pressure

production. Similar results were also reported by Vardoulakis and Graf (1985).

Biaxial experiments with shear band formation provide the following information: Equilibrium bifurcation is manifested in the form of a high porosity localization, as this can be seen for example by using X-ray radiography. This type of localized bifurcations occur at a state of maximum dilatancy for the homogeneously deformed specimen and are emerging out of small density disturb-ances inside the specimen or from small boundary imperfections. At their early stage these localized bifurca-tion have the form of a lens. In the biaxial test these "lenses" of high porosity do not intercept the whole

specimen. Porosity localizations as seen through X-ray radiography are interpreted as zones of extreme dilatan-cy. These dilatancy zones are sharp and progress rapidly inside the sand body very much like cracks. At some

Fig. 10. Plastic deviatoric strain rate in pure stress rotation paths


given state soon after a dominant porosity lens is first observable (e.g. by means of X-rays) its tips reach the boundaries of the specimen, leading to shear failure, meaning that the specimen at this point loses its integrity and becomes a mechanism. From this point the biaxial specimen reacts as a set of two rigid sand bodies which are sheared along a plane. Thus in the post-failure regime we may speak of a fully formed shear band which separates the specimen. The material behavior inside the shear-band is significantly affected by the sudden prin-cipal axes rotation that accompanies in general the transi-tion from a plane-strain rectilinear deformation prior to bifurcation to a planar direct shear deformation in the

post-failure phase.In order to explore the ramifications of these observa-

tions we consider here as an example the results from a biaxial test on medium dense Karlsruhe sand (Hettler and Vardoulakis, 1984; Vardoulakis and Graf, 1985). This a

uniformly grained sand, consisting of quartz grains with

a specific unit weight of 26.5 kN /m3. The grain-size

distribution of this sand is shown in Fig. 11. The

maximum and minimum porosity are 45% and 36%,

respectively. The initial dimensions of the specimen were

40 mm x 80 mm x 140 mm. During such a test the

following piston load, F(•} 0.2%), the axial displace-

ment, uv(•} 0.2%), the horizontal displacement of the

moving sled, uh( •} 0.2%), the lateral displacements at two

opposite phases of the specimen, u11(•} 0.2%) and u12

( •} 0.2%), and the confining pressure cc(•} 0.5%). As

shown in Figs. 12 and 13 during the post-failure phase of

such a test the Coulomb friction angle and the dilatancy

Fig. 11. Grain size distribution curve of the Karlsruhe sand (Hettler

and Vardoulakis, 1984)

Fig. 13. Typical evolution of the post-failure mobilized friction and

dilatancy angles as functions of shear displacement for a medium

dense sand (test BSE-5, after Vardoulakis and Graf, 1985) . The solid lines indicate averaging; the round symbols and the triangles

refer to the experimentally determined values of friction and

dilatancy angle, respectively

Fig. 12. The shear-band geometry in a biaxial test and the post-failure hodograph

STRESS-DILATANCY 69

angle of the sand participating in the deformation of the

shear-band are computed as follows

(57)

and are plotted against the total shear displacement,

(58)

According to these results, we have that during its early formation stage the shear-band is always contracting, although the sand was initially densely packed. In order to explore further the consequences from such complex behavior during the post-failure phase of shear-band deformation, let us consider the rate of deformation tensor in the 0(x' , y') system of the shear-band (Figs. 14 and 15),

(59)

In this expression v is the relative velocity of the two sliding rigid bodies, adjacent to the shear-band, and 2dB is the shear-band thickness. The shear-band thickness 2dB is estimated by optical method. For example, as can be seen in Fig. 16, this thickness can be measured from the extend of wrinkling that occurs locally at the rubber membrane. Notice that in the considered test the shear-band thickness was, 2dB ,,,, 4.3 mm.

In the coordinate system 0(x, y) of the principal axes

of initial stress we have the following matrix representa-

tions of the stress and of the rate of deformation tensors;

(60)

(61)

We assume that at states past the shear-band bifurcation

point elastic deformations are negligible as compared to plastic deformations, Dr; ,D,J, and that no stress discon-tinuities develop across the shear-band boundaries. The latter assumption is justified by the fact that in the consid-ered biaxial test the minimum stress is applied through a confining pressure on the vertical flexible boundaries of the specimen.

The angle of the principal axes of the rate of deforma-tion with respect to the axes of the initial stress is

Fig. 14. Shear-band in a biaxial test

Fig. 15. The Mohr circle of deformation at the incipient failure state

Fig. 16. Optical estimation of the shear-band thickness from the

wrinkling of the rubber membrane (Vardoulakis, 1989)


estimated as follows (Fig. 15)

(62)

In this expression OB is the measured inclination angle of the shear-band with respect to the major principal stressσ1(σ2<σ1<0) and θR is the so-called"Roscoe'"soiution;

(63)

As it can be seen from Figs. 13, 14 and 15 the initial

contraction of the shear-band is leading to a significant

and sudden rotation of the principal axes of plastic

strain-rate;e.g. forθB=570 and for Ys=-200 we get"

△ θ=220.

We emphasize that shear-banding would not involve

principal axes rotation only in this case, where the shear-band inclination would be equal to the Roscoe solution, i.e. if the shear-band direction coincided with the zero-extension direction of the strain-rate field at bifurcation. In that case we would get that the dilatancy angle of the material inside the shear-band to initially decrease very slowly, since shear-band bifurcation is occurring near

peak (i.e. at the state of maximum mobilized friction and dilatancy). However the experiment is disproving this theoretical possibility. It was first found experimentally by Arthur (1977) and it was validated later theoretically and experimentally by Vardoulakis (1980) that the shear-band inclination is approximately equal to the mean of the so-called "Coulomb" and "Roscoe" solutions5:

(64)

In these expressions 4B and WB are the Mohr-Coulomb mobilized friction and dilatancy angles of the material for homogeneous deformation at the point of bifurca-tion.

These observations allow us to conclude that shear banding will always be accompanied with some rotation of the principal axes of the plastic rate of deformation with respect to the principal axes of the stress tensor. Moreover, the experimental evidence is pointing towards a sequence of collapsing and restructuring of the granular microstructure, which is expressed macroscopically as an alternating behavior between volume contraction and dilatancy of probably stabilizing amplitude (Fig . 13).

In order to compute the rate of dissipation for simplici-ty we neglect here the effect of the intermediate stress and we are using the stress and strain-rate components in the

plane of deformation. According to these assumption we have

(65)

Thus the stress vector in the Ishihara plane lies always on the X-axis (Fig. 17). Similarly the rate of deformation is

given as

(66)

We observe that in the considered case, the Budiansky

angle for the deviatoric plastic rate of deformation in the

Ishihara plane is

(67)

With these assumptions and notations we can estimate

the rate of local dissipation as follows:

(68)

where we have introduced the following stress and dilatancy functions:

a) The mobilized Mohr-Coulomb friction coefficient ,

(69)

b) The mobilized Hansen-Lundgren dilatancy coefficient,

Fig. 17. Stress and deviatoric plastic-rate of deformation vector in the

Ishihara plane for post-failure deformation in biaxial testing

5The Coulomb or statical solution gives an upper bound for the inclination angle . It corresponds to the static failure criterion, namely that shear-

failure is taking place along planes of maximum stress obliquity angle (max (z/a)-planes). The Roscoe or kinematical solution gives a lower boundfor the inclination angle, and corresponds to the kinematic criterion , that failure lines are zero-extension lines.

STRESS-DILATANCY 71

(70)

We also emphasize that the non-coaxiality parameter in Eq. (68) takes the following form,

(71)

where in the considered test,

(72)

As it can be shown from Fig. 18 the deformation of the shear-band is initially strongly non-coaxial (0 < c< 1), turning gradually to a coaxial one (c 1) .

If we want to study the validity of the generalized Taylor rule, as this is introduced in the previous section, we must see closely the evolution of the dissipation function

(73)

From Fig. 19 we see that shear-banding starts with a catastrophic phase with a value for the dissipation, which is almost 2 times higher than the residual dissipation. This phase is associated with the sudden principal axes rotation (Fig. 15), which causes a collapse (contraction) of the granular fabric. Then the rate of the dissipation drops rapidly and assumes extremely small values. As it can be seen from Fig. 13, this phase is characterized by an enormous re-dilation of the granular fabric inside the

shear-band. Past this state and after some oscillations a state is reached, where the rate of dissipation is oscillating in a narrow band.

We notice in the considered setting the generalized Taylor rule Eq. (56) becomes,

(74)

In Fig. 20 we plotted the dilatancy function dm versus the modified stress-ratio function cfm, with the non-coaxiality

parameter according to Fig. 18. From this plot we obtain that the modified Taylor's rule, Eq. (74), holds only on the average and for

(75)

A plausible explanation for the oscillations in the shear-band dilatancy behavior could be drawn from micromechanical studies and simulations (Fig. 21). Indeed from the micromechanical point of view an im-

portant structure that appears to dominate localized deformation is the formation and collapse (buckling) of

grain columns, as this was demonstrated experimentally by Oda (1997) and theoretically by Satake (1998). These

Fig. 18. Evolution of the non-coaxiality factor during post-failure shear-band deformation

Fig. 19. The evolution of the normalized dissipation function

Fig. 20. Stress-ratio dilatancy relation


load-carrying columns belong to the so-called competent

grain fraction (Dietrich, 1976; Vardoulakis, 1989; Staron et al., 2001) and their current length reflects more or less to the current shear band thickness. We notice also that in order to account for these microstructural effects, grain rotation, the basic asymmetry of shear stress and micropolar effects must be accounted for (Miihlhaus and Vardoulakis, 1987; Bardet and Vardoulakis , 2000; Ehlers et al., 2003).

DYNAMIC STABILITY ANALYSIS OF UNDRAINEDSHEAR STRESS

In the previous section we demonstrated that biaxial tests on sand specimens indicate that in the post-failure regime the volumetric behavior of the sand inside the band is oscillating: At the point of equilibrium bifurca-tion the material is strongly dilatant, then at the point of incipient shear-band formation it turns suddenly to con-tractant in order to accommodate to the strong principal axes rotation of the rate of deformation tensor, which are associated with the formation of the shear band. This incipient contraction is followed by (oscillatory) dilation . In this section, we will show that this behavior grossly destabilizes shear-banding in case of rapid shearing of a water-saturated soil.

Let us assume an infinitely long shear-band , as shown in Fig. 22. Let the thickness of the shear-band be 2dB. By choosing the xl-axis to be parallel to the axis of the shear band, we assume that all field properties are independent of it (8/3x1E--_, 0).

Accordingly in the (x1, x2)-plane of deformation the components stress tensor are

(76)

The stress is decomposed into effective stress and pore

water pressure according to the operational definition of

Terzaghi,

(77)

The rate of deformation is given by

(78)

where v1) = vi(x,, t) are the components of the velocity of the solid phase. The total volumetric and shear-strain rate are,

(79)

The total rate of deformation is decomposed into an

elastic and a plastic part;

(80)

Let an and Tn > 0 be the normal effective stress and shear stress acting on planes parallel to the shear-band bounda-ries. Within the frame of elasto-plasticity theory in monotonous loading undrained stress paths we obtain the following evolution equations for the shear and the normal effective stress (cf. Vardoulakis , 1996a, 1996b),

(81)

where K and G are the elastic 2D compression and shear moduli of the soil skeleton; ,u and j3 are the mobilized-friction and dilatancy functions of the plastic shearing

strain intensity y P

(82)

h is the friction hardening modulus,

(83)

Fig. 21. A micromechanical model of the shear-band in granular

material: Columns of grains transmit the intergranular forces

constituting the competent fraction of the grains; these columns are

supported by the lesser loaded grains, called the frail grain fraction

(Vardoulakis and Sulem, 1995). The buckling of the granular columns results in the observed shear-band contraction. Softening

of the shear-band is partially due to the "roller-bearing" effect of rolling grains at shear-band boundaries

Fig. 22. Infinitely long layer under normal and shear stress

STRESS-DILATANCY 73

and

(84)

(85)

For a given set of material parameters, Eq. (84) can be integrated to yield the undrained stress path. From these evolution equations it follows that at a state CT, with x=

XT the shear stress assumes a local maximum, In = max!. As demonstrated experimentally by Han and Vardoulakis

(1991) in undrained biaxial tests on St. Peter sandstone sand the post-failure effective stress path shows such a local maximum for the shear stress for medium dense specimens, as indicated in Fig. 23 below. In Fig. 24 we

plotted the post-failure evolution of the friction and dilatancy functions for the considered globally undrained biaxial test. From this figure we obtain clearly that the oscillation of volumetric behavior inside the shear band in the post-failure regime is related to the dynamic stabil-ity since it allows for a transition from contractive

behavior, with zir>0, to dilatant behavior, with ;ft< 0. Since in the post-failure regime the mobilized friction is monotonously decreasing (x < 0) the critical condition, Eq. (84), is indeed met at the post-failure state CT of locally maximum deviator; past this state x-r < x < 0 Based on earlier papers by Rice (1975), Rudnicki (1984) and Vardoulakis (1985) the dynamic stability of the equilibri-um past this state CT was studied by Vardoulakis (1986) by using a simple 2D-plasticity model. This model results finally to the following set of constitutive equations for the effective stress rate

(86)

where

(87)

The governing equations of the considered problem are:1. Mass balance for the fluid phase as this is expressed

by the storage equation (Vardoulakis and Sulem, 1995):

(88)

where q, is the relative specific discharge vector; i.e. the

relative discharge of fluid with respect to the solid,

(89)

where v 2) is the velocity of the fluid.2. Balance of linear momentum for the solids-fluid

mixture and the fluid phase respectively:

(90)

In these expressions p1=(1-n)ps and p2=np, are the partial densities of the solid, and the aqueous phase, and

ρs and ρw are the densities of the solid and aqueous

constituent, respectively. The torcing term - fqi cor-responds to the Darcian fluid drag force acting on the skeleton phase.

As is shown by Vardoulakis (1986) a linear stability analysis of the above set of governing partial equations

yields finally to a 4th order differential equation for the dimensionless incremental volumetric strain Z = Z(x2)

(91)

with

(92)

and Z(iv), '(ii) are the fourth and second derivative of the

dimensionless incremental volumetric strain E with

respect to x2, respectively.

Fig. 23. Post-failure effective stress path for a medium dense St. Peter

sandstone sand specimen tested in the biaxial apparatus under

globally undrained conditions showing clearly the existence of a

post-failure peak deviator state CT (test DC-04 after Han and Vardoulakis, 1991)

Fig. 24. Post-failure evolution of the mobilized friction angle, the mobilized dilatancy angle and the (normalized) shear stress (test

DC-04 after Han and Vardoulakis, 1991)


Λ=det(Λij)

(93)

The parameter in the above expression is a small dimen-

sionless number which compares the hydraulic and dynamic properties of the shear-band

(94)

As an example for the material parameters listed in Table 1 we obtain the value 10-2.

In Eq. (93) f is the Lijapunov exponent of the instabil-ity: If Re (f) > 0, then the background equilibrium state is linearly unstable. This is because Re (f) > 0 implies that small perturbations out of the uniform deformation will grow exponentially in time. Thus Re (f) > 0 shows instability and the relation between the parameter , f and the actual behavior of the soil is reflected in Eq . 95, through the parameters x and XT, which in turn depend on the rate of frictional strain softening and the contrac-tant-to-dilatant transition. The latter allows for x-r to as-sume sufficiently small values, so that the critical condi-tion x = XI-, met at state CT, of maximum shear stress . It can be shown (Vardoulakis, 1986) that past the state of CT the instability grows exponentially as e' , where

(95)

(Fig. 25) and t* is the dimensionless time factor, such that

(96)

Here for the values listed in Table 1, the evolution of the Lijapunov exponent is plotted in Fig . 25. Notice that in this case the time scale given by Eq . (96) is, t* (447 sec -1)t, indicating the catastrophic nature of the evolu-tion of this instability.

SUMMARY OF MAIN RESULTS

From this study we conclude that Taylor's rule is strict-ly valid only for coaxial deformations. For smooth non-coaxial deformations, Taylor's rule must be modified by introducing a non-coaxiality correction factor as original-ly proposed by Gutierrez and Ishihara . However if in a deformation process an abrupt rotation of principal axes is imposed, as this is the case at the onset of localization , then the Gutierrez-Ishihara rule breaks down . During this transition phase the mobilized friction function f ff, decreases but little, whereas the dilatancy oscillates start-ing from a large negative value. It is also worth noticing that from Figs. 13 and 24, the dilatancy oscillations in biaxial tests do not seem to fade at large displacements . More importantly we observe that the mean value of the dilatancy angle assumes small positive values between 30 and 50. If we assume that the shear-band thickness is constant, this observation stands in contradiction to the critical-state concept, which in turn requires zero dilatan-cy. In order to remedy this we may assume that at large displacements a 'critical state' is reached but at the same time the shear-band boundaries are not stationary and move outwards. In other words the shear-band is eroding into the surrounding material. This shear-band thicken-ing will then explain the positive values of the 'apparent' dilatancy angle at large displacements (Otsuki , 1978; Waterson, 1986; Drescher, 1990).

Based on this observation it is shown here that in

globally undrained, displacement controlled tests this incipient post-failure contractancy to dilatancy oscilla-tion results in a dynamic instability . We should remark that, as shown in Vardoulakis (1996b) in a displacement-controlled ("slow") test, locally undrained shear-band-ing is rather impossible, since unloading of the outer region will supply the amount of fluid which is needed to accommodate the dilatancy of the shear-band .

The type of instability discussed here should be of

practical importance, since in most cases shear-band failure is driven rather by load control (Desrues and Georgopoulos, 2004). We expect in near future to be able to report results from an ongoing detailed experimental

Fig. 25. Evolution of the growth coefficient of the instability in the

post-failure (softening) regime and past the state CT of maximum deviator

Table 1. Material properties related to a medium dense St. Peter

sandstone sand specimen tested in the biaxial apparatus (test DC-04

after Han and Vardoulakis, 1991)

STRESS-DILATANCY 75

and theoretical study where these predictions will bc examined in more detail.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the EU project Degradation and Instabilities in Geomaterials with Appli-cation to Hazard Mitigation (DIGA) in the framework of the Human Potential Program, Research Training Networks (HPRN-CT-2002-00220) while the first author would like to acknowledge the Isaac Newton Institute for Mathematical Sciences for its support during his stay in September 2003 within the programmes "Flow Regimes, Transitions and Segregation in Granular and Particle-laden Flows" and "Flow and Failure of Dense Fluid-Infiltrated Granular Materials".

NOTATION

δW:incremental work

τ: shear stress

σ':normal effeCtiVe StreSS

uh: relative horizontal displacement

uv: relative vertical displacement

φs:mobilized Coulomb friction angle

ψs:mobilized dilatancy angle

δD: work dissipated by internal forces

φeq: eqUiValent friction angle

φcs: In0bilized friction. angle at critical state

φcv:mobilized friction angle at zero volumetric strain state

ds: dilatancy function

C: coefficient correction for Taylor,s dilatancy rule

Dij:deformation tensor

Dij: deforlnation rate tensor

Deij:elastic deformation tensor

De ij:elastic deformation rate tensor

Dpij: plastic deformation tensor

Dpij:Plastic deformation rate tens0r

ψ:non-decreasing plastic hardening parameter

Q: function

σij: StresS tenSor

Sij:deviatOric StreSS tenSOr

Sij:second power of the deviatoric stress tensor(Sij=s2ij)

δij:Kroneker symbol

I1σ: lst invariant of stress tensor

J2s: 2nd invariant of deviatoric stress tensor

J3s: 3rd invarianもof deviatoric stress tensor

T: shear stress intensity

θ:angle between major principal stress and x1-axis

βs: angle between normal unit vector in direction ofthe stress devi-

ator(mij)and stress rate deviatoric tensor(3ij)

mij: unit vector in direction of the stress deviator(sij)

nij: unit vector normal to the stress deviator(sij)

λ: scalar multiplier

βe,: angle between the stress deviator vector(sij)and the incremen-

tal plastic deviatoric strain vector(△elj)

F: function of the yield surface

Dpcij:plastic c0axial deformation tensor

Dpnij:plastic nOrmal deformation tensor

p:mean pressure(=1/3σkk)

uP:plastic volumetric strain rate

gP: plastic shear strain rate

D: first order rate of energy dissipation

φσ: stress ratio( :7「/-P)

us: total shear displacement

γ: shear strain rate

θB: inclination angle of shear band

θR: "Roscoe"solution angle

φB:Mohr-Coulomb mobilized friction angle at bifurcation point

ψB:mobilized dilatancy angle at bifurcation point

δ: dissipation fuitction

μ(1)j: velocity of solid phase

ε: total volumetric Strain rate

γ: total shear strain rate

σ'n: normal effective stress acting on planes parallel to shear-band

boundaries

τn: shear stress acting on planes parallel to shear-band boundaries

K,G:2Delastic compression and shear moduli of soil skeleton

γP: plastic shearing strain intensity

h:friction hardening modulus(=4α/dγp)

α=K/G

μ=tanφs

β=tanψs

ν: Poisson's elastic ratio

X=(|σ'n|/O)h

XT=-qμ β

qi: relative specific discharge vector-fqi: Darcian fluid drag force

ρ1: partial density of solid phase

ρ2: partial density of aqlJ}eous phase

Gs: specific gravity

υk: kinematic viscosity of water

n: poroSlty

Cc:Terzaghi compressi0n index

Cs: Terzaghi swelIing index

k: physical permeability

2dB: shear-band thickness

ζ: dimensionless parameter

f: Lijapunov exponent

t*: dimeRsionless time factor

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