## Abstract

Progressive and catastrophic failure in soils has been commonly associated with the phenomenon of shear-band propagation. This is a challenging topic, both in terms of understanding and simulation. Most existing studies have focused on the propagation of planar shear bands. This paper is an attempt to analytically model the rate of progressive shear-band propagation in shear-blade tests, especially designed to produce curved shear bands in dense silty sand. The simplified analytical solution is based on fracture mechanics energy balance and on limiting equilibrium approaches. The analytical solution is validated against experimental data. Comparison shows that, despite the simplifications, the energy-balance approach is a useful tool in modelling the rate of non-planar shear-band propagation, both qualitatively and quantitatively.

## 1. Introduction

Soil failure occurs when mobilized shear stresses exceed the shear strength of the soil. In sensitive soils with post-peak strain-softening behaviour, failure is associated with localized formation and progressive propagation of shear bands. Understanding and modelling of progressive and catastrophic shear-band propagation in soils are not, however, easy tasks. Discontinuities, moving boundaries and scale effects are just some factors complicating the analysis. It is not surprising, therefore, that numerical simulations of the phenomenon of shear-band propagation bear a number of intrinsic problems. The major problem is mesh dependency: the shear-band width appears to be at least the size of the mesh element. Higher order numerical formulations (such as Cosserat continuum or gradient elasto-plasticity) or introduction of interfaces along the shear-band propagation path claim to solve the mesh-dependency problem. In reality, however, they rely on a certain ‘internal length’ or ‘material length’, which is just another way to sneak in the width of the shear band. And this still does not guarantee that the propagation rate is modelled correctly.

Another problem of these simulations is that they make it difficult to understand what the true mechanical and physical mechanisms are behind shear-band propagation (and not just mathematical and numerical conditions necessary for it to take place).

An alternative to numerical modelling could be a simple analytical approach, based on clear mechanical principles and, naturally, free of the mesh-dependency problems. It could facilitate a better understanding of the phenomenon and serve as benchmarks for numerical simulations. In particular, Puzrin & Germanovich (2005) proposed to extend the fracture mechanics energy-balance approach of Palmer & Rice (1973) to model catastrophic shear-band propagation in an infinite slope in granular materials. Although the model is one dimensional, its experimental validation and numerical simulations are difficult owing to the catastrophic nature of the shear-band propagation, which is a dynamic problem. Nevertheless, this simple model helped to explain both quantitatively and qualitatively some peculiar phenomena related to evolution of Tsunamigenic landslides (Puzrin *et al.* 2004; Puzrin & Germanovich 2005). A more detailed and comprehensive literature review on this topic can be found in Puzrin *et al.* (2004).

In general, everything based on energy balance makes an impression of a fundamental approach, and it is relatively easy to forget that there are many rather restrictive assumptions involved. Needless to say, the ultimate proof is always the experimental validation. Unfortunately, for shear-band propagation, physical modelling is also a challenge: unstable catastrophic propagation takes place in a very short period of time, while stable progressive propagation has been observed only in very few experimental set-ups. Most existing experimental and numerical studies have focused on the propagation of planar shear bands (e.g. Vardoulakis *et al.* 1981; de Borst & Vermeer 1984; Tanaka & Sakai 1993; Saurer & Puzrin 2007), which are a predominant failure mechanism of shallow slides. Deeper slides, also called slumps, are associated with the formation of curved slip surfaces (e.g. Tappin *et al.* , 2001; Synolakis *et al.* 2002). Curvilinear shear bands have already been observed in cavity-inflation experiments, with particular focus on the state of stress needed for the shear-band formation, the shear-band orientation and the relation of surface instabilities to shear-band formation (Alsiny *et al.* 1992). Here, a novel test, the shear-blade test has been developed to simulate and study the rate of progressive propagation of curve-shaped shear bands in soils. This is the main focus of this paper.

The larger scope of this study is experimental validation of the energy-balance approach to shear-band propagation in soils. In particular, the purpose of the paper is to apply the energy-balance approach to the problem of a shear-blade test and obtain analytical solutions for cohesive and frictional-dilative material, with their initial validation against some experimental and numerical results.

In terms of the methodology, the purpose of the paper is not to build a comprehensive three-dimensional model of shear-band propagation in the shear-blade test, but to understand the fundamental mechanics of the phenomenon. This, in the authors’ view, justifies numerous simplifications and the elementary mechanics applied. Development of a more comprehensive model requires understanding of the basic mechanical principles, and the present study should be seen as a step in this direction.

## 2. Experimental programme

With the purpose of investigating the rate of shear-band propagation in frictional material, a novel test device, the shear-blade apparatus (figure 1), has been developed. The device allows for a shear band of approximately semi-circular shape to propagate progressively (in a stable manner), simulating the shape of a sliding surface in slumps. In this test, a blade is rotated, provoking shear bands to propagate in a curved shape from one tip of the blade towards the opposite one.

### (a) Test set-up

The device consists of a square box of 120×120 mm plan area and a height of 25 mm. A two-winged 80 mm long (*R*=40 mm) and 25 mm high (*t*=25 mm) shear blade is located in the centre of the box (figure 1). The rotational velocity of the blade is controlled by a step-motor, while the torque is measured continuously by a static torque sensor installed between the step-motor and the box.

The bottom plate of the box is made out of transparent acrylic glass (figure 1). The pressure on the stiff top plate can be varied, and is controlled by a pneumatic cylinder above the plate with an electronic pressure controller. For the observation of the surface of the material, tests are recorded using a progressive-scan charge-coupled device camera, with a maximum resolution of 1392×1040 pixels at the rate of two full frames per second.

### (b) Material properties

The experimental programme consisted of three shear-blade tests using dry silty sand with a mean grain size of *d*_{50}=0.06 mm (figure 2).

Standard-oedometer tests up to a vertical load of 200 kPa were performed. In figure 3, the constrained modulus *M* for loading is plotted against the vertical load only for stresses above 25 kPa (because of the frequent blocking of the oedometer loading plate at stresses smaller than 25 kPa). In addition, upper and lower bounds for these values have been generated using a regression curve of the form *M*=*a*(*σ*_{v})^{b}. The upper bound is given by *a*=189.73 and *b*=0.67, and for the lower bound, *a*=166.3 and *b*=0.62.

From direct shear tests at an initial vertical stress of 50 kPa, friction angles at peak and constant volume (critical state) have been determined as *φ*′_{p}=44^{°} and *φ*′_{cv}=38^{°}, respectively, where the latter has been confirmed by measurement of the angle of repose. Taking the angle of repose, which provides a rather high value for *φ*′_{cv}, seems justified for the case under study because of the very low normal stresses applied during the shear-blade test. The friction between the soil and both the plastic and glass boundaries has been determined as *φ*′_{s}=25^{°}, with no significant difference in friction and lubrication being observed. The samples for both the oedometer tests and the direct shear tests have been reconstituted by compaction of dry material, identically as in the shear-blade tests, up to a relative density of *D*_{r}=80–90%. A maximum dilation angle of 4.4^{°} has been measured in the direct shear tests, gradually dropping to zero as constant-volume shearing is reached. These values confirm the consistency of the results.

### (c) Sample preparation and testing procedure

The sample preparation was performed with the shear box facing upwards. The box was filled with four layers of dry soil, each layer being compacted by applying pressure via a compaction tool with a 6 cm^{2} contact area. Four rounds of compacting were applied to each layer, which allowed for the relative density of *D*_{r}=80–90% to be achieved. Radial marker lines were added onto the surface at the intervals of *π*/8 (rad) (figure 4*a*) and the box was covered with glass. After having applied a constant pressure of 15 kPa, which was maintained constant throughout the test, the apparatus was carefully turned upside down.

After these preparation steps, the shear blade in the sample was rotated at a constant velocity of 0.2 r.p.m., and the surface of the sample was filmed by the camera.

### (d) Test interpretation

The tests were interpreted as follows. After Vardoulakis *et al.* (1981), the shear-band tip represents a point where the shear strength in the band has dropped to the residual level. This is assumed to take place when the relative shear displacement has reached half of the shear-band width (which is about 20 mean grain sizes, i.e. 1.2 mm). After a certain rotation of the blade, on the passive side (in front of the blade), a circular shear band starts to propagate from the tips of the blade crossing one marker line after another (figure 4*b*). The progress of this shear band has been recorded and plotted as a function of the shear-blade rotation angle, corrected by the rotation of the material at the tip of the shear band. On the active side of the blade, i.e. behind the shear blade, a diffusive active failure zone was detected. The angle between the front of the blade and the active failure zone is defined as *η*_{LE}.

Results from the three experiments are plotted in figure 5. The angle between the blade and the shear-band tip *η* is plotted against the corrected rotation of the blade *α*_{0}, as defined in figure 6*b*.

Though no experimental parametric study has been carried out at this stage, the stability of the test results to small variations in the soil density and applied confining stresses has been confirmed.

## 3. Analytical modelling—cohesive material

For simplicity (and for completeness), we begin with an analytical solution for cohesive material. This implies that the shear strength of soil is independent of normal stresses and drops from its peak value *τ*_{p} to residual shear strength *τ*_{r} solely as a function of the relative displacement (figure 6*a*). The shear strength of *τ*_{s} at the top and bottom boundary plates is also independent of normal stresses.

### (a) Geometry and assumptions

Owing to the application mode of the pressure via a rigid plate, the real test set-up suggests a solution between generalized plane-stress and plane-strain boundary conditions. Derivation and validation of the plane-strain solution are given in Saurer (2009). Here, only the case of generalized plane-stress boundary conditions is derived. The following assumptions have been made:

— the shear band is assumed to be cylinder shaped;

— in the modelled test, we control the rotation

*α*_{0}of the blade of half length*R*and try to predict the corresponding angle*η*between the initial blade position and the shear-band tip in the material (figure 6*b*);— behind the blade, the active failure shear zone develops with a moving boundary, where angle

*η*_{LE}is the angle between the initial blade position and the front of the active failure zone (figure 6*b*);— the soil inside the sector bounded by the blade wing and the shear band is linear elastic with loading modulus

*E*and Poisson’s ratio*ν*. Although the soil does not behave purely linear elastic at larger strains, Young’s modulus has been selected to account for the elasto-plastic nature of the soil in an approximate manner. For clayey soils, this modulus is obtained from oedometer or triaxial tests during loading of normally consolidated samples. During unloading–reloading, another modulus should be used, defined from the tests on over-consolidated samples;— the soil outside this sector is rigid plastic;

— apart from the small end zone

*ω*; i.e. the process zone, where the softening takes place, the shear resistance*τ*along the shear band is equal to its residual value*τ*_{r}. Ahead of the tip of the shear band, as well as at any point outside the band, the maximal shear resistance is equal to its peak value*τ*_{p}. Within the end zones, called the process zones*ω*, the shear resistance*τ*decreases as a function of the relative displacement*δ*from its peak*τ*_{p}to its residual*τ*_{r}value (at the relative displacement*δ*_{r}, figure 6*a*);— the circumferential normal stress

*σ*_{θ}is assumed to be uniformly distributed along any radial line*θ*=const., with its value being equal to the average normal effective stress*σ*_{θ}acting on this plane; and— the shear strength at the boundary plates is

*τ*_{s}.

The initial pressure in the out-of-plane direction is *σ*_{z,0}=*p*_{v}. This generates a uniform initial stress state within the *r*–*θ* plane given by3.1where *K*_{0}=*ν*/(1−*ν*) is the earth pressure coefficient at rest. The secondary indices (zero) denote initial values.

### (b) Stresses and strains

Please note that owing to the rotational symmetry of the problem, only half of the model is considered for the derivations. The soil inside the sector bounded by the blade wing and the shear band is described by a one-dimensional elastic–plastic model with linear hardening that, in loading, is given by3.2
3.3and
3.4where *E* is Young’s modulus taken equal to the tangent elasto-plastic modulus of soil. Unloading is linear elastic governed by the unloading–reloading tangent modulus *E*_{ur}. However, because the soil inside the sector bounded by the blade wing and the shear band does not undergo any unloading during the tests, it can be assumed that its behaviour is linear elastic with elastic modulus *E* (the tangent elasto-plastic modulus of soil). Behind the shear band, the soil fails in the active mode.

From the initial stress state shown in equation (3.1), and since the generalized plane-stress condition implies that the stress in the out-of-plane direction does not change, it follows that:3.5Furthermore, because the average radial strain remains equal to zero owing to the confinement in the radial direction, we neglect radial strains caused by the rotation of the blade, so that from the linear elasto-plastic relationship, it follows that:3.6where *σ*_{θ}=*σ*_{θ,0}+Δ*σ*_{θ} and *σ*_{r}=*σ*_{r,0}+Δ*σ*_{r} are the circumferential and radial normal stresses, respectively, given by the initial value plus the increment owing to rotation. Note that initial stresses from equation (3.1) satisfy condition (3.6) automatically.

From condition (2.6), it follows that:3.7so that using condition (3.5), the circumferential and radial stress changes are related by3.8Therefore, using equations (3.5) and (3.6), the incremental elastic–plastic stress–strain relationship in a circumferential direction is given by3.9where the stiffness parameter *D* is defined as *D*=*E*/(1−*ν*^{2}) for generalized plane-stress conditions.

Moment equilibrium around the centre of an elementary wedge with opening angle d*θ* within the elastic part of the model (figure 7) can be written as3.10which can be simplified to3.11Note that *σ*_{r} and *σ*_{θ} are not principal stresses owing to the presence of the shear stresses *τ*_{rθ}=*τ*_{θr} acting on these planes. Owing to the kinematic constraints, however, these shear stresses are not likely to cause significant shear strains *γ*_{rθ}=*γ*_{θr} in the wedge body, and their contribution to the internal energy can be neglected.

Integration of equation (3.11) using the boundary condition *σ*_{θ}(0)=*p*_{0} results in the expression for the circumferential stress as a function of angle *θ*, normal stress at the blade *p*_{0} and the shear stresses at the boundaries of the elastic wedge3.12The circumferential displacement (i.e. rotation *α*(*θ*)) in the elastic part can be calculated by integration of the strains over the arc length of the corresponding sector. Using the assumption that the process zone is small compared with the total length of the shear band, we obtain the boundary condition *α*(*η*+*ω*)=*α*(*η*)=0. The rotation *α*(*θ*) is calculated using equations (3.9), (3.12) and (3.1),3.13For *θ*=0, we get the rotation of the blade as a function of the stress at the blade *p*_{0} and the length of the shear band *η*,3.14

### (c) Energy-balance propagation criterion

#### (i) Incremental propagation

Consider an incremental propagation of the shear band, which results in an increase of its angular length Δ*η*, at the normal stress at the blade *p*_{0} assumed to stay constant during this incremental propagation. This causes additional rotation of the shear blade Δ*α*_{0}, which is equal to the strain at the shear-band tip multiplied by this incremental propagation. This gives, using equation (3.9),3.15where denotes circumferential stress at the shear-band tip *θ*=*η*.

#### (ii) Work components

Initial propagation of the shear bands is modelled using the fracture mechanics energy-balance approach. This approach sets a condition for an incremental propagation: the incremental sum of work of external forces and released internal energy should be higher than the incremental sum of the elastically stored energy and the dissipated plastic work in the shear band, its process zone and along the boundaries.

Mathematically, this can be expressed as the following inequality:3.16where we consider the energy balance between the following five components:

— external work Δ

*W*_{e}done by the external force acting on the shear blade on the displacement of the shear blade;— internal elastic energy Δ

*W*_{i}stored or released owing to the changes of stresses in the soil inside the shear band;— plastic work Δ

*D*_{l}dissipated owing to residual friction within the shear band along the entire length of the shear band*l*;— plastic work Δ

*D*_{s}dissipated owing to friction between the soil wedge and the boundary plates; and— plastic work Δ

*D*_{ω}dissipated in the process zone*ω*of the shear band owing to the friction above residual.

Note that, owing to the assumption that for an incremental shear-band propagation, the stresses change only in the propagation zone between *η* and *η*+Δ*η*, the soil between the blade (*θ*=0) and the tip (*θ*=*η*) rotates as a rigid body by angle Δ*α*_{0}.

Using equation (3.15), the first term in equation (3.16), the external work, is given by the expression3.17which is the work owing to the pressure applied to the shear blade multiplied by its incremental rotation.

Because stresses and strains change only in the propagation zone between *η* and *η*+Δ*η*, change of internal elastic energy is limited to this area. With the help of equation (3.9), neglecting the contribution to the internal work of the shear stresses, which owing to the kinematic constraints are not likely to cause significant shear strains in the wedge body, we obtain3.18The sum of plastic work dissipated along the shear band and along the boundary plates can be calculated as3.19where the second equality is derived using equilibrium from equation (3.12).

Finally, the plastic work dissipated in the process zone (figure 6*a*) owing to the friction above residual is3.20where *r*_{s}=*τ*_{p}/*τ*_{r}; *α*_{r}=*δ*_{r}/*R* and *δ*_{r}=*δ*_{m}, assuming linear decrease of the shear strength.

#### (iii) Energy-balance criterion

Substitution of equations (3.17)–(3.20) into equation (3.16) gives, after some manipulation, the inequality3.21which represents the shear-band propagation criterion.

Substitution of equation (3.12) into equation (3.21) gives the condition for the shear-band propagation in a stress-controlled test,3.22Then, after substitution of equation (3.22), using *p*_{0}=*p*_{cr} into the equation of the rotation of the blade (equation (3.14)), the condition for the displacement-controlled test is given by3.23This relationship between the shear-band length and the shear-blade rotation can be written in a more compact way,3.24whereThe energy-balance criterion controls the shear-band propagation up to the state when the remaining sector between the tip of the shear band *η* and the current boundary of the active failure shear zone *η*_{LE} reaches the state of the limiting equilibrium. After that, further progressive and catastrophic propagation is driven by the limiting equilibrium criterion.

### (d) Limiting equilibrium approach

The failure will take place when the remaining sector between the tip of the shear band *η* and the current boundary of the active failure shear zone *η*_{LE} reaches the state of the limiting equilibrium. This will happen when the shear stresses along the shear-band path between the current tip of the shear band *η* and the current boundary of the active failure shear zone *η*_{LE} in figure 8 reach the peak strength of the material.

The stresses acting at the sector boundaries are: pressure at the boundary with the elastic region; the active earth pressure *p*_{a}=*σ*_{z,0}*K*_{a} at the boundary with the active failure zone; the peak strength of soil *τ*_{p} along the shear-band path; and the shear strength *τ*_{s} at the boundaries between the soil and the top and bottom plates. The limiting condition for the moment equilibrium can therefore be written as3.25where *K*_{a}=(*σ*_{z,0}−2*s*_{u})/*σ*_{z,0} denotes the active earth pressure coefficient. In cohesive normally consolidated material, the undrained shear strength can be assessed as *s*_{u}=0.25*σ*′_{v}, where, for the investigated case, *σ*_{v}′=*σ*_{z,0}, resulting in *K*_{a}=0.5.

Using equation (3.12) and defining the ratio , where ,we obtain the relation3.26which represents the criterion for the stress-controlled test. Substitution of equation (3.26) into equation (3.14) gives the solution for the displacement-controlled test3.27

## 4. Analytical modelling—frictional material

With the principal approaches to the shear-band propagation in the shear-blade test being established in §3 for a simpler case of a cohesive material, it is now possible to extend this method to a more general case of a frictional material, which would allow for comparison with the experimental results from §2. The main complications compared with the cohesive material are:

— dependency of the peak and residual shear strength on the normal stresses and

— dilation of the soil in the shear band.

### (a) Shear strength and dilation

Densely packed frictional material experiences dilation when sheared. According to Taylor (1948), the dependency between friction angle and dilation (figure 9*a*) can be calculated from4.1where *ψ* is the dilation and *φ*′_{cv} is the residual friction angle at constant volume (critical state), respectively, and the factor *K*_{d} lies in the range of 1≥*K*_{d}>0. Because of the assumption that the maximum volume is reached after a relative displacement of *δ*_{r} within the shear band, we can write a linear expression for the dilation angle *ψ*4.2for *δ* < *δ*_{r} and *ψ*=0 for *δ*≥*δ*_{r} (figure 9*b*).

The peak and residual shear strengths in the shear band depend on the radial stress *σ*_{r} and are calculated using equations (4.1) and (4.2) for *δ*=0 and *δ*=*δ*_{r}, respectively,4.3Owing to dilation, the cylinder-shaped shear band expands. The expansion of the shear band can be calculated by integration of the assumed linearly decreasing dilation angle over the relative shear displacement *δ*,4.4At the end of the dilation process, i.e. after the relative displacement of *δ*_{r}, the total radial displacement at the shear band is then4.5For simplicity, we assume that this expansion causes a homogeneous radial strain within the elastic sector bounded by the radius *R*. In reality, the total expansion of the shear band results in compression both inside and outside the sector, and will cause a smaller radial strain. This effect of the lack of constraint is taken care of by a factor *K*_{b}≤1,4.6

### (b) Assumptions

The following assumptions have been made:

— the shear band is assumed to be cylinder shaped;

— in the modelled test, we control the rotation

*α*_{0}of the blade of half length*R*and try to predict the corresponding angle*η*between the initial blade position and the shear-band tip in the material (figure 6*b*);— behind the blade, the active failure shear zone develops with a moving boundary, where angle

*η*_{LE}is the angle between the initial blade position and the front of the active failure zone (figure 6*b*);— the soil inside the sector bounded by the blade wing and the shear band is linear elastic with loading modulus

*E*and Poisson’s ratio*ν*. Like for cohesive material, Young’s modulus has been selected to account for the elasto-plastic nature of the soil in an approximate manner. In this study, Young’s modulus was obtained from the loading portion of the granular material in an oedometer test (see §2*b*and figure 3);— the soil outside this sector is rigid plastic;

— apart from the small end zone

*ω*, i.e. the process zone where the softening takes place, the shear resistance*τ*along the shear band is equal to its residual value*τ*_{r}. Ahead of the tip of the shear band, as well as at any point outside the band, the maximal shear resistance is equal to its peak value*τ*_{p}. Within the end zones, called the process zones*ω*, the shear resistance*τ*decreases as a function of the relative displacement*δ*from its peak*τ*_{p}to its residual*τ*_{r}value (at the relative displacement*δ*_{r}, figure 6*a*);— the circumferential normal stress

*σ*_{θ}is assumed to be uniformly distributed along any radial line*θ*=const., with its value being equal to the average normal effective stress*σ*_{θ}acting on this plane;— the shear strength at the boundary plates is

*τ*_{s};— the peak friction angle is assumed to be independent of the state of stress, therefore, in order to apply realistic angles, the value adopted for the model evaluation is defined in direct shear tests at the low stresses corresponding to those in the shear-blade tests; and

— the soil stiffness is assumed to be independent of the state of stress, therefore the values adopted for the model evaluation are obtained from the odometer tests in the range of the low stresses applied in the shear-blade tests.

### (c) Stresses and strains

Accounting for the initial stress state from equation (3.1) and considering that, under generalized plane-stress conditions, stresses in the out-of-plane direction do not change, Δ*σ*_{z}=0, the linear elastic constitutive equations (3.2)–(3.4) can be rewritten as4.7and4.8with the stiffness parameter defined as *D*=*E*/(1−*ν*^{2}), and the homogeneous radial strain caused by dilation .

Radial normal stress on the shear band will change depending on the level of dilation and can be calculated using the linear elastic relationship4.9For peak and residual shear strength, radial strains *ε*_{r}=0 and are considered, respectively, so that from equations (4.3) and (4.9),4.10and4.11In generalized plane-stress conditions, because Δ*σ*_{z}=0, shear resistance along the boundaries between the top and bottom plates is constant,4.12Stress distribution and deformations within the elastic part are derived in analogy to the cohesive material case (equation (3.11)) by calculating the equilibrium of moments acting on an elementary wedge, resulting in4.13Substitution of equations (4.11) and (4.12) into this expression results in the differential equation for the circumferential normal stress4.14Rewriting this expression using4.15gives4.16The solution of this differential equation is of the form4.17Using the boundary condition on the shear blade *σ*_{θ}(*θ*=0)=*p*_{0}, it follows that *c*_{1}=*p*_{0}+*b*/*a* and therefore4.18From the constitutive equations (4.7) and (4.8) follows the expression of circumferential strains as a function of the change in circumferential stress and dilation:4.19Integration over the elastic area yields the rotation as a function of the pressure along the blade and of the shear-band tip, respectively: the rotation *α*(*θ*) is calculated by integration of these strains similar to equation (3.13). For *θ*=0, we get the rotation of the blade as a function of the stress at the blade *p*_{0} and the length of the shear band *η*,4.20

### (d) Energy-balance criterion

#### (i) Incremental propagation

Similar to the cohesive material case, we consider an incremental propagation of the shear band, which results in an increase of its angular length Δ*η*, at the constant normal stress at the blade *p*_{0}. This causes additional rotation of the shear blade Δ*α*_{0}, which is equal to the strain at the shear-band tip multiplied by this incremental propagation. This gives4.21where denotes circumferential stress at the shear-band tip *θ*=*η*.

#### (ii) Work components

Energy terms in the energy-balance criterion equation (3.16) are derived in analogy to the cohesive material.

The external work component is4.22Calculation of the internally stored energy in the elastic wedge is more complex than in the case of the cohesive material, because of dilation, the stresses perform work not only on circumferential but also on radial strains. Therefore, the incremental elastic work should be calculated as an integral of the specific strain-energy function *U* over the area of the elastic wedge,4.23To define this specific strain energy *U*, which is a unique function of the strain state, we note that it serves as a potential for stresses producing the linear elastic law equations (4.7) and (4.8),4.24
4.25and
4.26Equations (4.24)–(4.26) are satisfied when the strain-energy function is given by the following expression:4.27Substitution of equation (4.27) into equation (4.23) gives the expression for the internally stored elastic energy component,4.28which, by using expressions from equation (4.19),4.29and the expression for derived from equations (4.5) and (4.6)4.30can be written asand further simplified to4.31Dissipated work along the shear band and at the boundary plates is given by4.32Substitution of the equilibrium equation (4.13) into this expression gives4.33After substitution of equation (4.21), this can be written as4.34Finally, the dissipated energy at the shear-band tip (process zone) is defined as4.35The shear strength in the strain-softening branch is given by4.36which, in combination with equations (4.1)–(4.3), gives4.37Normal radial stresses are obtained from equation (4.9),4.38which after substitution of equation (4.4) and applying correction *K*_{b} gives4.39The dissipated work in the process zone can be calculated by substituting equations (4.37) and (4.39) into equation (4.35),and integrating the resulting expression to producewhich can be rewritten as4.40

#### (iii) Energy-balance criterion

Substitution of the work increments equations (4.22), (4.31), (4.34) and (4.40) into the energy-balance criterion in equation (3.13) gives4.41which can be reduced to the following shear-band propagation criterion:and after solving the quadratic inequality4.42written in normalized form, this gives4.43where .

From equation (4.18), which, in the normalized form, becomes4.44where it follows that:4.45This is the condition for the stress-controlled test. Its substitution into equation (4.20) gives the condition for the displacement-controlled test, i.e. the relationship between the shear-band length and the shear-blade rotation,i.e.4.46where4.47

### (e) Limiting equilibrium criterion

The limiting equilibrium condition (4.25) holds for the frictional case as well, with a difference that the peak shear strength and the strength on the boundary-plate contacts are defined by equations (4.10) and (4.12), respectively. After the corresponding substitutions, the condition for the shear-band propagation according to the limiting equilibrium can be written as4.48which can be resolved with respect to the pressure applied by the shear blade using equation (4.18),4.49Substitution of the above inequality into equation (4.20) results in the expression for the shear-band propagation for the displacement-controlled test,4.50where from equation (4.47) and .

## 5. Parametric study

A sensitivity study has been performed in order to understand parameter dependency of the solution. Sensitivity has been checked for two parameters: the stiffness parameter *M* and the relative displacement *δ*_{r}, assuming a linear decrease of strength *δ*_{m}=*δ*_{r}, while all other parameters remained constant (table 1).

In figure 10, the shape of the curves with frictional material is plotted for constrained modulus *M* of 250, 500 and 1000 kPa. With increasing stiffness, the rate of propagation increases. The rate of the progressive shear-band propagation is defined here as the slope of the shear-band length versus the shear-blade displacement curve. In this study, the emphasis has been on the validation of the energy-balance approach, therefore, the active failure zone has been neglected, and the limiting equilibrium approach was not considered, focusing on the range of parameters in which the shear band is purely driven by the energy-balance criterion.

Another parameter that is difficult to define is the relative displacement *δ*_{r} needed to reach friction at constant volume at the shear-band tip (figure 6*a*). It has been assumed that this parameter is equal to half of the shear-band width, which is a multiple of the mean grain diameter. Correspondingly, figure 11 shows the dependency of the rate of the shear-band propagation on the relative displacement *δ*_{r} as a multiple of the mean grain diameter. With increasing relative displacement, the rate of the shear-band propagation slows down. This is because more energy is dissipated in the process zone of the shear band. The sensitivity of the shear-band propagation rate is, however, rather low.

## 6. Comparison of analytical curves with experimental data

Comparison between experiments and analytical curves is performed using two plots: in the first plot (figure 12), the propagation of both the analytical solutions and the experiments are shown. The analytical curves are fitted to provide the upper and lower bounds for the experimental dataset from figure 5 using the method of least-squared errors and optimizing the stiffness parameter *D* and the angle between the blade and the active failure zone *η*_{LE} for both the fastest (figure 12, curve B) and the slowest (figure 12, curve A) propagation path. Then, in a second plot (figure 13), the stiffness *D* obtained from the above regression analysis is compared with the experimental stiffness curves obtained from the oedometer tests. Because in the oedometer tests (figure 3), stresses are much higher than stresses applied in the shear-blade tests, regression curves have been calculated in order to be able to compare the values.

Coefficient *K*_{b}, defining the compression strain equation (4.6) caused by dilation within the shear band in the sector inside the shear band has been adopted as 0.5. In order to validate this parameter, a numerical simulation using Fast Lagrangian Analysis of Continua (FLAC; Itasca Consulting Group, Inc. 2005) has been performed to investigate the displacement inside and outside the shear band owing to a pressure increase along the shear band. It has been found that *K*_{b}=0.5 is an appropriate assumption. The remaining parameters used for the analytical solution are given in table 1.

From regression analysis, the angle *η*_{LE} has been found to be between 1.45 and 1.48 (rad). This value controls the transition between the energy balance and the limiting equilibrium criteria, and it is the same for both the criteria. For the tests performed here, propagation driven by the energy-balance criterion (the entire curve B and the lower part of curve A) has been found to be progressive, while the propagation driven by the limiting equilibrium is catastrophic, represented by the upper straight vertical part of curve A. This study focuses on modelling curve B and the lower part of curve A using the energy-balance approach.

From the plot shown in figure 13, it can be concluded that correct rates of propagation of the shear band are predicted using the stiffness from the lower range obtained in the oedometer tests.

## 7. Conclusions

A novel test for studying the progressive shear-band propagation of cylinder-shaped shear bands has been presented. An analytical model based on the fracture mechanics energy balance and the limiting equilibrium approaches has been derived for both cohesive and frictional-dilative materials. The rate of the shear-band propagation appears to be rather sensitive to soil stiffness and less sensitive to the rate of strain softening. Analytical solutions have been validated against experimental results. Results show that the method provides both qualitatively and quantitatively reasonable results, with correct rates of shear-band propagation being predicted using the soil-stiffness values from the lower stiffness range obtained in the oedometer tests.

This study confirms that the fracture mechanics energy-balance approach is a useful tool for modelling of shear-band propagation in cohesive and frictional-dilative soils.

The purpose of the study was not to simulate the real-life problem, but to understand and model the mechanics of the phenomenon in controlled laboratory conditions. All the theoretical assumptions are of a simplified nature and are clearly stated in the paper. The justification of the assumptions is the ability of the model to provide reasonable quantitative descriptions of the test results using realistic soil parameters.

## Acknowledgements

The authors are grateful to the ETH workshop for the design of the test apparatus and the support during the experiments. This work has been supported by the Swiss National Science Foundation (SNF) (grant no. 200021-109195).

- Received June 4, 2010.
- Accepted July 20, 2010.

- © 2010 The Royal Society