## Abstract

This paper is an attempt to apply the Palmer–Rice fracture mechanics approach to the shear band propagation in sands and normally consolidated clays. This approach, proposed 30 years ago for overconsolidated clays, had a tremendous advantage of treating a shear band evolution as a true physical process and not just as a sufficient mathematical condition for its existence. Extension of this approach to a wider variety of soils requires for non-elastic soil properties (e.g. isotropic hardening plasticity, strain softening, lack of tensile strength, dilatancy, active and passive failure modes, etc.) to be taken into account. This paper demonstrates how the energy balance and process zone approaches can be applied to the simple problem of the shallow shear band propagation in an infinite slope built of such a soil. The energy balance approach appears to be the most conservative one. It allows for catastrophic and progressive types of soil failure to be properly identified, and dramatically effects the results of the slope stability analysis.

## 1. Introduction

### (a) Background

In their classic 1973 paper, Palmer and Rice, hereafter designated as PR, proposed an approach for analysis of the growth of localized shear bands in the progressive failure of *overconsolidated clay*. This approach assumes a gradual decay of shear resistance within the end zone of the shear band, from peak to residual values, with increasing relative sliding displacement. The criteria for the shear band propagation were derived using fracture mechanics (FM) techniques, namely the J-integral (Rice 1968*a*,*b*). This approach was also utilised by Rice (1973), Rice & Cleary (1976) and Rice & Simons (1976) to study such factors governing the rate of propagation as local pore water suctions caused by dilation within the shear band, as well as viscoelastic creep of soil and bulk pore water diffusion owing to the load changes. An important advantage of this approach is that it treats the failure in soil as a process of shear band propagation as opposed to, for example, limiting equilibrium approach, which assumes that shear band evolves instantly.

Unfortunately, this approach has never been applied to other types of soils or to non-trivial multidimensional problems. There are two possible reasons for that. The first one is that *sands* and *normally consolidated clays* exhibit intrinsically non-elastic behaviour, so that the J-integral, in general, becomes stress path dependent. It still can be used for elastic–plastic materials when the corresponding boundary-value problem under study satisfies certain geometric condition formulated by PR. However, this has not been pursued much further and the attempts by Vallejo (1987, 1988, 1989), Saada *et al*. (1985) and Chudnovsky *et al*. (1988) to apply FM principles to unsaturated brittle clay and overconsolidated stiff clay were also based on linear elastic assumption. For ductile saturated normally consolidated clay, however, Saada & Bianchini (1992), Saada, Bianchini *et al*. (1994) and Saada, Liqun *et al*. (1994) demonstrated that the plasticity of the material dominated the clay behaviour around the shear band tip. These works, however, did not offer shear band propagation criteria for the elasto-plastic case.

Another possible reason for abandoning the PR approach is that, approximately at the same time, Rice (1973) proposed an alternative—mathematical bifurcation condition for initiation of shear bands in soils. This condition is obtained by combining the constitutive relationship, equilibrium and compatibility conditions on assumed localized shear band and is solved as an eigenvalue problem. Non-trivial solution of this problem is a necessary condition for the shear band existence and provides both the shear band orientation and deformation jump across the shear band. Rudnicki & Rice (1975) obtained solutions for realistic elasto-plastic constitutive relationships for granular soils, and ever since, this approach has entirely dominated the literature related to strain localization analysis in soil.

Needless to say, the great advantage of Rudnicki & Rice's (1975) approach is its ability to account for elasto-plastic soil behaviour as well as to provide automatically the shear band orientation and the deformation jump. However, there is a price to pay—the physical problem of the shear band evolution is substituted by the mathematical problem of its existence. As a result, the shear band appears in a body instantly, instead of gradually propagating in a progressive or catastrophic manner. Very few attempts of application of mathematical bifurcation approach to simulate the pseudo-static shear band propagation (e.g. Borja & Regueiro 2001) heavily depend on *ad hoc* assumptions of the utilised numerical procedure. These assumptions may produce meaningful results only if they are based on the proper physical analysis of the propagation problem, which mathematical bifurcation approach cannot provide.

### (b) Objectives

This paper is an attempt towards developing a general approach (based on FM principles) which would be applicable to the analysis of shear band propagation in a wide range of saturated and unsaturated soils—normally and overconsolidated clays as well as dense and loose sands. This would allow for formulation of quantitative conditions for progressive and catastrophic failure in these soils that can be further applied to practical geotechnical problems.

In terms of geometry, the problem of the shallow shear band propagation in an infinite slope, solved in this paper, is one of the simplest. The emphasis here is made on accurate modelling of such soil properties as elasto-plasticity, isotropic hardening, strain softening, lack of tensile strength, active and passive failure at compressive stresses, dilatancy, and so forth. Our goal is to demonstrate that certain FM concepts applied by PR to the shear band growth in overconsolidated clays can be further generalized for the analysis of sands and normally consolidated clays, exhibiting the properties listed above. The solution of the simple problem of an infinite slope in such a material may give an insight into the possible treatment of more complex boundary-value problems.

An additional advantage of starting with a simpler problem is that its solution will be obtained in a physically meaningful closed form, convenient for a parametric analysis. Practical application of this solution includes stability of slopes, in particular, of continental margins, where a landslide may be a major cause of devastating tsunami (e.g. Almagor & Wiseman 1977; Frydman & Talesnick 1988; Puzrin 1996).

### (c) Rationale

The hope that some FM concepts can be extended to describe the shear band growth in a wide variety of soils is based on the following four observations. The first one is the experimental evidence of the gradual shear band propagation process in soils. Experimental studies of strain localization pioneered in clays by Morgenstern & Tchalenko (1967) and in sands by Vardoulakis *et al*. (1981) demonstrated formation of the shear bands with thickness of about 200 and 20 mean grain diameters, respectively. Gradual propagation of shear bands has been observed in both normally and overconsolidated clays (e.g. Saada, Bianchini *et al*. 1994; Saada, Liqun *et al*. 1994; Lizcano *et al*. 1997). Although much more limited than in clay, experimental evidence in sands (Graf 1984; Gudehus *et al*. 1985; Tanaka & Sakai 1993; Desrues 1998) also suggests the shear band growth.

The second observation in favour of applicability of certain FM techniques to soils is that in some soils (e.g. unsaturated brittle clay and overconsolidated stiff clay), the linear elastic FM approach has produced meaningful results (e.g. Saada & Bianchini 1992). In other types of soils, these linear elastic FM techniques may be used as a benchmark towards the elasto-plastic solution.

The third observation is that although the J-integral is a convenient form for expressing the energy balance related to the shear band propagation (Palmer and Rice 1973), the fact that it may be path dependent in non-elastic particulate materials simply means that the energy balance will have to be calculated in a more complex manner. However, the energy balance propagation criterion may still be applicable, provided the energy dissipation in the end process zone is specified.

Finally, the fourth observation is that, when the end process zone has a finite length and/or direction of the shear band growth is unknown, similar to FM (e.g. Rice 1968*b*), the alternative propagation criteria may be developed from a proper analysis of the process zone at the shear band tip. This paper presents three examples of the application of the linear elastic fracture mechanics (LEFM), energy balance and process zone approaches, respectively, to the same problem of the shear band propagation in an infinite slope.

## 2. Shallow shear band propagation in an infinite slope

### (a) Formulation of the problem

The problem formulated below is closely related to the problem of a long slope inclined by angle *α* to the horizontal, into which a step of height *h* is cut, causing the shear band of the length *l* to propagate upward from the base of the cut, parallel to the slope surface (figure 1*a*). This problem was first addressed by Christian & Whitman (1969) and analysed approximately by PR using the J-integral with the assumption that *h* and *ω* are small in comparison to *l*. Here *ω* is the size of the end zone near the tip, beyond which the shear resistance *τ* is essentially equal to residual shear strength *τ*_{r}.

While applicable to overconsolidated clays, the problem illustrated by figure 1*a* is not really meaningful for sands and normally consolidated clays, because the stresses developed in the sliding part of the slope are tensile and cannot be sustained by soils with no tensile strength. Instead, we are going to consider a related problem (figure 1*b*) where all the stresses are compressive.

We are going to consider a thin linear discontinuity which is parallel to the slope and its length *l* is sufficiently larger than both its depth *h* within the slope and the length *ω* of its end zones (figure 1*b*). Apart from the two small end zones, the shear resistance *τ* along this discontinuity drops to its residual value *τ*_{r}. At the very tips of the shear band as well as at any point outside the band, the shear resistance is equal to its peak value *τ*_{p}. Within the end zones the shear resistance *τ* gradually decreases from its peak *τ*_{p} to its residual *τ*_{r} value as a function of the relative displacement *δ* (figure 2). We are interested in conditions, under which the initial shear band will grow parallel to the slope surface. In particular, we shall be seeking answers to the following questions:

What is the criterion for the shear band propagation?

Will the shear band growth lead to the shallow failure of slope?

### (b) Assumptions

Two rather strong assumptions have been made in the above formulation. The first one is with respect to the existence and location of the initial discontinuity along which the shear strength dropped to its residual value. This discontinuity may initially be caused by the presence of a weaker inclusion, methane hydrate decomposition, or local increase in the pore water pressure owing to cyclic seismic or storm wave loading. However, if the initial band forms in a material, which is absolutely homogeneous and intact, its depth cannot be predicted in advance.

The second assumption is with respect to the direction of the shear band propagation. In the majority of true 2D and 3D problems, the shear band will most probably deviate from its initial plane, and defining the direction of its propagation is probably the most challenging task of the current research. In our particular problem, however, of a shallow failure in a very long slope, the initial shear band will probably first grow parallel to itself, before finally propagating towards the surface. There is no sound theoretical justification for this assumption at the present, apart from the observations of numerous relatively shallow (10–100 m deep) and very long (10–100 km) submarine landslides that took place in normally consolidated sediments in the continental margins of California (e.g. Lee & Edwards 1986), Norway (e.g. Bugge *et al*. 1988), Israel (e.g. Almagor & Wiseman 1977) and so forth. One way to explain the origin of these slides is to assume that the shear strength dropped simultaneously over the length of up to hundreds of kilometres, which is hardly a realistic assumption. An alternative approach is to assume that the shear strength dropped over a limited length of the slope, but then the resulting shear band had to propagate within its own plane before reaching the surface.

### (c) Limiting equilibrium approach

First of all, let us assess stability of the slope using the conventional limiting equilibrium method. Within this approach the material is assumed to be rigid with failure taking place at the contact between rigid blocks. The most dangerous slip surface will coincide with the discontinuity, protruding from its tips towards the slope surface (figure 3). The lower and upper triangular blocks apply to the sliding thin layer and the average normal contact stresses *p*_{p} and *p*_{a}, respectively. These stresses can be derived from the limit equilibrium of the corresponding triangular blocks, depending on the particular failure criterion of soil in the layer, which is not going to be specified at this stage. In the coordinate system adopted in figure 1*b*, considerations of the static equilibrium of the infinite slope produce the following expressions for stresses(2.1)where *γ*_{t} is the total unit weight of soil.

The safety factor for stability of the sliding thin layer can be defined as the ratio between the supporting and sliding forces(2.2)where . The critical length of the shear band leading to the global failure is obtained for *F*_{s}=1(2.3)so that the safety factor (2.2) can be rewritten as:

Because , so that , for very shallow depths of the shear band *h*≪2*l*tan *α*, the safety factor reduces to(2.4)where the first term gives the safety factor for an infinite slope, while the second represents addition caused by the finite nature of the shear band. Clearly, for any *l*≤*l*_{f} the slope is stable, unless for some reason, the shear band starts growing and reaches the failure length. Non-fracture mechanics attempts to treat the growth problem within the finite elements or finite differences frameworks (Hansbo *et al*. 1984; Bernander *et al*. 1989; Wiberg *et al*. 1990) utilising the linear elasticity assumption and producing shear band propagation criteria dependent on the numerical procedure with no strong physical basis. In the following sections we are going to search for more reliable criteria.

## 3. Linear elastic fracture mechanics approach

If the soil was linearly elastic, it would be possible to derive a shear band propagation criterion by using conventional FM techniques. The last expression (2.1) indicates the fact that the normal stress *σ*_{x} acting parallel to the slope cannot be found from equilibrium considerations alone. In fact, in our analysis we will just be interested in an average value of this stress over the thickness *h* of the layer:(3.1)

Let us assume that in an intact slope this average normal stress would be equal to . The exact value of *p*_{0} is not important since it does not appear in the final expressions, which depend upon only stress perturbation from the initial state. Existence of the shear band where the shear resistance *τ*_{r} is lower than the gravitational shear stress *τ*_{g}, will cause the stress field to deviate from its initial value. As a result, in the layer at the top end of the shear band decreases, while at the bottom end it decreases. Therefore, somewhere along the band, the average normal stress is still equal to its initial value , and this is the point (figure 1*b*) which we shall choose for the origin of the *x*-axis. The bottom and top tips of the shear band are located at the distance *x*_{1} and *x*_{2} from the origin, respectively, so that *x*_{1}+*x*_{2}=*l*. In this case, in the layer is distributed along the shear band according to the static equilibrium of the layer:(3.2)where , while in the far field along the slope, the average stress asymptotically returns to the initial value . In the linear elastic case, when the material behaviour is similar in loading and unloading, it can be shown that the infinite slope assumption results in *x*_{1}=*x*_{2}=*l*/2.

Following LEFM, PR assumed that the shear band propagates if the corresponding energy release rate is sufficient to maintain the propagation. Unfortunately, in spite of its apparent simplicity (figure 1), the exact closed form solution for this problem is not available and it should be either solved numerically (e.g. Grekov & Germanovich 1998) or asymptotically (in the case of *h*≪*l*, which is of interest here).

The general asymptotic solution for long shallow cracks parallel to the surface of the linear elastic half plane was presented by Dyskin *et al*. (2000). They computed J-integral by matching the outer and inner asymptotics, that is, outer beam approximation with Zlatin & Khrapkov's (1986) inner solution for a semi-infinite crack parallel to the half-space boundary. As a result, they obtained the following asymptotic expression for the J-integral(3.3)where *E*_{u} is the plane strain modulus in unloading; *δ*_{k} is a constant representing a combination of contour integrals computed by Zlatin & Khrapkov (1986; with the accuracy of three decimals, *δ*_{k}=0.620) and(3.4)for the lower and upper tips of the shear band, respectively.

Using equilibrium of the layer above the shear band, condition *x*_{1}=*x*_{2}=*l*/2 and the thin layer approximation, we derive(3.5)or, keeping the leading order term:(3.6)

PR suggested that for a propagating shear band, the J-integral should exceed the value of equal to the shadowed area in figure 2. This gives the following propagation criterion: the shear band will propagate if its length *l* exceeds the critical value of *l*_{c}(3.7)where(3.8)is the gravitational shear stress ratio(3.9)is the characteristic length(3.10)is the characteristic shear displacement in the process zone. Inequality (3.7) can be interpreted as a sufficient condition for the shear band growth in linear elastic soil. In some types of soils (e.g. unsaturated brittle clay and overconsolidated stiff clay), the linear elastic assumption may produce meaningful results. However, in cohesion-less soils and lightly overconsolidated clays, the linear elastic techniques may be only used as a benchmark towards more realistic elasto-plastic solution (see below). In the following we are searching for such a solution.

## 4. Energy balance approach (small end zones)

### (a) General

The elasto-plastic relationship between the average normal stress in the layer along the shear band and the average linear strain is schematically illustrated in figure 4 and given by the following formulae(4.1)where *p*_{a} and *p*_{p} are the average values of active and passive lateral pressure in the layer, respectively, while *E*_{l} and *E*_{u} are the loading and unloading plane strain moduli, respectively, co that *E*_{l}<*E*_{u}. Nonlinear stress–strain relationships can be easily utilised within this framework given a proper definition of the average linear strain . However, in spite of being bi-linear, the constitutive relationship (4.1) reflects such important properties of plastic soil behaviour as:

isotropic hardening (it has different stiffness for loading and unloading);

active and passive failure modes under entirely compressive stresses.

In comparison, the constitutive model adopted by PR for the problem in figure 1*a* considered only unloading branch, all normal stresses were tensile and no failure criterion for the normal stress was specified.

Rice (1973) demonstrated how the J-integral-based propagation criterion for problem in figure 1*a* can be interpreted in terms of the energy balance. In this section we shall attempt derivation of the shear band propagation criterion for the problem in figure 1*b* directly from the energy balance, without making use of the J-integral which may or may not be stress path dependent.

The assumptions made in this analysis are those adopted by PR with some amendments to account for elasto-plastic constitutive behaviour of soil. Under condition of *l*≫*h* most of the energy transfer during shear band propagation will take place owing to:

external work made by gravitational forces on downslope movements of the layer above the shear band;

internal work made by the normal stress acting parallel to the slope surface on deformations of the layer caused by changes in these stresses;

plastic work dissipated on the shear band;

dissipated plastic work of dilation in the band against the normal stresses perpendicular to the band (ignored here and treated separately in the Appendix A).

Two extreme assumptions can be made with respect to the length *ω* of end zones of the shear band: *l*≫*h*≫*ω* or *l*>*ω*≫*h*. First, we shall consider the former case (implicitly assumed by PR), where the end zones are small compared with the thickness of the sliding layer. The latter case will be studied in §4*b*.

Equation (3.2) gives the distribution of the average normal stress in the layer along the shear band, which is derived from equilibrium considerations (figure 5*a*). Because behaviour is not elastic, condition *x*_{1}=*x*_{2}=*l*/2, used in §3, does not hold here. Distribution of the average linear strain in the layer (figure 5*b*) is calculated using the constitutive law (4.1). Following PR, displacements below the shear band are neglected, therefore average displacements in the soil layer parallel to the slope surface will be equal to the relative slide *δ* on the shear band and given by integration of the average strain (figure 5*c*):(4.2)

At the tips of the shear band the relative slide *δ* is zero, so that *δ*(−*x*_{1})=0, which is satisfied by expression (4.2) automatically, while condition *δ*(*x*_{2})=0 results in the following constraint on the average strain distribution:(4.3)

Having made all the necessary assumptions we may proceed to derivation of the energy balance criterion for the shear band propagation. This criterion requires that the energy surplus produced in the body by incremental propagation of the shear band should exceed the work required for this incremental propagation. Mathematically, this can be expressed as the following inequality(4.4)where(4.5)is external work made in our case by gravitational forces on downslope movements of the layer above the shear band(4.6)is the internal work made by the normal stress acting parallel to the slope surface on deformations of the layer caused by changes in these stresses(4.7)is the plastic work dissipated on the shear band, which is required to overcome the residual shear resistance along the band(4.8)is the plastic work dissipated in the shear band during its propagation, which is required to overcome the shear resistance in excess of residual in the end zones of the band. (Note that *δ*(*x*)>*δ*_{r} and everywhere within the band outside the small end zones. Within the end zones, however: *δ*(*x*)≤*δ*_{r} and but their contribution is small, so that the upper bound of provided by inequality (4.8) is actually rather accurate).

Assuming that during the shear band propagation the increments of its length are *dx*_{1} and *dx*_{2} for the lower and upper ends, respectively, the corresponding increments of work are given by the following expressions:(4.9)

(4.10)

(4.11)

(4.12)

Expressions (4.9)–(4.12) after being substituted into inequality (4.4) yield the sufficient shear band propagation condition for a general form of constitutive law(4.13)where and .

Three successive stages of the shear band propagation are to be considered next:

stage I: the shear band is not sufficiently long to cause an active or/and passive failure in the parts of the layer adjacent to its ends;

stage II: either active or passive failure has occurred in the part of the layer adjacent to one of the shear band ends;

stage III: both active and passive failure has occurred in the corresponding parts of the layer adjacent to the both ends of the shear band.

### (b) Stage I: shear band propagation before local failure in the layer (figure 1*b*)

The average strains for this case (figure 5*b*) are given by substitution of equation (3.2) into (4.1):(4.14)which being substituted into equation (4.2) gives the distribution of the relative displacements along the shear band (figure 5*c*). Substitution of (4.14) into expression (4.3) yields the following restriction on the shear band dimensions, as well as on their changes in the course of the shear band propagation:(4.15)

By substituting equations (4.1) and (4.14) into inequality (4.13) and utilising expressions (4.15), after certain mathematical elaboration we obtain the propagation criterion for the shear band(4.16)where *r* and *l*_{u} are defined by expressions (3.8)–(3.10). For comparison, the criterion obtained by PR for the problem in figure 1*a* is given by:(4.17)

As expected, in the slope which was undercut, propagation of a shear band requires a smaller stress ratio than in an intact slope. In particular, even if *τ*_{g}=τ_{r}, it is still possible (PR) that the energy recovered by relief of the initial pressure *p*_{0} in the undercut slope could be adequate to drive the shear band. In the intact slope this is impossible.

### (c) Stage II: shear band propagation after the local slope failure (figure 6)

The propagation criterion (4.16) is applicable to the shear band that is not sufficiently long to cause an active or/and passive failure in the parts of the layer adjacent to its ends. Eventually, owing to the shear band growth, the decreasing average normal stress at the upper end of the shear band will reach the critical value of and cause the active failure in the layer above. Alternatively, the increasing average normal stress at the lower end of the shear band may reach the critical value of first, and cause the passive failure in the layer below. The question is at which end of the shear band will the failure take place first, or does it happen simultaneously?

From equation (3.2) it follows that the active failure will take place when the length *x*_{2} reaches the critical value of(4.18)while the passive failure will take place when the length *x*_{1} reaches the critical value of(4.19)

The ratio between these two critical values is given by(4.20)the latter inequality being a well-known fact in soil mechanics. Let us assume that the passive failure was reached first, that is, *x*_{1}=*x*_{1p}. Then at the verge of the failure the dimensions of shear band are related by equation (4.15):(4.21)

However from inequality (4.20) it follows that *x*_{2a}≤*x*_{1p}. Therefore, from inequality (4.21) it can be concluded that for passive failure to occur first, the length *x*_{2} should reach a value in excess of its critical value for active failure *x*_{2a}. This proves that the active failure should occur first, which is supported by the field evidence of ‘tension’ cracks and slump scars appearing at the top part of slopes some time before the failure.

When the active failure takes place, the top end of the shear band effectively deviates from its plane, cuts the sliding layer and reaches the surface. Further propagation *dx*_{1} of the shear band takes place at its lower end only. The energy balance criterion is obtained from expression (4.13) by setting *dx*_{2}=0 and adding to external work a term describing the work of active force on corresponding displacements:(4.22)

Substitution of equations (4.1) and (4.14) into inequality (4.22) yields the propagation criterion for the shear band after active failure at its top end(4.23)where(4.24)is the length of the band at which the active failure takes place for the first time.

### (d) Stage III: the global failure of the slope (figure 3)

The lower end of the shear band will keep propagating till the critical value of *x*_{1}=*x*_{1p} is reached (equation (4.19)). At this point and the passive failure takes place in the layer at the lower end of the shear band. The bottom end of the shear band deviates from its plane, cuts the sliding layer and reaches the surface after which the slope fails globally. The length of the shear band at global failure is obtained by summing up equations (4.18) and (4.19), and is given in equation (2.3).

### (e) Summary

Let us now summarize the results of the energy balance approach. Equations (4.16) and (4.23) represent energy-based criteria for propagation of the shear band, before and after the failure of the soil layer at its top end, respectively. Condition (4.16) requires that for propagation of the shear band before the failure of the layer takes place at its top end, the length of the shear band should exceed the value of *l*_{cr}:(4.25)

Shear band will propagate and reach the length la defined by equation (4.24). At this length the active failure takes place in the soil layer at the top end of the band and condition (4.23) requires that for further propagation of the shear band its length should exceed the value of *l*_{b}:(4.26)

However, because *l*_{a}≥*l*_{cr} by definition, from inequality (4.25) it follows that , which after substitution into (4.26) yields *l*_{b}≤*l*_{a}. This result makes condition (4.26) redundant, because this condition is only applicable for *l*≥*l*_{a}, i.e. after the active failure takes place. Therefore, inequality (4.25) is the sufficient energy-based condition for the shear band to propagate both before and after active failure, till the global failure of the slope takes place at *l*=*l*_{f} given by expression (2.3).

## 5. Process zone approach (finite end zones)

In the previous section we imposed the requirement *l*≫*h*≫*ω* on the length *ω* of the end zones of the shear band. This assumption allowed for the energy balance to be easily calculated leading to the simple closed form criterion for the shear band propagation. In this section we are going to weaken the above requirement by assuming that the length *ω* is not necessarily small compared with the length of the shear band *l*. Instead, we shall assume it being large compared with the thickness of the sliding layer: *l*>*ω*≫*h*. Energy balance approach in this case is still applicable, but the necessity of accounting accurately for the end zones in equations (4.5)–(4.12), makes it rather cumbersome, presenting a good case for demonstration of an alternative ‘process zone’ approach.

Within the end zones, the effect of the shear resistance *τ*(*x*) on distribution of the average normal stress in the layer can be assessed using equilibrium considerations. Starting from the tip of the end zone, the shear resistance *τ*(*x*) gradually decreases from the peak *τ*_{p} to residual *τ*_{r} value as a function of the relative displacement *δ* (figure 2). The equilibrium equation follows from the analysis of the elementary slice of the sliding layer (figure 7):(5.1)

Let us first consider the upper end zone *ω*_{2} of the shear band (figure 1*b*). The soil in the layer above this zone is unloading, so that from equations (4.1) and (4.2) it follows that(5.2)which after being substituted into equation (5.1) yields differential equation for the normalized displacement :(5.3)where *r* and are given by expressions (3.8) and (3.10), respectively, and(5.4)is the normalized form of the constitutive relationship in figure 2. Equation (5.3) can be rewritten using the non-dimensional coordinate (5.5)where *l*_{u} is the characteristic length given by equation (3.9). Boundary conditions at the tips of the end zone are(5.6)where and .

Equations (5.5) and (5.6) represent a boundary-value problem, with a solution that can be expressed in a general way(5.7)which after being substituted into equation (5.4) gives distribution of the shear resistance along the end zone(5.8)

But what about the length of the end zone, is it possible to give an estimate of *ω*_{2}? Condition of continuity of the average normal stress across the boundary *x*=*x*_{2}−*ω*_{2} of the end zone (figure 8) implies a constraint on the length of the zone *ω*_{2}.

Outside the end zone, the average normal stress on the boundary *x*=*x*_{2}−*ω*_{2} is defined using equation (3.2):(5.9)

Inside the end zone, the average normal stress on the boundary *x*=*x*_{2}-*ω*_{2} is defined using equations (4.2), (4.1) and (5.7):(5.10)

Continuity condition therefore can be expressed as(5.11)

Substitution of equation (5.7) into (5.11) gives an algebraic equation for the length of the end zone *ω*_{2}:(5.12)

The procedure described above is applicable for any form of the constitutive relationship (5.4). The relationship adopted in this work is schematically illustrated in figure 9 and given by the following formula:(5.13)

Which after substitution into (5.5) yields a linear differential equation:(5.14)

Solving this equation with boundary conditions (5.6), where , we obtain(5.15)as well as the distribution of the shear resistance along the end zone of the band:(5.16)and the distribution of the average linear strain in the layer above the end zone(5.17)

Figure 10 shows this distribution of the shear resistance along the end zone of the band for the whole range of 0≤*r*≤1, as described by equation (5.16) for a case of *τ*_{p}/τ_{r}=2. The lower solid line in the plot corresponds to *r*=1, while the upper solid line—to *r*=0. The dashed line represents the linear distribution as assumed by PR, which is surprisingly close to the middle solid line of *r*=0.5. With decreasing all the curves get closer to the linear distribution.

Finally, substitution of equation (5.15) into continuity condition yields equation for the end zone length(5.18)

Let us now consider the lower end zone *ω*_{1} in figure 1*b*. For this zone, the differential equation for is still given by equation (5.5), however, its non-dimensional coordinate becomes . The boundary conditions become(5.19)where and , while the continuity condition becomes(5.20)

After relationship (5.13) is adopted, solution of differential equation (5.14) is given by(5.21)the shear resistance distribution is similar to that in figure 10(5.22)and the distribution of the average linear strain in the layer above the end zone:(5.23)

The length of the lower end zone is given by the equation which is algebraically identical to that of the upper zone (5.18):(5.24)

At this stage, the mechanical picture of the end zones is complete: stresses, strains and displacements both inside and outside the upper and lower end zones are given by equations (4.1), (5.15)–(5.18) and (5.21)–(5.24). They can be substituted into the work equations (4.5)–(4.8) and the energy-based propagation criterion can be derived after rather elaborate calculations following the pattern of the previous section.

However, because the stress–strain state of the end zone in this case is known explicitly, all the elaborate calculations of work may be avoided by applying the so-called ‘process zone’ criterion. In terms of FM, the portion of the shear band where *δ*(*x*)>*δ*_{r} and *τ*(*x*)=τ_{r} (i.e. between the end zones) is analogous to a shear crack. The end zones, where *δ*(*x*)≤*δ*_{r} and *τ*_{p}≥*τ*(*x*)≥*τ*_{r}, are then analogous to the process zones of the shear crack. The sufficient condition of the shear crack propagation requires that the virtual incremental propagation of the external tip of the process zone should result in a positive increment in the shear crack length. Mathematically, this ‘process zone’ criterion can be expressed as: , where and (*i*=1, 2). When applied to equations (5.18) and (5.24) this ‘process zone’ criterion becomes . Equations (5.18) and (5.24) express the factored normalized length of the shear band as a periodic function of the normalized length of the end zone . Therefore, may have an infinite number of solutions for a given set of and *r*. However, we are interested only in those solutions where and . These two inequalities are always satisfied within the interval of where the solutions are unique (figure 11). The lower thick line in the plot in figure 11 corresponds to *r*=0, while the upper thick line—to *r*=1. The thin line represents approximation by(5.25)

The fact that the length of the end zone decreases with the shear band propagation possibly indicates that for long shear bands the end zones are small.

Substitution of inequality into equation (5.24) yields:(5.26)

After substitution of condition into equation (4.2) we obtain(5.27)

Substituting equations (4.12) into expression (5.27) and considering equations (5.18) and (5.24) the following restriction on the shear band dimensions can be derived:(5.28)

Therefore, the process zone criterion (5.26) can be rewritten via the length *l*=*x*_{1}+*x*_{2}:(5.29)

## 6. Discussion

In the above sections we applied three different FM approaches to the problem of the shear band propagation in an infinite slope. Each approach required its own assumptions and resulted in a propagation criterion summarized in table 1.

Comparison between different criteria appears to be rather straightforward. Approximate linear elastic FM criterion is applicable for shallow shear bands in brittle unsaturated and overconsolidated clays. It also successfully serves as a benchmark for a particular case *E*_{u}=*E*_{l} of the energy balance criterion, which is not surprising, because they both utilise energy principles and the same geometric assumptions. However, because in soils *E*_{u}≥*E*_{l}, the linear elastic FM criterion in our problem appears to be non-conservative and should not be utilised in the slope stability analysis.

Another important observation is that, because 0≤*r*≤1, the energy balance criterion also provides a lower bound for the process zone propagation criterion. Therefore, the energy-based propagation criterion *l*≥*l*_{cr} has been found for this particular problem to be the most conservative one and, therefore, it is the one to be used in the slope stability analysis.

This brings us back to the limiting equilibrium failure criterion. According to expression (2.4), the slope is going to be stable for any *l*≤*l*_{f}. However, according to the energy balance criterion, if *l*_{cr}≤*l*≤*l*_{f}, the shear band will grow until its length reaches *l*=*l*_{f} and the global landslide takes place. The safety factor (2.4) cannot reflect this fact, because for *l*=*l*_{cr} it is still larger than unity. Clearly, expression (2.4) has to be modified in order to account for the phenomenon of the shear band propagation, and the new safety factor definition proposed here is given by(6.1)where the first term again gives the safety factor for the infinite slope, while the second one represents an addition restricted by the shear band propagation criterion (4.25). As in the case of equation (2.4), the safety factor expression (6.1) can be applied only to very shallow shear band depths: .

## 7. Applications: safety factors for slope stability

The results of this work will be best illustrated if we consider an infinite slope made of dry dense sand, so that application of linear elastic FM approaches cannot be justified. The sand behaviour is characterized by the following mechanical properties: *ϕ*′_{p} and *ϕ*′_{cs} are the peak and critical state effective angles of internal friction, respectively; *G*_{u} and *G*_{l} are the average shear moduli in unloading and loading, respectively; *ν* is Poisson's ratio, so that, at this stage we assume that average shear moduli are independent of depth:(7.1)

Shear strength is defined by Coulomb's law of dry friction(7.2)and the average active and passive pressures are computed Chu (1991) from figure 12 for *ϕ*′=*ϕ*′_{p}, i.e. assuming that the failure takes place at the peak strength:(7.3)

Substitution of expressions (7.1) and (7.2) into energy balance criterion (4.25) gives(7.4)that is, the critical length of the shear band is independent of depth. Substitution of expressions (7.2) and (7.3) into equation (2.3) produces(7.5)

From equations (7.4) and (7.5) it follows that(7.6)that is, beyond certain depth of the shear band, the critical length ratio of the shear band becomes smaller than unity, therefore, the limit equilibrium safety factor (2.4) will not be conservative, and will become even less conservative with increasing depth of the shear band slip surface.

For typical values of strength parameters for dense dry sand *ϕ*′_{p}=40° and *ϕ*′_{cs}=30°, the critical length ratio *l*_{cr}/*l*_{f} is plotted in figure 13 versus normalized depth of the shear band *h*/*l*_{cr} (limited by the assumption *h*≪*l*) for a range of different gravitational shear stress ratios *r*.

*Example*:

For dense dry sand the typical values of parameters are *ϕ*′_{p}=40°, *ϕ*′_{cs}=30°, *G*_{0u}=32 MPa, *G*_{0l}=8 MPa, *p*_{r}=100 kPa, *ν*=0.3, and *γ*_{t}=18 kPa. In a slope with *α*=32° (*r*=0.182), the critical length of the shear band at the depth *h*=10 m predicted by the energy balance method (7.4) is *l*_{cr}=79.0 m≫*h*, while the limit equilibrium (7.5) predicts the global failure length *l*_{f}=307.7 m. For the shear band of initial length *l*=80 m, the applicability condition for both the safety factor equations (2.4) and (6.1) is satisfied, and the corresponding slope stability safety factors are given in table 2:

In table 2, the infinite slope safety factor is given for comparison assuming that we do not know anything about the shear band existence. In the limit equilibrium approach (2.4) we know about the band, but the calculated safety factor raises no concern about stability of the slope. All FM approaches (table 1) predict rather close values of the critical length of the shear band. However, only the energy balance approach (6.1) predicts the failure of the slope.

Strictly speaking, the energy balance, linear elastic FM and the process zone approaches can be applied only if the corresponding assumptions with respect to the size of the end zone (table 1) are satisfied. For the energy balance and linear elastic FM approaches, PR suggested the following estimate for the upper bound of end zone length(7.7)which for our example becomes(7.8)justifying the assumption *l*≫*h*≫*ω* of the small end zone used in the energy balance and linear elastic FM approaches.

For the process zone approach, the upper bound for the end zone length (figure 11) is given by(7.9)which for our example yields:(7.10)

Because in this case the length of the end zone is comparable with the thickness of the sliding layer *h*=10 m, assumption *l*>*ω*≫*h* of the process zone approach is not satisfied. Therefore, the energy balance approach gives not only a more conservative result, but also the most reliable result.

Finally, after substitution of expressions (7.2) into expressions (A 7) and (A 8) in Appendix A, we can assess effects of dilation on the energy balance propagation criterion:(7.11)

For the parameters of the above example, assuming that *ψ*_{p}=0.5*ϕ*′_{p}=20°, the critical length becomes *l*_{cr}=122.1 m, resulting in a safety factor (7.10) of *F*_{s}=1.040. As is seen, dilatancy may have a significant effect on the slope stability calculations, and an overestimation of dilatancy may lead to unconservative analysis and design.

## 8. Conclusions

Application of the FM techniques to analysis of the shear band propagation in sands and normally consolidated clays, where no conventional tensile fracture can take place, requires rather careful treatment. However, once accomplished, this approach has a number of advantages. First of all, the shear band evolution is treated as a true physical process and not just as a sufficient mathematical condition for its existence.

Another important advantage of this approach is that it for different types of failure to be clearly distinguished. In *progressive failure*, propagation of the shear band is stable in the sense that it requires work of additional external forces. In *catastrophic failure*, propagation of the shear band is unstable and takes place under existing external forces. In our example of the infinite slope, the shear band propagates catastrophically once its length exceeds the critical value. The term *delayed failure*, often used in geotechnical literature, is in fact a particular case of catastrophic failure, in which propagation of the shear band is unstable, but delayed in time owing to, for example local pore water suctions caused by dilation within the shear band, or viscoelastic creep of soil and bulk pore water diffusion.

While this issue is relatively well understood in FM (e.g. Cherepanov & Germanovich 1998), in soil mechanics literature there is no clear distinction between catastrophic, delayed and progressive failure. We find it important to distinguish between them, because for the long-term stability, the delayed failure is as much a concern as the catastrophic failure. Progressive failure under certain conditions may become catastrophic, while the delayed failure may become progressive. All these and other subtleties prove the importance of the consistent classification of failure types. This classification becomes possible under the energy-based approach to the shear band propagation.

From the practical point of view, it has been shown that catastrophic shear band propagation may cause a failure under collapse loads much smaller than those predicted by conventional plasticity analysis. In these cases the corresponding energy-based propagation criteria should serve as a basis for the conservative design and analysis.

This paper is an attempt to apply the generalized FM techniques to shear band propagation in sands and normally consolidated clays. Extension of this approach to non-homogeneous soils and undrained loading conditions, as well as its application to the analysis of the evolution of tsunamigenic submarine landslides, will be published in the companion papers. A more challenging task, however, is to extend this approach to 2D and 3D shear band propagation, and in our opinion, the efforts in this direction should be strongly encouraged.

## 9. Uncited references

## energy balance propagation criterion for dilating soils

Dilatancy manifests itself in a volume increase in drained shear of overconsolidated clays and dense sands, so that a tangential displacement *δ* along the shear surface is accompanied by the normal displacement *δ*tan *ψ*, where *ψ* is the angle of dilation. In the presence of dilatancy, the energy balance criterion for the shear band propagation (4.4) has to be modified to include the incremental dissipation Δ*D*_{d}(A1)where(A2)is the plastic work done by the dilation in the shear band against the normal stresses perpendicular to the band. The inner integral in (A 2) reflects the fact that the angle of dilation is not constant during the strain-softening stage of the post-peak behaviour: it decreases from its peak value *ψ*=*ψ*_{p} at the peak shear stress *τ*_{p} to *ψ*=0 (critical state) at the residual shear stress *τ*_{r}. Assuming, in the first approximation, that the angle of dilation decreases linearly with the shear stress(A3)and substituting equation (A 3) into (A 2), we obtain(A4)

Note that *δ*(*x*)<*δ*_{r} and everywhere within the band outside the small end zones. Within the end zones, however: *δ*(*x*)≤*δ*_{r} and , but their contribution is small, so that the upper bound of *D*_{d} provided by the inequality (A 4) is rather accurate.

Assuming that during the shear band propagation the increments of its length are *dx*_{1}and *dx*_{2} for the lower and upper ends, respectively, the corresponding increment of work is(A5)

Expressions (4.9)–(4.12) and (A 5) after being substituted into inequality (A 1) yield the sufficient shear band propagation condition for a general form of constitutive law. By substituting equations (4.1) and (4.14) into this general criterion and utilising expressions (4.15), after certain mathematical elaboration we obtain the propagation criterion for the shear band in dilating soil(A6)where *r* and *l*_{u} are defined by expressions (3.8)–(3.10);(A7)

Condition (A 6) requires that for propagation of the shear band in a dilating material, the length of the shear band should exceed the value of *l*_{cr}:(A8)

As is seen here, it differs from the original propagation criterion (4.25) by a factor of , which depends solely on soil friction and dilation parameters.

## Footnotes

- Received July 29, 2003.
- Accepted July 2, 2004.

- © 2005 The Royal Society