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The Pre-failure Deformation

  • Luigi Coppola
Chapter

Abstract

The analysis of phenomena of instability of slopes must be carried out based mainly upon the observation of the natural cracking of the soils and upon the interpretation of the mechanisms that induced them. This analysis is far more important when the final purpose of the research is the prediction and prevention of disasters induced by the hydrogeological instability of the slopes. In this chapter, the subjects are the Apennines range and, in part, the Argentinian Pre-Andes of the Provinces of Salta and Jujui, that are the areas on which our experience is founded onto.

4.1 Introduction

The analysis of phenomena of instability of slopes must be carried out based mainly upon the observation of the natural cracking of the soils and upon the interpretation of the mechanisms that induced them.

This analysis is far more important when the final purpose of the research is the prediction and prevention of disasters induced by the hydrogeological instability of the slopes. In this chapter, the subjects are the Apennines range and, in part, the Argentinian Pre-Andes of the Provinces of Salta and Jujui, that are the areas on which our experience is founded onto.

In a progressive-sequential deformation system of thrust tectonics, the Apennines chain was affected, from Upper Tortonian onwards, by subsequent over-sliding of large lithic masses advancing towards the Adriatic Foreland (Fig. 4.1). In an invariant compressive stress regime, in this context, exists the problem of the non-nativity of lithic masses, which, in a process of deconstruction, have become the place for numerous and wide areas of hydrogeological instability (Fig. 4.2).
Fig. 4.1

Southern Apennine—lithologic scheme of the areas subjected to continuous monitoring because of the risk of landslides

Fig. 4.2

Hydrogeological instability in Marche region (Italy) (CNR 1989)

Indeed, the current orogenetic arrangement of the Apennines lithological units is the result of compressive deformation mechanisms that have acted through a series of events from the Lower Miocene to the present making the soil structure chaotic, heterogeneous and anisotropic; layering is generally not identifiable, while the lithic mass is intimately fractured and folded. Therefore, the knowledge of the initial conditions and at the boundary of the soil, of the applied stresses, of the post-failure behaviour of the slope, which so much influence induces on the damage caused by landslide events, represents the necessary and preliminary condition for the development of tools of calculations capable of providing an adequate and credible response for the protection against hydrogeological risks. In general, the mechanical properties of the deposits with prevalent clayey component depend on both the mineral and granulometric characteristics and on the lithological discontinuities. The presence of a few discontinuities is sufficient to ensure that the overall mechanical behaviour of the soil becomes completely different from that of the soil matrix.

The evaluation of the mechanical characteristics of a rock mass (considered in the broad sense) can therefore only be derived from an accurate description of the discontinuities present and from a careful study of their properties. In addition, discontinuities indicate an oriented “structure” within the rock mass, even when this is intrinsically made up of isotropic rock; the discontinuities cause an obvious anisotropy of the mechanical characteristics and only in limit conditions of extreme fracturing it is possible to obtain again an isotropic structure, but only from a statistical point of view.

Particular attention must be paid to fissured overconsolidated clays. Actually, the presence of cracks and discontinuities affects both the in situ behaviour, both the response upon sampling and during laboratory testing.

Since it can be imagined that the resistance to shearing of a large sample of clay is mainly governed by the strength along the discontinuities, it can also be expected that this decreases as the sample size increases. In fact, the resistance obtained by means of large-plate loading tests (865 mm) clearly shows (Marsland 1974) that increasing the volume of soil tested, both the results dispersion and the resistance reduce considerably. From these considerations, the following indications can be made:
  • when the soil has a macrostructure characterized by fissures, laboratory results are scarcely representative of the behaviour that a volume of soil may have on site; referring, instead, to in situ tests, representative values can be obtained, provided that the use of tests involve a significant portion of the ground;

  • when comparisons are carried out on the results of various types of tests, the influence of several factors must be taken into account, among which the most important is the volume of soil tested and the phenomenon of progressive failure; furthermore, it still remains uncertain, according to the current state of knowledge, whether it is possible or not to speak properly of undrained behaviour in the case of fissured material (Simpson et al. 1979).

Given that at the origin of every technical investigation is needed a careful geological survey of the site, in order to establish the lithological application field, and a geomorphological survey, in order to classify and characterize the phenomena of slope transformation, for the choice of the methods of analysis, the study of rock fissuring should be directed essentially to the definition of a geomechanical model useful, for other professionalisms, for design analyses or specific assessments. For this purpose, the method of study and interpretation of fracturing must correspond to the following operations: (1) monitoring of deformations (in the wider sense) and measurements of the corresponding geometric elements; (2) geometric and kinematic interpretation of the observed deformations.

For the purpose of the geometric interpretation it is necessary to distinguish all the structures belonging to the same deformation phase, in order to obtain a model representative of the entire deformation system. For the kinematic interpretation, then, separate structures belonging to different systems must be separated to establish a relative time history of the corresponding deformation phases.

In this chapter, it is attempted, in a basic manner, to provide those useful notions that are in accordance with the laws of mechanics in order to arrive to this second phase of the study.

In geological practice, the analysis of rock fissuring embraces a vast field. It is essential for underground works, such as mining tunnels and tunnels for civil purposes or for water mains purposes, as well as for exploration and uptake of groundwater, for the underground storage of fluids and wastes and even for low-depth works, such as foundations, excavations, open-cast mines and, finally, surface works such as dams, coal or nuclear power plants, etc.

From the point of view of the stability of slopes or of mass movements the fissuring of rocks affects the characteristics of deformability and strength of the rock mass, which depends predominantly on geometry, spacing, orientation and mechanical-physical characteristics of the planes of discontinuity. Therefore, the first step towards the analysis of the stability of a slope is to find a relationship between systems of discontinuities and possible kinematics of instability. Relationship to be obtained from the examination:

  1. (a)

    of discontinuities, unstable rock volume and possible failure surfaces;

     
  2. (b)

    of the slope and of the mechanical characterization of the discontinuities.

     

4.2 Development of Cracks Under Compression1

It is very interesting to describe the development of cracks under monoaxial compression, i.e. without lateral stresses.

Considering an open, elliptical, very flattened crack, contained in a rock sample subjected to axial compression σ1, it is possible to distinguish several successive steps with the increase of σ1 before achieving the failure (Fig. 4.3a).
Fig. 4.3

Failure of a sample with a pre-existing internal crack under monoaxial compression (from Bles and Feuga 1981)

  1. 1.

    Initially, a closure of the fissure is obtained (Fig. 4.3b).

     
  2. 2.

    A slight relative sliding of the two sides of the fissure is then induced (Fig. 4.3c).

    This movement, initially elastic, is not entirely reversible because there is dissipation of energy, becoming movement.

     
  3. 3.

    By increasing σ1, crack tends to develop because of the formation, at its ends, of tensile cracks, which extend, initially, perpendicularly to the initial crack, but then tend to become parallel to the direction of σ1 (Fig. 4.3d). This indicates that under monoaxial compression it is obtained the development of tensile stresses which produce secondary extensional cracks. These are always parallel to σ1.

     
  4. 4.

    If σ1 increases beyond a certain threshold, the complete failure of the sample is obtained because of the unstable development of extensional cracks.

     
  5. 5.

    The new tension cracks do not propagate, if not at short distance from the main elliptic crack, until the value of σ1 does not overtake a certain failure threshold. When, instead, σ1 exceeds this threshold, a horizontal stress (σ3), orthogonal to σ1, appears within the sample because of the confining stress of the lithic mass, that opposes to the trend of the increase in volume of extensional cracks. This is how the rupture of Fig. 4.3e is achieved. The brittle deformation of a rock, therefore, always occurs under triaxial conditions (σ1 > σ2 > σ3). The length of cracks is function of the ratio σ3/σ1 and of the length of the initial crack.

     
Under triaxial conditions, starting from the development of the stable extensional crack of Fig. 4.4(1), there are small, almost capillary, extension cracks at the extreme ends of the initial crack. By increasing the triaxial stress state, the closure of the crack is initially obtained and then the sliding of the two parts of the rock along the initial penetration plane (Fig. 4.4(2)). The frictional resistance at failure between the two sides of the original crack is defined by Coulomb’s law (1773).
Fig. 4.4

Possible failure model under triaxial compression (from Bles and Feuga 1981)

$$\tau = \tau_{0} + \sigma_{n} {tag} \cdot \varphi$$
where
$$\sigma_{n} = \left( {\frac{{\sigma_{1} + \sigma_{3} }}{2}} \right) - \left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}} \right) \cos \cdot 2\vartheta$$
$$\tau = \left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}} \right) \sin \cdot 2\vartheta$$

The failure corresponds to a value \(\tau = \left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}} \right)\) which is obtained when \(\upvartheta = 45^{\circ}.\)

If after the formation of stable extension cracks (Fig. 4.3d) the shear resistance is exceeded, failure is achieved due to shear movement along the initial failure plane (Fig. 4.4(2)). However, the cohesion of the sample or the roughness of the shearing surfaces, can resist to failure and stop the development of extensions cracks parallel to σ1, located at the end of the original elliptical crack, and cause a tensile process with sliding and opening of the latter (Fig. 4.4(3)).

Alternatively, failure can be produced from the combination of the two previous processes according to a shear surface which is intermediate between the cracks parallel to σ1 and the initial failure plane.

During the deformation, changes take place in both the stress field and the mechanical properties of the material. In this case the fissuring becomes stable (Fig. 4.4(4)). An increase in σ1 is therefore necessary for the fissuring to develop further.

4.3 Analysis of Brittle Deformation

The rocks that are currently present on the surface of Earth have deformations that can be of two different types: bending, if the deformation is of plastic type, or cracking, if the deformation is rigid.

These deformations mainly develop under the effect of a tectonic thrust that influences the lithological nature and the mechanical characteristics of the rocks.

Geologists use the term “competent” for those rocks with a rigid or brittle behaviour; thus, they are those rocks that, under tectonic actions, develop failure without presenting a plastic-like behaviour, such as limestones, sandstones and granites.

Non-competent rocks, such as clays, are characterized by plastic behaviour and thus induce phenomena of bending. This useful distinction, however, does not take into account the conditions of pressure and temperature associated with the depth at the time when the rocks are stressed by tectonic forces. In effects, a rock which is competent when at surface can become non-competent at depth and vice versa. For example, limestones are competent on the surface as well as clays are non-competent, but from a certain depth the limestones become less competent than clays.

Under conditions of increasing pressure and temperature, i.e. when moving from surface to depth of an area of the Earth’s crust, a stratified and homogeneous rock may subsequently undergo the following types of global deformation:

  1. 1.

    cracking;

     
  2. 2.

    cracking and bending;

     
  3. 3.

    bending;

     
  4. 4.

    bending and flattening;

     
  5. 5.

    flattening;

     
  6. 6.

    flattening and melting;

     
  7. 7.

    melting.

     

This indicates that with depth the deformation of a rock always assumes a more plastic behaviour.

For the same tectonic domain, the mechanisms of deformation remain globally the same within large volumes, but these mechanisms vary vertically with depth; therefore, the notion of structural levels is created.

There are three major structural levels, that are identified, from top to bottom (Fig. 4.5a, b), as:
  • higher structural level, characterized by the predominance of brittle behaviour of most of the rocks, thus showing an important fracturing by shearing or discontinuous deformation. However, very plastic rocks are subject to types of deformations by bending, that do not exclude that the same rocks can be equally fractured;

  • intermediate structural level, corresponding, in general, to deformation by bending of stratified levels. Isopach folds are the result of this bending and are often accompanied by fractures and faults, especially in the most competent rocks, that make up the skeleton of the rock volume. Incompetent rock levels bend in disordered or disharmonious modes in comparison to the isopach folds of competent rocks;

  • lower structural level, characterized by the development of schistosity accompanied by metamorphism and subsequently by a flow because of the fusion of rocks.

Fig. 4.5

a Block diagram of a portion of Earth’s crust showing the superposition of several structural levels and the style of the corresponding structures (from Mattauer 1973). b Block diagram of a symmetric mountain range with small shortening and with distribution of the corresponding structural levels (b1). It is evidenced that, following to lithologic contrast between the sediments and their crystalline substrate, shear can take place at the base of the zone subjected to bending (from Mattauer 1973)

The development of schistosity takes place generally on large thicknesses and can be then distinguished from top to bottom:

  1. (a)

    schistosity zones from cracks and cracks/folds (cleavage, strain-slip cleavage or crenulation cleavage);

     
  2. (b)

    schistosity zone by melting (fluid-cleavage);

     
  3. (c)

    flaking zone (schistosity)

     

This simplified scheme is applied to a tectonised zone affecting the thickness of sedimentary soils.

It is clear that this ideal breakdown of structural levels can be edited or varied according to whether sedimentary coverage or granitic mass or other type of soil is considered as well as deformation also depends on the intensity of tectonic stress.

An important factor in the deformation of rocks at great depth is fluid pressure or interstitial pressure.

The presence of important interstitial pressure can induce the development of extension fractures or produce a rigid behaviour in those rocks that are usually subjected to plastic deformations.

The deformation is thus produced during an epoch during which the rocks under discussion are at different depths and therefore subjected to different tectonic stresses according to the structural level the rock belongs to. In the same epoch, different deformations can be obtained according to the structural level the rock belongs to. The deformation however has the same orientation whatever is the structural level of belonging of the deformed rock. An epoch that has the same orientation of structures corresponds to a certain tectonic phase; more tectonic phases correspond to an orogeny or to an orogenic cycle such as the Ercinico cycle or the Alpine cycle.

More tectonic phases involve overlapping of deformation effects.

4.4 Definition and Description of the Various Types of Fractures

The term fracture is taken in the sense of tectonic discontinuity. It therefore incorporates diaclases, fentes or extension fractures, stylolitic joints and faults of various sizes.

Schistosity and foliations are not comprised in this term as, obviously, are not the different types of stratification.

The geometric description of the fractures can be made by defining, for each of them, type, morphology, extension, continuity, thickness and possibly the nature of the infill as well as the degree of opening.

It is not very easy to quantify these parameters, but their systematic description is often needed in applied geology or civil engineering works.

In this section it will be attempted to describe the morphology of fractures in order to interpret their genesis and thus to identify the generating stress field so as to obtain a reconstruction of the stress history of the rock mass and therefore to evaluate the difference in mechanical behaviour between a soil tested in the laboratory and investigated on site.

4.4.1 Diaclases and Joints

Diaclases and joints are lithological discontinuities that do not show any relative movement of the two detached parts. Typically, the terms diaclases or joints are used when discontinuities are perpendicular or oblique to the directions of stratification or schistosity (Fig. 4.6a, b).
Fig. 4.6

Examples of diaclases (a) and of joints (b) (from Bles and Feuga 1981)

Furthermore, stratification joints or schistosity joints are defined in the case of discontinuity planes parallel to these two types of structures.

Diaclases and joints are often arranged in groups of two, three or four directional families.

These discontinuities are generally planar or slightly wrinkled: their linear dimensions may vary from a few decimetres to a few metres or to a few decametres.

The original thickness of diaclases is null by definition, but following to later tectonic movements, subsequent to their formation, they can open of a few millimetres and then getting infilled with quartz or calcite.

4.4.2 Extension Fractures (Fentes)

Fractures presenting lenticular-like cross sections are often observed in limestones or siliceous rocks. Their edges are open at the centre of the slit because of the tensile action perpendicular to the initial failure plane, the thickness of these extension fractures is generally centimetric and can reach a few decimetres (Fig. 4.7a–c).
Fig. 4.7

Morphology (a) and en énchelon arrangement (b and c) of tension cracks (Bles and Feuga 1981)

The length, often of the order of the decimetre, can vary from a few centimetres to more than a meter and sometimes to a decametre. Longitudinal continuity must be significantly superior to cross-sectional continuity. Extension fractures are often infilled with crystallized calcite or quartz and show a structure made up of fibres perpendicular or slightly oblique to the failure plane. They are often arranged “ en echelon “ in the same way as shown in Fig. 4.7b. Figure 4.7c shows the different cross-sectional patterns of extension fractures that can be found on the ground.

4.4.3 Faults

A fault is a fracture where the two blocks show the sliding of one part over the other in a direction parallel to the failure plane. The sliding of one block over the other often produces the presence of striations on the failure plane that indicate the direction of movement. Faults often develop in two groups with different orientations called conjugated planes. The movement of a plane is opposite to that of the conjugated plane, but these two directions of motion show the same kind of deformation (Fig. 4.8a–c).
Fig. 4.8

ac Different types of faults

There are usually three types of faults, according to the orientation and direction of movement of the two blocks in relation to the horizontal plane:
  • Normal faults, oblique to the horizontal plane, that present the lowering of one block with respect to the other (Fig. 4.8a);

  • Reverse faults, equally oblique on the horizontal, that show the oversliding of one block on the other (Fig. 4.8b);

  • Strike-slip faults, that present vertical planes and cause horizontal movements of one block with respect to the other; such movement may be dextral (right-lateral strike-slip fault) or sinistral (left-lateral strike-slip fault) with respect to the observation plane (Fig. 4.8c).

In ideal cases of Fig. 4.8, the striations are perpendicular to the intersections of conjugated faults. For normal faults and reverse faults, the striations are thus parallel to the maximum slope of the failure surface, while for strike-slip faults, the striations are horizontal.

Actually, in nature, if there is a plan of pre-existing discontinuity, it can be seen that the striations can have any position on the fault plane; so, there are intermediate positions of striations corresponding to the four major movements of the faults. They are spaced every 45° (Fig. 4.9):
Fig. 4.9

Denomination of faults according to orientation and direction of motion (from Bles J.L. and Feuga B.). i reverse fault; n normal fault; d dextral strike-slip; s sinistral strike-slip; is reverse-sinistral fault; si sinistral-reverse fault; ns normal-sinistral fault; sn sinistral-normal fault; nd normal-dextral fault; dn dextral-normal fault; id reverse-dextral fault; di dextral-reverse fault

  • sinistral-reverse faults;

  • reverse-sinistral faults;

  • reverse faults;

  • reverse-dextral faults;

  • dextral-reverse faults;

  • dextral strike-slip faults;

  • dextral-normal faults;

  • normal-dextral faults;

  • normal faults;

  • normal-sinistral faults;

  • sinistral-normal faults;

  • sinistral strike-slip faults.

The movement, measured in the direction of striations, corresponds to the real rejection of the fault.

The movement, measured according to random sections, corresponds to apparent rejections.

Normal faults cause a volume increase along the horizontal axis and a shortening in the vertical direction. They are related to distension phenomena that establish in tectonic regions.

Reverse faults and strike-slip faults correspond to a shortening along the horizontal axis, while the elongation for reverse faults takes place according to the vertical. These two major types of faults are related to horizontal compression actions during the period of the tectogenesis.

The size of faults is extremely variable, as they are produced on all scales; from small faults, of decimetric or decametric length, to large faults, of more than hundreds of kilometres. In particularly brittle materials, the faults can be found within areas of intense fracturing and the total rejection in the fault zone corresponds to the sum of all the rejections of the small faults that compose it.

4.5 Interpretation of the Mechanisms of Fractures Formation

4.5.1 Introduction

The study of fractures in terms of stresses is not theoretically possible, except in simple and ideal cases involving a continuous, homogeneous and isotropic medium, or considered as such, and in which faults appear according to planes where the shear stress is maximum.

However, rocks exhibit pre-existing fractures or discontinuities, in general, which result in planar anisotropies that disturb the distribution of the tensors of a stress field close to each of these discontinuities.

In these cases, theoretically, it is impossible to define a stress field tensor that is the same at all points of the rock mass. That is why Arthaud (1970) proposed to study fracture populations in terms of deformations. Usually, deformations (intended in the mechanical sense) are commonly distinguished between continuous and discontinuous. In the continuous deformation, two points initially close remain close (Goguel 1965). Continuous deformations, such as shrinkage, are penetrative and affect all the parts of the rock.

In discontinuous deformation, there is an important sliding of the parts that were initially close. Faults, as well as all other fractures (diaclases, joints), are not penetrative elements of the rocks at the scale of the sample or of the outcrop.

Fracturing is therefore considered to be a discontinuous deformation that has three main deformation axes generating the ellipsoid of deformations of the stress field: σ1 is the elongation axis, σ2 is the direction of the intermediate axis, σ3 is the direction of the shortening axis. These three directions are perpendicular to each other (Fig. 4.10a–c).
Fig. 4.10

The initially underformed (a) soil element subjected to continuous (b) and discontinuous (c) deformation

4.5.2 Formation of Diaclases and Joints

It is difficult to understand how diaclases and joints are formed because, by definition, these deformations do not show traces of movement. Moreover, diaclases and joints do not seem to be related to other types of deformations, such as large faults or folds.

However, the presence of diaclases or joints very often increases in the proximities of a fault and, moreover, it is understood that at least one family of them has the same orientation as the fault; in this case, it is logical to suppose that there is a relationship of continuity between minor discontinuities and fault resulting from the same stress field.

Sopena and Soulas (1973) highlighted the symmetry relationships between sub-vertical families of diaclases and the directions of global discontinuous deformation. Price (1959) showed that joints and diaclases are formed at the beginning of the deformation generated by tectonic stresses. These discontinuities can only be formed under the effect of the decompression the rocks are subjected to when these are brought from depth to surface, with a decrease in the lithostatic load, that tends to become null.

4.5.3 Formation of Extension Fractures and Fentes

The morphology of fentes or extension fractures indicates that they were formed under the effect of tension in the same way as fractures in direct or indirect tensile tests developed in the laboratory.

When fentes are filled with crystallised fibres, the orientation of fibres corresponds to the direction of the elongation (σ1). When fentes do not involve crystallisation or when the infilled crystallization does not have a fibrous structure, the elongation direction σ1 may be indicated for morphological irregularities of the fracture surfaces and these irregularities are sufficiently important to allow a reconstruction of the stress state before the opening. The alignment en échelon (Fig. 4.11(1–5)) of the extension fractures corresponds to deformed zones under the effect of shear stress. The development of these shear zones can be achieved in two different ways depending on whether the rock contains pre-existing fractures or not.
Fig. 4.11

15 Genesis and development of tension cracks en échelon

  1. (a)

    formation of extension fractures en échelon in rocks without pre-existing fractures

     
In this case, three main stages can be distinguished starting from a primitive element subjected to shear stress:
  1. 1.

    Fentes are positioned in a medium subjected to shear and their orientation is 45° with respect to the direction of movement (Fig. 4.11(2)).

     
  2. 2.

    Fentes develop by extending their ends and are always oriented at 45° with respect to the sliding direction. However, in their intermediate part, which continues to open, they undergo a torsion or rotational deformation that can be considered as a simple plastic deformation movement (Fig. 4.11(3, 4)).

     
  3. 3.

    If the plastic deformation continues, a failure may occur on a plane parallel to the direction of movement and generally passing through the centre of the sliding area. Thus, a discontinuous deformation state is obtained with the formation of a micro-fault (Fig. 4.11(5)).

     
The plastic deformation due to simple sliding is not always located to the centre of the sliding movement as shown in Fig. 4.11. On the contrary, it may develop near the ends of the extension fentes and cause them to stretch (Fig. 4.12(4)). Torsion of the two ends of fentes is much less frequent than the torsion of their central part; this case was observed by Shainin (1950), Goguel (1965) and Roering (1968).
Fig. 4.12

Torsion of tension cracks en échelon (Roering 1968), 1 Initial tension cracks; 2 and 3 torsion of the middle part of the cracks; 4 torsion of the ends of the cracks

New rectilinear fentes overlapping the previously curved ones often appear on the rocks. In the case shown in Fig. 4.13(1–3), it is likely the formation of several generations of fentes in the same shearing zone; this is a phenomenon originated by the overlapping of several contiguous extension fractures and is schematized in Fig. 4.13(3); there is therefore no doubt that the oldest (F1 and F2) fentes underwent a rotation induced by the combined effect of tensile stresses that are different over time.
Fig. 4.13

Development of several generations of tension cracks within a shearing zone (Durney and Ramsay Durney and Ramsay 1973)

  1. (b)

    formation of extension fractures en échelon in rocks with discontinuous fractures

     
During a tectonic phase, the maximum compression axis (σ1) can lie obliquely on pre-existing fractures; it will create a shear stress along the plane of these. If the fractures have pre-existing discontinuity surfaces with very accentuated roughness , materialized by the angles of faces σ1 and σ2 with the median plane of the discontinuity, there are two possible directions of movement corresponding to the previously discussed laws, but which provide different results depending on the sliding that is produced on the edges with angles α1 and α2. Figure 4.14a, b represent discontinuities where σ2 > σ1. If we assume that ϕP and ϕR are the same for the two shearing directions, it will be obtained:
Fig. 4.14

Torsion of the tension cracks en échelon

  1. 1.

    for sliding along the peaks,

     
$$\tau_{1} = \sigma \; tg\left( {\upphi_{R} + a_{1} } \right)\quad{\text{and}}\quad \tau_{2} = \sigma \; tag\left({\upphi_{R} + a_{2} } \right)$$
  1. 2.

    at the failure of the peaks,

     
$$\tau_{1} = Ca_{1} + \sigma tg\;\upphi_{P} \quad{\text{and}}\quad \tau_{2} = Ca_{2} + \sigma tg\;\upphi_{P}$$

Existing discontinuities may have a cohesive behaviour when there are:

  • a clay deposit in the stratification joints;

  • clay infilled in the fault planes;

  • mineralization or crystallisation of minerals (calcite, quartz or mica) between fracture or schistosity surfaces;

  • existence of lenses of heterogeneous materials between diaclases planes, shrinkage fractures or decompression joints.

Adherence to the walls of a diaclasis of cohesive material results in an upward movement of the τ and σ curves corresponding to the peak strength (Fig. 4.15a–c). In these figures is the behaviour of discontinuities containing lenses of cohesive material with heterogeneous roughness.
Fig. 4.15

Behaviour of discontinuities in presence of cohesive material

If a shear stress is applied to a pre-existing discontinuity consisting of a rock intercalation within a cohesive material, the movement due to shearing along the discontinuity plan is very fast, because a large amount of load induced stress is applied to it. This stress transmission is induced by the formation of tensile stresses at the ends of the existing discontinuities and by the emergence of secondary tension fractures which are oriented parallel to the direction of σ1 induced by the shearing stress (Fig. 4.15b).

The shearing of a discontinuity with heterogeneous roughness is very similar to that of the case just described: areas with large roughness have a similar role to that of Fig. 4.15b (see Figs. 4.15c).

4.5.4 Formation of Faults

Distribution and morphology of faults may be different if the rock, though cohesive, has been affected by discontinuities before the tectonic phase.

The formation of faults will first be studied in rocks without pre-existing discontinuities and subsequently in those rocks that have pre-existing discontinuities.

  1. (a)

    Formation of faults in rocks without pre-existing discontinuities

     
When rocks have no existing planar discontinuities, it is possible to use the results of laboratory experiments to interpret their deformation as a fault.

From mechanical laboratory experiments, the value of the angle θ between the failure plane and the main axis σ1 is in direct relation with the internal friction angle of the rock φ: θ = 45° − φ/2. The angle φ varies with the value of minimum principal stress σ3. When σ3 acts in compression, this can be tested using the triaxial apparatus. The value of θ varies from about twenty degrees to about forty degrees, and for elevated σ3 values (from 100 to 300 bars) angles 2 θ ranging from 60° to 80° can be obtained.

From experiences on triaxial tests, when the stress field can be represented by a revolution ellipsoid (σ2 = σ3), a series of fractures are obtained that are approximately tangent to 2 cones opposed to a failure point (Fig. 4.16); if this result is transmitted to faults formed in rocks considered isotropic, under the effect of a stress field in which the main stresses σ1 > σ2 > σ3, the faults should appear at less than 45° from axis σ1. These can be subdivided according to one or two symmetrical systems in relation to σ1. Two faults are defined as conjugated if their intersection is parallel to σ2. The elongation, i.e. relaxation, takes place in the direction of σ3 while σ1 becomes the bisector of the dihedral angle between the two faults. In the ideal case of fault type fracturing, a relative movement of the blocks is produced. This movement often produces striations on the failure planes and when these are visible they are perpendicular to the intersection of the conjugated faults (Fig. 4.17).
Fig. 4.16

Model of the formation of conjugated faults in rocks without pre-existing discontinuities

Fig. 4.17

Scheme of simple and ideal case of conjugated faults

There are two systems of intersection of conjugated faults (Fig. 4.18): in the first case the two faults may occur at the same time; at the point of intersection then numerous fractures are produced that tend to approach the central fracture point (Fig. 4.18(1)). In the second case, the two faults may occur during the same tectonic phase; the conjugate faults may therefore not be exactly contemporary: one lowers the other (Fig. 4.18(2)). When they occur, the first fault locally modifies the initial stress state and this modification may result in the formation of the second fault (Fig. 4.18(2b)).
Fig. 4.18

Two modes of the intersection of conjugated faults. 1 Simultaneous faults with trend of closing the gaps (from Muller 1963). 2 Non-simultaneous faults with movement of the one (a) towards the other (b)

  1. (b)

    Formation of faults in rocks with pre-existing planar discontinuities

     
In a rock subjected to tectonic stresses, existing planar discontinuities such as stratification, schistosity, or a set of diaclases may be the location of failures by sliding. Mechanical tests carried out in the laboratory show that the resulting faults can provide, depending on σ1, angles ϑ ranging from fifteen degrees to about sixty degrees.
These experiences also show that failure by sliding is more easily produced when the pre-existing planes form an angle θ with σ1 of 30° or 35°. If the angle θ is less than 30° (15° < θ < 30°), the sliding of the parts according to the existing discontinuities is often accompanied by extensional failure of the rock mass located between the failure axes and the main compression axis σ1. The faults thus created will not have planar surfaces but stepped surfaces where one face will undergo a sliding motion and the second an extensional movement (Fig. 4.19).
Fig. 4.19

Collapse of rocks containing a plane of discontinuity

If the angle θ is greater than 30° or 35° (35° < θ < 60°), axis σ3 and axis σ1 of the stress field are respectively stronger and less strong than those applied on the planes rotated 30° or 35° from σ1. The compression stress required for the failure therefore must be necessarily greater.

In the rocks that involve two families of discontinuities of the same nature, the direction of axis σ1, applied to them by a new tectonic phase, will no longer be, in general, the bisector of the dihedral defined for the two families. It is therefore foreseeable that shearing failure will be more easily produced according to one of the two families of discontinuities for the sole reason of having a similar orientation. If the two families of discontinuities are of a different nature (e.g. stratification and schistosity), the ease of failure, always function of the orientation of the pre-existing planes, will be accentuated or opposed by the mechanical characteristics of the plane of each fault (internal friction angle φ′ and cohesion cu) (see Figs. 4.19 and 4.20).
Fig. 4.20

Failure mode for simple compression of a rock containing two sets of pre-existing discontinuities (Masure 1970). a Shear along stratification planes; b and c shear along schistosity planes; d anisotropic extension and formation of new shear cracks

  1. (c)

    Formation of faults with non-planar pre-existing discontinuities

     
Natural discontinuities are not strictly flat, especially those along the shear surface. At the scale of the outcrop, for example, discontinuities considered as flat may have, instead, rather regular roughness due to an arrangement in succession. To interpret these deformation figures observed on the surface of natural faults, it is practical to use a model of discontinuities with regular and dissymmetric roughness used in the study of rock mechanics. For example, let us consider a fracture F with two opposite failure surfaces, F1 and F2, having respectively angles i1 = 15° and i2 = 30° with the median plane of fracture F (Fig. 4.21). In cross-section, F1 surfaces are about twice as long as F2.
Fig. 4.21

Model of failure with regular and asymmetric roughness (profile perpendicular to the crack and to the roughness and parallel to the direction of movement) (from Bles and Feuga 1981)

This failure model is in common for the two directions of movement, which can be defined as “dextral” and “sinistral” movements in Fig. 4.21. For each of these movements, there may be two cases: sliding on surfaces F1 and sliding on surfaces F2 according to the orientation of the applied thrust in relation to the median plane of fracture F. In this figure, the planes corresponding to F1 have much greater areas than those of F2. However, these solutions are dependent on the direction of the maximum thrust tensor; therefore, according to the movement that is considered, extension or shortening will be obtained for surfaces F2. In fact, the dimensions of surfaces F2 are defined by the sliding of surfaces F1, that are produced for a sinistral movement in the fault model F of Fig. 4.21.

This sinistral movement with sliding on F1 is only possible when the angle between σ1 and F1 has a value around 15°, i.e. when the angle between σ1 and the median plane of F is between 0° and 45° (Fig. 4.22a).
Fig. 4.22

Scheme of the evolution of a non-planar discontinuity with slipping faults, diverging faults and smoothing of roughness (from Bles and Feuga 1981)

The movement can develop as shown in Fig. 4.22b, c; i.e. when:
  1. 1.

    at the beginning, shearing is applied to plane F1 and at the same time an extension according to planes F2 is produced (Fig. 4.22b). This is due to the fact that, during sliding, F1 surfaces can be striated, while the space generated from the opening on F2 can be infilled with calcite or quartz crystals, depending on the rock being calcareous or siliceous.

     

Calcite or quartz crystals often have a fibrous structure parallel to the direction of sliding and therefore parallel to the striations of F1 planes. In this case, it is likely that crystallization occurs during sliding. The arrangement of striations allows to define the direction of movement. This movement is accompanied by dilatancy perpendicular to the median plane of fracture F and therefore the fault is defined as stepped fault (Fig. 4.22b).

If the rock is made of soft cohesive soil, generally shallow-depth soil, this fracturing pattern is indistinguishable; however, by means of a careful analysis of the ground, it is possible to find dissymmetric gaps in succession along a section having an angle of about 45° with respect to σ1 and a material often at the plastic state.
  1. 2.

    During the sliding, the two contact surfaces become more distant and this results in an increase in the normal intermediate tensor. The breaking of the edges delimitating the F1 and F2 planes can therefore be achieved (Fig. 4.22c). Summarising, the surfaces of each block can undergo the sequences: striated rock → fibrous crystallization → crushed rock, indicating the direction of movement (Fig. 4.22d).

     

When the breaking of the edges occurs at the beginning of the movement, crystallization may be absent and replaced by rock fragments of the edges themselves of the block above.

Let us now analyse the case of faults in which the sliding of the blocks causes a shortening of surfaces F1 and F2 (Fig. 4.23a–d). Sliding on surfaces F1 is possible if tensor σ1 forms with such surfaces an angle between 15° and 60° or if σ1 is inclined of between 30° and 75° with reference to the average plane of fracture F (Fig. 4.23a).
Fig. 4.23

Scheme of the evolution of a non-planar discontinuity with slipping and contraction (Bles and Feuga 1981)

Contraction of F2 and sliding on F1 can occur only when rock dissolution is possible, which implies a relatively high porosity value (Sellier 1976; Sellier and Morlier 1976). If this dissolution is possible, the sliding on F1 is accompanied by the formation of dissymmetrical stylolithic joints developed from surfaces F2 (Fig. 4.23b, c). This structure corresponds to a contraction of the two surfaces in the perpendicular direction to the median plane of fracture F and therefore the faults defined as “shortening faults” (Arthaud and Mattauer 1969).

When rock dissolution is no longer possible, a stop in the movement takes place at stage b or c (Fig. 4.23). In this case, a breaking of the most angular parts that exist in the median plane of the fracture is possible. If rock dissolution can go on indefinitely, the state of the deformation is likely to be similar to that of Fig. 4.23d.

Rocks having porosity compatible with the development of dissolution under tectonic stress are essentially limestones, sandstones, silts and clays.

In siliceous, non-porous, rocks, crushing of the rocks of F2 surfaces of the fault is produced, with shortening of the fault itself. Crushing can be accompanied by deep striations marked on surface F1 and, in the case of very high tectonic thrust, crushing can evolve and constitutes small “mylonitised“ areas (Petit 1976; Blès and Gros 1980).

4.6 The Stress Field

4.6.1 Identification of the Stress Field

Geologists often ask the question of where the stress field comes from. It can be said that this is originated as a “reaction” of bodies to external forces that are applied to them.

The rocks of the Earth’s crust are permanently subjected to forces. These may vary considerably according to the fact that they are subjected to compression, relaxation or at rest in an undeformed plate.

The in situ analysis, for the purpose of the prediction, the prevention and the control of hydrogeological instability, must take into account of the structural setup of the local orogeny and therefore of the processes of deformation and shearing. In fact, the pre-failure deformation is due to the combined effect of the stresses induced by the stress field and the regime of neutral pressures that develops in the ground. Nature and intensity of such deformations depend on the pre-existing stress field while the depth of the sliding surface is a function of the orientation of the maximum thrust tensor. From this it follows that if the fractures have a predominant orientation approximately parallel to the slope, the sliding surface is shallow; on the contrary, if the fractures are almost perpendicular to the slope, the shearing surface of the landslide is very deep; furthermore, in the first case the time occurring for the achievement of the deformation at failure is always lower than the other case. Obviously, the propagation mechanism in the shear area of a landslide in cohesive soils is much more complex than how it is generally described here for a number of variables such as those linked to consolidation and/or viscous ground deformations that may cause a redistribution of the stress field.

Let us consider a small element in the Earth’s crust with infinitesimal surface dS of random orientation, rotating around a point M (Fig. 4.24). The surface is subjected to a force dF. It is called tensor in M the limit of the ratio σ = dF/dS when dS tends to zero (actually, a stress is a force divided by an area). Let us consider further elements of surface S, passing through the same point M but having different orientations: two cases can be produced:
Fig. 4.24

Sketch taken from Mattauer (1973)

  1. (a)
    tensor (σ) remains constant whatever is the orientation of dS. In this case the tensor is termed as hydrostatic because it is the condition found within liquids. In fact, if using a manometric membrane at a given point of a liquid and allowing it to assume all the possible orientations it is found that the pressure is always the same. This tensor, which is called isotropic, can be represented by a sphere with radius σi (Fig. 4.25a).
    Fig. 4.25

    a and b Representation of an isotropic (sphere) and anisotropic (ellipsoid) stress field (from Mattauer 1980)

     
  2. (b)

    the tensor varies in intensity and direction, while the orientation of dS varies. At point M there are therefore different possible values of σ.

     

When the rock is homogeneous and continuous, the locus of the end of the vector originated from M, having σ for length and dF as the direction is an ellipsoid, called ellipsoid of tensors or tension field where the axes are generally known as σ1, σ2, σ3 and called the main axes of the stress field, respectively the main axis, the intermediate axis, the minimum axis; therefore, the stress field is triaxial (Fig. 4.25b) and defined by an ellipsoid of revolution.

The state of stress of rocks in the Earth’s crust is given by:
  1. (a)

    Rocks at rest in a stable plate (very deep)

     
In this case, a rock element located, for example, at depths of 5 km, is subject to the weight of the overlying rocks, which can be easily calculated if the natural density (γ) of the rock is known; if the density γ = 25 kN/m3, the pressure at 5 km depth is 125 MPa. Now it can be observed that this value does not change significantly if pressure measurements are made in the various directions. In short, it can be considered that in that portion of the Earth’s crust a lithostatic stress field is observed, in which σ1 ≈ σ2 ≈ σ3 and which increases in intensity with depth. This system is obviously extremely viscous and deformational effects only appear in the long-term (Fig. 4.25a).
  1. (b)

    Rocks subjected to stress fields

     
The stress field is of triaxial type and is characterized by σ1 > σ2 > σ3 (Fig. 4.25b). It can be considered as due to the overlapping of the lithostatic load σi and the tectonic thrust σt. Triaxial compression can occur in a continuous medium (intact rock) or in a discontinuous medium (fractured rock).

4.6.2 Stress Field in a Continuous Medium

Let us start from the case of intact rock (continuous medium) (Fig. 4.26(1–3)). Solutions that may occur are:
  1. 1.

    σ1 is vertical: a relaxation regime takes place, whatever the position of σ2 and σ3. The striations have the same direction of σ1 and pitch ϑ = 90°.

     
  2. 2.

    σ 2 is vertical: strike-slip regime. Strike-slip can be both right-lateral and left-lateral. Striations have the same direction of σ 1 and pitch ϑ = 0.

     
  3. 3.

    σ 3 is vertical: reverse compression regime, whatever the arrangement of σ 1 and σ 2 . Striations have the same direction as σ 1 and pitch ϑ = 90°.

     
Fig. 4.26

13 Cases for intact rock (continuous medium)

It should be noted that, in a continuous medium, striations always have the same direction of σ 1 . The direction of movement of blocks is therefore always parallel to that of σ 1 . The pitch is ϑ = 0 or ϑ = 90°. The rejection (R) is always of pure type.

If rock P is intensely fractured (strongly discontinuous) it can behave as an isotropic material; especially if the rock mass contains several fracture planes. In this case, the determination of the stress regime can be reconducted to the type of the intact rock.

As it can be observed, the type of fault within a continuous material is a function of the orientation of the stress field, i.e. it is function of the direction of the tensors in the space. In addition, the direction of the movement is always the one of σ1 and the blocks always move towards σ3.

4.6.3 Stress Field in a Discontinuous Medium

If the rock is already fractured, i.e. it contains previous surfaces of discontinuity (discontinuous medium), then it is necessary to distinguish which of the three stress axes lie on the fracture plane and which are oblique to it (Fig. 4.27—cases A, B, C).
Fig. 4.27

AC Cases for fractured rock (discontinuous medium)

When a tensor lies on a pre-existing fault plane it becomes null; the movement of the blocks and then the direction of the striations will be given by the resultant of the two remaining stress components. The deformation is logically applied to a two-dimensional plane. In the case, instead, of pre-failure plane oblique to a pre-existing shear surface, then the stress field is three-dimensional and the motion will be the one deriving from all of the three components, σ1, σ2, σ3. Therefore, all of the cases that may occur are shown in Fig. 4.27A–C.

  • CASE A—The stress field consists of σ1 = vertical, σ2 and σ3 = horizontal:

A1—Case where σ1, vertical, lies on an existing shear surface; σ1 = 0 and the stress field becomes two-dimensional with σ2 that assumes the role of the prevailing tension axis. The shear deformation will be, therefore, a strike-slip fault, dextral or sinistral according to the position of σ2 on the horizontal plane. Sub-horizontal striations with pitch ϑ = 0 can be observed (Fig. 4.27A, 1a, b).

A2—Case where σ2, horizontal, lies on the pre-existing failure plane. A normal pure fault is obtained, in the sense that the movement of the block free to move is vertical and therefore the striations have a pitch ϑ = 90° whatever the position of σ3 on the horizontal plane (Fig. 4.27A, 2a, b).

A3—The pre-existing rupture plane contains the stress axis σ3. Tensor σ2, in this case, becomes the one with lower intensity and therefore the movement follows an orientation in which a pure normal fault (pitch ϑ = 90°) is obtained whatever the position of σ2 (Fig. 4.27A, 3a, b).

A4—The pre-existing failure surface is oblique to all of the three stress axes. In that case, a transtensive normal movement of the block free to move is obtained. The pitch ϑ will assume a value n, resulting from the intensity of σ2 and σ3, and is between 30° < n < 90° (Fig. 4.27A, 4a).

  • CASE B—The stress field is composed of σ2 = vertical, σ1 and σ3 lie on the horizontal plane. The regime is of a strike-slip type. However:

B1—If σ1 lies on the pre-existing failure plane, the vertical σ2 component prevails as the axis of major tectonic thrust and therefore a pure normal motion is obtained in the sense that pitch ϑ = 90° (Fig. 4.27B, 1a, b).

B2—The pre-existing failure plane contains σ2, vertical; a pure strike-slip movement with pitch ϑ = 0 is obtained (Fig. 4.27B, 2a, b).

B3—If the pre-existing failure plane contains σ3, once again there is a two-dimensional stress field where σ2, vertical, is the axis with lower tectonic thrust. In this case, a pure reverse fault is obtained i.e. the pitch ϑ = - 90° (Fig. 4.27B, 3a, b).

B4—The pre-existing failure surface is oblique to all of the three stress axes. The movement may be of transtensive normal or transpressive reverse type according to the position of σ1 lying on the horizontal plane (Fig. 4.27B, 4a, b).

  • CASE C—The stress field consists of σ3 vertical, σ1 and σ2 lie on the horizontal plane. The regime is of reverse type. However:

C1—If the pre-existing failure surface contains σ1, horizontal, a pure reverse motion (pitch ϑ = −90°) is obtained. The striations direction is in line with that of axis σ2 (Fig. 4.27C, 1a, b).

C2—In the similar case where σ2, horizontal, lies on the failure surface, the movement of the block is reverse with pitch ϑ = −90° (Fig. 4.27C, 2a, b).

C3—If the pre-existing failure plane contains σ3 a pure strike-slip movement is obtained, in the sense that the pitch ϑ = 0 (Fig. 4.27C, 3a, b).

C4—The pre-existing failure surface is oblique to the three stress axes. In this case the movement is transpressive reverse and the pitch will be −90 < ϑ < −30 (Fig. 4.27C, 4a).

In summary: Tables 4.1 and 4.2.
Table 4.1

Type of faults and characteristics in function of the stress regime

Stress regime

Types of shear deformations

Tensors on pre-existing planes

Pitch

Ellipsoid

Notes

Triaxial distension

σ1 = vertical

1 < R < ∞

30 < f(R) < 90

Pure normal

Pure normal

Strike-slip

Transtensive normal

Not existing

σ2

σ1

Oblique

90°

90°

0

ϑ = n

σz > σx > σy

σz > σx > σy

σz > σx > σy

σz > σx > σy

Does not allow reverse faults

Does not allow transpressive faults

Triaxial compression

Strike-slip regime

σ2 = vertical

0 < R < 1

−30 < f(R) < 30

Strike-slip

Strike-slip

Pure normal

Pure reverse

Transtensive normal

Transpressive reverse

σ2

σ1

σ3

Oblique

Oblique

0

0

90°

−90°

ϑ = n

ϑ = −n

σy > σz > σx

σy > σz > σx

σy > σz > σx

σy > σz > σx

σx > σz > σy

σy > σz > σx

Allows all the solutions

Triaxial compression

Reverse regime

σ3 = vertical

−∞ < R < 0

−90 < f(R) < −30

Pure reverse

Pure reverse

Strike-slip

Transpressive

Reverse

σ1

σ3

Oblique

−90°

−90°

0

ϑ = −n

σy > σx > σz

σy > σx > σz

σy > σx > σz

σy > σx > σz

Does not allow transtensive faults

Does not allow normal faults

Radial

distension

σ1 = vertical

R = ∞

Pure normal

Pure normal

Pure normal

σ2

σ3

90°

90°

90°

σz > σy = σx

σz > σy = σx

σz > σy = σx

Normal faults arranged in radial direction

Does not allow strike-slipping

Uniaxial distension

σ1 = vertical

R = 1

Pure normal

Pure normal

Trascorrente

σ2

σ1

90°

90°

0

σz = σy > σx

σz = σy > σx

σz = σy > σx

Does not allow transtensive normal faults

Radial

compression

σ3 = vertical

R = −∞

Pure reverse

Pure reverse

Pure reverse

σ2

σ1

−90°

−90°

−90°

σy = σx > σz

σy = σx > σz

σy = σx > σz

Reverse faults arranged in circular pattern

Does not allow strike-slip faults

Uniaxial

compression

σ3 = vertical

R = 0

Pure reverse

Pure reverse

Strike-slip

σ2

σ3

−90°

−90°

0

σy > σx = σz

σy > σx = σz

σy > σx = σz

Small reverse faults

Does not allow normal faults

Table 4.2

Characteristics of the deformation for continuous and discontinuous materials

Discontinuous material

 

Continuous material

 

Vertical axis

σ1

σ2

σ3

Strike-slip

ϑ = 0

Faults containing

σ1, σ2, σ3

σ1

σ2

σ3

Rejection always of pure type. Striations with same orientations of σ1 except when σ1 is contained within the fault

σ2 vertical

Pure normal

ϑ = 90°

σ2

σ1

σ1 vertical

σ3

Pure reverse

ϑ = −90°

σ1

σ3 vertical

σ3

σ2

Transtensive

ϑ = n

Oblique faults

σ1, σ2, σ3

σ1

σ2vert.

σ1=X

Striations always oblique

−ϑ = n

 

Transpressive

ϑ = −n

σ2vert.

σ1=y

σ3vert.

  1. (a)

    In a continuous medium the failure plane will always be determined by σ1 and the direction of the striations will be parallel to that of the axis of maximum thrust, whether it is vertical or horizontal; the pitch can therefore always assume a pure value, which is ϑ = 90° if σ1 is vertical and ϑ = 0 if it is horizontal while with horizontal σ1 and σ3 vertical ϑ = −90°, i.e. assumes a reverse regime.

     
  2. (b)

    In a discontinuous medium, the failure plane will always be determined by σ1 as well as the direction of the striations, except when the maximum thrust axis lies on the pre-existing failure plane.

     
  • A strike-slip movement is obtained when the vertical component is that contained in a pre-existing discontinuity plane. This means that all strike-slip structures in a discontinuous medium admit a pre-existing failure surface and thus can show a double striation system, of which the most recent one is superimposed to the older one. In the Apennines, generally, the striations showing a strike-slip movement are always overlaid by pre-existing normal faults (in Jurassic limestone).

  • In the case of pre-existent failure surfaces oblique to the three axes of the stress field, the strike-slip movement (σ1 horizontal) is never pure (ϑ = 0) but only transtensive or transpressive.

  • A transtensive normal regime is obtained when σ1 is vertical whatever the orientation of σ2 and σ3; or when σ2 is vertical and σ1 is horizontal and corresponds to the axis of the X.

  • In no cases with σ3 vertical is obtained a transtensive normal regime.

  • A transpressive reverse regime is obtained when σ3 is vertical or when σ2 is vertical and σ1 corresponds to the Y axis.

  • In no cases with σ1 vertical is obtained a transpressive reverse fault.

It is therefore very important to take into account the conclusions obtained using the model by Bott (1971). The direction of movement on a fault plane is function of the directions of the principal tensor but at the same time they depend on the relationship between the values of the tensors themselves. This relationship is called tensors ratio (R) and corresponds to the shape of the ellipsoid of the stress field which defined the regime of tectonic forces to which the rock is subjected. In fact, getting back to the study of the behaviour of already fractured anisotropic rock, it should be considered that the direction of the striations corresponds to that of the movement of the block, that is to that of the tangential component (τ) of a tensor (σ) applied to the plane of a failure surface (Fig. 4.28). The direction of motion is defined at failure by an average tensor, resulting from the three principal tensors of the field where the deviation due to the pre-existing plane can be calculated by measuring the striations of a set of fractures. In Fig. 4.28, therefore, σ is an average tensor of the stress field acting on a pre-existing surface.
Fig. 4.28

p = line projection (a) perpendicular to plane. tg θ = direction of vector t producing striae l, m, n = direction cosines of P with reference to axes x, y, z

Let us now consider the relationship existing between the values of the principal tensors and the direction of the striations on a pre-existing fracture plane.

By means of the principal directions of stress tensors, all the sliding directions can be obtained on the same shear plane by varying the values of the three principal tensors. The diagram is that of Fig. 4.9 for the definition of faults. In this diagram it can be observed that the direction of motion is independent from the arrangement in the space of the stress field, since it is function of the ratio (R) among the values of the three tensors.

Thus, since the striations give the direction of the tangential component (τ) of the average tensor (σ) on the fracture plane, the pitch of the striations, instead, is dependent solely on the ratio
$$R = \frac{{\sigma^{2} - \sigma_{x} }}{{\sigma_{y} - \sigma_{z} }}$$
among the three principal tensors.
This relationship is valid only if the failure plane is oblique with respect to the three principal axes of the stress field, i.e. with respect to the σx, σy, σz tensors. In fact, by choosing a reference axis system x, y, z parallel to the principal axes of σx, σy, σz, tensors for any sliding surface, oblique to the axes, where the pole is located, from direction cosines l, m, n, it can be determined the direction of the sliding through the formula
$$tg\,\delta = \frac{n}{lm}\left\{ {m^{2} - \left( {l - n^{2} } \right)\frac{{\sigma_{2} - \sigma_{x} }}{{\sigma_{y} - \sigma_{z} }}} \right\}$$
(Bott’s Formula, 1971).

Pole P is the projection of the straight-line perpendicular to the plane (in stereographic projection).

By means of this relationship it can be seen that, when the value of one of the direction cosines varies, so does the direction of tg δ, i.e. the movement of the blocks, as long as the failure plane is oblique to the axes of the stress field. The only surfaces that escape this rule are those on which lie the principal axes (σx, σy, σz). In fact, if a failure surface contains the z axis (Fig. 4.28), i.e. it is vertical and in agreement with the movement, axis σ will be perpendicular to it and will be perpendicular to z; in this case, the angle formed by the straight line (σ) and the z axis will be 90°. The direction cosine “n” of this angle will therefore be n = cos δ = cos 90° = 0 and the whole expression will be equal to 0, or tg δ = 0 (Fig. 4.29).
Fig. 4.29

Director cosine orthogonal to z

The result is similar if the plane is parallel to axes x or y. This proves that if a pre-existing plane contains one of the tensors (σx, σy, σz), the direction tg δ of the component τ is no longer dependent on the ratio of the tensors
$$R = \frac{{\sigma_{2} - \sigma_{x} }}{{\sigma_{y} - \sigma_{z} }}$$

In this case τ and y are equal to 0.

The pitch is independent from the regime of R (n = cos 90° = 0). Thus, it is equal to 0. Therefore, the striations will have an arrangement parallel to the direction of the pre-existing shear plane, condition that can occur only when the movement of the blocks becomes pure strike-slip.

If the plane contains σx or σy, i.e. it is perpendicular to σz, the pitch will be 90°, i.e. the striations will be perpendicular to the direction of the plane, condition that can only occur when the movement is purely reverse or purely normal.

Let us now observe the practical implications of this theoretical model.

First of all, it should be noted that all this analytical process is aimed at determining the relationship between the striations and the triaxial stress field. Actually, up to now we have dealt with this topic only for the relationship between movement (striations) and acting tensor. It is important to understand that all the operations so far expressed lead to a fact, irrefutable, that striations are the effect of a particular movement. Starting from this certainty, Bott sought a relationship between the striations produced and the acting stress field. The result is extremely reliable and therefore the structural interpretation is based on objective, incontrovertible evidence. There is therefore a correspondence among the values of the ratio of the tensors and the different tectonic regimes. The value of the ratio of the tensors of a stress field has a precise tectonic meaning, very useful for the study of a deformation and for the detection of the stress field established in the ground and sometimes still acting.

4.6.4 Method of the Minimum Dihedral (or Quick Method)

In the case of sub-vertical microfractures with strike-slip rejections, where the vertical component is negligible, an acute angle can be determined, the smallest possible between the two rupture planes.

The bisector of this angle gives the direction of σ1.

This is a method for determining the axes of shortening and of elongation, also known as “quick method”. It can only be used in the presence of small, strike-slip fractures, with sub-vertical planes. The method cannot be used if a normal or reverse rejection is associated to strike-slip surfaces, i.e. there is rejection with vertical component. The method consists in positioning the shortening axis along the bisector of the smallest angle between two strike-slip surfaces, dextral and sinistral. At 90° there will be the elongation axis.

The results obtained are comparable to those deriving from tension fractures and from the corresponding stylolithic peaks, respectively corresponding to the directions of the elongation and of the shortening.

The method has its validity as it is based on movements of planes close to the vertical and can be considered very often as neo-formed.

4.7 The Analysis Process

The combined effect of variations in the total stress state and of the neutral pressure regime due to fracturing, erosion, meteoric precipitation and seismic action results in the decrease of the intensity of the effective stresses field in predominantly clayey soils. At present, it is not yet very clear what are the major effects that result in a progressive reduction in the shear strength and in the size of the yield surface in pre-failure processes if not assigning to time an influence on the combined action of the factors mentioned in previous paragraphs. Picarelli (1999), quoting Eigenbrod et al. (1992), writes that they “carried out specific laboratory investigations on the behaviour of slightly overconsolidated clays of medium plasticity subjected to cyclic oscillations of the back pressure, simulating in that way fluctuations in the neutral pressure regime. They observed that the effects produced are similar to those due to creep induced by monotonic increments of the deviatoric stress and consist in the accumulation of irreversible deformations of such an entity depending on the number of cycles, that means depending on time. In principle, also groundwater excursions could cause deformation and movements in the ground, especially when considering the additional effect of time on soil properties (mechanical decay)” (Fig. 4.30a–c).
Fig. 4.30

ac Hypothetic evolution of horizontal movement (a) of the pre-failure stress state (b) and of post-failure stress state (c) at a point of a natural slope subjected to fluctuations of the superficial groundwater table (from Picarelli 1999)

This problem presents some typical aspects. First of all, for a natural period, the stability analysis carried out using the Global Limit Equilibrium Method (Terzaghi 1943 and Taylor 1948) is always a verification of the existence of a limit condition of equilibrium. Only definitely unstable slopes are analysed, on which evident mass movements have already occurred, which later increase over time in relation to the evolution of some factors external to the slopes themselves. Consequently, the analysis of the limit equilibrium state of a slope cannot proceed by determining a coefficient of safety to which is attributed a supposed equilibrium condition of the whole unstable mass. In fact, if the analysis is correct, the safety coefficient is equal or very close to one. In addition, the variations in the stress state that can occur in a natural slope because of the evolution of external factors are always very limited, therefore variations in soil strength are modest and, considering also the relative value of mechanical parameters c and φ′, it can be obtained that φ′ has slight influence on the variations in soil strength. The deduction of strength parameters of cohesive soils from stability analyses is a problem that is not easy to be sorted out because it often translates into the definition of implicit relations not correctly set and, above all, not considering the evaluation of the time factor. In addition, the evaluation in the laboratory of the “effective cohesion” c′, unlike of the angle of shearing φ′, varies considerably with the size of the samples tested as it is strongly influenced by the mesostructures existing in the lithic units present in the ground.

The analysis of unstable slopes can be carried out much more easily when failure occurs in terms of total stresses and, with reference to Figs.  2.21 and  2.25, where ϑ = 45° for τmax, so that the cohesive material assumes a failure envelope with φ = 0 and the failure resistance is solely supported by the undrained cohesion cu
$$\tau_{max} = c_{u};$$
condition that can only happen after tensile tests or under saturated conditions of cohesive soils.

Unfortunately, parameters φ = 0 and τmax = cu are not the physical characteristics of the material but only mechanical parameters that describe a certain kind of behaviour. This is to say that the limited variation in the tension state, which characterizes the problem, is an artifice that allows to overcome the difficulty of describing the behaviour of the soil in terms of effective stresses and makes the adopted model more similar to the real behaviour of the soil in order to solve more easily practical issues. However, on one hand increase the difficulties arising from the use of a strength parameter, cu, heavily influenced by mesostructural, heterogeneous and anisotropic soil characteristics; on the other hand, the possibility to assess correctly the effect of the variations in the effective stresses on soil strength and slope stability conditions ceases to exist.

In conclusion, strength parameters expressed in terms of effective stresses are better defined than those in total stresses; but cohesion c’ has a great influence on the stability of the slopes, it depends significantly on the mesostructures, in particular on discontinuities and swelling phenomena, while the advantage of the higher reliability of the values of friction angle φ′ is often cancelled by the uncertainty in the values of interstitial pressure. On the contrary, solutions in terms of total stresses, which offer significant advantages in terms of analytical and experimental simplicity, always require the support of a wide and extended experience on the mechanical behaviour of the soils involved, which allows to establish reliable empirical correlations between undrained cohesion cu and the actual in situ strength. However, they are really valid only in sufficiently homogeneous soils.

The difficulty in defining appropriate values of the strength parameters and of the factors related to slope stability analysis according to the current Global Limit Equilibrium Method is, therefore, one of the problems in geotechnical engineering presenting greater difficulty and uncertainty in the results, especially if the research is aimed to the determination of predictions and prevention conditions of the hydrogeological risk of a slope.

The problems of the instability of slopes in clayey lithotypes are therefore to be addressed through a detailed analysis of the geological and structural characteristics of the deposits as well as of their deformational behaviour by means of long-term monitoring, with particular reference to phenomena that accompany the development of progressive failure, as they are the ones that affect the mechanics of failure either along pre-existing shearing surfaces or along newly formed surfaces. Therefore, obtaining a preventive indication of the hydrogeological risk of a slope, the essential condition for starting the calculation stage, is at least to know beforehand the real depth of the sliding surface in the ground before it is activated, both in case of a pre-existing landslide and in the case of a recent landslide.

Footnotes

  1. 1.

    The Content of Sects. 4.14.4 is Inspired by the Manual “La Fracturation des roches” by Jean-Louis Bles–Bernard Feuga (1981)

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of BasilicataPotenzaItaly

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