Rock damage mechanics

Damage mechanics of rocks is a concept that requires a comprehensive presentation as it involves multiple physical and sometimes chemical phenomena. Mechanical damage is inextricably linked to cracking and the resulting loss of mechanical properties. A collateral effect of damage is the alteration of the fluid transport characteristics of rocks that can have geoenvironmental consequences. This chapter outlines damage characterization at the scale of the microstructure of a rock specimen and then presents different analytical approaches available to mathematically describe the evolution of the mechanical properties. These approaches essentially rely on Fracture Mechanics and Continuum Damage Mechanics. Finally some examples of numerical modeling are presented to illustrate their application in the field of rock engineering. The chapter also presents a brief review of the techniques that can be adopted to integrate results of damage mechanics to fluid transport phenomena.

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Rock damage mechanics

1 Introduction

Damage is a consequence of physical degradation that can impair or progressively weaken a structure. In geological formations, damage can affect the rock matrix, the rock discontinuities or the assembly of both: the rock mass. In rock engineering, any damage can be of concern to the serviceability and stability of structures such as rock slopes, underground openings, deep wellbores, etc.

Environmental issues related to transport and diffusion of harmful substances (radionuclides, CO₂, etc) may arise due to the development of damage zones in rock formations. Damage to rock or engineered structures unavoidably generates micro to macro-cracks, which increases the rock permeability. This has been extensively investigated over the last few decades for deep geological disposal of radioactive waste (Hudson et al., 2009). Rock formation damage is also an important feature for the extraction of unconventional resources such as shale gas (Zimmerman, 2010), or geothermal energy (Li et al., 2012).

Rock damage can occur due to either monotonic or repeated mechanical action (alteration of the in situ stress field), changes in temperature, transformation resulting from chemical processes or a variation in the pore pressure.

Today, several different methods have been adopted to address damage from a mechanical point of view. This chapter aims to clarify the advantages and disadvantages of the approaches currently used to study mechanically-induced damage.

2 Experimental evidence

2.1 Investigations at the laboratory scale

The problem of damage will first be addressed from an engineering point of view. Let us imagine an intact rock specimen, void of any defects, subjected to a uniaxial state of stress in compression. Initially, the specimen will store the energy provided by the loading system. When this energy exceeds the specific potential energy associated with the inter-atomic bonds of the material, these will be progressively broken and micro-cracks will appear. The final damage stage occurs when the rock specimen can no longer sustain additional stress and the cracks coalesce to form a large macro-crack. Between the incipient loading and the final collapse, the specimen undergoes successive transitional states of damage, from slight to moderate and finally severe damage.

Figure 1a illustrates a typical stress–strain curve for a rock specimen axially loaded in compression. Measurement of the axial strain ε₁ and lateral strain ε₃ allows the computation of the volume variation of the specimen, referred to as the volumetric strain, ε_v. After the crack closure (σ_cc), the onset of damage is evidenced by the loss of linearity in the variation of the lateral deformation. This corresponds to the nucleation of cracks, which results in a slower decrease in volume change (σ_ci, for crack initiation). The specimen is then slightly damaged. With a further increase in the loading, the crack will grow steadily until the energy provided by the loading system leads to unstable crack propagation. The transition point between these two propagation modes (σ_ci, for crack damage) is indicated by the change of the volume variation passing from compaction to dilation. At this time, the material is severely damaged. The loading process then continues until total failure of the specimen is observed (σ_cf, for crack failure). Figure 1b shows a thin section extracted from a shale rock specimen after failure. Macro-crack networks as well as micro-cracks are clearly visible.

The damage process described above has been reported and interpreted by several authors (e.g. Martin, 1997; Patterson & Wong, 2005) since the pioneering works of Brace et al. (1966, 1972) and Bieniawski (1967). At the present time, standard laboratory tests can be supplemented by additional measurements such as the Acoustic Emission activity (Stanchits et al., 2006) or changes in ultra-sonic wave velocity induced by the degradation of the specimen (Fortin et al., 2007; Pellet & Fabre, 2007). Indeed, as shown by Keshavarz et al. (2009), such data can also help to delineate the different damage thresholds.

(a) Typical stress–strain curve for a rock specimen axially loaded with characteristic thresholds defining different damage intensity; (b) Macro-crack and micro-crack networks in a shale specimen (80 mm in height) loaded in mono-axial compression (Pellet,

Figure 1 (a) Typical stress–strain curve for a rock specimen axially loaded with characteristic thresholds defining different damage intensity; (b) Macro-crack and micro-crack networks in a shale specimen (80 mm in height) loaded in mono-axial compression (Pellet, 2015).

2.2 Physical interpretation of cracking of rock at the micro-scale

Rocks are heterogeneous materials with many flaws and micro-defects such as micro-pores or micro-cracks. The latter play a significant role in the changes that occur in the mechanical properties of a rock during the loading process. Micro-defect characteristics have been widely studied; for example, Kranz (1983) distinguished four types of cracks:

– Cracks associated with grain boundaries that separate two alike or different crystals or crystals and cement.
– Inter-crystalline cracks (or inter-granular) initiated at the contact between the grains and pass along one or more grains.
– Intra-crystalline cracks (or intra-granular) which propagate in the grain; the length of these cracks is often much shorter than the grain size and the opening can be very small (˂ 10 μ).
– Cleavage cracks are particular intra-crystalline cracks separating the cleavage planes of a crystal; they occur as parallel networks to the symmetry planes of the mineral.

Further information on the rock fabric and petro-physical characteristics is given by Davis and Reynolds (1996).

During the loading process, micro-cracks will nucleate or propagate, eventually coalescing to form macro-cracks (Figure 1b). In order to be considered a macro-crack, several grains or crystals have to be involved. The images presented in Figure 2, taken with an optical microscope, show different types of cracks. Figure 2a illustrates cracks in the clay matrix of a Callovo-Oxfordian marl showing the debonding of calcite crystals (Fabre & Pellet, 2006). Figure 2b shows trans-granular cracks in a gabbro specimen subjected to very high stresses, up to 1.7 GPa (Pellet et al., 2011).

Figure 2 Crack damage in rocks specimens: (a) Inter-granular cracks in Callovo-Oxfordian marl (Fabre & Pellet, 2006); b) Trans-granular micro-cracks in plagioclase crystals of a gabbro (Pellet et al. 2011). Note that the development of micro-cracks crosses the original crystal twinning.

The theoretical strength of a crystal, R_th, is related to the inter-atomic bonding forces (Figure 3). It can be computed using the following expression (Dorlot et al., 1986):

R_{t h} = 2 \sqrt{\frac{E γ}{a_{0}}}

with:

E the Young’s modulus of the crystal,
γ, the energy required to debond surfaces per unit area,
a₀, the equilibrium distance between atoms.

The theoretical strength is between one-third and one-tenth of the Young’s modulus E. However, due to the presence of flaws and micro-defects, the usual values for the tensile strength of rocks are much lower, in the order of 1/100 to 1/1000 of the Young’s modulus.

Figure 3 Schematic of crystal atomic bonding.

In summary, since most rocks are formed by an assemblage of various types of crystals, fracturing can occur at different scales. At the scale of the micro-structure, damage could result from inter-granular bonds breaking or sometimes rupturing within the crystal. This type of rupture is referred to as dislocation mechanics (McClintock & Argon, 1966).

3 Constitutive Modeling

3.1 Introduction

Constitutive modeling of damage evolution has motivated numerous studies in recent decades. Basically, there are two approaches: The first is based on the theories of fracture mechanics while the second is derived from continuum damage mechanics. In the following sections the main aspects of both approaches will be reviewed and the domain of validity of each method will be identified.

3.2 Linear fracture mechanics

3.2.1 Energy balance consideration

It is important to review the basic concepts involved in fracture mechanics. Griffith (1924) formulated a rupture criterion for a material based on energy balance. He acknowledged that defects (pore spaces, cracks or structural defects such as dislocations in the crystal lattice or grain boundaries) can exist in any material and stress amplification at such defects will govern the failure stress.

Considering an elementary volume subjected to a uniaxial tensile stress (Figure 4), and based on the principles of thermodynamics, equilibrium is achieved when the total potential energy of the system is at a minimum. The energy balance of the system leads to the expression of the total energy, W, as the sum of the energy of the external forces, the elastic strain energy, and the energy dissipated during the cracking process:

W = W_{e x t} + W_{e l} + W_{s}

with:

W: total energy
W_ext: energy due to external forces
W_el: elastic strain energy
W_s: dissipated energy

During extension, the crack length increases from 2c to 2(c + dc) and the conservation of energy gives:

\frac{\partial W}{\partial c} = \frac{\partial W_{e x t}}{\partial c} + \frac{\partial W_{e l}}{\partial c} + \frac{\partial W_{s}}{\partial c} = 0

The energy release rate G is therefore defined as:

G = - \frac{1}{2} \frac{\partial}{\partial c} (W_{e x t} + W_{e l})

The energy dissipated by the crack propagation is assumed to vary linearly with the increase in crack length, i.e.:

d W_{s} = 4 γ d c

where γ is the specific energy of the material.

By substituting the expressions 4 and 5 in Equation 3, the crack static equilibrium is obtained when:

G = 2 γ

Consider the particular case of an elastic plate of unit thickness with a through crack of length 2c and subjected to a uniaxial tensile stress (Figure 4). The change in elastic energy caused by the introduction of the crack in the elastic plate (Jaeger et al., 2008) is given by:

W_{e l} = \frac{π σ^{2} c^{2}}{E}

where E is the Young’s modulus of the plate material and c is the crack half-length.

Figure 4 Elementary volume of rock with a plane crack loaded in tension.

According to the virtual work principle and for a constant load, it can be shown that:

W_{e x t} = - 2 W_{e l}

Thus, according to Equations 4 and 7, the rate of energy release G (Equation 4) can be expressed as a function of the stress and the crack length as follows:

G = - \frac{1}{2} \frac{\partial W_{e l}}{\partial c} = \frac{π c σ^{2}}{E}

The equilibrium condition (Equation 6) is then used to calculate the critical failure condition (Equation 10) and the critical stress, σ_c is given by:

π c σ^{2} = 2 γ E

σ_{c} = \sqrt{\frac{2 γ E}{π c}}

Note that the critical failure stress depends not only on the intrinsic material properties (Young’s modulus E, and the specific surface energy γ) but also on the crack length, 2c. In particular, the critical value of the stress is lower when the length of the crack is larger. The expression of the critical stress in the case of an arbitrarily oriented crack in a biaxial tensile stress field is presented by Eftis (1987) and extensive discussions are given in many texts on crack and fracture mechanics and in the volume by Sih (1991).

Griffith’s analysis was extended to the case of an inclined crack in a biaxial stress field taking into account the friction on the crack faces (Jaeger et al., 2008). Under these conditions, the critical failure condition becomes:

π c (| τ | - μ σ_{n}^{})^{2} = 2 γ E

where σ_n and

| τ |

are, respectively, the normal stress and shear stress acting in the crack and μ is the friction coefficient between the two faces of the crack.

In this analysis, Griffith states that the energy dissipated by the propagation of the crack is solely due to the extension of the crack surface. In fact, this analysis reflects crack initiation rather than crack propagation. Other forms of energy dissipation must be taken into account during the crack propagation; for example, plastic energy dissipation, which is associated with high stress concentrations at the crack tip, and kinetic energy, related to the acceleration of the crack propagation. In the case of a purely brittle process, the plastic deformation energy can be neglected. On the other hand, it seems incorrect to neglect the effects of kinetic energy dissipation even in the case of quasi-static loading. This will be discussed in the next paragraph.

3.2.2 Steady and unsteady crack propagation

Extending Griffith’s analysis, considering energy dissipation due to inertia, leads to the calculation of the total energy (Equation 2), by including a term for the kinetic energy, W_cin:

W = W_{e x t} + W_{e l} + W_{s} + W_{c i n}

Crack propagation will be steady if the kinetic energy remains constant during the cracking process:

\frac{\partial W_{c i n}}{\partial c} = 0

From the equilibrium equation (Equation 13) and the definitions provided by Equations 4 and 5, we obtain:

G = 2 γ

We thus recover the Griffith equilibrium condition for a static crack (Equation 6). However, the significance of Equation 15 is that it represents the condition of crack propagation at a constant rate; but crack propagation will be unsteady if the kinetic energy increases:

\frac{\partial W_{c i n}}{\partial c} > 0

Therefore, from the balance Equation 13 and the definitions (4) and (5):

G > 2 γ,

Resuming the system of uniaxial tensile elementary volume (Figure 4), the axial deformation is then expressed by:

ε = \frac{σ}{E_{eff}}

with E_eff, the effective Young’s Modulus:

E_{eff} = \frac{E}{1 + 2 π c^{2} / b t}

ε = \frac{σ}{E} (1 + 2 π c^{2} / b t)

By introducing the stability condition (15) into Equation 20, we obtain the stability criterion in the stress–strain plane:

ε = \frac{σ}{E} + 8 \frac{γ^{2} E}{π b t σ^{3}}

Representation of Equation 21 in the stress–strain plane is shown in Figure 5 (curve AB). For the undamaged material (a plate without a crack), Hooke’s law is satisfied with a modulus of elasticity E (line OA). For a plate with a crack half-length c = c₁, the material also follows Hooke’s law but with a lower modulus E_eff (Equation 19) than the virgin material (line OP’, Fig. 5). When the stability curve AB is reached, the crack steadily grows to a point P″ where unloading is achieved. According to Equation 19, unloading shows a weaker effective modulus of elasticity (line P″O). The surface area between segments OP′, P″O and P′P, corresponds to the energy expended for crack propagation.

Figure 5 Griffith’s stability criterion for an elementary volume in the stress–strain plane.

It should be noted that if the initial half-length of the crack is less than c = bt/6π (corresponding to the point where the tangent to the curve is vertical, point V), then the propagation of the crack will be initially unstable (line PQ). The excess energy represented by the area PQV, is then transformed into kinetic energy. Part of this energy enables the crack to propagate beyond the stability limit (point Q) to point F ′. The area QF′F represents the surface energy needed to propagate the crack, although some of the energy is dissipated as heat. The remaining kinetic energy is emitted in the material as elastic waves. In addition, for a stress-imposed loading, the crack propagation can be unstable since stability is only possible if there is a decrease in the stress. For a strain-imposed loading, two types of post peak behavior are observed depending on the initial length of the crack. The first (Type I) occurs when the crack length is such that c < c_c, and requires an increase in the energy transmitted to the specimen to obtain a steady propagation. In contrast, when the initial crack length is such that c > c_c, energy has to be extracted from the specimen to stabilize the crack propagation (Type II). Point V, where c = c_c, represents the case where the energy of the specimen is in equilibrium with the energy necessary for crack propagation. Wawersik and Fairhurst (1970) experimentally observed the presence of these two post-peak behavior patterns by controlling failure using a servo-controlled testing machine. In the case of a behavior of type II, known as “snap–back”, only the control of transverse deformation allows the control of rupture.

The case of overall compressive stress was studied by Cook (1965). The stability curve has the same characteristics as that shown in Figure 5. Following Cook’s approach, Martin and Chandler (1994) found, using Lac du Bonnet granite, that the change of stress marks the initiation of unstable crack propagation during progressive damage of the Griffith criterion type.

The Griffith global energy approach shows that the essential phenomena governing the behavior of a cracked body lie near the crack tip. Fracture mechanics investigates issues of the initiation and development of cracks by analyzing the stress in the vicinity of the crack. In the case of a brittle elastic solid, the presence of cracks leads to stress singularities. The study of these singularities allows the definition of stress intensity factors that correspond to the particular kinematics of the crack propagation. These stress intensity factors control the behavior of the crack.

When the stored elastic energy is close to the specific surface energy (G= 2γ), cracks propagate sub-critically (Atkinson, 1984); in other words, the propagation rate is lower than the sonic velocity. In contrast, when the stored elastic energy is much higher, cracks propagate super-critically.

In all cases, the cracks develop in the direction of the minor compressive stress or perpendicular to the major principal stress.

3.2.3 Griffith crack initiation criterion

In the case of complex loadings, stress analysis at the vicinity of the crack tip is required to determine the crack propagation criterion, the length increase and the orientation change. For this purpose, the discipline of Fracture Mechanics has been developed and widely used since Griffith (1924).

In order to study the stress field around an elliptical cavity in a plate subjected to a uniaxial tensile stress, as shown in Figure 6a, the solution for the stress distribution at the periphery of the crack has to be calculated using the theory of elasticity (Inglis-Kirsch type solution) (Timoshenko & Goodier, 1970; Little, 1973; Barber, 2010; Selvadurai, 2000a).

Referring to Figure 6, the stress σ_yy is largest at the wall of the cavity, where the ellipse curvature radius is the smallest (point C). Thus:

σ_{y y (c)} = 1 + 2 \sqrt{c / ρ}

Subsequently, the stress σ_yy gradually decreases to the lowest value corresponding to the applied field, σ.

The stress σ_xx increases rapidly to reach a maximum near the end of the ellipse and then gradually decreases to approach zero. The area of influence of the cavity is of the order of the length c and the stress gradients are very high in the length zone of the curvature radius of the cavity ρ. According to Equation 22 for a semi-axis cavity of half-length c, the greater the radius of curvature the smaller the stress.

Figure 6 (a) Geometry of the plate loaded in uniaxial stress; (b) Elastic stresses distribution in the crack tip area.

3.2.4 Stress field and fracture modes

Consider plane problems (2D) for which all the components of the tensors of stresses and deformations depend only on two Cartesian or polar coordinates. The crack is assumed to be in a homogeneous medium, which is isotropic linear elastic.

Recall that two basic modes of fracture exist (in fact, there are three fracture modes but the third is not relevant to massive bodies) (Figure 7):

The extension mode (Mode I) corresponds to a discontinuity of the normal displacement field to the plane of the crack. The plane shear mode (or Mode II) corresponds to a shift of the edges of the crack parallel to the plane of the crack along a sliding direction normal to the crack front. For any load, several basic fracture modes may overlap. This is referred to as a mixed fracture mode.

Figure 7 The three main fracture modes; in rock mechanics only Mode I and Mode II are relevant.

Solving the equilibrium equations with the compatibility conditions leads to a bi-harmonic equation for a stress function. The exact solution of this problem was established by Muskhelishvili (1953) based on the solution by Westergaard (1939) and expresses the stress fields and displacement in the crack for each of the three modes in coordinate systems.

The solution introduces the notion of stress intensity factors K_Ic and K_IIc for Mode I and Mode II fracture, respectively. Knowledge of these stress intensity factors will allow the determination of the stresses and displacements in the fissured structure. Conversely, if we know the stresses and displacements, it is possible to determine the stress intensity factors. For example, in the simple case of a plane infinite medium containing a crack of length 2c loaded in Mode I by a stress σ, the stress intensity factor can be expressed in the form:

K_{I} = σ \sqrt{π c}

The stress intensity factor is measured in units N.m^–3/2 or MPa.m ^1/2.

A similar approach can lead to the establishment of the stress intensity factor K_II for Mode II fractures (Irwin, 1957).

3.2.4.1 Failure criterion in the open mode (Mode I)

In the tensile mode, stress at the tip of the crack is infinite. Therefore, a crack initiation criterion can be introduced based on the concept of a critical threshold for the factor K_I. This criterion postulates that:

K_{I} \leq K_{I c} No crack propagation

K_{I} = K_{I c} Crack propagation onset

The critical value, K_Ic, is the toughness, a physical characteristic of the material (Atkinson and Meredith, 1987). Toughness values obtained from the literature (Table 1) show a significant scatter, which results from the high parameter sensitivity to the investigative method used and the environmental conditions (pressure, temperature, fluid). Note that toughness increases rapidly with the pressure applied.

In terms of energy, crack growth requires that the released energy is higher than the specific energy, which leads to the establishment of the following equation:

W = W_{e l} - W_{s}

Therefore, when

\frac{\partial W}{\partial c} \geq 0

, stress at the crack tip is:

σ \geq \sqrt{\frac{2 γ E}{π c}} \geq σ_{t}

Remember that the energy release rate G controls the crack behavior:

G > 2γinstability
G = 2γcontrolled steady cracking (the kinetic energy does not increase)

Table 1 Toughness of different types of rocks, after Atkinson and Meredith (1987).

Rock Type	K_Ic [MPa.m ½]
Berea Sandstone	0.28
Carrara Marble	0.64 – 1.26
Westerly Granite	0.60 – 2.50

3.2.4.2 Failure criterion in the mixed mode

Now we consider the same plate as previously, but loaded in compression on all four sides (Figure 8a). The development of the crack is no longer along its plane and the contact of the crack surfaces can therefore withstand stress. The direction of the extension of the crack must be defined and the work dissipated by friction must be taken into account.

In the mixed mode, the definition of a failure criterion is therefore more difficult. However, linear fracture mechanics makes it possible to find a relationship between the stress intensity factors and the rate of energy release, G, when the crack grows in extension. This relationship is known as the Irwin formula:

24)Inplanestrain

G = \frac{(3 - 4 ν) + 1}{8 G} (K_{I}^{2} + K_{I I}^{2})

where G is the shear modulus.

The Griffith onset crack propagation criterion (Equations 25 and 26) thus generalizes the original criterion:

\begin{array}{l} {(σ_{1} - σ_{3})}^{2} - 8 \cdot σ_{t} \cdot (σ_{1} + σ_{3}) = 0 & if σ_{1} > 3 \cdot σ_{3} \end{array}

\begin{array}{l} σ_{3} = - σ_{t} & if σ_{1} < 3 \cdot σ_{3} \end{array}

Note, however, that the growth assumption of fissure in extension is a serious limitation to the application of the Irwin formula. In fact, this assumption is only valid in the case of a failure mode in pure opening (Mode I); in the case of a mixed mode, the crack deviates from its plane. Other onset failure criteria in the mixed mode are available in the literature, namely the criterion of maximum normal stress, the energy criterion of minimal elastic deformation, the criterion of maximum energy of restitution, etc.

Figure 8 Griffith onset crack propagation criterion in the principal stress plane (adapted from Jaeger et al., 2008).

3.3 Continuum damage mechanics

Continuum damage mechanics was first developed for structural engineering (Kachanov, 1958), with a special emphasis on structural elements loaded in tension. This approach has attracted particular interest in the design of concrete structures (Chaboche, 1988; Bažant, 1991; Lemaitre, 1996; Selvadurai, 2004).

The motivation for this approach was based on the evidence that, under certain loading conditions, the development of micro-cracks that spread throughout the material does not necessarily cause a macroscopic fracture. However, gradual deterioration of physical properties such as strength and stiffness are observed, sometimes well before rupture occurs. A comprehensive understanding and modeling of these phenomena that precede rupture is therefore of great practical interest.

Continuum Damage Mechanics was developed within the framework of Continuum Mechanics. This method enables the analysis of how the development of micro-defects influences the overall behavior of the material using a phenomenological approach, which describes the evolution of the material structure with internal state variables.

Let us suppose that the effects of micro-debonding in a Representative Elementary Volume (REV) of a solid body can be described using mechanical variables called damage variables. Since such damage variables are macroscopic variables that reflect the state of the loss of mechanical integrity, they can be considered as internal state variables from a thermodynamic point of view. With these assumptions, damage problems can be analyzed based on the principles of thermodynamics of irreversible processes. Therefore, the phenomenological damage model can be defined by:

– A law of transformation,
– A principle of equivalence,
– The nature of the damage variable,
– The evolution law of the damage variable.

3.3.1 Law of transformation

Phenomenological models appeal to the effective macroscopic quantities whose role is to define the damage state of the material. Thus, a heterogeneously damaged material obeys the same mechanical constitutive laws as its homogeneous undamaged equivalent; however, the states of stress and strain undergo changes from the nominal configuration. This change is made via a law of transformation, defined by the tensors

{\overset{= 4 =}{M}}_{σ}

and

{\overset{= 4 =}{M}}_{ε}

, linking the effective variables (

\tilde{σ}, \tilde{ε}

) to nominal values (

σ, ε

) as follows:

\tilde{σ} (σ, D) = {\overset{= 4 =}{M}}_{σ} (D) : σ

\tilde{ε} (ε, D) = {\overset{= 4 =}{M}}_{ε} (D) : ε

The tensors

{\overset{= 4 =}{M}}_{σ}

and

{\overset{= 4 =}{M}}_{ε}

are linear operators that apply a symmetric tensor of the second order on itself and which are functions of the damage variable, formally denoted by D. When there is zero damage, the actual magnitudes coincide with the nominal values, and the transport tensor coincides with the unity.

Zheng and Betten (1996) demonstrated, by analyzing the general form of the transformation law, that the transport tensor must be an isotropic function of its arguments. Assuming a damage variable as a tensor of order two, this mathematical condition is:

\begin{array}{l} \overset{= 4 =}{M} (Q D Q^{T}) = Q \overset{= 4 =}{M} (D) Q^{T} & \forall Q \in Θ \end{array}

where Θ is the full group of orthogonal transformations Q.

3.3.2 Principle of equivalence

The principle of equivalence can be expressed in terms of deformation, stress or energy. The principle of equivalence in deformation implies that the effective deformation is equal to the nominal deformation, thus:

\tilde{ε} = ε

\tilde{σ} = {\overset{= 4 =}{M}}_{σ} (D) : σ

More explicitly, it means that any representation of the deformational state of a damaged material can be described by the constitutive laws of an undamaged medium by simply replacing the nominal stress by the effective stress. This leads to the concept of the effective stress, which is not to be confused with the Terzaghi effective stress, which can be defined through reciprocal analysis, as the stress to be applied to the undamaged volume element to achieve the same strain as that caused by the nominal stress applied to the damaged volume element.

The principle of equivalence in stress is defined by assuming that the effective stress,

\tilde{σ}

, is equal to the nominal stress, σ, i.e., the effective deformation causes the same stress as the application of the nominal strain to the damaged volume:

\tilde{σ} = σ

\tilde{ε} = \overset{= 4 =}{M_{ε}} (D) : ε

Finally, the equivalence energy principle is established assuming that the effective elastic energy density,

\tilde{W}

, is equal to the nominal density of elastic energy, W. This assumption is expressed by:

\tilde{σ} = {\overset{= 4 =}{M}}_{σ} (D) : σ

\tilde{ε} = {\overset{= 4 =}{M}}_{ε} (D) : ε

\tilde{W} = \frac{1}{2} \tilde{σ} : \tilde{ε}

W = \frac{1}{2} σ : ε

\tilde{W} = W

3.3.3 Damage variable

Since the work of Kachanov (1958), a new variable is available to describe the internal state of a damaged material in the context of continuum mechanics. As indicated before, the notion of damage is closely linked to micro-structural aspects. However, the damage variable introduced by Kachanov described the presence of micro-defects in a comprehensive manner. The advantage of this homogenized approach is that it allows the indirect determination of the influence of the state of damage on the overall strength of the material through simple measurements.

Although the authors have not detailed the physical significance of this variable, it can be easily understood by using the concept of a Representative Elementary Volume (REV). Let us consider a REV in a damaged solid (Figure 9). Assuming that S_total is the area of a section of a volume element indicated by the normal

\vec{n}

, and S_flaws is the area of all micro-cracks and micro-pores, the classical damage variable can then be expressed by:

D_{\vec{n}} = \frac{S_{f l a w s}}{S_{t o t a l}}

From a physical point of view the damage variable,

D_{\vec{n}}

, is the ratio between the area of distributed micro-defects on the total area of the element in the plane normal to the direction,

\vec{n}

Figure 9 Damage in rocks: a) Definition of the damage parameter Dn; b) Elastic modulus for different states of damage. E0 is the initial elastic modulus; the elastic modulus decreases (E0> E1> E2> E) as damage increases D0 = 0 <D1 <D2 <D3.

D_{\vec{n}}

allows the various states of damage to the material to be defined:

–
$D_{\vec{n}} = 0$
corresponds to the undamaged material,
–
$D_{\vec{n}} = 1$
corresponds to the material volume that has failed into two parts along the plane normal to,
$\vec{n}$
–
$0 < D_{\vec{n}} < 1$
characterizes the intermediate damaged state.

3.3.4 Isotropic and anisotropic damage

In the case where the orientation of the defects is assumed to be randomly distributed in all directions, the variable is independent of the orientation

\vec{n}

and the scalar D completely characterizes the state of damage:

\begin{array}{l} D_{\vec{n}} = D & \forall \vec{n} \end{array}

The hypothesis of the equivalence principle in deformation directly leads to the definition of effective stress,

\tilde{σ}

\tilde{σ} = \frac{σ}{1 - D}

\tilde{σ}

is the stress related to the section that effectively resists forces and the law of transformation takes the particular form:

{\overset{= 4 =}{M}}_{σ} (D) = \frac{1}{1 - D} \overset{= 4 =}{I}

The most general form of the transformation law to describe isotropic damage was given by Ju (1990):

{\overset{= 4 =}{M}}_{σ} = α \overset{= 4 =}{I} + β I \otimes I

where α and β are two independent or dependent scalar variables, with

α \geq 1

and

- 1 / 3 \leq β / α \leq 0

to ensure positivity of the stiffness tensor. The interpretation of isotropic damage does not necessarily imply a scalar variable.

The existing isotropic damage models with a scalar variable have the undeniable advantage of being simple to use. However, many experimental results (Tapponnier & Brace, 1976; Wong, 1982 for granite; Gatelier et al., 2002 for sandstone; Lajtai et al., 1994 for rock salt) have demonstrated that the mechanically-induced damage is anisotropic regardless of whether the intact rock is initially isotropic or anisotropic. In other words, material symmetries change during the loading process (Figure 10). In order to describe this phenomenon, the introduction of a tensor damage variable is needed. In the relevant literature we encounter second order variables (Kachanov, 1993; Murikami, 1988, Pellet et al., 2005), or sometimes fourth order variables (Chaboche, 1979; Lubarda & Krajcinovic, 1993).

Figure 10 Isotropic damage β =1 (right hand side) and anisotropic damage β=0 (left hand side), after Pellet et al., 2005.

3.4 Discussion of Fracture Mechanics (FM) vs. Continuum Damage Mechanics (CDM)

Fracture Mechanics (FM) is an approach that aims to describe the rupture of solid bodies based on energy consideration at the crack scale. This method allows the analysis of how the development of micro-defects influences the overall behavior of the material. Despite the small number of assumptions, closed form solutions are often complex and difficult to establish for boundary values problems. Although the origin of linear fracture mechanics dates back to the early 20^th century (Griffith, 1924), the development of this discipline has accelerated in recent years. We now speak about nonlinear fracture mechanics when the effects of plasticity or viscosity and dynamic fracture propagation are taken into account.

Continuum Damage Mechanics (CDM) relies on the concept of effective stress. It aims to describe the overall behavior of a Representative Elementary Volume (REV) by decreasing the material stiffness (elastic moduli) with the damage states. This is a phenomenological approach, which is easier to incorporate in computational codes.

Returning to phenomenological considerations, it may be said that FM is more appropriate for describing the propagation of a large single micro-crack. In contrast, CDM could be more suitable for describing the behavior of materials with multiple micro-cracks randomly spread in the body. Figure 11 illustrates with respect to scale the optimum applicability domain of both methods.

Schematic distinction between the representation of damage by Continuum Damage Mechanics and Fracture Mechanics (adapted from Chaboche,

Figure 11 Schematic distinction between the representation of damage by Continuum Damage Mechanics and Fracture Mechanics (adapted from Chaboche, 1988).

4 Other Types of Loading

4.1 Time-dependent damage

Studying time-dependent damage requires rate-dependent constitutive models (visco-plastic or viscoelastic) to be considered in Continuum Damage Mechanics. The first such attempts were proposed by Kachanov (1958) and Rabotnov (1969) for metallic alloys.

Based on Lemaitre’s works, Pellet et al. (2005) developed a constitutive model to account for anisotropic damage and dilation of rock specimens. The strain rate is expressed by the following equation:

{\dot{ε}}^{v p} = \frac{\partial Ω}{\partial σ_{i j}} = \frac{3}{2} \frac{1}{1 - D} {[\frac{σ_{e q}}{(1 - D) K p^{1 / M}}]}^{N} \frac{S}{σ_{e q}}

where Ω is the visco-plastic potential; K, N and M are the viscoplastic parameters of the model; σ_eq is the von Mises stress; S is the stress deviator and p is a variable that represents strain hardening.

It is also necessary to associate a law for the evolution with respect to time of the damage parameter. For uniaxial loading, the latter is expressed as follows:

\dot{D} = {[\frac{σ}{A (1 - D)}]}^{r}

where r is the damage exponent, q is the damage progression parameter and A is the tenacity coefficient.

From a physical point of view, this constitutive model is realistic as it accounts for volumetric strain (i.e. contraction and dilation) and damage-induced anisotropy through a second-rank tensor. Moreover, taking time into account allows numerical regularization and ensures the uniqueness of the solution.

Using this model, it is possible to predict delayed rupture by determining the time to failure, as observed in creep tests shown in Figure 12.

Comparison between the strain–time curves obtained from the model and experimental results from a creep test performed on marble, Singh (

Figure 12 Comparison between the strain–time curves obtained from the model and experimental results from a creep test performed on marble, Singh (1975), after Pellet et al. (2005).

4.2 Damage under cyclic loading

The gradual weakening of rock properties can also be highlighted by performing cyclic loading tests. There are two types of cyclic tests: type 1, where the loading is cycled between two prescribed limits, and type 2, where the loading is increased from one cycle to the next. Most of the published data on type 1 tests are aimed at producing S–N curves, that relate the maximum stress, S, applied to a specimen to the number of cycles prior to specimen failure N. It has been shown (Costin & Holcomb, 1981) that cycling decreases rock strength, possibly by a combination of cyclic fatigue and stress corrosion. Over the last few decades several experimental programs have been performed to characterize rock behavior under static and cyclic behavior. The objective is to characterize the progressive development of damage in rocks under cyclic loading (Erarslan & Williams, 2012).

Gatelier et al. (2002) presented an extensive laboratory investigation of the mechanical properties of sandstone, which exhibits transversely isotropic behavior. Particular attention was paid to the influence of the structural anisotropy on the progressive development of pre-peak damage. Uniaxial and triaxial cyclic tests were performed for several orientations of the isotropy planes with respect to the principal stress directions in order to quantify the irreversible strains and the changes of oriented moduli with the cumulative damage. Two main mechanisms are involved throughout the loading process: compaction and micro-cracking.

Compaction is active at all stress levels. In uniaxial tests, both mechanisms were shown to be strongly influenced by the inclination of loading with respect to the planes of isotropy. However, with increased confining pressures, the influence of anisotropy is significantly reduced.

For an unconfined compression test carried out on sandstone specimens, Gatelier et al. (2002) showed that during the cycling, progressive damage is accompanied by a change in the volumetric strain. Figure 13 clearly shows that, at the early stage, the rock specimen tends to contract whereas later in the cycling process, it exhibits dilation. This observation is useful when developing an appropriate constitutive model.

Figure 13 Axial stress as a function of volumetric strain for an unconfined cyclic compression test (from Gatelier et al. 2002, with permission from Elsevier).

4.3 Thermo-mechanical damage

In many situations, rock formations can be subjected to high temperatures that can lead to drastic changes in their mechanical properties. For example, in energy and environmental engineering, such as geothermal energy production, deep geologic disposal of high-level radioactive waste or tunnels that have experienced fire accidents, special attention has to be paid to thermal damage. In all these examples, the question that needs to be addressed is the same: how do changes in temperature influence the physical and mechanical properties of rocks?

Keshavarz et al. (2010) showed that thermal loading of rocks at high temperatures induces changes in their mechanical properties. In this study, a hard gabbro was tested in the laboratory. Specimens were slowly heated to a maximum temperature of 1,000°C. Following this thermal loading, the specimens were subjected to uniaxial compression. A drastic decrease of both unconfined compressive strength and elastic moduli was observed (Figure 14). The thermal damage to the rock was also highlighted by measuring elastic wave velocities and monitoring acoustic emissions during testing. The micro-mechanisms of rock degradation were investigated by analyzing thin sections after each stage of thermal loading. It was found that there is a critical temperature above which drastic changes in mechanical properties occur. Indeed, below a temperature of 600°C, micro-cracks start to develop due to a difference in the thermal expansion coefficients of the crystals. At higher temperatures (above 600°C), oxidation of Fe and Mg atoms, as well as bursting of fluid inclusions, are the principal causes of damage.

Damage of thermally altered gabbro specimens: top fig.) Stress-deformation curves from uniaxial compression tests. Axial and lateral deformations at failure increase with the loading temperature. A noticeable increase in both deformations occurs at temperature higher than 700°C. Lower fig.) Normalized elastic wave velocities

Figure 14 Damage of thermally altered gabbro specimens: top fig.) Stress-deformation curves from uniaxial compression tests. Axial and lateral deformations at failure increase with the loading temperature. A noticeable increase in both deformations occurs at temperature higher than 700°C. Lower fig.) Normalized elastic wave velocities Vp, Vs (After Keshavarz et al., 2010).

4.4 Rock joint damage

Thus far, we have focused on damage to the rock material (rock matrix). However, it is well known that discontinuities (joints, faults, etc.) are of the utmost importance in analyzing rock mass behavior (Vallier et al., 2010). Traditionally, the mechanical behavior of rock discontinuities is analyzed with shear tests in different loading conditions: Constant Normal Load (CNL), Constant Normal Stiffness (CNS) or Constant Volume (CV). The two latter tests require advanced testing equipment to allow control of the displacements in the 3 spatial directions (Boulon, 1995) (see also Selvadurai & Boulon, 1995; Nguyen & Selvadurai, 1998).

Jafari et al. (2004) studied the variation of the shear strength of rock joints due to cyclic loadings. Artificial joint surfaces were prepared using a developed molding method that used special mortar; shear tests were then performed on these samples under both static and cyclic loading conditions. Different levels of shear displacement were applied to the samples to study joint behavior before and during considerable relative shear displacement. It was found that the shear strength of joints is related to the rate of displacement (shearing velocity), number of loading cycles and stress amplitude. Finally, based on the experimental results, mathematical models were developed for the evaluation of shear strength under cyclic loading conditions.

Figure 15a presents the evolution of rock joint dilation during cycling. It is shown that dilation progressively decreases as asperities damage is developed. In Figure 15b the associated decrease of shear strength is represented with respect of the number of cycles for both peak strength and residual strength.

Figure 15a Cyclic shear test on rock discontinuity: Normal displacement versus shear displacement while cycling; (Jafari et al., 2004).

Figure 15b Cyclic shear test on rock discontinuity: Normalized shear strength as a function of the number of cycles (Jafari et al., 2004).

5 Permeability Evolution with Damage

During the development of damage in a rock, the permeability characteristics can be altered through either the development of discrete fractures or the development of continuum damage in the form of micro-cracks and micro-voids. The consequences of such damage development not only cause a reduction in the stiffness characteristics of the rock but also contribute to enhanced fluid flow through the accessible pore space of the rock. The development of enhanced fluid flow through rocks is of particular interest to geoenvironmental endeavors that focus on deep geologic disposal of hazardous nuclear wastes (Selvadurai & Nguyen, 1995, 1997; Nguyen et al., 2005) and contaminants (Testa, 1994; Apps & Tsang, 1996; Selvadurai, 2006), geologic sequestration of greenhouse gases (Pijaudier-Cabot & Pereira, 2013; Selvadurai, 2013) and other energy extraction endeavors. In particular, excavation damaged zones in repositories used for storage of hazardous waste or in resources extraction activities can experience permeability alterations that will alter the fluid transport characteristics and consequently the potential for the enhanced migration of hazardous substances from repositories. The alteration of permeability of rocks during damage evolution is therefore of considerable importance to geoenvironmental applications. The influence of micro-crack generation and damage on the evolution of hydraulic conductivity of saturated geomaterials has been discussed by Zoback and Byerlee (1975), who present results of tests conducted on granite that indicate increases of up to a factor of four in the magnitude of the permeability. Results by Shiping et al. (1994) on sandstone indicate that for all combinations of stress states employed in their tests, the permeability increased by an order of magnitude. In triaxial tests on anisotropic granite Kiyama et al. (1996) presented results that also indicate increases in the permeability characteristics. Coste et al. (2001) present the results of experiments involving rocks and clay stone; their conclusions support the assumption of an increase in the permeability, of up to two-orders of magnitude, with an increase in deviator stresses. Focusing on the behavior of cementitious materials, the experimental results presented by Samaha and Hover (1992) indicate an increase in the permeability of concrete subjected to compression. Results by Gawin et al. (2002) deal with the thermo-mechanical damage of concrete at high temperatures. In their studies, empirical relationships have been proposed to describe the alterations in the fluid transport characteristics as a function of temperature and damage; here the dominant agency responsible for the alterations is identified as thermally-induced generation of micro-cracks and fissures (see also, Schneider & Herbst, 1989; Bary, 1996). Also, Bary et al. (2000) present experimental results concerning the evolution of permeability of concrete subjected to axial stresses, in connection with the modeling of concrete gravity dams that are subjected to fluid pressures. Gas permeability evolution in natural salt during deformation is given by Stormont and Daemen (1992), Schulze et al. (2001), and Popp et al. (2001). While the mechanical behavior of natural salt can deviate from a brittle response that is usually associated with brittle rocks, the studies support the general trend of permeability increase with damage evolution that leads to increases in the porosity. The article by Popp et al. (2001) also contains a comprehensive review of further experimental studies in the area of gas permeability evolution in salt during hydrostatic compaction and triaxial deformation, in the presence of creep. Time-dependent effects can also occur in brittle materials due to the sub-critical propagation of micro-cracks, a phenomenon known as stress corrosion (Shao et al., 1997). In a recent study, Bossart et al. (2002) observed an alteration in permeability of an argillaceous material around deep excavations, a phenomenon attributed to the alteration of the properties of the material in the EDZ.

Souley et al. (2001) describe the excavation damage-induced alterations in the permeability of granite of the Canadian Shield; they observed an approximately four-orders of magnitude increase in the permeability in the EDZ. It is noted, however, that some of this data is applicable to stress states where there can be substantial deviations from the elastic response of the material as a result of the generation of localized shear zones and foliation-type extensive brittle fracture, which may be indicative of discrete fracture generation as opposed to the development of continuum damage. The data presented by Souley et al. (2001) has recently been re-examined by Massart and Selvadurai (2012, 2014), who used computational homogenization techniques to account for permeability evolution in rock experiencing dilatancy and damage.

It should also be noted that not all stress states contribute to increases in the permeability of geomaterials. The work of Brace et al. (1978) and Gangi (1978) indicates that the permeability of granitic material can indeed decrease with an increase in confining stresses. These conclusions are also supported by Patsouls and Gripps (1982) in connection with permeability reduction in chalk with an increase in the confining stresses. Wang and Park (2002) also discuss the process of fluid permeability reduction in sedimentary rocks and coal in relation to stress levels. A further set of experimental investigations, notably those by Li et al. (1994, 1997) and Zhu and Wong (1997) point to the increase in permeability with an increase in the deviator stress levels. These investigations, however, concentrate on the behavior of the geomaterial at post-peak and, more often in the strain softening range. Again, these experimental investigations, although of considerable interest in their own right, are not within the scope of the current chapter that primarily deals with permeability evolution during damage in the elastic range. The experimental studies by Selvadurai and Głowacki (2008) conducted on Indiana Limestone, similarly indicate the permeability reduction of a rock during isotropic compression up to 60 MPa. The studies also indicate the presence of permeability hysteresis during quasi-static load cycling. The internal fabric of the rock can also affect the enhancement of permeability of rocks even during isotropic compression. The experimental work conducted by Selvadurai et al. (2011) on the highly heterogeneous Cobourg Limestone indicates that the permeability can increase during the application of isotropic stress states. In this case the external isotropic stress state can result in locally anisotropic stress states that can lead to permeability enhancement through dilatancy at the boundaries of heterogeneities.

5.1 Modeling saturated porous media with continuum damage

The modeling of fluid-saturated porous media that experience continuum damage can be approached at various levels, the simplest of which is the modification of the equations of classical poroelasticity developed by Biot (1941) (see also Rice & Cleary, 1976; Detournay & Cheng, 1993; Selvadurai, 1996; Wang, 2000; Cheng, 2016; Selvadurai & Suvorov, 2016) to account for damage evolution in the porous fabric and the enhancement of the permeability characteristics of the fluid-saturated medium by damage evolution. In the case of an intact poroelastic material, considering Hookean isotropic elastic behavior of the porous skeleton, the principle of mass conservation and Darcy’s law applicable to an isotropic porous medium, the coupled constitutive equations governing the displacement field

u (x, t)

and the pore pressure field

p (x, t)

in a fluid-saturated body void of body or inertia forces can be expressed in the form

μ \nabla^{2} u + \frac{μ}{(1 - 2 ν)} \nabla (\nabla . u) + α \nabla p = 0

and

κ β \nabla^{2} p - \frac{\partial p}{\partial t} + α β \frac{\partial}{\partial t} (\nabla . u) = 0

In (46) and (47)

\begin{array}{l} α = \frac{3 (ν_{u} - ν)}{\tilde{B} (1 - 2 ν) (1 + ν_{u})} & ; & β = \frac{2 μ (1 - 2 ν) {(1 + ν_{u})}^{2}}{9 (ν_{u} - ν) (1 - 2 ν_{u})} \end{array}

ν and μ are the “drained values” of Poisson’s ratio and the linear elastic shear modulus applicable to the porous fabric,

ν_{u}

is the undrained Poisson’s ratio and

\tilde{B}

is the pore pressure parameter introduced by Skempton (1954),

κ (= k / γ_{w})

is a permeability parameter, which is related to the hydraulic conductivity k and the unit weight of the pore fluid

γ_{w}

and ∇, ∇ and

\nabla^{2}

are, respectively, the gradient, divergence and Laplace’s operators. When damage occurs in a general fashion the elasticity and fluid transport characteristics of the rock can result in the development of properties that can be anisotropic. If full anisotropy evolution with damage is considered, the elasticity parameters increase to 21 constants and the fluid transport parameters increase to six. Therefore the generalized treatment of damage-induced alterations to coupled poroelasticity effects can result in an unmanageable set of parameters and damage evolution laws. This has restricted the application of damage mechanics to the study of coupled poroelasticity problems. The prudent approach is to restrict attention to isotropic damage evolution that can be specified in terms of the effective stress defined by (41) and the strain equivalent hypothesis. The alterations in the elastic stress–strain relationships for the damaged isotropic elastic skeleton can be represented in the form

σ = 2 (1 - D) μ ε + \frac{2 (1 - D) μ ν}{(1 - 2 ν)} (\nabla . u) I + α p I

where D is the isotropic damage parameter and implicit in (49) is the assumption that Poisson’s ratio remains unaltered during the damage process and alterations to the isotropic Lamé and other elastic constants and the permeability parameters maintain the energetic requirements

μ (D) > 0; 0 \leq \tilde{B} (D) \leq 1; - 1 < ν (D) < ν_{u} (D) \leq 0.5; κ (D) > 0

This assumption makes the incorporation of influences of damage on the poroelastic response more manageable. In addition to the specification of the constitutive relations for the damaged geomaterial skeleton, it is also necessary to prescribe damage evolution criteria that can be postulated either by appeal to micro-mechanical considerations or determined by experiment. Examples of the evolution of the damage parameter with either stress or strain have been presented in the literature and references to these studies are given by Mahyari and Selvadurai (1998), Selvadurai (2004) and Selvadurai and Shirazi (2004, 2005); the experimental information that describes the variation of the poroelastic parameters with damage is generally scarce. Also, the extent of damage necessary to create substantial alterations to permeability evolution is generally small. By and large, the porous skeleton of the fluid-saturated rock can remain elastic with a small change in the elasticity parameters during which the permeability of the rock can change substantially. Based on this observation, Selvadurai (2004) introduced the concept of “Stationary Damage” in poroelastic solids where the skeletal elasticity properties remain at a constant value after the initiation of elastic damage and alterations in the hydraulic conductivity characteristics are determined at the damage level corresponding to the stationary damage or elastic estimates. Zoback and Byerlee (1975) have documented results of experiments conducted on granite and Shiping et al. (1994) give similar results for tests conducted on sandstone. They observed that the permeability characteristics of these materials can increase by an order of magnitude before the attainment of the peak values of stress and they can increase up to two-orders of magnitude in the strain softening regime where micro-cracks tend to localize in shear faults. Kiyama et al. (1996) also observed similar results for the permeability evolution of granites subjected to a tri-axial stress state. This would suggest that localization phenomena could result in significant changes in the permeability in the localization zones. It must be emphasized that in this study the process of localization is excluded from the analysis and all changes in permeability are at stress states well below those necessary to initiate localization or global failure of the material. Furthermore, in keeping with the approximation concerning scalar isotropic dependency of the elasticity properties on the damage parameter, we shall assume that the alterations in the permeability characteristics also follow an isotropic form. This is clearly an approximation with reference to the mechanical response of brittle geomaterials that tend to develop micro-cracking along the dominant direction of stressing, leading to higher permeabilities in orthogonal directions. The studies by Mahyari and Selvadurai (1998) suggest the following postulates for the evolution of hydraulic conductivity as a function of the parameter ξ_d:

\begin{matrix} k^{d} = (1 + \tilde{α} ξ_{d}) k & ; & k^{d} = (1 + \tilde{β} ξ_{d}^{2}) k \end{matrix}

where k^d is the hydraulic conductivity applicable to the damaged material, k is the hydraulic conductivity of the undamaged material,

ξ_{d}

is the equivalent shear strain, which is related to the second invariant of the strain deviator tensor, and

\tilde{α}

and

\tilde{β}

are material constants. This approach has enabled the application of isotropic damage mechanics concepts to examine the time-dependent effects that can materialize in fluid-saturated poroelastic media where the porous fabric undergoes mechanical damage with an attendant increase in the hydraulic conductivity characteristics.

5.2 Application of the concept of “Stationary Damage”

As an illustration of the application of the concept of “Stationary Damage” to problems in poromechanics, we specifically consider the problem of the indentation of a poroelastic halfspace by a rigid circular indenter with a flat smooth base. This is a celebrated problem in contact mechanics; the elastic solutions were first presented in the classic studies by Boussinesq (1885) and Harding and Sneddon (1945). Computational techniques are applied to examine the influence of elastic damage-induced fluid transport characteristics on the time dependent indentational response of the Boussinesq indenter.

Indentation and contact problems occupy an important position in both engineering and applied mechanics. Solutions derived for classical elastostatic contact problems have been applied to examine the mechanics of indentors used for materials testing, mechanics of nano-indentors, tribology, mechanics of foundations used for structural support, biomechanical applications for prosthetic implants and, more recently, in the area of contact mechanics of electronic storage devices. We consider the problem of the frictionless indentation of a poroelastic material by a rigid circular punch with a flat base (Figure 16), which is subjected to a total load P₀, which is in the form of a Heaviside step function of time. The associated classical elasticity solution was first given by Boussinesq (1885), who examined the problem by considering the equivalence between the elastostatic problem and the associated problem in potential theory.

Figure 16 Indentation of a damage susceptible poroelastic halfspace.

Harding and Sneddon (1945) subsequently examined the problem in their classic paper that uses Hankel transform techniques to reduce the problem to the solution of a system of dual integral equations. The procedure also resulted in the evaluation of the load-displacement relationship for the indentor in exact closed form. In a subsequent paper, Sneddon (1946) presented complete numerical results for the distribution of stresses within the halfspace region. The classical poroelasticity problem concerning the static indentation of a poroelastic halfspace and layer regions by a rigid circular indentor with a flat smooth base and related contact problems were considered by a number of authors including Agbezuge and Deresiewicz (1974), Chiarella and Booker (1975), Gaszynski (1976), Gaszynski and Szefer (1978), Selvadurai and Yue (1994), Yue and Selvadurai (1994, 1995a, b) and Lan and Selvadurai (1996), using differing boundary conditions related to the pore pressure at the surface of the halfspace both within the contact zone and exterior to it. These authors also use different computational schemes for the numerical solution of the resulting integral equations and for the inversion of Laplace transforms. Of related interest are problems associated with the dynamic problem of a rigid foundation either in smooth contact or bonded to the surface of a halfspace (Halpern & Christiano, 1986; Kassir & Xu, 1988; Philippacopoulos, 1989; Senjuntichai & Rajapakse, 1996), where, in certain circumstances, the static transient poroelasticity solution can be recovered. The former studies will form a basis for a comparison with the modeling involving stationary damage; here we will consider only changes in the hydraulic conductivity characteristics, which will be altered corresponding to the initial elastic strains induced during the loading of the indenter. Also, the load applied is specified in the form of a Heaviside step function in time. In order to determine the stationary spatial variation of hydraulic conductivity properties within the halfspace region, it is first necessary to determine the distribution of the equivalent shear strain ξ_d in the halfspace region. Formally, the distribution ξ_d(r, z), is determined by considering the stress state in the halfspace region associated with the elastic contact stress distribution at the indenter-elastic halfspace region, which is given by

σ_{z z} (r, 0) = \frac{P_{0}}{2 π a \sqrt{a^{2} - r^{2}}}; r \in (0, a)

and the classical solution by Boussinesq (1885) for the problem of the action of a concentrated normal load at the surface of a halfspace region (see also Selvadurai, 2000a, 2001). The displacement distribution at the surface of the halfspace region is given by

u_{z} (r, 0) = {\begin{matrix} Δ \\ \frac{2 Δ}{π} \sin^{- 1} (\frac{a}{r}) \end{matrix} \begin{matrix} ; & r \in (0, a) \\ ; & r \in (a, \infty) \end{matrix} .

The stress state in the halfspace region is given by

\begin{array}{l} σ_{r r} (ρ, ζ) = - \frac{P_{0}}{2 π a^{2}} [J_{1}^{0} + 2 \tilde{ν} {J_{1}^{0} - J_{0}^{1}} - ζ J_{2}^{0} - \frac{1}{ρ} {(1 - 2 \tilde{ν}) J_{0}^{1} - ζ J_{2}^{1}}] \\ σ_{θ θ} (ρ, ζ) = - \frac{P_{0}}{2 π a^{2}} [2 \tilde{ν} J_{0}^{1} + \frac{1}{ρ} {(1 - 2 \tilde{ν}) J_{0}^{1} - ζ J_{2}^{1}}] \\ σ_{z z} (ρ, ζ) = - \frac{P_{0}}{2 π a^{2}} [J_{1}^{0} + ζ J_{2}^{0}] \\ σ_{r z} (ρ, ζ) = - \frac{P_{0}}{2 π a^{2}} [ζ J_{2}^{1}] \end{array}

where

\tilde{ν}

is Poisson’s ratio for the elastic solid and the infinite integrals

J_{n}^{m} (ρ, ζ)

are defined by

J_{n}^{m} (ρ, ζ) = \int_{0}^{\infty} s^{n - 1} \sin (s) \exp (- s ζ) J_{m} (s ρ) d s .

As has been shown by Sneddon (1946), these infinite integrals can be evaluated in explicit closed form as follows:

\begin{array}{l} J_{1}^{0} (ρ, ζ) = \frac{1}{\sqrt{R}} \sin (\frac{ϕ}{2}); J_{0}^{1} (ρ, ζ) = \frac{1}{ρ} (1 - \sqrt{R} \sin (\frac{ϕ}{2})) \\ J_{1}^{1} (ρ, ζ) = \frac{Ψ}{ρ \sqrt{R}} \sin (θ - \frac{ϕ}{2}); J_{2}^{1} (ρ, ζ) = \frac{ρ}{R^{3 / 2}} \sin (\frac{3 ϕ}{2}) \\ J_{2}^{0} (ρ, ζ) = \frac{Ψ}{R^{3 / 2}} \sin \frac{3}{2} (ϕ - θ) \end{array}

where

\begin{array}{l} \tan θ = \frac{1}{ζ}; \tan ϕ = \frac{2 ζ}{(ρ^{2} + ζ^{2} - 1)}; Ψ^{2} = (1 + ζ^{2}) \\ \begin{matrix} R^{2} = [{(ρ^{2} + ζ^{2} - 1)}^{2} + 4 ζ^{2}] & ; ρ = \frac{r}{a}; & ζ = \frac{z}{a} . \end{matrix} \end{array}

The principal stress components are determined from the relationships

\begin{matrix} \begin{matrix} σ_{1} \\ σ_{3} \end{matrix}} = \frac{1}{2} [(σ_{r r} + σ_{z z}) \pm \sqrt{{(σ_{r r} - σ_{z z})}^{2} + 4 σ_{r z}^{2}}] & ; & σ_{2} = σ_{θ θ} \end{matrix}

and the equivalent shear strain

ξ_{d}

can be expressed in the form

ξ_{d} = \frac{1}{2 \sqrt{3} \tilde{μ}} {[{(σ_{1} - σ_{3})}^{2} + {(σ_{3} - σ_{2})}^{2} + {(σ_{2} - σ_{1})}^{2}]}^{1 / 2}

where

\tilde{μ}

is the linear elastic shear modulus for the elastic solid. In these general elasticity solutions, the elastic constants

\tilde{μ}

and

\tilde{ν}

can be assigned values corresponding to their values at time

t = 0

, to reflect the undrained behavior of the poroelastic solid.

An examination of both (52) and (54) indicates that the elastic stress state is singular at the boundary of the rigid indenter. This places a restriction on the rigorous application of the stress state (54) for the determination of damaged regions. By definition, damaged regions are assumed to experience only finite levels of isotropic damage that would maintain the elastic character of the material. The singular stress state can result in either plastic failure of the rock (Ling, 1973; Johnson, 1985) or even brittle fracture extension in the halfspace region (Selvadurai, 2000b). Such developments are assumed to be restricted to a very limited zone of the halfspace region in the vicinity of the boundary of the indenter region. Also, in the computational modeling of the contact problem, no provision is made for the incorporation of special elements at the boundary of the contact zone to account for the singular stress state that can be identified from mathematical considerations of the contact problem. In the computational modeling, the mesh configuration is suitably refined to account for the sharp stress gradients that will result from the elastic stress state (Figure 17).

Figure 17 Finite element discretization of the damage susceptible spherical poroelastic region for the cylindrical indenter problem.

The distribution of equivalent shear strain is accounted for by assigning the values of the equivalent strains to the integration points within the elements. These in turn are converted to alterations in the hydraulic conductivity characteristics of the medium through the use of the expressions (51) that relate the hydraulic conductivity to the equivalent shear strain. In the study by Selvadurai (2004), the computational modeling was performed using the general purpose finite element code ABAQUS, although any computational code that is capable of examining poroelasticity problems for inhomogeneous media can be adopted for the purpose.

An 8-noded isoparametric finite element is used in the modeling and the integrations are performed at the nine Gaussian points. The displacements are specified at all nodes and the pore pressures are specified only at the corner nodes.

The application of the “Stationary Damage Concept” to the examination of the poroelastic contact problem where the porous skeleton can experience damage due to stress states that will not induce plastic collapse or extensive plastic failure in the rock involves the following steps:

– Apply the loadings to the elastic domain and determine the equivalent shear strain
$ξ_{d}$
in the domain.
– Determine the distribution of permeability alteration in the region by appeal to the prescribed variation of permeability with
$ξ_{d}$
. This will result in a poroelastic domain with a permeability heterogeneity.
– Solve the poroelasticity problem where the loads are applied at a prescribed time variation, where the elasticity and permeability characteristics are as implied by the previous steps.

We consider the computational modeling of the indentation problem where the pore fluid that saturates the porous elastic solid is assumed to be incompressible. The indenter is subjected to a load with a time variation in the form of a Heaviside step function i.e.:

P (t) = P_{0} H (t); t > 0

where

P_{0}

is the magnitude of the total load acting on the indentor. Since the saturating fluid is assumed to be incompressible, the undrained behavior of the fluid-saturated poroelastic medium at time

t = 0

corresponds to an elastic state with

\tilde{ν} = 1 / 2

. The initial elastic strains that induce the spatial distribution of damage during the indentation are evaluated by setting

\tilde{ν} = 1 / 2

in the principal stresses computed, using the stress state (52), relevant to the smooth indentation of the halfspace with the porous rigid circular cylinder.

The finite element discretization of the halfspace domain used for the analysis of the indentation of the poroelastic halfspace by the porous rigid circular smooth indenter, of radius

a = 2 m

, is shown in Figure 17. To exactly model a poroelastic halfspace region it is necessary to incorporate special infinite elements that capture the spatial decay in the pressure and displacement fields (see e.g. Simoni & Schrefler, 1987; Selvadurai & Gopal, 1989). These procedures were, however, unavailable in the computational modeling software. The poroelastic halfspace region is modeled as a hemispherical domain, where the outer boundary is located at a large distance (60 a) from the origin (Figure 17). This external spherical boundary is considered to be rigid and all displacements on this boundary are constrained to be zero. The boundary is also considered to be impervious, thereby imposing Neumann boundary conditions for the pore pressure field at this spherical surface. (It is noted here that computations were also performed by prescribing Dirichlet boundary conditions on this surface. The consolidation responses computed were essentially independent of the far field pore pressure boundary condition.) The accuracy of the discretization procedures, particularly with reference to the location of the external spherical boundary at a distance 60 a, was first verified through comparisons with Boussinesq’s solution for the indentation an elastic halfspace region by a cylindrical punch. The value of the elastic displacement of the rigid circular indenter can be determined to an accuracy of approximately three percent. The computational modeling of the poroelastic indentation problem is performed by specifying the following values for the material parameters generally applicable to a geomaterial such as sandstone (Selvadurai, 2004): hydraulic conductivity

k = 1 \times 10^{- 6} m / s

; unit weight of pore fluid

γ_{w} = 1 \times 10^{4} N / m^{3}

; Young’s modulus.

E = 8.3 G P a

; Poisson’s ratio

ν = 0.195

; the corresponding coefficient of consolidation, defined by

c = \frac{2 μ k}{γ_{w}} = 0.6946 m^{2} / s .

Figure 18 illustrates the comparison between the analytical solution for the time-dependent variation in the rigid displacement of the circular indentor given by Chiarella and Booker (1975) and the computational results obtained for the indentation of a poroelastic region with an external boundary in the shape of a hemisphere. The comparison is between an estimate for a halfspace region and a region of finite extent and there is reasonable correlation between the two sets of results. Since the influence of “Stationary Damage” will be assessed in relation to the computational results derived for the indentation of the hemispherical region, the accuracy of the computational scheme in modeling the indentation process is considered to be acceptable. The normal stress distribution at the contact zone can be determined to within a similar accuracy except in regions near the boundary of the indenter where, theoretically, the contact stresses are singular. In the computational treatments, the stationary damage-induced alteration in the hydraulic conductivity is evaluated according to the linear dependency in the hydraulic conductivity alteration relationship given by (51) and the parameter α is varied within the range

α \in (0, 10^{4})

. Figure 19 illustrates the results for the time-dependent displacement of the rigid cylindrical smooth indentor resting on a poroelastic hemispherical domain that displays either stationary damage-induced alteration in the hydraulic conductivity or is independent of such effects. These results are for a specific value of the hydraulic conductivity-altering parameter,

α = 10^{3}

. Computations can also be performed to determine the influence of the parameter α and the stationary damage-induced alterations in the hydraulic conductivity on the settlement rate of the rigid indenter. The results can be best illustrated through the definition of a “Degree of Consolidation”, defined by

Figure 18 Comparison of analytical results and computational estimates for the cylindrical rigid indenter problem.

Figure 19 Influence of stationary damage on the displacement of the rigid cylindrical indentor [α= 1000].

U_{C} = \frac{Δ_{C} (t) - Δ_{C} (0)}{Δ_{C} (\infty) - Δ_{C} (0)}

The value

Δ_{C} (t)

corresponds to the time-dependent rigid displacement of the indenter. Both the initial and ultimate values of this displacement, for the purely poroelastic case, can be evaluated, independent of the considerations of the transient poroelastic responses since the poroelastic model allows for purely elastic behavior at

t = 0

and as

t \to \infty

, with

ν = 1 / 2

and

\tilde{ν} = ν

respectively: i.e.

Δ_{C} (0) = \frac{P_{0}}{8 μ a}; Δ_{C} (\infty) = \frac{P_{0} (1 - ν)}{4 μ a} .

Figure 20 illustrates the variation of the degree of consolidation of the rigid indenter as a function of non-dimensional time and the parameter α that defined the alteration of the hydraulic conductivity with the equivalent shear strain

ξ_{d}

Influence of stationary damage and hydraulic conductivity alteration on the consolidation rate for the rigid cylindrical indenter.

Figure 20 Influence of stationary damage and hydraulic conductivity alteration on the consolidation rate for the rigid cylindrical indenter.

6 Numerical Modeling of Damage

We have seen in Section 3 that, from a theoretical point of view, two main approaches exist to model rock damage. These are based either on Continuum Mechanics, or on Fracture Mechanics.

In terms of numerical modeling, the continuum approach is classically handled using the Finite Element Method whereas fracturing is more difficult to deal with numerically. However, relatively recently, methods have been developed to incorporate the nucleation and propagation of cracks in an originally continuous medium. These methods belong to the family of Extended Finite Element Methods (XFEM). An alternative approach is the Finite Discrete Element Method (FDEM), which is also a recent computational tool that combines the Finite Element Method with the Discrete Element Method (DEM).

Currently, these methods are difficult to use in an engineering context because of the complexity of rock structures and the lack of knowledge of the boundary and initial conditions. For rock engineering problems, the computational time required is still cumbersome.

6.1 Modeling a loading test with CDM

Identification of model parameters is based on experiments such as creep tests, relaxation tests and quasi-static tests. Figure 21 shows the numerical modeling of a compression test performed with a low strain rate of loading (Fabre & Pellet, 2006). Both axial strain and lateral strain are well reproduced by the model. Additionally, the calculated change in volumetric strain (contraction-dilation) is consistent with the observed location.

Figure 21 Modeling of a uniaxial compressive test performed on shale (Fabre & Pellet, 2006).

6.2 Excavation Damage Zone (EDZ) around underground openings

When an underground opening is excavated in a stressed rock mass, short-term instabilities may occur during or right after operation. Collapses in the gallery may also take place many years or decades after the completion of the work. Indeed, the time-dependent behavior of rock has always been an issue for underground construction. This question has been recently studied, both experimentally and theoretically, by several authors for different types of rock (Blümling et al., 2007; Hudson et al., 2009).

A 3D numerical simulation of the mechanical behavior of deep underground galleries with a special emphasis on time-dependent development of the Excavation Damage Zone (EDZ) was presented by Pellet et al. (2009). In this study, the rock mass behavior is modeled by a damageable viscoplastic constitutive model (see section 4.1) in which both viscous and damage parameters are taken into account. Finite-element analysis investigates the evolution of near field stresses, progressive development of the damage zone as well as delayed displacements during the sequential construction process of the gallery. The influence of the orientation of in situ stresses with respect to the gallery axis is also highlighted. The effect of a support system is to reduce the damage zone and the displacements around the gallery (Figure 22).

Excavation Damage Zone (EDZ) and stress distribution. In the centre diagram the greyish areas represent the extension of the EDZ, 300 years after the tunnel construction, for a supported tunnel (left hand side) and for an unsupported tunnel (right hand side). The tangential stresses in the rock mass at different times are also represented (adapted from Pellet

Figure 22 Excavation Damage Zone (EDZ) and stress distribution. In the centre diagram the greyish areas represent the extension of the EDZ, 300 years after the tunnel construction, for a supported tunnel (left hand side) and for an unsupported tunnel (right hand side). The tangential stresses in the rock mass at different times are also represented (adapted from Pellet et al., 2009).

7 Conclusion

Damage to rock depends largely on its geological nature and the state of stress it is subjected to. The physico-chemical identification of rocks is therefore a prerequisite that requires special attention. This involves a precise characterization of the mineralogical content and a detailed description of the rock fabric. Using these observations, the petrophysical properties of the rock (porosity, specific gravity, water content and degree of saturation ...) will be estimated. There have been considerable advances in the digital treatment of experimental data over the last two decades, associated with the development of new testing equipment (Scanning Electron Microscope, Magnetic Resonance Imaging, Acoustic Tomography, etc.) that has greatly facilitated the identification of the rock type and its in situ state.

Laboratory testing can then be used to determine the mechanical properties of the rock under investigation; these include the strength and deformability. Here, too, modern techniques allow complex mechanical tests involving high pressure-high temperature, to be combined with physical measures such as acoustic activity or the velocity of elastic waves. 3D visualization by X-ray computed tomography (CT) of the pore space helps to localize the areas of deformation and cracking.

Constitutive models can only be developed based on an understanding and knowledge of the detailed characterization of the state of the rock both in its intact state and after mechanical testing. We have seen that there are two main approaches employed in Rock Mechanics for damage characterization: Continuum Damage Mechanics and Fracture Mechanics. The first, formulated in the context of the thermo-dynamics of irreversible processes, lends itself to numerical modeling although certain hypotheses (shear failure mode, small deformations, etc.) must be invoked. For example, processing softening behavior inevitably requires regularizing the solution by introducing a time variable to establish a rate-dependent model. If the model is rate-independent, such an approach should be reserved for the damage suffered before the peak stress is achieved.

The second approach, Fracture Mechanics, is older and also more physical in that it addresses the failure mode by extension (Mode I: crack opening), which is frequently observed experimentally. However, this fails to meet the assumption of a continuum and is therefore inconvenient to implement, particularly if multiple crack generation, crack branching and discrete fragmentation is involved. However the introduction in the last ten years of the XFEM technique (eXtended Finite Element Method) allows the two approaches to be combined. The engineering applications of these advances are limited; in particular, it is difficult to account for the merging of several networks of cracks.

The use of a constitutive model for full-scale processes requires many adaptations that go beyond the simple consideration of the scale effect. The need to account for damage discontinuities is paramount and often the time effects are equally important. Under certain loading conditions the effects of repeated actions (cyclic and/or dynamic) can be the most important consideration: loadings of a thermo-hydro-mechanical origin are often related to damage through permeability changes.

Over the past two decades, great progress has been made on the experimental characterization of damage in rocks and rock masses and numerical methodologies have improved substantially. Future developments should particularly focus on the development of numerical tools for the identification of the parameters to be used in constitutive models by means of inverse analysis of measurements made on real structures.

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In This Chapter

Rock damage mechanics

Rock Mechanics and Engineering

Abstract

Rock damage mechanics

1 Introduction

2 Experimental evidence

2.1 Investigations at the laboratory scale

2.2 Physical interpretation of cracking of rock at the micro-scale

3 Constitutive Modeling

3.1 Introduction

3.2 Linear fracture mechanics

3.2.1 Energy balance consideration

3.2.2 Steady and unsteady crack propagation

3.2.3 Griffith crack initiation criterion

3.2.4 Stress field and fracture modes

3.2.4.1 Failure criterion in the open mode (Mode I)

Table 1 Toughness of different types of rocks, after Atkinson and Meredith (1987).

3.2.4.2 Failure criterion in the mixed mode

3.3 Continuum damage mechanics

3.3.1 Law of transformation

3.3.2 Principle of equivalence

3.3.3 Damage variable

3.3.4 Isotropic and anisotropic damage

3.4 Discussion of Fracture Mechanics (FM) vs. Continuum Damage Mechanics (CDM)

4 Other Types of Loading

4.1 Time-dependent damage

4.2 Damage under cyclic loading

4.3 Thermo-mechanical damage

4.4 Rock joint damage

5 Permeability Evolution with Damage

5.1 Modeling saturated porous media with continuum damage

5.2 Application of the concept of “Stationary Damage”

6 Numerical Modeling of Damage

6.1 Modeling a loading test with CDM

6.2 Excavation Damage Zone (EDZ) around underground openings

7 Conclusion

References

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