ENS 080312 en JZ Notes Chapter 3

Chapter 3 Principe of Mechanics

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CHAPTER 3 PRINCIPLES OF MECHANICS This Chapter is intended as a review on the mechanics principles that relevant to rock mechanics. Reader who is not familiar with mechanics should start with one of the many textbook on mechanics, listed at the end of this Chapter. 3.1 Stress, Strain and Deformation Modulus 3.1.1 Normal and Shear Stresses Stress is commonly defined the internal distribution of force per unit area that balances and reacts to external loads applied to a body. It is a second-order tensor with nine components, but can be fully described with six components due to symmetry in the absence of body moments (Wikipedia). Stresses can be divided down into normal stresses and shear stresses. Normal stress is defined as stress perpendicular to the plane where the stress is act on. Shear stress is the stress parallel to the plane where the stress is act on. They are perhaps best represented by the diagram shown in Figure 3.1.1a. Figure 3.1.1a Normal and shear stresses on an infinitesimal cube aligned with the Cartesian axes. (Hudson 35) The stresses act on an infinitesimal cube aligned with the Cartesian axes has nine components: three normal stresses and six shear stresses. The three normal stresses are σxx, σyy and σzz; and the six shear stresses are τxy, τxz, τyx, τyz, τzx, and τzy. Often, the nine stress components are often conveniently represented in matrix form, where the row representing the stresses on a plane and the column representing the stresses in a direction, σxx τxy τxz σ = τyx σyy τyz (3.1.1a) τzx τzy σzz By considering the state of equilibrium, each pair of shear stresses are equal, hence, τxy = τyx, τxz = τzx, τyz = τzy (3.1.1b) The stress components in the matrix can be reduced from nine to six, and the stress matrix can be written as,


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σxx τxy τxz σ = τxy σyy τyz (3.1.1c) τxz τyz σzz 3.1.2 Principal Stresses Usually, there are nine stress components acting in three directions on three planes. However, if the planes where the stresses act on are rotated, there is a position that the magnitude of shear stresses is zero. Hence, the stresses acting on the three planes are only normal stresses. In this situation, the three normal stresses are termed as principal stresses. σ 1 0 0 σ = 0 σ 2 0 (3.1.2a) 0 0 σ 3 Figure 3.1.2a shows the principle of stress transformation on a 2D element subjected to normal and shear stresses. Considering a plane a-b, the normal and shear stresses on the a-b plane is σab and τab. Figure 3.1.2a Stress transformation and Mohr’s circle representation. Summing the components of forces acting on the element in the direction of a-b plane, it gives, in the normal direction, σab (ab) = σx sinθ (ab) sinθ + σy cosθ (ab) cosθ + 2 τxy (ab) sinθ cosθ or σab = σx sin2

θ + σy cos2θ + 2 τxy sinθ cosθ (3.1.2b)

or

σy + σx σy – σx σab = 2 + 2 cos2θ + τxy sin2θ (3.1.2c)

here, σy is assumed to be greater than σx. Again, in the shear direction along a-b plane, the force equilibrium can be written as, τab (ab) = -σx cosθ (ab) sinθ + σy sinθ (ab) cosθ – τxy cosθ (ab) cosθ + τxy sinθ (ab) sinθ


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or τab = σy sinθ cosθ – σx sinθ cosθ – τxy (cos2

θ – sin2θ)

or

σy – σx τab = 2 sin2θ – τxy cos2θ

To make the a-b plane that only principal stress exist, i.e., τab = 0, then,

2 τxy tan2θ = σy – σx

The above equation gives two values of θ that are 90° apart. This means that there are in fact two planes at right angles to each other on which shear stress is zero. These planes are the principal planes and stresses acting on these planes are principal stresses. By substituting the above equations, the values of the principal stresses can be found,

σy + σx σy – σxσab = σ1 = 2 + [( 2 )2 + τxy2 ]½

as the major principal stress, and,

σy + σx σy – σxσab = σ2 = 2 – [( 2 )2 + τxy2 ]½

as the minor principal stress. The normal stress and shear stress acting on any plane can also be determined by using the Mohr’s circle, as shown in Figure 3.1.2a. On the Mohr’s circle, the points where the circle intercept the normal stress axis, where shear stress is zero, are the values of principal stresses. The Mohr’s circle can also be used to determine normal and shear stresses acting on a plane of a given angle to the given principal stresses value and direction. This will be illustrated later in dealing with the Mohr-Coulomb strength criterion. The concept of principal stresses is important in rock mechanics. For example, in excavation, the excavated face is free from shear stress and normal stress, therefore, the rock at this position is subjected to principal stresses and one (normal to the excavated face) of the principal stresses is in fact zero. 3.1.3 Deformation and Strain


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When a material is subject to stress, the material deforms. The deformation occurs not only in the direction of the stress, but also in the other two perpendicular directions. Again, refer the cube in to Figure 3.1.3a, with the normal stress, σxx, the cube deforms (contracts) in the direction of x-x, and expends in the directions of y-y and z-z. Figure 3.1.3a Deformations due to stress on an infinitesimal cube. Strain is defined as the ratio of deformation to the original dimension, i.e., ε = δ / l (3.1.3a) It therefore has no dimension. Often, it is expressed in term of percentage. Strain can also expressed in matrix form: εxx γxy γxz ε = γxy εyy γyz (3.1.3b) γxz γyz εzz εxx, εyy and εzz are normal strain, and γxy, γxz and γyz are shear strain. Strain is a mechanical property of the material, it only occurs when a stress is applied, directly or through other means. In another word, when there is a strain, there must be stress accompanying the strain. Often, volumetric strain is used in rock mechanics. It is defined as the ratio of volume change to the original volume, i.e., εv = Δv / v (3.1.3c) 3.1.4 Poisson’s Ratio When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. Poisson's ratio is a measure of this tendency. Poisson's ratio the ratio of the contraction strain (normal to the applied load) divided by the extension strain (in the direction of the applied load). Refer to Figure 3.1.3, with a given normal stress in x-x direction, strains occur in the direction of x-x, as well as in the directions of y-y- and z-z. The ratio of strain in y-y- or z-z direction to strain in x-x direction is defined as the Poisson’s ratio, ν = εyy / εxx, ν = εzz / εxx (3.1.4a) For a perfectly incompressible material, the Poisson's ratio would be exactly 0.5. Most rock materials have ν between 0.2 and 0.4. The Poisson’s ratio is a mechanical property


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of the material. For a homogeneous material, the Poisson’s ratios in y-y and z-z directions are equal. 3.1.5 Deformation Modulus Deformation modulus is a mechanical property indicating the rate of change of strain with change of stress. For an elastic material, it is termed as Modulus of Elasticity, and often referred as the Young’s Modulus. It is defined as the ratio, for small strains, of the rate of change of stress with strain. This can be experimentally determined from the gradient of a stress-strain curve obtained from loading tests conducted on a sample of the material. E = Δσ / Δε (3.1.5a) The Young’s Modulus, E, has the same unit as the stress, σ. When a material is subject to a stress, it undergoes strain. The stress-strain relationship can often to be represented by the stress-strain curve, typically shown in Figure 3.1.5a. The deformation modulus at a specific stress level is the gradient of the stress-strain curve at point of that specific stress level. For an elastic material, the stress-strain is a straight line, and the modulus is therefore the gradient of the line. Figure 3.1.5a Determination of deformation modulus from stress-strain curves. Stress-strain relationship can be expressed in matrix, εxx S11 S12 S13 S14 S15 S16 σxx

εyy S21 S22 S23 S24 S25 S26 σyy

εzz S31 S32 S33 S34 S35 S36 σzz

γxy = S41 S42 S43 S44 S45 S46 σxy

(3.1.5b)

γyz S51 S52 S53 S54 S55 S56 σyz

γzx S61 S62 S63 S64 S65 S66 σzx

ε = S σ (3.1.5c) σ = D ε (3.1.5d) εxx 1 –ν –ν 0 0 0 σxx

εyy –ν 1 –ν 0 0 0 σyy

εzz 1 –ν –ν 1 0 0 0 σzz

γxy =

E 0 0 0 2(1+ν) 0 0 σxy(3.1.5e)

γyz 0 0 0 0 2(1+ν) 0 σyz

γzx 0 0 0 0 0 2(1+ν) σzx


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3.1.6 Plane Stress and Plane Strain For a thin wall plate, a 3D problem can be treated as a 2D plane stress problem. The stresses and strains are expressed as σxx τxy 0 σ = τxy σyy 0 (3.1.6a) 0 0 0 εxx γxy 0 ε = γxy εyy 0 (3.1.6b) 0 0 εzz εzz can often temporarily removed from analysis to make all stress and strain only the in-plane terms, and effective becomes a 2D analysis. For a long tunnel, a 3D problem can be approximated by a 2D plane strain problem. The strains and stresses are expressed as: εxx γxy 0 ε = γxy εyy 0 (3.1.6c) 0 0 0 σxx τxy 0 σ = τxy σyy 0 (3.1.6d) 0 0 σzz σzz is needed to maintain εzz being zero. σzz stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3D problem to a much simpler 2D problem. 3.2 Strength and failure criteria 3.2.1 Basic Definitions When a material is subjected to a stress, the material deforms. When the stress increases to a certain state, the material starts to yield and subsequently will either loss the strength substantially or will undergo large strain without additional stress, as shown typically in Figure 3.2.1a.


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Figure 3.2.1a Stress-strain curve with yield point, peak strength, post-peak ductile and brittle behaviour. Strength of a material is the limit states of stress (peak or yield) the material can sustain. It is often considered in terms of compressive strength, tensile strength, and shear strength, which are the limit states of compressive stress, tensile stress and shear stress respectively. There are various definitions of strength, some represented by yield and some by peak. In rock mechanics, common definition is by peak strength. Strain occurred at the peak strength is often defined as strain at failure. This is a useful measure in rock mechanics and is associated with the brittleness of the material. Hardness is another commonly used term describing rock material. It is the characteristic of a solid material expressing its resistance to permanent deformation. Toughness is the resistance to fracture of a material when stressed. It is defined as the amount of energy that a material can absorb before rupturing, and can be found by finding the area (i.e., by taking the integral) underneath the stress-strain curve. 3.2.2 Post-Peak Stress-Strain Behaviour Post-peak behaviour is the stress-strain behaviour after the peak strength. When a material has reached the limit of its strength, it usually has the option of either fracture or deformation, approximately resulting to two typical post-peak behaviours: brittle and ductile, as shown in Figure 3.2.1a. Brittle behaviour indicates a substantial reduction of stress-carrying capacity, reflecting fracturing of the material under stress. At post-peak region, the material may have residual strength but that residual strength is generally substantially lower than the peak strength. A brittle material usually has undergoes small strain before failure, i.e., small strain at failure. A brittle failure is generally associated with fracture by tension rather than shear, and there is little or no evidence of plastic deformation before failure. Ductile behaviour is represented by being capable of sustaining large deformations with losing stress carrying capability in the post-peak region. Ductility is the physical property of being capable of sustaining large plastic deformations without fracture. Post-peak behaviour can be influenced by pressure and temperature. This happens as an example in the brittle-ductile transition zone at an approximate depth of 10 km in the Earth's crust, at which rock becomes less likely to fracture, and more likely to deform ductilely.


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Residual strength is the stress carrying capacity at post-peak region. This strength indicates that even after failure, rock often has a (much smaller) load carrying capacity, which can have significant effects on rock engineering. 3.2.3 Yield Strength and Criteria Yield strength, or the yield point, is defined in engineering and materials science as the stress at which a material begins to plastically deform. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible. Knowledge of the yield point is vital when designing a component since it generally represents an upper limit to the load that can be applied (Wikipedia). A yield criterion is a hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles. Since stress and strain are tensor qualities they can be described on the basis of three principal directions. The most common yield criteria are briefly outlines below. Details of those criteria are readily available in textbooks of mechanics and strength. (a) Tresca-Guest criterion The Tresca-Guest criterion is the most simple yield surface. In principal stresses it is expressed as follows: max(|σ1 – σ2|, |σ2 – σ3|, |σ3 – σ1|) = σ0 The Tresca-Guest criterion in three dimensional space of principal stresses is a prism of infinite length and six sides, as illustrated in Figure 3.2.3a. This means that material remains elastic when all three principal stresses are roughly equivalent (a hydrostatic pressure), no matter how much compressed or stretched. But when the material is subject to shearing, one of principal stresses becomes smaller (or bigger), then the yield surface is crossed and material enters plastic domain. In two dimensional space, it is a cross diagonally cut section of the prism (Figure 3.2.3b). Figure 3.2.3a Tresca-Guest criterion in 3D spaces of principal stresses. Figure 3.2.3b Tresca-Guest and Mises criteria in 2D spaces of principal stresses. (b) Mises criterion


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The von Mises criterion is expressed as follows:

Also it can be expressed in non-principal stresses as below:

The von Mises criterion in three dimensional space of principal stresses is a circular cylinder of infinite length, with the same angle to all three axes, as seen in Figure 3.2.3c. In two dimensional space of principal stresses, it is an ellipse that diagonally cut cross the cylinder (Figure 3.2.3b). Figure 3.2.3c Mises criterion in 3D spaces of principal stresses. (c) Mohr-Coulomb criterion The Mohr-Coulomb criterion is a first two-parametric yield surface, for the maximum compression and tension. The model is the first one that takes shearing into account. It is expressed as follows:

or

The parameters are Rc and Rr which are the maximum values for compression and tension for the given material. it should be noted that the criterion considers the maximum difference between the major and the minor principal stresses only, and does not take the intermediate principal stress in the strength criterion.


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The Mohr-Coulomb criterion in three dimensional space of principal stresses is represented by a conical prism (Figure 3.2.3d). If K = 0 then it becomes Tresca-Guest criterion, thus K determines the inclination angle of conical surface. In two dimensional space of principal stresses, it is a cross section of the conical prism (Figure 3.2.3e). Figure 3.2.3d Mohr-Coulomb criterion in 3D space of principal stresses. Figure 3.2.3e Mohr-Coulomb and Drucker-Prager criteria in 2D space of principal stresses. The Mohr-Coulomb strength criterion can be represented graphically, by Mohr’s circle. Most of the classical engineering materials, including rock materials, somehow follow this rule in at least a portion of their shear failure envelope. Application and discussion of the Mohr-Coulomb strength criterion in rock mechanics is given in Chapter 4. (d) Drucker-Prager criterion The Drucker-Prager criterion is expressed as follows:

The Drucker-Prager criterion in three dimensional space of principal stresses is a regular cone (Figure 3.2.3f). If α = 0 then it becomes the Mises criterion. In two dimensional space of principal stresses, it is a cross section of this cone, which produces an ellipsioidal shape (Figure 3.2.3e). Figure 3.2.3f Drucker-Prager criterion in 3D spaces of principal stresses. (e) Unified Strength Criterion


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The Unified Strength Criterion by Yu has the following characteristics: (i) It is able to reflect the fundamental characteristics of brittle materials (including rock

and concrete), i.e., different tensile and compressive strengths, hydrostatic pressure effect, the effect of intermediate principal stress and its zonal change and material dependence.

(ii) It has a clear physics and mechanics background, a unified mathematical model, and a simple and explicit criterion, which includes all independent stress components and simple material parameters.

(iii) It is also suitable for different types of brittle materials under various stress states, and is consistent with the triaxial test results.

(iv) It can be easily applied to analytical and numerical modeling. The Unified Strength Criterion expressed in terms of three principal stresses is as follows: σ1 – α (b σ2 + σ3)/(1 + b) = σt, when σ2 ≤ (σ1 + α σ3)/(1 + α) (σ1 + b σ2)/(1 + b) – α σ3 = σt, when σ2 ≥ (σ1 + α σ3)/(1 + α) where σt is tensile strength, α is the strength ratio of tensile to compressive strength (σt/σc), b is the intermediate principal stress parameter. When b = 0, the Unified Strength Criterion becomes the Mohr-Coulomb (σ1 – α σ3 = σt). The Unified Strength Criterion can produce a full spectrum of new criteria when the value of b varies between 0 and 1 (0 ≤ b ≤ 1), to reflect the characteristics of various different materials. The Unified Strength Criterion is especially versatile in reflecting the σ2 effect to different extents for different materials. In general, the failure surface of the Unified Strength Criterion is a dodecahedral-shaped cone about the hydrostatic axis. The failure surface in the plane perpendicular to the cone axis is shown in Figure 3.2.3g. It is quite obvious that the surface is convex when 0 ≤ b ≤ 1. When b = 0 or 1, the dodecahedral-shaped cone reduces to hexagonal. When α=1, the Unified Strength Criterion is simplified to the Unified Yield Criterion applicable to these materials with the same yield stress in tension and compression, and the three-fold symmetric limit surfaces are simplified to six-fold symmetric yield surfaces. Figure 3.2.3g Failure surfaces of the Unified Strength Criterion on deviatoric plane. 3.3 Fracture Mechanics 3.3.1 Fracture Initiation and Propagation


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Fracture mechanics is a sub-division of solid mechanics studying the failure of a material containing a crack, in order to understand the initiation and propagation of cracking. It uses methods of analytical solid mechanics to calculate the driving force on a crack and methods of experimental solid mechanics to characterize the material's resistance to fracture. Fracturing of a material has to start somewhere within the material. It usually starts at a location of stress concentration. In most materials, they cannot be perfectly homogeneous and there are often existing pores and cracks within the material formed at its creation or through various processes. For example, all rock materials contain micro-cracks and pores. When loaded, those existing cracks and pores causes stress concentration at their tips and from there, new cracks will be initiated and further propagated. There are three types of crack initiation and propagation (Figure 3.3.1a): Mode I crack – Opening mode (a tensile stress normal to the plane of the crack), this mode is most relevant to rock mechanics. Mode II crack - Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front); Mode III crack - Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front). Figure 3.3.1a Three modes of crack initiation and propagation. 3.3.2 Griffith Crack Theory The Griffith crack theory is a theory of brittle fracture, using elastic strain energy concepts. It describes the behaviour of crack propagation of an elliptical nature by considering the energy involved. The equation basically states that when a crack is able to propagate enough to fracture a material, that the gain in the surface energy is equal to the loss of strain energy, and is considered to be the primary equation to describe brittle fracture. Because the strain energy released is directly proportional to the square of the crack length, it is only when the crack is relatively short that its energy requirement for propagation exceeds the strain energy available to it. Beyond the critical Griffith crack length, the crack becomes dangerous. For the simple case of a thin rectangular plate with a crack perpendicular to the load Griffith’s theory becomes: G = π σ2 a / E Gc = π σf

2 a / E where G is the strain energy release rate, σ is the applied stress, a is half the crack length, and E is the Young’s modulus. The strain energy release rate can otherwise be understood


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as the rate at which energy is absorbed by growth of the crack. When G is greater than the critical value Gc, crack will begin to propagate. This concept is often referred as Griffith’s energy instability concept. This concept was applied in rock mechanics to develop the Griffith strength criterion, as outlined in te next chapter. 3.3.3 Stress Intensity and Fracture Toughness A modification of Griffith’s theory was made by Irwin. In the modified criterion, strain energy release rate is replaced by stress intensity (KI)and surface energy is replaced by fracture toughness (Kc). Both of these terms are simply related to the energy terms in the original Griffith theory: KI = σ (π a)½ Kc = (E Gc) ½ for plane stress Kc = [E Gc /(1 – ν)]½ for plane strain where ν is the Poisson’s ratio. It is important to note that Kc has different values when measured under plane stress and plane strain conditions Fracture occurs when KI ≥ Kc. For the special case of plane strain deformation, Kc becomes KIc and is considered a material property. The subscript I refers to mode I cracking. 3.4 Mechanics of Discontinuity 3.4.1 Discontinuity and Global Continuum Law Discontinuity in a continuous material is defined a plane feature where continuity of the materials is no longer hold. They are usually significant mechanical breaks in a material. Continuity may continue to exist after the discontinuity. Typical discontinuities are fractures, interfaces and joints. In rock, discontinuities are fractures, joints, faults and bedding plane. While the rock materials are often considered as continuous materials, rock fractures and joints are commonly treated as discontinuities. While the mechanics at discontinuity is discontinuous, certain laws of continuum are still hold. 3.4.2 Stress and Strain at Discontinuity


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Stresses are often disturbed by a discontinuity. For example, at non-fully-contacted fracture, opening exists. Normal stress on the opening is zero and there are stress concentrations on the contact points of the fracture surface. The stress field is no longer the same as in the continuous material. The only exception is a fully-contact plane perpendicular to a uniaxial load. In this case, the stress field is continuous although strain may not. Similarly, displacement at discontinuity is not continuous. For example, at a fracture plane, sliding or shear displacement may occur. There may be much greater normal displacement at fracture than those of the material. Discontinuities can range from a fully-contacted interface to an opening containing different material. The mechanics of are vary different. For a fully-welded interface between two different materials, the interface actually has the continuities both is stress and displacement. The discontinuity is presented in the change of mechanics of the two materials on each side of the interface. For a fully-contacted smooth interface, the interface is subjected to shear displacement. For a locally-contacted fracture, i.e., there are void between the two sides, both stress and displacement discontinuous are expected. 3.4.3 Mechanics of Discontinuities for Normal Stress and Deformation Normal stress and displacement of fully-contact discontinuity is continuous and therefore can be dealt with continuum approach. For a locally-contacted fracture shown in Figure 3.4.3a, i.e., there are void between the two sides, stress-displacement function is discontinuous. For an idealised pillar-like locally-contacted fracture shown below, assume the ratio of contact area to intact area (A) is k (k=0~1), i.e., contact area at fracture is kA. Figure 3.4.3a Idealised fracture with local contact of constant area. Assume the normal stress in the intact portion is N and the total load is NA. The normal stress in the contact area is NA/mA = N/m. Since 0<m≤1, stress in the contact area is always higher than that in the intact portion. Assume the material modulus (E) is the same for the intact portion and the contact area, then strain of the contact area is N/kE. If there is no damage of the contact element, the contact area will remain the same with change of load. Therefore, the load-deformation of the contact zone is linear. Stiffness (k) of the contact zone is a constant, is given as,


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Load N A m A Ek = Deformation = (N / m E) d = d (3.4.3a)

For a prism shape locally-contacted fracture shown in Figure 3.4.3b, the contact area has an initial contact area of A0, and a thickness of d. The contact area in this case is no longer constant; it in fact increases with increasing displacement (closure) of the contact zone. i.e., ΔA∝Δh2. Figure 3.4.3b Idealised fracture with local contact of changing area. With increase of loading, and discontinuity closes, the contact area increases. Therefore, at beginning of loading, the contact area is small, rate of deformation is large. At higher load, the contact area increases, the rate of deformation becomes smaller. Clearly the load-displacement of the contact zone is no longer linear. This non-linear load-displacement characteristics is typical for rock fractures, as shown in Figure 3.4.3c. Figure 3.4.3c Typical non-linear load-displacement characteristics of rock fractures. 3.4.4 Mechanics of Discontinuities for Shear Stress and Deformation The most common known shear phenomenon of a discontinuity is the sliding between two contact surfaces (Figure 3.4.4a), i.e., the friction theory. It gives the relationship between the friction angle φ, the normal force (N) and shear force (S), as S = N tanφ. Figure 3.4.4a Sliding between two horizontal contact surface. When slipping at the surface of contact is about to occur, the maximum static frictional force is proportional to the normal force. When slipping is occurring, the kinetic frictional force is proportional to the normal force, as shown in Figure 3.4.4b. Figure 3.4.4b Mobilisation of shear stress with applied shear load. If the contact surface is at an inclined up angle (i), shown in Figure 3.4.4c, from the force diagram, along the sliding direction, the normal force is N cos(i) + S sin(i), the shear force is S cos(i) – N sin(i). Figure 3.4.4c Sliding between two contact surface at an inclination.


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By friction theory, S cos(i) – N sin(i) = [N cos(i) + S sin(i)] tanφ, S – N tan(i) = N tanφ + S tanφ tan(i), S = N [tanφ + tan(i)] / [1 + tanφ tan(i)], S = N tan(φ+i) 3.5 Flow Mechanics of Porous Material and Parallel Plates 3.5.1 Flow in Porous Materials Consider a cylindrical sample of porous material (e.g., soil or rock) under the different water pressure heads h1 and h2, shown in Figure 3.5.1a. In one dimension, steady water flows through the fully saturated sample without affecting the structure of the soil or rock, in accordance with Darcy's flow law, Q = A k i (3.5.1a) where Q = volume of water flowing per unit time, A = cross sectional area of sample corresponding to the flow, k = coefficient of permeability, i = hydraulic gradient = (h1–h2)/L, and L = length of sample. Figure 3.5.1a Darcy's flow experiment on porous material. Hence

Q Q L k = A i = A (h1 – h2) (3.5.1b)

The coefficient of permeability k is a constant. Experiments have shown, however, that its value depends not only upon the character of the material, but also upon the properties of the fluid percolating through it. The value of k is inversely proportional to fluid kinematic viscosity, ν, which can be expressed as,

K g k = ν (3.5.1c)

where g = gravitational acceleration, K = the intrinsic permeability, and is a property of the material only, with dimension of L2.


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3.5.2 Flow between Parallel Plates For flow of a viscous fluid through a narrow interspace between two closely spaced parallel plates (e.g., rock fractures), shown in Figure 3.5.2a, the Darcy’s flow law is applicable when the flow is laminar. Figure 3.5.2a Darcy's flow experiment on parallel plates. The intrinsic permeability for laminar flow between parallel plates is (Todd 1959; Verruijt 1970),

d2 K = 12 (3.5.2a)

where d is the thickness of the fluid lamina, i.e., the aperture of the two parallel plates. Hence for laminar flow through smooth parallel plates, flow equation can be written as

g d2 k = 12 ν (3.5.2b)

This is often called the "parallel plate theory" in the flow mechanics of rock joints. A laminar flow through smooth parallel plates is a potential flow with its Reynolds number ≤2300. By combining the above equation with Darcy’s equation, it gives

A i g d2 Q = 12 ν (3.5.2c)

Since the area A is the flow passage which is equal to the width w times to the aperture of the parallel plates, d, the above equation may be written

w i g d3 Q = 12 ν (3.5.2c)

The above equation is identified as the "cubic flow law" which is widely used to describe the flow of fluid through parallel plates. It is also used to describe flow in rock joints. For a planar array of parallel smooth openings, the equivalent permeability parallel to this array is given as,


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d3 k = 12 b (3.5.2d)

where b is the spacing between openings. The above equations show that the flow rate and permeability are extremely sensitive to the aperture of the opening. 3.6 Empirical Approaches 3.6.1 Use of Empirical Equations Empirical approaches are often used when theoretical approaches are limited due to many reasones, for example, the complexity of the material which leads to non-conformable to theory. Empirical equations then are obtained usually based on extensive experiment results, by regression. New regression techniques, e.g., neural network, have also been applied to analysis large amount variables and data. Empirical criteria commonly used in rock mechanics are the Hoek-Brown criterion for both rock material and rock mass, JRC-JCS shear strength equation for rock fractures (Figure 3.6.1a) and several others. Figure 3.6.1a Empirical JRC-JCS shear strength criterion. 3.6.2 Linear and Multiple Regression 3.6.3 Neural Network and other New Methods Further Readings Gere JM, Timoshenko SP, Mechanics of Materials, 2nd Edition. PWS-Kent, Boston, 1984. Jaeger JC, Cook NGW, Fundamentals of Rock Mechanics, 3rd Edition. Chapman and Hall, London, 1979.

1

σxτyx

τxx

σz

σy

τzy

τxz

τxy

τyz

xy

z

3.1.1a

σx, -τxy

σy, τxy

σ2 σ1O C 2θ

½ (σx + σy) √[½(σx+σy)]2+ τxy2

S M

θ

σn

τ

a

bσx

τxy

τxy

σy

τab

σab

c

3.1.2a

2

σx

3.1.3a

δxδy

δz

ε1

σ1

E

1

3.1.5a

3

σ

ε

3.2.1a

Yield Point

Peak Strength

Ductile

Brittle

3.2.3a

4

3.2.3b

3.2.3c

5

3.2.3d

3.2.3e

6

3.2.3f

3.2.3g

7

3.3.1a

Mode I Mode II Mode III

3.4.3a

Stress N, Area A, Modulus E

Contact area mA, thickness d

8

3.4.3b

Stress N, Area A, Modulus E

Initial contact area Ao, thickness d

3.4.3c

9

3.4.4a

N S

3.4.4b

F

PN

PF

FsFk

F=PN

10

3.4.4c

S

iN

L

A

h1

h2

Q

3.5.1a

11

h1

h2

L

Q

w

d

3.5.2a

3.6.1a

τ = σn tan[JRC JMC log(JCS/σn+φr)]

Documents

ENS 080312 en JZ Notes Chapter 3