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Stress and Strain1

A load applied to a material results in a stress at any point within it. Stresses may be:

• Compressive or squeezing • Tensile or stretching • Shear or tearing

In rock mechanics and soil mechanics, compressive stresses are positive by convention and tensile stresses are negative. Shear stresses are positive if the implied rotation is anticlockwise. STRESS is defined as force per unit area. The units of stress are Pascals . Soil stresses are typically in the kilopascal range (kPa). Rock stresses are usually measured in Megapascals (MPa). STRAIN is defined as change in length per unit length (normal strain) or change in angle between lines in a deforming body (shear strain). Strain is dimensionless and in rocks and soils is usually counted in microstrains. STRAIN RATE is change in strain per unit time. In rock deformation strain rates are usually given in microstrains/sec. In explosive deformation, strain-rates can be be very high. For geological processes, strain-rates are extremely low. In Situ Stresses How do we estimate the stress acting on a material? The vertical stress acting on a rock or soil at a depth z beneath the surface is simply the depth x the unit weight of the soil or rock. Unit weight is the force equivalent of density or the density x g (the acceleration due to gravity). Typical soil unit weights are around 18 kN/m3. Rocks have higher values, typically 25 kN/m3. At a depth of 10 metres below the surface in a typical soil the vertical stress will be 10 x 18 = 180 kN/m2 or 180 kPa. For comparative purposes, atmospheric pressure is normally 101.3 kPa.

1 from the University of Saskatchewan Department of Geological Sciences,: http://duke.usask.ca/~reeves/prog/geoe118/geoe118.032.html

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For rock at a depth of 1000 metres (the depth of Saskatchewan potash mines), the stress will be about 1000 x 25 = 25,000 kN/m2 or 25 MN/m2 or 25 MPa. The increasing rock pressure with depth in the crust is called the lithostatic gradient, typically 25 MPa/km. If the rocks are saturated with water, the water pressure increases with depth at a rate known as the hydrostatic gradient, typically 10 MPa/km.

The ratio of horizontal to vertical stresses is usually between 0.3 and about 3.0. Horizontal stresses greater than vertical stresses occur incompressive tectonic zones. Low H:V stress ratios occur in zones of crustal extension. If the horizontal and vertical stresses are equal, the stress field is said to be lithostatic and no shear stresses are present. Shear is generated by differences between vertical and horizontal stresses called deviatoric stresses. Geological Controls The engineering properties of rocks are influenced by a large number of geological factors. Mineralogy and particle-contacts control strength on a small scale; tectonic deformation, igneous activity aand metamorphism all result in substantial changes in the mechanical behaviour of rocks through recrystallization and fracturing.

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Burial and erosion of sediments results a series of consistent and predictable changes. The increase in sediment load during burial combined with cementation and filling pores results in:

• increased strength • decreased porosity • decreased permeabilty

Stripping away sediment by erosion and the consequent unloading and weathering results in the development of joints leading to:

• decreased strength • increased porosity • increased permeabilty

In general rocks become stronger and less porous and permeable as they get older. Recent sediments are normally weaker than ancient rocks with similar lithology and mineralogy. Rocks and soils with a level of compaction corresponding to their present burial depth re said to be normally consolidated. Where erosion has occured, rocks may be compacted much more than expected for their current depth of burial. These rocks and soils are said to be overconsolidated. Rocks that have not compacted to the expected extent for their depth of burial, perhaps because fluids could not escape, are said to be underconsolidated. Underconsolidated rocks are often associated with high fluid pressures (overpressure). An overpressure is a pressure in excess of the pressure predicted from the normal hydrostatic gradient. Laboratory tests Rock strength is measured by laboratory testing. Strengths are very different depending on the stress field applied to the rock. All rocks and soils are very much stronger in compression than in tension. The two common laboratory tests to determine the compressive strength of rock are:

• Uniaxial Unconfined Compression Test - A cylindrical rock core is loaded axially until it fails.

• Triaxial Confined Compression Test - A cylindical rock core is placed in a

cell, subjected to all around (confining) pressure by hydraulic oil acting through a thin impermeable membrane, and loaded axially to failure.

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There are a variety of tests to determine the tensile strength of rock:

• Direct Pull Test - A cylindrical rock core sample is anchored at both ends and stretched.

• Brazilian Test - A relatively thin disk is load across the diameter until it splits.

• Beam Flexure Test - A thin slab of rock is loaded vertically when supported at three or four points along it's length.

The Hoek cell is a specially designed triaxial cell that can be used with regular hydraulic jacks. This makes it highly portable and suitable for field i operation. The hydraulic pressure is applied radially through a thick membrane. The axial pressure is applied directly to the sample endcaps.

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All the tests mentioned so far are used to determine the properties of intact rock. Such properties are rarely representative of rock mass behaviour because of the presence of numerous discontinuities (fractures, joints, faults, etc) in virtually all rocks close to the Earth's surface.

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The portable field shear box is used to directly measure the shear resistance of joint surfaces. A block with a natural fracture is cemented into a split-box mould with the fracture surfaces in contact. When the cement or plaster is cured, normal and shear loads are applied using hydraulic jacks. The displacement is meassured with a dial-gauge. The shear-box provides information on the shear strength and shear stiffness of jointed rock masses. Mechanical Behaviour The mechanical behaviour of rocks depends on the pressure, temperature and strain-rate. At rapid strain-rates and low temperatures and pressures, most rocks behave in an elastic manner. A spring is a simple mechanical analaogue. All strain is recovered when the load is removed. The slope of the stress vs strain curve is the elastic modulus of the rock.

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At lower strain-rates and elevated temperatures and pressures, rocks behave in a plastic manner, showing large deformations at stresses beyond a critical stress level (called the yield stress. Strain is not recoverable, deformation is permanent. A block sliding on a plane is a simple mechanical analogue.

Sometimes the deformation of rocks becomes similar to that of a flowing fluid. The rate of deformation or strain-rate is controlled by the applied stress. Such behaviour is called viscous. The slope of the stress v strain-rate curve is the fluid viscosity. A dashpot is a simple mechanical analogue. Strain is not recovered. Viscous elements are introduced into models of rock behaviour to incorporate time-dependency.

The elasto-plastic model of material behaviour is often used to charcterize rocks. It combines elastic and plastic aspects of stress-strain behaviour. A block on a plane attached to a spring is the mechanical analogue. Strain is only recovered for the elastic part of the deformation.

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Strength of Rock Rocks fail in different ways depending on the temperature and pressure. At low temperatures and high strain rates rocks are brittle-elastic. They deform elastically at stresses up to about 70% of their strength then crack propagation becomes dominant and eventually the rock fails as cracks coalesce to form a large fracture or failure surface.

At low confining pressures, shallow depths or near free surfaces, vertical splitting (1) is the usual failure mode. At higher confining pressures (deeper) a single shear plane develops (2). At even higher confining pressures, a network of inclined shears develops (3). At low strain-rates, elevated temperatures and very high confining pressures the stress strain curve does not have a distinct maximum to indicate failure. Samples show the continuous deformation under load characteristic of ductile-plastic materials. Failed cores have a characteristic "barrel" shape. The transition from brittle-elastic behaviour to ductile-plastic behaviour is favoured by:

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• increasing pressure • increasing temperature • increasing fluid (pore) pressure

The change in behaviour is called the brittle-to-ductile transition. For most rocks it occurs at temperatures and pressures outside the normal range of engineering. However, some shales, fine grained limestones (chalk) and most evaporites (rocksalt, potash, gypsum etc) show ductile behaviour in near-surface, low-temperature environments. Creep In a conventional strength test, the axial stress on a rock core is gradually increased until the rock fails. Rocks will also deform under constant stress by a process called creep. Creep is time dependent deformation under constant stress.

The conventional test (red line) shows an increase in strain proportional to the applied stress up to the peak strength when the rock fails. In a creep test, (black lines) the stress is raised rapidly and held constant. The rock strains until it fails. Notice that the nearer the creep stress is to the peak strength, the smaller the amount of creep strain to failure.

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The strain-time curve for a creep test has a very characteristic form. Initially, as the load is applied the elastic strain occurs (virtually instantaneously). As time passes under constant stress, the rate of strain reduces. This period of decelerating strain-rate is called primary creep. The primary creep phase is followed by an extended period of slow (almost steady-state) deformation called secondary creep . At the end of this stage, the strain-rate begins to accelerate and the material rapidly fails. The final stage of accelerating deformation is called tertiary creep. Creep in rock masses is associated with crack propagation. During the primary creep phase the rock "aclimatises" to the applied stress and crack propagation slows to a stable, almost constant rate. During the "steady" secondary creep stage, the material is damaged more and more until finally, in the tertiary stage, uncontrolled accelerating crack propagation leads to failure. Creep is important at low temperatures and pressures only in a few rock types: shales, soft chalks and evaporite rocks. It is a major factor in the design and construction of potash mines in Saskatchewan. Geological structure is the study of the permanent deformation and rock failure created by the changes in stress through geologic time. It is by far the most important aspect of geology for the engineer to understand. Tectonic processes are responsible for the many discontinuity planes (fractures, faults, joints) that permeate rock masses controlling their strength, stress-strain characteristics and the transmission and storage of fluids. Structures may be conveniently subdivided into two groups:

• brittle structures - recording the brittle-elastic failure of rocks in the past. Faults and joints fall in this broad category.

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• ductile structures - preserving the permanent viscoplastic deformation of rock throughout geologic time. Folds and metamorphic foliations are the expression of this type of structure.

The most striking features of rocks as engineering materials is that they are not simple, isotropic, elastic and continuous but very complex, strongly anisotropic, anelastic discontinuous. It is virtually impossible to deduce the stress history of rocks from their observed deformation. There are always many ambiguous deformation paths that could have been followed to produce what is observed. The study of structure involves the careful recording of the orientation of lines and planes in rock masses in order to deduce the three-dimensional geometry of the distorted crust.

Stress-Strain Relations2 For linearly elastic materials, the relationship between stresses and strains is governed by Hooke's Law:

or

where C and D are the elasticity and compliance tensors, respectively, and

Cijkl-1 = Dijkl. One can show that

Cijkl = Cjikl = Cijlk = C jilk = Cklij Similar symmetrics also hold for Dijkl. The tensorial form of generalized Hooke's Law can then be rewritten in the following matrix form:

2 (from The University of Tennessee-Knoxville, College of Engineering http://www.engr.utk.edu/~cmc/528/chapter1 )

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where are engineering shear strains. It is very often that one regards the four indices as 2 pairs and denotes

. Thus Hooke's Law can be re-written by

where . Also,

where . Similarly, one can find

. It is noted that (matrix elements) = 4 Dijkl (tensorial components) for

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Material Symmetry Isotropic Materials The material properties of an isotropic material are independent of orientations. Consequently, the stiffness matrix reduces to

where

E = Young's modulus ν = Poisson's ratio G = shear modulus The compliance matrix of an isotropic material can be expressed by

where

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It is noted that only two independent elastic constants exist for isotropic materials. The tensorial expressions for stiffness and compliance of isotropic materials are given by

where

Transverely Isotropic Materials There is one and only one plane of isotropy (say, x1-x2 plane) in transverely isotropic materials. Thus, the stiffness and compliance matrices become

where E = E1 = E2

G3 = G23 = G31

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Also, note that

Conversely,

where

= C44