Lecture Practical LECTURE 7 STRESS AND STRAIN

Transcription

1 LECTURE 7 STRESS AND STRAIN LECTURE PLAN ) STRESS a) The relationship between stress and force b) Stress Components c) Principle stresses d) Rock Failure 2) STRAIN a) Components of strain b) The strain ellipse c) Rotational and non-rotational strain d) Finite and Infinitesimal Strain e) Special Types of Strain ) STRESS a) The relationship between stress and force Force is mass times acceleration, or F = Ma Forces acting on the outside of a body are applied forces (or surface forces). e.g. a vice exerts a force on a piece of wood. An applied force moves an object Position Position 2 A body force (e.g. gravity) moves an object Position Position 2 Forces acting at every point within a body are body forces. e.g. gravity. Lecture Practical Course Homepage Contact Staff

2 Forces acting on a body generally cause motion or acceleration. But if some or all the forces are absorbed by the body instead of moving it, then the body becomes stressed. The forces cause particle displacements within the body so that the body becomes deformed. σ σ2 = greatest = intermediate σ σ2 Force causes stress; and stress causes deformation (i.e. strain). Stress (σ) is the concentration of force per unit area. σ = F/A σ3 = least σ3 A commonly used unit is kg/cm 2. Therefore, the same force will do greater damage to a body if the area over which it is applied is smaller (e.g. a stilleto heel versus a flat heel). b) Principle stresses A cylinder of rock between two metal plates experiences an applied stress and then fails producing strain For any deformation event, orthogonal axes exist which have no shear stresses associated with them. These are the Principal compressive stress directions σ, σ2 and σ3. Rocks fail when the sigma and sigma 3 stresses differ.

3 Plate Convergence and thrusting Where plates are converging, Sigma is controlled by plate motion and the stress is larger than the stress caused by gravity (Sigma 3). Sigma 2 is parallel to the strikes of thrusts. Plate divergence and extension Plate strike-slip motion Where plates are diverging, the gravitational force is greater than the inward force produced by plate motion. So the gravitational force is Sigma and that produced by plate motion is Sigma 3. Sigma 2 is parallel to the strikes of normal faults. At strike-slip plate boundaries, Sigma, which is caused by plate motion at 45 o to the fault, is horizontal as is Sigma 3. Sigma 2 is vertical. Sigma - Maximum Principal Compressive Stress Sigma 3 - Minimm Principal Compressive Stress Sigma 2 - Intermediate Principal Compressive Stress (shown as a dot where coming out of the page).

4 b) Stress Components - Stress across a plane (such as a fault plane) can be resolved into a component normal to the plane and one parallel to it. The component normal to the plane is the normal stress (σ n ). The component along the plane is the shear stress (σ s ). Both can be quantified by plotting the orientation and length of the stress vectors on a scaled map. Thus, magnitudes of stress can be directly measured from the map. Resolving an applied stress into shear and normal stresses failure plain s Scale = 0 kg/cm 2 n to plane (friction) - normal stress, to plane promotes motion -shear stress

5 d) Rock Failure and regional orientations of stress A common question which is asked is: what levels of σ and σ3 will cause failure in a rock? By conducting a series of rock deformation experiments at different confining pressures (σ3), we can see what levels of σ are needed to cause failure. The method is: a) First to apply a confining pressure, usually by means of enclosing the sample in high pressure fluid (Thus, σ2 = σ3). b) Apply a force using a hydraulic ram to one side of the sample. This is σ. We can plot the results for a number of experiments on a Mohr Diagram. The result is a series of circles which increase in size towards the right of the diagram. This reflects the fact that (a) higher levels of differential stress, and (b) higher values are σ are needed to induce fractures as the confining pressure is raised. (Differential stress is the difference between the maximum and minimum principle normal stresses, σ - σ3). A cylinder of rock between two metal plates and within a pressurised fluid experiences an applied stress and then fails producing strain σ s kg/cm Pressure vessel Hydraulic ram failure envelope 3 3 No failure A Mohr envelope (failure envelope) can be constructed by linking the tangents to all the circles. The failure envelope separates failed from un-failed regions of the diagram, with the un-failed region lying within the envelope. If a circle lies completely within the failure envelope, the rock will not fail. Mohr Stress Diagram σ3 σ σ n

6 Andes AB Sub-Andes Amazon Basin PB Arabian Eurasian African Indian Antarctic Pacific Hawaii 88 σ North American 28 Yellowstone σ σ 3 99 σ Antarctic Cocos 26 6 South MacDonald 94 Nazca American Martin 80 Vaz Constructive plate margins Destructive plate margins -20 Directions and rates (mm/yr) of relative motion of the Earth's major plates. Circled numbers represent absolute motions (mm/yr) of the plate relative to a stationary hot-spot reference frame Paraná Basin 200m Stress Map of S. America with red lines showing the orientation of the maximun compressive stress σ The directions of stress can be measured in the earth from the focal mechanisms of earthquakes, from the deformation of boreholes in the crust (bore-hole breakouts), and perhaps from the orientations of joints and striated faults. Shown opposite is map of stress orientations for S.America compiled from such data. Note how the S. America is under c. E-W compressive stress due to ridge-push forces from the Mid-Atlantic ridge and subduction of the Nazca Plate beneath the Andes. Note how the stress orientations are parallel to plate motions

7 Tien Shan Xizang (Tibet) Plateau Tension Axes Shanxi Liaoning Jiangsu Zhejiang African 8 Arabian Indian 6 Eurasian 49 Antarctic Pacific Hawaii 88 σ North American 28 Yellowstone σ σ 3 99 σ Antarctic Cocos 26 6 South MacDonald 94 Nazca American Martin 80 Vaz σ 3 Vertical Less reliable km 300km Constructive plate margins Destructive plate margins Directions and rates (mm/yr) of relative motion of the Earth's major plates. Circled numbers represent absolute motions (mm/yr) of the plate relative to a stationary hot-spot reference frame. Tien 80 Shan Compression Axes Shanxi Liaoning Shown opposite is map of stress orientations for the Himalayas compiled from earthquake, borehole and fault data. Note how the Himalayas are under c. N-S compressive stress due to ridge-push forces from the Carlsberg Ridge Spreading Centre and collisional processes. The stress orientations form a fanning pattern due to the existence of a weak, low-resistance edge formed by the subduction zones of the western Pacific. Xizang (Tibet) Plateau Jiangsu Zhejiang Plate boundary Mean σ trajectory Mean σ 3 trajectory Boundary of plateau Lake Baikal Eurasian Plate SUMMARY OF HIMALAYAN STRESSES Pacific Plate σ Vertical Less reliable km 300km 20 0 Indian Ocean Plate Xizang (Tibet) Plateau Shanxi Graben East China Sea Philippine Sea Plate South China Sea

8 2) STRAIN a) Components of strain Homogenous Strain Non-rigid body motions involving changes in size or distortion of shape are referred to as strain. Strain can either be homogenous or heterogenous. Homogeneous deformation- 2 rules:- ) Straight lines that exist within the body before deformation remain straight after the deformation. 2) Parallel lines that exist within the body before deformation remain parallel after the deformation. If these rules are not met, then the deformation is described as heterogeneous deformation. Heterogenous Strain Worm burrow orientations In places, the burrows also show more intense shear across bedding planes Close-ups of sheared worm burrows in the footwall of the Arnaboll Thrust, Loch Eriboll, N.W. Scotland. Overthrusting of Pre-Cambrian Gneisses over Cambrian bioturbated sandstones involving simple shear has deformed the worm burrows. Homogenous strain in the centre of the bed is revealed by the parallel worm burrows. Heterogenous strain is revealed across a localised thrust shear zones along a bedding plane by sigmoidal burrow shapes.

9 Strain involves:- Dilation a) Dilation- internal points of reference within a body spread apart or pack together in such a way that the line lengths between points become longer or shorter while the shape remains the same. To quantify dilation we can use the following: - Extension (e). This value can be calculated as the deformed length minus the original length, divided by the original length. e.g. Something 5 cm long is deformed to a length of 8 cm. e = (8-5)/5 = 0.6 The positive value implies extension. (To calculate the percentage elongation, take the e value and multiply it by 00). - Stretch (s) (s) = l /l 0 (Where l is the deformed length and l 0 is the original length). - Quadratic elongation. The quadratic elongation is the square of the stretch. Photomicrograph of calcite veins producing dilation of a limestone. The dilation is discontinuous because the increase in volume has occurred across discrete fractures. Nonetheless, the limestone sample has increased in volume due to the random opening directions across the veins as shown below. Deformed geometry Sketches of the veins shown above Original Deformed lγ = (l /l 0 ) 2 The number has no units as it is a ratio. It is useful because it allows the strains of lines of originally different lengths to be compared. Restored geometry Original Deformed Comparison of deformed and restored lengths across the vein show a clear dilation with a greater strain in the direction of the red line. The restoration is not quite perfect, probably because of movement in and out of the plane of section.

10 Distortion- the changes in spacing of points within a body are such that the overall shape of the body is altered. Can be described by measuring the degree to which two original perpendicular lines are deflected from 90 o (known as the angular shear. 2 (a) Z X Distortion of an object through simple shear. Note how the x z directions have rotated within the zone of shear. g Thus, by quantifying dilation and distortion we can quantify strain. Our markers could be the original diameter lines of the strain ellipse. b) The strain ellipse A useful way of visualising the two dimensional properties of strain is through homogeneous distortion of a circle. If a body containing a perfectly circular reference marker undergoes homogeneous deformation, the elliptical form of the reference marker will be transformed into a perfect ellipse. 2 X Z The principle axes (X & Z) of a strain ellipse are mutually perpendicular, parallel to the directions of maximum and minimum elongation within the deformed body. The X line has undergone the maximum amount of extension within the ellipse, whilst the line Z, has undergone the least (negative) amount of extension (negative elongation). In three dimensions, deformation of a sphere produces a strain ellipsoid, the 3rd strain axis being termed Y. Distortion of an object through pure shear. Note how the x z directions have not rotated within the zone of shear.

11 c) Rotational and non-rotational strain During the incremental increase in strain which occurs during the progressive deformation of a body, the strain axes may rotate. This type of deformation is termed simple shear which is common to shear zones. i.e. material lines (e.g. the trace of bedding or a dyke in a cross-section) within the deforming body rotate relative to the orientation of the strain axes. As rotations occur within the shear zone, lines may first become extended and then shortened. 2 (y) Distortion of an object through simple shear. Note how the x z directions have rotated within the zone of shear. Worm burrow orientations Z X Distortion of an object through pure shear. Note how the x z directions have not rotated within the zone of shear. Pre-Cambrian Gneiss 2 X Z If the strain axes do not rotate and the deforming body simply becomes flattened, then the deformation is known as pure shear. In this instance, material lines within the deforming body do not rotate relative to the orientation of the strain axes. d) Finite and Infinitesimal Strain 2 3 Worm burrow orientations Worm burrow orientations Bedded Cambrian sandstones Mylonite 2 3 Close-up locations View and close-ups of sheared worm burrows in the footwall of the Arnaboll Thrust, Loch Eriboll, N.W. Scotland. Overthrusting of Pre- Cambrian Gneisses over Cambrian bioturbated sandstones involving simple shear has deformed worm burrows. The angle between bedding and the burrows decreases with proximity to the thrust, revealing a spatial variation in shear strain. Smaller, localised thrust shear zones occur along some bedding planes. Strain ellipses are not formed in an instant, but develop during progressive deformation where infinitesimal strain (a tiny increment of strain) is added to the total finite strain (the total strain which has accumulated up to that point in the deformation). Thus, during progressive deformation involving simple shear, the infinitesimal strain axes rotate relative to the initial directions in which strain accumulated, until the positions of the finite strain axes are achieved. Spatial variations in strain also occur, so strain varies temporally and spatially. In places, the burrows also show more intense shear across bedding planes In order for a body to undergo 60 degrees angular shear it undergoes a large number of smaller shear increments through time. The final strain is known as the finite strain, whilst each strain increment is an increment of infinitesimal strain Time (Numbers refer to the angular shear)

12 e) Special Types of Strain and regional strains from GPS Original object In the general case X>Y>Z. Prolate strain However, the following special cases exist:- a) Axially symmetric extension (X>Y=Z) Uniform extension in the X direction and equal shortening in all directions at right angles to it. Produces prolate or rugby ball shaped strain ellipsoid. This will produce L Tectonites which contain a strong preferred orientation of elongate minerals. No foliation develops. This is a linear fabric. (view from top) Oblate strain (side view) b) Axially symmetric shortening (X=Y>Z) Shortening in the Z direction and equal extension in all directions at right angle to it. This produces an oblate or pancake type of strain ellipsoid. This will produce S Tectonites where minerals are contained within a foliation. No strong linear fabric is present. This is a planar fabric. LS Tectonite Strong foliation and aligned elongate crystals Combinations of these two strain states produce LS Tectonites which have a strong foliation which contains elongate minerals aligned into a linear fabric. S Tectonite Strong foliation but no alignment of elongate crystals Strong alignment of elongate crystals but no foliation L Tectonite

13 It is now possible to use Global Positioning Satellites (GPS) to pin-point positions on the ground to a few millimetres accuracy, and then track how they move due to tectonic deformation by re-locating sites after a few years. Using observations of motions of the ground from GPS, Wang et al. (200) showed how velocities of points on the ground relative to Eurasia are greatest close to the frontal thrust of the Himalayas, decreasing to the north. England and Molnar (997), using an earlier database, showed how such motions can be used to calclate stress and strain in the Himalayas. Fig. 2. GPS velocity vectors (mm/year) with respect to the stable Eurasia, plotted on a shaded relief map of the Asia topography (gtopo30). The ellipses denote the region of - error. The polygons define three regions in which station velocities are used to formulate profiles shown in Fig. 3 and Supplementary figs. and 2 (26). From Wang et al. 200, Science, 294, overview Fig. 3. GPS velocity profileacross thetibetan Plateau in the direction of N2 E, parallel to the predicted direction of Indian-Eurasian collision. The red diamonds represent the N2 E component of velocity, which is parallel to the direction of the profile, and the green diamonds mark the N E component of velocity, which is perpendicular to the profile. The N2 E component shows a general linear trend of velocity gradient except for a high gradient across the Himalaya at the southern margin of the plateau. The N E component depicts eastward movement of the Central Tibetan Plateau withrespect to bothindia and Mongolia. From Wang et al. 200, Science, 294, overview From England and Molnar 997, Science, 278, From England and Molnar 997, Science, 278, overview overview Fig.. Field of crustal velocities in Asia [from figure 9 of (9)]. Arrows show crustal velocities calculated by integrating estimates of strain rates within triangular regions, derived from Quaternary slip rates on faults. Velocity is shown relative to undeforming Eurasia. Strain rates of triangular regions in Asia [from figure 9 of (9)], converted to dimensionless stress Eq. 7, with n 3. Bars show principal horizontal stresses; black bars correspond to contractional and white bars to extensional stress. Lengths of symbols are proportional to the magnitude of the respective stresses (arbitrary scale).

14 FURTHER READING AVAILABLE FROM THE ELECTRONIC LIBRARY Arthur Goldstein, Jonathan Knight and Kari Kimball, 998. Deformed graptolites, finite strain and volume loss during cleavage formation in rocks of the taconic slate belt, New York and Vermont, U.S.A., Journal of Structural Geology, 20,