Earth 103 Midterm Review PDF

Structural geology: geologic structures produced by deformation \*what do structural geologists do? -geometric description of structures -reconstruct sequence of deformational events -apply principles of physics, engineering, math, and materials science -concepts of stress and strain -problem...

Structural geology: geologic structures produced by deformation \*what do structural geologists do? -geometric description of structures -reconstruct sequence of deformational events -apply principles of physics, engineering, math, and materials science -concepts of stress and strain -problem solving using geometry, algebra, trig \*a fault has a long term slip rate of 2.0 mm/yr ad a total slip of 1.0 km. how long has the fault been active? \ [\$\$time = \\frac{\\text{total\\ slip}}{\\text{slip\\ rate}}\$\$]{.math.display}\ \ [\$\$time = \\frac{1000\\ mm}{2.0\\ mm/yr} = 500\\ kyr\\ \$\$]{.math.display}\ \*Paleo seismic studies indicate the fault slips episodically- each slip event produces an average of 3.3 m of displacement and a M 7.2 earthquake. How frequent do earthquakes occur? \ [\$\$recurrence = \\frac{\\text{slip\\ per\\ earthquake}}{\\text{slip\\ rate}}\$\$]{.math.display}\ \ [\$\$recurrence = \\frac{3300\\ mm}{2.0\\frac{\\text{mm}}{\\text{yr}}} = 1650\\ yrs\\ \$\$]{.math.display}\ \*how and why did this fault form and why is it behaving the way it is? -The fault likely formed due to tectonic forces, accumulating stress over time until it reached a critical threshold resulting in large episodic earthquakes. The recurrence suggest it follows a stick slip pattern rather than a continuous creep. Deformation: changes in the shape, size displacements and rotation that occur in rocks. -start in undeformed state, apply sufficient stress and en up in deformed stress Homogenous deformation: -**distortion** (strain) -rigid body **translation** -rigid body **rotation** \ [*deformation* = *distortion*(strain) ± *rotation* ± *translation*]{.math.display}\ A diagram of different types of lines AI-generated content may be incorrect. \*why does deformation(strain) happen? -it's a response to stresses acting on rocks. Geologic structure examples: -fractures and faults \- folds -fabrics (foliations, lineation) [Brittle vs ductile deformation ] Brittle deformation: includes fracturing and faulting -implies loss of cohesion, abrupt displacements -related to forces that are present -invokes concepts of elastic deformation, yield stress, strength, and brittle failure -brittle behavior comes from experimental fracture mechanics Ductile deformation: -implies no loss of cohesion -occurs gradually -implies higher temperatures (metamorphic conditions) -results in tectonite fabrics (foliations/lineations) -related to strain (changes in shape) **Mode of deformation** and **rock strength** depends on 6 things -temperature -rock composition and grain size -confining pressure (depth) -differential pressure or stress -deformation rate -pore fluid pressure Rock strength: defined as maximum differential stress that a rock can support before it fails [How are geologic structures analyzed?\ ]geometric description: statement of what can be objectively seen, minimal interpretations Kinematic description and analysis: interpretation about how a structure developed through time Mechanical or dynamical modes: models [Three principal types of plate margins:] -convergent margins (subduction and collision zones) -divergent margins (mid ocean ridges, continental rifts) -transform margins (strike slip boundaries) [Structure and compositions of the lithosphere ] Lithosphere: is crust + upper most mantle -0-150 km thick Oceanic lithosphere: oceanic crust (6km)+ mantle lithosphere (0-100 km) Continental lithosphere: -15-80 km Scalars: only have magnitude -T, mass, length, density Vectors: have magnitude and direction -velocity, displacement, acceleration, force Tensor: infinite array of vectors -stress, permeability, magnetic susceptibility, conductivity, speed of light in a crystal \*how do you quantify stresses within the earths crust? \ [*F* = *m* \* *a*]{.math.display}\ Unit=N \ [\$\$1\\ N = \\frac{1kg}{m/s\\hat{}2}\\ \$\$]{.math.display}\ [Forces on rocks ] Internal: inter atomic forces in crystal lattice -determine material properties of rocks (strength, rigidity) External: Body forces: act on each particle of mass in a volume (gravity) Surface forces: one body acting on another (plate colliding with another) Traction: a force applied to an area \ [\$\$Ʃ = \\frac{F}{A}\$\$]{.math.display}\ The unit of stress is pascals (1 Pa= 1N/m\^2) 1 Mpa =10\^6 pa =10 bars 100 Mpa= 1 kbar  [2-D stress ] -stress ellipse defined by max and least principlal dtresses -rocks fail when applied stress exceeds their strength -max differential stress is [σ 1 -- *σ* 3]{.math.inline} A blue background with black text and white text AI-generated content may be incorrect.  Lithostatic stress: stress produced by the overlying rock column [Mohr diagram] If we know the state of stress, we can use the Mohr circle to calculate sigma n and sigma s for any plane passing through that point, as long as we know the orientation of that plane relative to the principle stresses States of stress terminology -Hydrostatic pressure: all principal stresses are equal -Uniaxial compression -Uniaxial tension -axial compression -axial tension -triaxial stress -pure shear stress -deviatoric stress -differential stress -effective stress Vertical stress \ [ = *pgh*]{.math.display}\ Brittle deformation -can be classified by joints or faults [Modes of fracturing ] Mode I- extension fractures -relative motion perpendicular Mode II- shear fractures -relative motion perpendicular [Fault terminology ] -fault plane, hanging wall, footwall -displacement -high angle, low angle, listric [Fracture criterion ] -mathematical description of fracture conditions depending on critical stresses required to produce failure and orientation of the fracture plane -represents an idealized model for fracture [Fracture mechanics ] Based on laboratory experiments performed on rocks at various P,T, strain rates Depend on: -confining pressure -fluid pressure -preexisting weaknesses in rocks Controls: magnitude of sigma 1 and 3 Records: sigma 1 and 3 and theta [Tensile failure criteria ] -tensile strength of a rock : -10 to -40 Mpa -represented as a vertical line on Mohr space -Mohr circle that touches this line is a critical Mohr circle (depends on sigma 3) -fracture angle where circle is tangent to failure line **[Shear failure ]** -linear approximation to tangent line for a set of critical Mohr circles -diameter increases linearly with mean normal stress -slope of line is related to predicted angle of failure [Mohr coulomb fracture criterion]: -predicts that fault normals will always be at about 60 to the maximum compressive stress sigma 1 -fault surface will be about 30 to sigma 1 Conjugate shear fractures : two fault surfaces about 30 on either side of sigma 1 [Shear failure criteria] -function of mean normal stress and cohesion of rock (20-80 Mpa) -C= cohesion (intercept) (20-100 Mpa) \- μ= coefficient of internal friction (0.58 +/- 0.12) \- φ = angle of internal friction (30 +/- 5) -all Mohr circle within the space bound by these lines are stable critical stress state, one of two possible shear fractures will develop \- αf \~ 30° (45- φ/2) , -θf \~ 60°(45+ φ /2) -make acute angle with sigma 1 \*why do fractures tend to form at 30 to sigma 1 ? -sigma s/ sigma n ratio is maximized at 30, or theta f at 60 Mode I: joints fissures, veins, dikes Mode II: faults, deformation bands, shear zones Mode IV: stylolite, anti cracks, closing fractures Andersonian mechanics assume: -one of the principal stresses is always vertical (the other two must be horizontal) -rocks fail by either tensile failure or coulomb failure criterion -predicts what the general orientations of all faults and extension fractures should be -problems: principal stresses do not need to be vertical, does not account for non coulomb behavior of some faults, and does not consider pre existing weaknesses [Griffeth crack theory] -rocks are a lot weaker than theoretical predictions -they are riddled with microscopic elliptical cracks that act to strongly to concentrate stresses -provides a general theory for both tensile and shear failure -stresses at tips of cracks-more elliptical=higher stress concentration [How shear fractures actually form:] -linking up of tiny extension fractures -minor volume increase -wing cracks at terminations and edges Pore fluids and effective stress -Pf has enormous weakening effect on rock strength -effective stress= applied normal stress -- Pf (equivalent to shifting Mohr circle to the left by and equal amount to Pf) -shear stresses not affected  A white paper with black text AI-generated content may be incorrect. Week 1 review questions: 1. What are some of the different investigative approaches taken in structural geology (SG) and what are some of the advantages and disadvantages of these different approaches? What is stress and why is it important in SG? What are its units. What are some typical values of stresses within the crust? Stress causes deformations in the earth. -the units are N= 1kg/(m sec\^2) -Mpa -bar/kbar What is a tensor and how does it differ from a scalar and a vector? Can you give other examples of physical properties or variables that are tensor properties? Tensors are an infinite array of vectors. (ex: stress, permeability, magnetic susceptibility, conductivity, speed of light in a crystal) -scalars have magnitude only (T, mass, length, density) -vectors have magnitude and direction. (velocity, displacement, acceleration, force) What are 4 different ways we can portray the 2-D state of a stress at a point? -stress ellipse -two principal stresses -normal and shear stresses on any 2 perpendicular planes -tensor notation (4 components)  What are the fundamental 2-D stress equations that describe the normal and shear stresses acting on any plane whose orientation is known relative to the principal stresses A math problem with black text AI-generated content may be incorrect. What is meant by the lithostatic pressure gradient? Demonstrate why the vertical stress σ~v~ = ρgh, where ρ = the density in gm/cm^3^; g = acceleration of gravity in m/sec^2^; and h = depth in km. Lithostatic stress is the stress produced by the overlying rock column. \ [\$\$\\sigma\_{v} = \\frac{\\text{pghA}}{A}\$\$]{.math.display}\ \ [*σ*~*v*~ = *pgh*]{.math.display}\ What is the Mohr Circle for stress -- what does it represent? What are the axes of the Mohr diagram? How does physical space relate to Mohr space? How can you use a Mohr circle to determine the stresses acting on any plane relative to the principal stress axes? How is the 3-D state of stress illustrated on a Mohr diagram. -Mohr circle is a complete representation of the state of stress at a point. -the axes are [*σ*s and *σ*n ]{.math.inline} -angles in a physical space are doubled in Mohr space. What are some of the possible states of stress in the earth? -hydrostatic pressure -unixaial stress -uniaxial tension -axial compression -axial extension -triaxial stress -pure shear stress -deviatoric stress -differential stress -effective stress What are some of the ways that we determine the orientations and magnitudes of stresses in the earth? -borehole breakout -overcoring -hydraulic fracturing -volcanic vent and fracture alignments. **[Some sample stress problems]** What is the amount of stress (pressure) at the bottom of a 10 m deep swimming pool? How about at the bottom of the ocean on the abyssal plains (\~ 5 km deep)? \ [*σ*~*v*~ = *pgh*]{.math.display}\ \ [\$\$= \\frac{2.5g}{cm\^{3}}\*\\frac{10m}{s\^{2}}\*0.01km =.25\\ Mpa\$\$]{.math.display}\ 5km: \ [\$\$\\frac{2.5g}{cm\^{3}}\*\\frac{10m}{s\^{2}}\*.005\\ km =.125\\ Mpa\$\$]{.math.display}\ What is the vertical stress at 10 km depth in continental crust with an average density of 2.7 gm/cm^3^ \ [\$\$\\sigma\_{v} = 2.7\\frac{g}{cm\^{3}}\*\\frac{10m}{s\^{2}}\*10\\ km = 270\\ Mpa\$\$]{.math.display}\ In a mine 1500 m below the surface we establish that the maximum horizontal stress is one of the principal stresses and we use a stress gauge to determine it has a value of 44 MPa and is oriented exactly N-S. In an E-W direction, we determine that the stress has a value 34 MPa, and we further establish that the average density of the rocks from surface to the bottom of the mine is 2.6 gm/cc. *What are the magnitudes and orientations of the 3 principal stresses? What is the mean normal stress? What is the differential stress? What type of stress state is this? Illustrate this stress state in tensor notation and on a Mohr diagram. What would the normal and shear stresses be that are acting on a vertical plane whose strike is 045?* If sigma one is vertical and = 60 MPa, and sigma 3 = 40 MPa and is horizontal and oriented E-W, what is the normal and shear stress on a N-S striking plane that dips 60° to the east? A N-S striking vertical plane has a normal stress of 42 MPa and a shear stress of 6 MPa. An E-W striking vertical plane has a normal stress of 24 MPa and shear stress of -6 MPa. *What are the magnitudes and orientations of the maximum and minimum horizontal principal stresses?* Solve this problem both algebraically and with a Mohr circle. Be sure to draw a physical space diagram and show your work What are the two principal modes of fracture? How are they defined based on sense of movement?\ What are some examples of each fracture mode in nature? mode I: tensile failure \- forms extension fractures like joints, veinds, dikes -walls move apart from each other Mode II: shear failure -forms shear fractures (faults) -walls move parallel to each other How do the two modes of fracture relate to the orientation of the principal stresses? Mode I: fractures form perpendicular to sigma 3 Mode II: fractures form at an acute angle (30) to sigma 1 What exactly is meant by fracture criterion? Do you understand the fracture criterion for forming mode I fractures. Do you understand the fracture criterion for forming Mode II fracture? How are these failure criterion determined?\ -fracture criteria are an empirically derived equation that specifies the stress conditions required to form a particular type of fracture, as well as indicating what the orientation of that fracture will be relative to the principal stress axes. -tensile failure says a rock will fail if the normal stress drops to a value equal to the tensile strength of the rock and the fracture will be perpendicular to sigma 3. -coulomb failure says that the rock will fail if the effective shear stress on the plane reaches a critical value for a given normal stress and has the form of a linear equation that define two lines in Mohr space representing failure envelopes. Predicts tow potential planes of failure that form an acute angle with sigma 1. What are the meanings of T0, C, μ, ø and what are some typical values of each? -T0- tensile strength (-10- -20 Mpa) -C= cohesion, intercept of Mohr coulomb failure criterion (40-50 Mpa) \- μ=coefficient of internal friction= slope of Mohr failure criteria envelope (0.58) \- ø= angle of internal friction (30 degrees) In a typical experimental fracture mechanics apparatus, what variables can be controlled, and what is recorded at the end of each experiment? -controlled: confining stress (sigma 3), axial load (sigma 1) and Pf. Rock sample, temperature -recorded: sigma 1, sigma 3 differential stress at time of failure, orientation and sense of displacement of fracture. What is illustrated by a Mohr coulomb failure envelope- how does it derive and plotted, and what is represented by various fields? What is meant by a "critical Mohr circle or stress state"? -failure envelope is derived by experiments. -enveloped plotted using all critical Mohr circles and examining best fit tangent lines to these circles on either side of x axis. -critical Mohr circle illustrates the state of stress at the point of failure. -diameter of circle is the rock strength for confining pressure or mean normal stress. If rock strength is defined as the maximum differential stress that a rock can support prior to failure, then how does coulomb failure criteria predict rock strength should change as a function of increased mean normal stress? -rock strengths will increase linearly as a function of increased mean normal strength. What is meant by effective stress and why do we worry more about effective stress rather than applied stresses when it comes to rock failure? -effective stress stipulates that the rocks tendency to fail is a function of effective stress present in the rock, not the actual stresses. -presence of a pore fluid in the rock reduces all the normal stresses present by an amount equal to the pore fluid pressure, while not having any influence on the shear stresses -Pf shifts the Mohr circle to the left -true stresses are not changed but probability of rock failure increased due to effective shift. How do we explain the uniquity of extension fractures within the crust even though we believe most stress regimes within the crust are entirely compressive?\ -a way to cause tensile failure is to increase pore fluid pressure such that the pore fluid pressure is greater than sigma 3 by an amount equal to the tensile strength. -if you increase the Pf pressure enough, you will shift the effective circle left until sigma 3 becomes tangent to the tensile failure envelope. What is meant by conjugate shear fractures and what does their orientation tell you about the orientation of the principal stresses? -predicts that you will have two equally probable potential planes of shear failure, where the planes form an angle of 30° to sigma 1. -you will expect to find shear fractures in two orientation that make an acute angel of about 60° to each other and with opposite sense of slip. -we can assume that sigma 1 was oriented that it bisected the acute angle between two sets of fractures, sigma 3 was oriented that it bisected the obtuse angle between the fractures and sigma 2 was oriented parallel to the line of intersection of the two fractures. What is meant by Andersonian mechanics? What are the two fundamental assumptions that it makes? What does it predict specifically about the orientation and types of faults and extension fractures on earth? What are some of the reasons that these predictions might not hold true? -rocks fail y either tensile failure or coulomb shear failure -one of the principal stresses is always vertical -predicts that we should see extension fractures that are either vertical or horizontal -vertical strike slip faults (when sigma 2 vertical) -normal faults (when sigma 1 is vertical) -thrust faults (when sigma 3 is vertical) -may not be true: one of the principal stresses need not be vertical, does not account for non-tensile or non-coulomb failure of faults -does not account for preexisting weaknesses