today’s plan
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Today’s plan. Continue Force, Stress, and Strength (Chapter 3) [continue reading Chapter 3] Be sure to cross reference the lecture and the textbook. Force, Stress, and Strength. Introduction Dynamic analysis Force Tractions (stress simply) Stress Relationships between stress and strain - PowerPoint PPT PresentationTRANSCRIPT
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GLG310 Structural Geology
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Today’s plan
• Continue Force, Stress, and Strength (Chapter 3) [continue reading Chapter 3]
• Be sure to cross reference the lecture and the textbook
April 21, 2023 GLG310 Structural Geology
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Force, Stress, and Strength
April 21, 2023 GLG310 Structural Geology
• Introduction
• Dynamic analysis
• Force
• Tractions (stress simply)• Stress
• Relationships between stress and strain
• Deformation experiments
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April 21, 2023
Stress simplySimple definition of stress (for the moment) = pressure =Force/Area
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Stress simplyStress = Force/Area = N/m2 = Pascal = Pa
Blaise Pascal, was a French mathematician, physicist, inventor, writer and Christian philosopher. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum. Pascal also wrote in defense of the scientific method. Born: June 19, 1623, Clermont-FerrandDied: August 19, 1662, Paris(Wikipedia)
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Stress and deformation• Use your stress and deformation words
correctly:• Stress Deformation (strain)• Tension Extension• Compression Contraction or
shortening -watch out for Art Sylvester!
http://www.geol.ucsb.edu/faculty/sylvester/
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April 21, 2023 GLG310 Structural Geology
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https://www.azpm.org/s/5840-a-visit-to-the-ray-mine
Ray Mine AZ: rocks come from depth “overburden”
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Tractions and stress
Rock quarried from the earth
Replace effect of adjacent rocks with forces
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April 21, 2023
Tractions and stress
A more strict definition of stress
•Traction is stress relative to a surface through a point p.
•Stress tensor is the field of tractions acting over a point p.
•Stress field is the entire collection of stress tensors in a body.
GLG310 Structural Geology
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April 21, 2023
Tractions and stress
• Start with equilibrium (Newton’s 3rd Law):
• Resolve each into its components in the coordinate directions
GLG310 Structural Geology
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April 21, 2023 GLG310 Structural Geology
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Tractions and stress
• Total torque must vanish
GLG310 Structural Geology
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April 21, 2023 GLG310 Structural Geology
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Tractions and stress
• Sign Conventions: compression positive
-Fundamentals of Rock Mechanics by Jaeger and Cook (3rd ed., p. 10)
In many engineering situations, we consider tension as positive and compression negative. This is not usually the case in structural geology.
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Tractions and stress
GLG310 Structural Geology
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Normal vector: unit length, perpendicular to plane
A polygon and two of its normal vectors
A normal to a surface at a point is the same as a normal to the tangent plane to that surface at that point.
A vector field of normals to a surface
http://en.wikipedia.org/wiki/Normal_(geometry)
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Tractions and stress
(a) “Traction” or “Stress vector”
(b) Equilibrium“Cauchy’s relationship
(c) Components of the Traction Vector
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April 21, 2023 GLG310 Structural Geology
Textbook notation!
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April 21, 2023 GLG310 Structural Geology
Baron Augustin-Louis Cauchy (21 August 1789 – 23 May 1857) was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics. More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy).
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April 21, 2023 GLG310 Structural Geology
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Equilibrium
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April 21, 2023 GLG310 Structural Geology
Multiple sources of stress
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Stress tensor
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April 21, 2023 GLG310 Structural Geology
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Stress tensor
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April 21, 2023 GLG310 Structural Geology
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April 21, 2023 GLG310 Structural Geology
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3D Stress tensor
Note that the opposite sides have the stress components too!
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Stress tensor
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April 21, 2023 GLG310 Structural Geology