508L1.24.1.07

508L1.24.1.07 - Mechanics of Earthquakes and Faulting...

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Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/Courses/Geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure criteria Wednesday 24 Jan. 2007

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Continuum Mechanics, Historical. (See Love, 1926). 1638: Theory of elasticity starts with Galileo and his work on beams. For a beam extending from a wall, how long can it be before it breaks when loaded by: its own weight, a mass at the end? 1660: Hooke’s law (published as an anagram in 1678: ceiiinosssttuv: Ut tensio sic vis) 1821: Navier’s general equations (of motion for e lastic materials) --also known by Cauchy’s name. 1860: Young (Lord Kelvin) Concept of modulus introduced. Hooke’s law in simple form: F = k x, where F is force, k is stiffness and x is displacement. This was later generalized to σ = E ε , where σ is stress, E is Young’s modulus and ε is linear strain. In Hooke’s time the generality of strain was not understood in terms of linear and shear components. Strain was simply referred to as “tension,” probably reflecting the difficulty of separating the application of stress and strain in the laboratory.
Continuum Mechanics. 200 years later, Young posited the notion of modulus –in a way that made it seem to have dropped from the sky. In Young’s words: modulus is defined by the following statement. …a column of the same substance capable of producing a pressure at its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length. σ = F/A ε = dl/L E = σ / ε It’s a statement that you have to read over a few times in order to get the gist of… But with these words, Young introduced for the first time a definite physical concept associated with the coefficient of elasticity. Pressure at the base is ρ g l l ρ

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Continuum Mechanics. Think about how modulus differs from stiffness. Consider a simple experiment. Consider two identical springs of equal length l. Cut one in two, so that it’s sections are of length l/2. Although each of the three sections are of identical material, the longer one will deform to a greater extent under a given load. The same could be said for any material –including lengths of granite and columns of concrete. Stiffness is a useful concept, but it is not a material property. Seismic moment, the passage of elastic waves, the strain field around a fault, and the velocity of a propagating rupture all depend on modulus, not stiffness. To make things confusing, we sometimes refer to a generalized modulus as the ‘stiffness’ tensor, as in:
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508L1.24.1.07 - Mechanics of Earthquakes and Faulting...

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