Earth and Space Sciences - Midterm Study Questions | ESS 411, Study notes of Environmental Science

Download Earth and Space Sciences - Midterm Study Questions | ESS 411 and more Study notes Environmental Science in PDF only on Docsity! Earth and Space Sciences 411/511 Geophysical Continuum Mechanics Fall 2008 Mid-term Study Questions: The Mid Term will be a take-home test with a limit of 2 hours. It should be taken without access to notes or other people or books except for our text by Mase and Mase. The exam will consist of 4 questions that will be selected from the questions below, perhaps after slight modifications. Problems 1 – 5. Mase and Mase: 3.4, 3.16, 3.19, 3.22, 3.29 Problem 6. Mohr’s circles for stress Explain in prose (use of diagrams to illustrate your prose is OK.): (a) what Mohr’s circles for stress represent about a stress state, (b) how they can be constructed, and (c) why they are useful for investigating fault failures. Problem 7. Tensors and Index notation (a) Using index notation, show that the dot product of two vectors pi and qj is independent of the coordinate system. (b) Show that a stress state is a pure-shear state if its first and third invariants are zero. Find the magnitude σ0 of the stresses in this pure shear state in terms of the second stress invariant IIT . Hint: consider the characteristic equation for eigenvalues. Problem 8. More Mohr’s circles Assume that σ1 and σ3 are both negative, σ2 is halfway between σ1 and σ3, σ1 > σ3 (engineering convention in Mase and Mase) and internal friction exists. (a) Use Mohr circles to show why • rocks at depth do not fracture under lithostatic pressure alone and • the deviatoric stress needed for fracture increases at greater depth. (b) Suppose a rock under compression in all directions (e.g. deep in the Earth) is stressed close to its brittle limit. Show graphically which change is more likely to make the rock fracture: • decreasing the magnitude of σ1 or • increasing the magnitude of σ3 by the same amount.