EMA 622 - Fall 2003
Homework

Textbook "Intro to Continuum Mechanics," THIRD edition by Michael Lai, David Rubin and Erhard Krempl, published by Butterworth-Heineman (http://www.bh.com).

Here's what Prof Drugan expects you to read and solve : (i.e. better make sure you know this if you want to take a course from him such as Theory of Elasticity)

Study Sections	Homework Problems
1.1, 1.2, 2A1-2A5	2A(6, 7, 8, 12)
2B1-2B15	2B(1, 2, 4, 7)
2B17-2B20, 2C1-5	2B(9, 12, 13, 15, 30, 34)
2D(1-3), 3.1-3.5, 2B16, 3.6-3.11	2B(36, 37, 40), 2C3, 2D1
3.12-3.15, 3.18-3.27, 3.29	3.(5, 11, 23, 27, 30, 39, 44, 46, 54)
next week	3.(61, 66, 68, 69)
4.1-4.7, 3.28, 4.9-4.11	4.(3, 11, 13, 30)
7.1-7.6, 7.9-7.10	4.(5, 18, 41), 7.(2, 3, 13)
4.12-4.15, 5.32-5.38	5.(93-95, 97, 99)
5.1-5.5, 5.12-5.16	5.1-5.3, 5.25, 5.60
6.1-6.7, 6.9-6.13	6.(6, 10, 21)

This is a minimum set that may be collected from time to time. You should try out more exercises, especially if you have difficulties with any of these. But you know that. Here's a list section by section with some bold extra problems from me

• 2A(6, 7, 8, 12)
• 2B(1, 2, 4, 7, 9, 12, 13, 15, 26, 29, 30, 34, 36, 37, 40, 43)
  More extras:
  1. Show that det(Q) = epsilon_ijk Q_i1Q_j2Q_k3= epsilon_ijk Q_1iQ_2jQ_3k . Interpret this geometrically (think Det=mixed (or "triple scalar') product)
  2. Show that epsilon_ijk Q_ilQ_jmQ_kn= epsilon_lmn det(Q)
  3. Note: these expressions can be generalized to higher dimensions using the permutation tensor: epsilon_{j1 ...jn} = +/-1 depending on whether j₁ ...j_n is an even or odd permutation of 1...n.
  4. If c is the cross product of vectors a and b and H=I-2 nn is a Householder reflection tensor about the plane perpendicular to n , show that H . c is minus the cross product of H . a and H . b . Convince yourself (and hopefully, more skeptical people) that a reflection changes a right-handed frame into a left-handed frame. [Hints: This is "obvious" if you reason with your right and left hands, to visualize cross product and reflection. Mathematically, show that c = (c . n) n + n x (c x n) for any vector c and unit vector n. Interpret both components geometrically. Now, show that if c= a x b then H.c changes the sign of the first component, while H.a x H.b changes the sign of the second component. The result is H.a x H.b= - H.c, so the cross product changes sign under reflection.]
• 2C3, 2C6c,d,e
• 2D1, 2D3, 2D5
• Flex your indicial muscles and show by direct calculation that the momentum eqns transform as claimed in class (and in the book) when written in terms of the original configuration and the 1st Piola-Kirchoff stress tensor. You need to use the formula for derivative of a determinant that we derived in class (exercise 3.69). With that in hand, you can quickly show that F^-1_kj d/dX_k [F_jl/det(F)] vanishes, where F^-1_kj = dX_k/dx_j and F_jl = dx_j/dX_l so F^-1_kj F_jl = delta_kl.