# Chapter4 - AE 321 Practice Problems Chapter 4 Material...

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AE 321 – Practice Problems Chapter 4: Material Behavior 1. For a linearly elastic isotropic material starting from ! ij = 2 μ " ij + #\$ ij kk , where λ , μ are the Lamé moduli, show that E ij = 1 + ( ) # ij \$ "% ij kk , where E , ν are the Young’s modulus and Poisson’s ratio, respectively. 2. (a) Show that the principal axes for stress and strain coincide for a linearly elastic isotropic material. (b) Under what condition do the principal axes for stress and strain coincide for an orthotropic solid? 3. Show that for a linear, elastic, isotropic homogeneous solid, strain energy density W can be expressed as: (a) W = ij ij + 2 kk ( ) 2 (b) W = 1 + 2 E ij ij # 2 E kk ( ) 2 (c) W = 1 2 E Q 1 2 ! 2 1 + ( ) Q 2 [ ] where Q 1 and Q 2 are the first and second stress invariants, and the other symbols have their usual meaning. 4. For steel E = 207 GPa and = 0.3. Assume that the strain in an (x 1 , x 2 , x 3 ) frame at a given point is

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! ij [ ] = 0.004 0.001 0 0.006 0.004 sym . " 0.001 # \$ % % ( ( . Find the normal and shear components of traction acting at that point on a surface with normal (1/ 3, 1/ 3, -1/ 3). 5. For steel
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## This note was uploaded on 09/16/2009 for the course AE AE321 taught by Professor Lambros during the Spring '09 term at University of Illinois at Urbana–Champaign.

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Chapter4 - AE 321 Practice Problems Chapter 4 Material...

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