5.a - Ae 315 Homework 5 Solutions - 2008/2/8 We have...

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Ae 315 Homework 5 Solutions - 2008/2/8 We have studied all the elements of a theory of elasticity. Here we put everything together with the simplest problems: We know the displacements and we want to confirm the other equations. Preliminaries These are just some items that set-up or describe the operating environment of this program. In[1]:= $Version Out[1]= 6.0 for Mac OS X x86 H 32 - bit L H April 20, 2007 L In[2]:= DateList @D Out[2]= 8 2008, 2, 7, 14, 54, 56.002228 < In[3]:= Off @ General:: spell D ; Off @ General:: spell1 D ; Off @ Solve:: svars D ; Off @ ParametricPlot:: ppcom D ; Off @ ParametricPlot3D:: ppcom D In[8]:= Needs @ "Units`" D Needs @ "PhysicalConstants`" D Needs @ "BarCharts`" D ; Needs @ "Histograms`" D ; Needs @ "PieCharts`" D Needs @ "VectorAnalysis`" D Common functions for all problems Here you are given a dispalcement field and you need to interperet what it means. ü reference region. This is the undeformed configuration. Here I have to assume some numerical values for some of the parameters so that the plotting can be accomplished. Here the radius "a" is 1.0, the length "c" is 5. I do not plot the inside radius. Also, note that I make use of polar coordinates to parameterize the "surface". In[12]:= aValue = 1.0; cValue = 5.0; surface = 8 aValue Cos @ q D , aValue Sin @ q D , z < ; Note that I plot the "z" direction first so my cylinder comes out nearly horizontal rather than vertical. I do this with the "Rotate- Right" command. NOTE: I am only plotting 1/2 of the section where x 3 >= 0. The equations that are provided only work on this sub-region.
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In[15]:= ParametricPlot3D @ RotateRight @ surface D , 8 q , -p , p < , 8 z, 0, cValue <D Out[15]= ü displacement First, lets define the displacement field: (Note I define the displacements here in such a manner so I can automate certain opera- tions.) Here I'm going to try andd make things a bit more clear in how the displacements are defined. In[16]:= u @ 1 D@8 x1 _ , x2 _ , x3 _ < , 8 a_ , b_ , g_ , d_ < , 8 m _ , n_ , a _ , c _ <D : = 1 2 a I x3 2 - n x2 2 + n x1 2 M - n b x1 - g x3 x2 + d n 2 H c - x3 L I x1 2 - x2 2 M - x3 3 6 + c x3 2 2 ; u @ 2 D@8 x1 _ , x2 _ , x3 _ < , 8 a_ , b_ , g_ , d_ < , 8 m _ , n_ , a _ , c _ <D : = n a x1 x2 - n b x2 + g x3 x1 + d n H c - x3 L x1 x2; u @ 3 D@8 x1 _ , x2 _ , x3 _ < , 8 a_ , b_ , g_ , d_ < , 8 m _ , n_ , a _ , c _ <D : = -a x1 x3 + b x3 - d - 3 4 + n 2 a 2 x1 + 1 4 I x1 3 - 3 x1 x2 2 M + x1 x2 2 + x3 x1 c - x3 2 ; Here I have separated out each symbol so its easier to determine what the equation is doing. Here "x" is a vector of the coordi- nates and "p" is a column matrix representing the parameters " g ", " d " and " n ", "c", and "a" respectively. To check this out: In[19]:= u @ 1 D@8 x 1 , x 2 , x 3 < , 8 a , b , g , d < , 8 m, n , a, c <D Out[19]= -b n x 1 - g x 2 x 3 + 1 2 a I n x 1 2 - n x 2 2 + x 3 2 M + d 1 2 n I x 1 2 - x 2 2 M H c - x 3 L + c x 3 2 2 - x 3 3 6 In[20]:= u @ 2 D@8 x 1 , x 2 , x 3 < , 8 a , b , g , d < , 8 m, n
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5.a - Ae 315 Homework 5 Solutions - 2008/2/8 We have...

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