visualization friday, 9/6/2002. hedgehog method matlab functions plot example x = 0:pi/100:2*pi; y =...
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Visualization
Friday, 9/6/2002
Hedgehog Method
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σ xx σxy
σ yx σyy
⎡
⎣ ⎢ ⎤
⎦ ⎥ =1 2
2 1⎡
⎣ ⎢ ⎤
⎦ ⎥
MATLAB functions
plot
Example
x = 0:pi/100:2*pi;y = sin(x);plot(x,y)
Bean-Bag Method
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σ11 σ12 σ13
σ21 σ 22 σ 23
σ31 σ 32 σ 33
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ =
1 2 3
2 2 −1
3 −1 1
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
MATLAB functions
Examples:
[x,y]=meshgrid(-10:10,-10:10);
[r,theta]=meshgrid(0:5:10,0:pi/20:pi);
z=sqrt(x.^2+y.^2);mesh(x,y,z)
meshgridmesh
Sphere Scanning
theta = linspace(0,2*pi,ntheta);phi = linspace(0,pi,nphi);[x,y] = meshgrid( theta, phi );for i=1:nphi for j=1:ntheta n = [ sin(phi(i))*cos(theta(j)), ... sin(phi(i))*sin(theta(j)), ... cos(phi(i)) ]'; x(i,j) = … … ; y(i,j) = … … ; z(i,j) = … … ; endendmesh(x,y,z)
Hole Problem
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σr =σ0 1−a2 r2( )
σθ =σ0 1+a2 r2( )
Stress Field
Radial stress
Tangential stress
Cymbals
Stress Field Around Crack Tip
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τzx=τzy =0
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σzx ≈0Thin plate
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σy =K
2πrcos
θ2
1+sinθ2
sin3θ2
⎛ ⎝
⎞ ⎠
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σx =K
2πrcos
θ2
1−sinθ2
sin3θ2
⎛ ⎝
⎞ ⎠
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τxy=K
2πrsin
θ2
cosθ2
cos3θ2
Von Mises’ Yield Criterion
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σe =2
2(σx−σy)
2 +(σy−σz)2 +(σz−σx)
2 +6(τxy2 +τyz
2 +τzx2 )2
[ ]1/ 2
Perfect plasticity
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σe =σys, if σ e >σ ys
Crack Tip Stress Field
Mountain Stream
A mountain stream comes trickling in the sunlight down the hill,And gathers volume until it has strength to run the mill;It happily continues then, upon its useful way,Turns other mills still further down, until it joins the bay.
Dislocation Emissions
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