Visualization

Friday, 9/6/2002

Hedgehog Method

€

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

€

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

€

σr =σ0 1−a2 r2( )

σθ =σ0 1+a2 r2( )

Stress Field

Radial stress

Tangential stress

Cymbals

Stress Field Around Crack Tip

€

τzx=τzy =0

€

σzx ≈0Thin plate

€

σy =K

2πrcos

θ2

1+sinθ2

sin3θ2

⎛ ⎝

⎞ ⎠

€

σx =K

2πrcos

θ2

1−sinθ2

sin3θ2

⎛ ⎝

⎞ ⎠

€

τxy=K

2πrsin

θ2

cosθ2

cos3θ2

Von Mises’ Yield Criterion

€

σe =2

2(σx−σy)

2 +(σy−σz)2 +(σz−σx)

2 +6(τxy2 +τyz

2 +τzx2 )2

[ ]1/ 2

Perfect plasticity

€

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