plot strain ellipse and principal stress axes using eigenvalues and eigevectors.

Hello,

I am trying to plot strain ellipses and principle stress axes (longest and shortest axes of the ellipse) on a grid (x_new,y_new) according to followig instructions. I would be really greatful if someone could help me complete this code.

Thank you.

x = -333.6:5:831.4; %(1x234)
y = 0:5:500; %(1x101)

[xx, yy] = meshgrid(x, y);
dx = 5;
dy = 5;

u = flow values of each grid node in x direction.

v = flow values of each grid node in y direction.

%% Plot Principle axes

pick the 4, center most u and v values. Using this 4 grid nodes u and v values we can compute ux, uy, vx, and vy values for the center point.

then using computed ux,uy, vx, vy values, ( e.g. ux=(u1+u2/dx) )

    
    strain_tensor = [ux uy; vx vy];
    
    [V,D]=eig(strain_tensor);

Now I want to use this eigenvalues and eigenvectors to plot strain ellipeses at the center of each four grid nodes along with quiver arrows inside the ellipses.

[Matt J]

What is supposed to be the relationship between the eigenvalues/vectors and the geometry of the ellipse?

[Anitha Limann]

maximum eigen value is the length of the longest axis and minimum eigenvalue is the length of shortest axis. attached below is a theory slide for this.

Answer 1

[Matt J]

a=10; %long axis
b=3;  %short axis
theta=30;  %rotation angle

p=rotate( scale(nsidedpoly(1000),[a,b])  ,theta);

plot(p)

[Anitha Limann]

Hello,

Thank you for your answer.

I wrote my code as below but i get the following error with that.

>>

Unable to use a value of type polyshape as an index.

Error in Markers_Script (line 111)

p=rotate(scale(nsidedpoly(1000),[major_axis,minor_axis]),theta1);

a = max(max(m)); % longest axis
b = min(min(m)); % shortest axis

theta1 = atan2(mv(1,2), mv(1,1))*180/pi; % angle between x-axis and largest eigenvalue
theta2 = atan2(mv(2,2), mv(2,1))*180/pi; % angle between x-axis and smallest eigenvalue

scale =2;

p=rotate(scale(nsidedpoly(1000),[a,b]),theta1);

figure
plot(p)

[Matt J]

It's not coming from anything in the code you've shown. You have variables elsewhere in your workspace that are interfering. You need to clear them,

clearvars -except mv m

a = max(max(m)); % longest axis
b = min(min(m)); % shortest axis

theta1 = atan2(mv(1,2), mv(1,1))*180/pi; % angle between x-axis and largest eigenvalue
theta2 = atan2(mv(2,2), mv(2,1))*180/pi; % angle between x-axis and smallest eigenvalue

scale =2;

p=rotate(scale(nsidedpoly(1000),[a,b]),theta1);

figure
plot(p)

[Anitha Limann]

Hello,

I am so sorry to trouble you. but I still get the same error msg. Below is my complete code.

clearvars -except mv m

m=[0.0428602731501996	0; 0	-0.0159426987862691];
mv=[0.973953757211399	0.139025444512699; 0.226746728341998	0.990288809276388];

a = max(max(m)); % longest axis
b = min(min(m)); % shortest axis

theta1 = atan2(mv(1,2), mv(1,1))*180/pi; % angle between x-axis and largest eigenvalue
theta2 = atan2(mv(2,2), mv(2,1))*180/pi; % angle between x-axis and smallest eigenvalue

scale =2;

p=rotate(scale(nsidedpoly(1000),[a,b]),theta1);

Unable to use a value of type polyshape as an index.

figure
plot(p)

[Matt J]

The line scale =2 serves no apparent purpose and should probably be removed. Also, your [a,b] should both be positive numbers.