Need to find "Knees" in Stress Strain curves where slope change abruptly

I have stress strain curve which is basically peicewise linear. I need to find everytime there is a sudden change in the slope, the point at which it occurs and the corresponding magnitude. If it were a true local maximum, I can use some tool like "peakFinder" and get the information. But nto all are true peaks but some of them are slope changes. How do I find those. I am attaching a picture here. Thanks in advance.

[Mathieu NOE]

hello

try use gradient (twice) to get first and second derivatives

the knees should correspond to a peak value in the second derivative

[Pappu Murthy]

There is a slight problem with the approach you suggested. If I have a linear segment then the second derivative is zero.

[Star Strider]

If the intent is to get the maxima or minima of the second derivative, the other values will not matter.

[Pappu Murthy]

The intent is really to find where the y values changed abruptly from its normal linear course The attached picture clarifies this point.

[Mathieu NOE]

hello

can you share the corresponding data ?

[Pappu Murthy]

I surely can. May be a little bit later, I will add the file here.

[Pappu Murthy]

Attached text file contains the data. First column is Strain and Second column is Stress. If you plot them you will see a picture like the one attached earlier.

Answer 1

[Mathieu NOE]

hello

this would be my first suggestion

of course the results may be sensitive to the threshold used in my method (based on the area of a triangle made of 3successive points => we are looking at the points where the triangle hasan area above a given threshold , meaning the two successive segments have a certain angle)

clc
clearvars

data = readmatrix('Test.txt'); %  First column is Strain and Second column is Stress

Strain = data(:,1);
Stress = data(:,2);
[A,curv] = compute_curv(Strain,Stress);
% ind = find(curv>1e-5);
ind = find(A>8e-4);
% remove first and last indices which are obviously not to keep
ind = ind(2:end-1);

figure(1)
plot(Strain,Stress,'-+',Strain(ind),Stress(ind),'dr');

function [A,curvature] = compute_curv(x,y)
% run along the curve and find the radius of curvature at each location.
numberOfPoints = length(x);
curvature = zeros(1, numberOfPoints);
for t = 1 : numberOfPoints
  if t == 1
    index1 = numberOfPoints;
    index2 = t;
    index3 = t + 1;
  elseif t >= numberOfPoints
    index1 = t-1;
    index2 = t;
    index3 = 1;
  else
    index1 = t-1;
    index2 = t;
    index3 = t + 1;
  end
  % Get the 3 points.
  x1 = x(index1);
  y1 = y(index1);
  x2 = x(index2);
  y2 = y(index2);
  x3 = x(index3);
  y3 = y(index3);
 
  
   a = sqrt((x1-x2)^2+(y1-y2)^2); % The three sides
   b = sqrt((x2-x3)^2+(y2-y3)^2);
   c = sqrt((x3-x1)^2+(y3-y1)^2);
   A(t) = 1/2*abs((x1-x2)*(y3-y2)-(y1-y2)*(x3-x2)); % Area of triangle
   curvature(t) = 4*A(t)/(a*b*c); % Curvature of circumscribing circle
   
end


end

[Pappu Murthy]

Excellent. Worked very well for my problem. This is exactly what I wanted. Just one question though. Do we need the variable "curv" at all since we are using the Area to threshold? Again thanks!

[Mathieu NOE]

hello

no , "curv" can be removed , but maybe it would be great to keep it for another problem / task

you can simply comment the line and remove it from the output variables of the function

all the best

[Pappu Murthy]

Thanks for the clarification. I thought so as well. Also, I realized there is a "polyarea" function available with in the Matlab and one can use that to compute area of triangles instead of going through the actual process of finding sides etc. But if you need "curv" as well then your approach is the best. Again thanks for all the help. I have learnt some useful tricks.