I'm not skilled at writing articles. I'm sure that the pictures and their captions can be placed better, I just don't know how to do it better than it currently is. Please make it better if you can :)

Janek Kozicki 17:24, 19 November 2006 (UTC)Reply

Oh, and I don't have the formulas for Brestler-Pister and Willam-Warnke expressed in principal stresses (). If I find it, I will be able to plot it, just like all the other surfaces. If you know this formula, plese write it here, so later I will see your addition, and I will plot the surface.

Janek Kozicki 07:51, 20 November 2006 (UTC)Reply

OK, I couldn't find in the literature if the formula for Bresler - Pister yield surface is correct. But using this formula, I was able to plot the surfaces, and I added the images of this surface. Someone please verify this if you can. Janek Kozicki (talk) 17:40, 17 May 2008 (UTC)Reply

Willam-Warnke surface

edit

To plot the Willam-Warnke surface in   space you will need  ,   and   in terms of  . From the article on Stress we have

 

To find   in terms of   we note that

 

In matrix form (in terms of the principal stresses)

 

Therefore,

 

This leads to the needed expression for   in terms of  

 

Next you can compute the Haigh-Westergaard coordinates using the relations

 

Then compute

 

followed by

 

Finally, plug into the expression for the Willam-Warnke condition

 

A (untested) piece of code to generate a scalar field (of which the 0 level set is the yield surface) is:

void viewer::generateScalarField()
{
  using namespace std;
  for(int i=0;i<sizeX;i++) {
    for(int j=0;j<sizeY;j++) {
      for(int k=0;k<sizeZ;k++) {

	// calculate s1,s2,s3 from the position in 3D array scalarField
	// the origin of the coordinate system is in the center (sizeXYZ * 0.5)
	// and the 3D array scalarField represents a cube from -1.0 to 1.0

	long double s1 = (long double)i/(long double)sizeX*2.0-1.0;
	long double s2 = (long double)j/(long double)sizeY*2.0-1.0;
	long double s3 = (long double)k/(long double)sizeZ*2.0-1.0;

        long double I1 = s1+s2+s3;
        long double J2 = 1.0l/6.0l*(pow(s2-s3,2.0l)+pow(s3-s1,2.0l)+pow(s1-s2,2.0l));
        long double J3 = 1.0l/27.0l*(2.0l*s1-s2-s3)*(2.0l*s2-s1-s3)*(2.0l*s3-s1-s2);
 
        long double xi = 1.0l/sqrt(3.0l)*I1;
        long double rho = sqrt(2.0l*J2);
        long double theta = 0.0000001l;

        // calculate theta
        double t4 = J2*J2*J3;
        double t5 = 1.5*sqrt(3)*J3;
        double t6 = sqrt(t4);
        bool flag = true;
        if (fabs(t6) <= fabs(t5)*1.0e-16) {theta = 0.0000001; flag = false;}
        if (fabs(t6) < 1.0e-16 && flag) {theta = 0.0000001; flag = false;}
        double t7 = t5/t6;
        double ta1 = fabs(t7-1.0);
        double ta2 = fabs(t7+1.0);
        if (ta1 < 0.1 && flag) {theta = 0.0000001/3.0; flag = false;}
        if (ta2 < 0.1 && flag) {theta = 3.1415921/3.0; flag = false;}
        if ((t7 > 1.0 || t7 < -1.0) && flag) {theta = 3.1415921/3.0; flag = false;}
        if (flag) {
          theta = 1.0/3.0*acos(t7);
          flag = false;
        }

        // Willam-Warnke
        long double sc = 1.0l;
        long double st = 0.2l;
        long double sb = 1.5l;

        long double r_c = sqrt(1.2l)*sb*st/(3.0l*sb*st + sb*sc - st*sc);
        long double r_t = sqrt(1.2l)*sb*st/(sc*(2.0l*sb+st));

        long double g_numer = 2.0l*r_c*(r_c*r_c-r_t*r_t)*cos(theta) + r_c*(2.0l*r_t - r_c)*
           sqrt(4*(r_c*r_c - r_t*r_t)*pow(cos(theta),2.0l) + 5.0l*r_t*r_t - 4.0l*r_t*r_c);
        long double g_denom = 4.0l*(r_c*r_c - r_t*r_t)*pow(cos(theta),2.0l) + pow(r_c-2*r_t,2.0l);
        long double g = g_numer/g_denom;
 
        long double lambda = sqrt(0.2l)/g;
        long double B = 1.0l/sqrt(3.0l)*(sb*st)/(sc*sb-st);

        scalarField[i][j][k] = lambda*rho + B*xi - sc;
      }
    }
  }
}

Bbanerje (talk) 02:59, 25 May 2008 (UTC)Reply

Pictures:

   

Problems:

So I calculated this again. I changed the code (shown above) to use high precision - long double (the "l" after numbres say that they are long double also, instead of double). Now we can see that the holes are rather not due to marching cubes mistake, but rather there is some linear discontinuity on the "edges".

 

Janek Kozicki (talk) 19:24, 25 May 2008 (UTC)Reply

You are running into problems at the locations where   and  . Also the yield surface should not be cylindrical.
I've updated the Willam-Warnke page with a slightly different version of the model.
 
But now
 
and
 
with
 
The new version of the Willam-Warnke criterion is
 
The updated version of the code is
void viewer::generateScalarField()
{
  using namespace std;
  for(int i=0;i<sizeX;i++) {
    for(int j=0;j<sizeY;j++) {
      for(int k=0;k<sizeZ;k++) {

	// calculate s1,s2,s3 from the position in 3D array scalarField
	// the origin of the coordinate system is in the center (sizeXYZ * 0.5)
	// and the 3D array scalarField represents a cube from -1.0 to 1.0

	long double s1 = (long double)i/(long double)sizeX*2.0-1.0;
	long double s2 = (long double)j/(long double)sizeY*2.0-1.0;
	long double s3 = (long double)k/(long double)sizeZ*2.0-1.0;

        long double I1 = s1+s2+s3;
        long double J2 = 1.0l/6.0l*(pow(s2-s3,2.0l)+pow(s3-s1,2.0l)+pow(s1-s2,2.0l));
        long double J3 = 1.0l/27.0l*(2.0l*s1-s2-s3)*(2.0l*s2-s1-s3)*(2.0l*s3-s1-s2);
 
        long double xi = 1.0l/sqrt(3.0l)*I1;
        long double rho = sqrt(2.0l*J2);
        long double theta = 0.0000001l;

        // calculate theta
        double t4 = J2*J2*J3;
        double t5 = 1.5*sqrt(3)*J3;
        double t6 = sqrt(t4);
        bool flag = true;
        if (fabs(t6) <= fabs(t5)*1.0e-16) {theta = 0.0000001; flag = false;}
        if (fabs(t6) < 1.0e-16 && flag) {theta = 0.0000001; flag = false;}
        double t7 = t5/t6;
        double ta1 = fabs(t7-1.0);
        double ta2 = fabs(t7+1.0);
        if (ta1 < 0.1 && flag) {theta = 0.0000001/3.0; flag = false;}
        if (ta2 < 0.1 && flag) {theta = 3.1415921/3.0; flag = false;}
        if ((t7 > 1.0 || t7 < -1.0) && flag) {theta = 3.1415921/3.0; flag = false;}
        if (flag) {
          theta = 1.0/3.0*acos(t7);
          flag = false;
        }

        // Willam-Warnke
        long double sc = 1.0l;
        long double st = 0.2l;
        long double sb = 1.5l;

        long double r_c = sqrt(3.0l)*sc*(sb-st)/((sc+st)*sb-sc*st);
        long double r_t = sqrt(3.0l)*(sb-st)/(2.0l*sb-st);

        long double u_theta = 2.0l*r_c*(r_c*r_c-r_t*r_t)*cos(theta);
        long double v_theta = r_c*(2.0l*r_t - r_c)*
           sqrt(4*(r_c*r_c - r_t*r_t)*pow(cos(theta),2.0l) + 5.0l*r_t^2 - 4.0l*r_t*r_c);
        long double w_theta = 4.0l*(r_c*r_c - r_t*r_t)*pow(cos(theta),2.0l) + pow(r_c-2*r_t,2.0l);
 
        long double lambda = sqrt(2.0l/3.0l)*(u_theta+v_theta)/w_theta;
        long double B = 1.0l/sqrt(3.0l)*(sb*st)/(sb-st);

        scalarField[i][j][k] = rho + lambda*(xi - B);
      }
    }
  }
}
Bbanerje (talk) 00:53, 26 May 2008 (UTC)Reply

Potential mistake in Drucker Prager equation

edit

Hi, I believe there is a mistake in the equation for the Drucker-Prager yield surface. At present the equation contains the following:

(m-1)/2 and (m+1)/2

Based on the definitions given on the main page for the Drucker Prager yield criterion (https://en.wikipedia.org/wiki/Drucker%E2%80%93Prager_yield_criterion), I believe the coefficients should be

(m-1)/2m and (m+1)/2m.

Can somebody please check and confirm.

Enenen10 (talk) 22:14, 22 August 2018 (UTC)Reply