Symmetric Examples: Kriz [3,2,1] and Harting ______________________________________________________ determination of eigenvalues by jacobi's method, where n = 3 itmax = 50 eps1 = 0.100E-09 eps2 = 0.100E-09 eps3 = 0.100E-04 the starting matrix a(1,1)...a(n,n) is 0.300E+01 0.000E+00 0.000E+00 0.000E+00 0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 convergence has occured, where iter = 1 s = 0.140E+02 sigma2 = 0.140E+02 ____________________ eigenvalues eigen(1)...eigen(n) are 0.300E+01 0.200E+01 0.100E+01 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 Final stress tensor reconstructed by superposition of the principal stress state plus the transformation of effective moment/volume tensor as off-diagonal shear stress terms 0.300E+01 0.000E+00 0.000E+00 0.000E+00 0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 ------------------------------------------------- Results of symmetric stress tensor transformation -- for eigenvalue = 1, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 -- for eigenvalue = 2, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 -- for eigenvalue = 3, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 eigenvalues eigen(1)...eigen(n) are 0.300E+01 0.200E+01 0.100E+01 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 tensor transformation matrix, a(i,j) is 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 angles (degrees) between rotated-unrotated axes ( X->x' Y->y' Z->z' ) 0.000000 0.000000 0.000000 euler angles (degrees) ( x y z ) 0.000000 0.000000 0.000000 ___________________________________________________________ CHECK: reconstruct principal stress state by a second order tensor transformation. Transpose eigenvectors = transformation matrix: e.g. off-diagonal shear stress components of the transformed stresses go to zero and the diagonal components recover the eigenvalues 0.300E+01 0.000E+00 0.000E+00 0.000E+00 0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 ______________________________________________________ determination of eigenvalues by jacobi's method, where n = 3 itmax = 50 eps1 = 0.100E-09 eps2 = 0.100E-09 eps3 = 0.100E-04 the starting matrix a(1,1)...a(n,n) is 0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 convergence has occured, where iter = 1 s = 0.140E+02 sigma2 = 0.140E+02 ____________________ eigenvalues eigen(1)...eigen(n) are 0.300E+01 -0.200E+01 0.100E+01 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 Final stress tensor reconstructed by superposition of the principal stress state plus the transformation of effective moment/volume tensor as off-diagonal shear stress terms 0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 ------------------------------------------------- Results of symmetric stress tensor transformation -- for eigenvalue = 1, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 -- for eigenvalue = 2, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 -- for eigenvalue = 3, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 eigenvalues eigen(1)...eigen(n) are 0.300E+01 -0.200E+01 0.100E+01 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 tensor transformation matrix, a(i,j) is 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 angles (degrees) between rotated-unrotated axes ( X->x' Y->y' Z->z' ) 0.000000 0.000000 0.000000 euler angles (degrees) ( x y z ) 0.000000 0.000000 0.000000 ___________________________________________________________ CHECK: reconstruct principal stress state by a second order tensor transformation. Transpose eigenvectors = transformation matrix: e.g. off-diagonal shear stress components of the transformed stresses go to zero and the diagonal components recover the eigenvalues 0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 ______________________________________________________ determination of eigenvalues by jacobi's method, where n = 3 itmax = 50 eps1 = 0.100E-09 eps2 = 0.100E-09 eps3 = 0.100E-04 the starting matrix a(1,1)...a(n,n) is -0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 convergence has occured, where iter = 1 s = 0.140E+02 sigma2 = 0.140E+02 ____________________ eigenvalues eigen(1)...eigen(n) are -0.300E+01 -0.200E+01 0.100E+01 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 Final stress tensor reconstructed by superposition of the principal stress state plus the transformation of effective moment/volume tensor as off-diagonal shear stress terms -0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 ------------------------------------------------- Results of symmetric stress tensor transformation -- for eigenvalue = 1, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 -- for eigenvalue = 2, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 -- for eigenvalue = 3, row sum = 0.0 -- -- confirm: for row =1 sum = 0.000E+00 -- confirm: for row =2 sum = 0.000E+00 -- confirm: for row =3 sum = 0.000E+00 eigenvalues eigen(1)...eigen(n) are -0.300E+01 -0.200E+01 0.100E+01 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 tensor transformation matrix, a(i,j) is 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 angles (degrees) between rotated-unrotated axes ( X->x' Y->y' Z->z' ) 0.000000 0.000000 0.000000 euler angles (degrees) ( x y z ) 0.000000 0.000000 0.000000 ___________________________________________________________ CHECK: reconstruct principal stress state by a second order tensor transformation. Transpose eigenvectors = transformation matrix: e.g. off-diagonal shear stress components of the transformed stresses go to zero and the diagonal components recover the eigenvalues -0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 ______________________________________________________ determination of eigenvalues by jacobi's method, where n = 3 itmax = 50 eps1 = 0.100E-09 eps2 = 0.100E-09 eps3 = 0.100E-04 the starting matrix a(1,1)...a(n,n) is -0.690E+02 -0.740E+02 0.210E+02 -0.740E+02 -0.180E+02 -0.410E+02 0.210E+02 -0.410E+02 0.274E+03 iter = 1 sigma1 = 0.802E+05 sigma2 = 0.954E+05 iter = 2 sigma1 = 0.954E+05 sigma2 = 0.954E+05 convergence has occured, where iter = 3 s = 0.954E+05 sigma2 = 0.954E+05 ____________________ eigenvalues eigen(1)...eigen(n) are -0.122E+03 0.264E+02 0.282E+03 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.811E+00 0.578E+00 0.918E-01 0.585E+00 -0.796E+00 -0.157E+00 0.175E-01 -0.181E+00 0.983E+00 Final stress tensor reconstructed by superposition of the principal stress state plus the transformation of effective moment/volume tensor as off-diagonal shear stress terms -0.122E+03 0.000E+00 0.000E+00 0.000E+00 0.264E+02 0.000E+00 0.000E+00 0.000E+00 0.282E+03 ------------------------------------------------- Results of symmetric stress tensor transformation -- for eigenvalue = 1, row sum = 0.0 -- -- confirm: for row =1 sum = -0.493E-13 -- confirm: for row =2 sum = 0.119E-12 -- confirm: for row =3 sum = -0.681E-12 -- for eigenvalue = 2, row sum = 0.0 -- -- confirm: for row =1 sum = 0.311E-14 -- confirm: for row =2 sum = 0.133E-13 -- confirm: for row =3 sum = -0.711E-14 -- for eigenvalue = 3, row sum = 0.0 -- -- confirm: for row =1 sum = -0.572E-12 -- confirm: for row =2 sum = -0.377E-12 -- confirm: for row =3 sum = -0.764E-13 eigenvalues eigen(1)...eigen(n) are -0.122E+03 0.264E+02 0.282E+03 eignevectors are arranged respectively in columns. matrix is transposed when calculating Euler angles 0.811E+00 0.578E+00 0.918E-01 0.585E+00 -0.796E+00 -0.157E+00 0.175E-01 -0.181E+00 0.983E+00 tensor transformation matrix, a(i,j) is 0.811E+00 0.585E+00 0.175E-01 0.578E+00 -0.796E+00 -0.181E+00 0.918E-01 -0.157E+00 0.983E+00 angles (degrees) between rotated-unrotated axes ( X->x' Y->y' Z->z' ) 35.797536 142.748652 10.465577 euler angles (degrees) ( x y z ) 9.019288 5.330664 -36.297615 ___________________________________________________________ CHECK: reconstruct principal stress state by a second order tensor transformation. Transpose eigenvectors = transformation matrix: e.g. off-diagonal shear stress components of the transformed stresses go to zero and the diagonal components recover the eigenvalues -0.122E+03 0.107E-13 -0.686E-12 0.103E-13 0.264E+02 -0.711E-14 -0.686E-12 -0.142E-13 0.282E+03 ______________________________________________________ Max= 0.500E+00 Min= 0.500E+00 Scale Factor= 0.200E+02 x(1)= 0.000E+00 y(1)= 0.000E+00 z(1)= 0.000E+00 x(2)= 0.100E+02 y(2)= 0.000E+00 z(2)= 0.000E+00 x(3)= 0.000E+00 y(3)= 0.000E+00 z(3)=-0.100E+02 x(4)= 0.100E+02 y(4)= 0.000E+00 z(4)=-0.100E+02