Symmetric Examples: Kriz [3,2,1] and Harting ______________________________________________________ determine eigenvalues by IMSL:EVCRG/EPIRG when n= 3 the starting matrix S(1,1) ... S(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 -- 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(l)...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 ______________________________________________________ determine eigenvalues by IMSL:EVCRG/EPIRG when n= 3 the starting matrix S(1,1) ... S(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 -- 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(l)...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 ______________________________________________________ determine eigenvalues by IMSL:EVCRG/EPIRG when n= 3 the starting matrix S(1,1) ... S(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 -- 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(l)...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 ______________________________________________________ determine eigenvalues by IMSL:EVCRG/EPIRG when n= 3 the starting matrix S(1,1) ... S(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 -- for eigenvalue = 1, row sum = 0.0 -- -- confirm: for row =1 sum = -0.172E-04 -- confirm: for row =2 sum = 0.381E-05 -- confirm: for row =3 sum = -0.668E-05 -- for eigenvalue = 2, row sum = 0.0 -- -- confirm: for row =1 sum = 0.464E-04 -- confirm: for row =2 sum = 0.570E-04 -- confirm: for row =3 sum = -0.668E-05 -- for eigenvalue = 3, row sum = 0.0 -- -- confirm: for row =1 sum = 0.286E-04 -- confirm: for row =2 sum = -0.477E-06 -- confirm: for row =3 sum = -0.381E-05 eigenvalues eigen(l)...eigen(n) are 0.282E+03 -0.122E+03 0.264E+02 eignevectors are arranged respectively in columns matrix is transposed when calculating Euler angles 0.918E-01 0.811E+00 -0.578E+00 -0.157E+00 0.585E+00 0.796E+00 0.983E+00 0.175E-01 0.181E+00 tensor transformation matrix, a(i,j) is 0.918E-01 -0.157E+00 0.983E+00 0.811E+00 0.585E+00 0.175E-01 -0.578E+00 0.796E+00 0.181E+00 angles (degrees) between rotated-unrotated axes X->x' Y->y' Z->z' 84.735428 54.221004 79.583794 euler angles (degrees) ( x y z ) -52.748646 -72.621414 -15.009850 ___________________________________________________________ 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 0.282E+03 -0.119E-04 -0.381E-05 -0.134E-04 -0.122E+03 0.178E-04 -0.763E-05 0.188E-04 0.264E+02 ______________________________________________________ 3D Sij gradients of glyphs scaled for adequate spacing 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