Nonsymmetric Examples: Kriz [3,-2,1,2,2,2] ______________________________________________________ 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 0.200E+01 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 s(2,1) > s(1,2) and emo(3)= 0.200E+01 Nonsymmetric stress tensor detected. Stress tensor decomposed into a symmetric stress tensor and an equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) The symmetric stress tensor ss(i,j): 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 The equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) 0.000E+00 0.000E+00 0.200E+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 ------------ Nonzero off-diagonal shear stresses are the transformed equivalent moment/volume vector using the same transposed eigenvector matrix used for the stress transformation 0.000E+00 0.000E+00 0.200E+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 0.200E+01 -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 0.200E+01 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.000E+00 0.100E+01 s(1,3) > s(3,1) and emo(2)= 0.200E+01 Nonsymmetric stress tensor detected. Stress tensor decomposed into a symmetric stress tensor and an equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) The symmetric stress tensor ss(i,j): 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 The equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) 0.000E+00 0.200E+01 0.000E+00 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 ------------ Nonzero off-diagonal shear stresses are the transformed equivalent moment/volume vector using the same transposed eigenvector matrix used for the stress transformation 0.000E+00 0.200E+01 0.000E+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 0.300E+01 0.000E+00 0.200E+01 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 0.200E+01 0.100E+01 s(3,2) > s(2,3) and emo(1)= 0.200E+01 Nonsymmetric stress tensor detected. Stress tensor decomposed into a symmetric stress tensor and an equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) The symmetric stress tensor ss(i,j): 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 The equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) 0.200E+01 0.000E+00 0.000E+00 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 ------------ Nonzero off-diagonal shear stresses are the transformed equivalent moment/volume vector using the same transposed eigenvector matrix used for the stress transformation 0.200E+01 0.000E+00 0.000E+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 0.300E+01 0.000E+00 0.000E+00 0.000E+00 -0.200E+01 0.000E+00 0.000E+00 0.200E+01 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 0.200E+01 0.200E+01 -0.200E+01 0.000E+00 0.000E+00 0.200E+01 0.100E+01 s(2,1) > s(1,2) and emo(3)= 0.200E+01 s(1,3) > s(3,1) and emo(2)= 0.200E+01 s(3,2) > s(2,3) and emo(1)= 0.200E+01 Nonsymmetric stress tensor detected. Stress tensor decomposed into a symmetric stress tensor and an equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) The symmetric stress tensor ss(i,j): 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 The equivalent nonsymmetric off-diagonal shear stress moment/volume first order tensor (vector) 0.200E+01 0.200E+01 0.200E+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 ------------ Nonzero off-diagonal shear stresses are the transformed equivalent moment/volume vector using the same transposed eigenvector matrix used for the stress transformation 0.200E+01 0.200E+01 0.200E+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 0.200E+01 0.200E+01 -0.200E+01 0.000E+00 0.000E+00 0.200E+01 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 ______________________________________________________ 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