Yukun’s FEAP(Version 8.5) Reference Note

d(*)： Material Parameters

\(\begin{array}{rll} \hline \textrm{Parameter} & \textrm{Name} & \textrm{Description} \\ \hline 1 & E & \textrm{Young's modulus} \\ 2 & \nu & \textrm{Poisson ratio} \\ 3 & \alpha & \textrm{Thermal expansion coefficient} \\ 4 & \rho & \textrm{Mass density} \\ 5 & - & \textrm{Quadrature order for arrays} \\ 6 & - & \textrm{Quadrature order for outputs} \\ 7 & a & \textrm{Mass interpolation} (a = 0: \textrm{Diagonal}; a = 1: \textrm{Consistent})\\ 8 & q & \textrm{Loading intensity (plates/shells)} \\ 9 & T_0 & \textrm{Stress free reference temperature} \\ 10 & \kappa & \textrm{Shear factor (plates/shells/beams)} \\ 11 & b_1 & \textrm{Body force/volume in 1-directions} \\ 12 & b_2 & \textrm{Body force/volume in 2-directions} \\ 13 & b_3 & \textrm{Body force/volume in 3-directions} \\ 14 & h & \textrm{Thickness (plates/shells)} \\ 15 & \texttt{nh1} & \textrm{History variable counter} \\ 16 & \texttt{stype} & \textrm{Two dimensional type: 1 - plane stress; 2 - plane strain; 3 - axisymmetric} \\ 17 & \texttt{etype} & \textrm{Element formulation: 1 - displ; 2 - mixed; 3 - enhanced} \\ 18 & \texttt{dtype} & \textrm{Deformation type: <0 - finite; >0 - small} \\ 19 & \texttt{tdof} & \textrm{Thermal degree-of-freedom} \\ 20 & \texttt{imat} & \textrm{Non-linear elastic material type} \\ 21 & d_{11} & \textrm{Material moduli} \\ 22 & d_{22} & \textrm{Material moduli} \\ 23 & d_{33} & \textrm{Material moduli} \\ 24 & d_{12} & \textrm{Material moduli} \\ 25 & d_{23} & \textrm{Material moduli} \\ 26 & d_{31} & \textrm{Material moduli} \\ 27 & g_{12} & \textrm{Material moduli} \\ 28 & g_{23} & \textrm{Material moduli} \\ 29 & g_{31} & \textrm{Material moduli} \\ 30 & C & \textrm{Fung pseudo elastic model modulus} \\ 31 & \psi & \textrm{Orthotropic angle} x_1 \textrm{principal axis 1} \\ 32 & A & \textrm{Area cross section (beam/truss)} \\ 33 & I_{11} & \textrm{Inertia cross section (beam/truss)} \\ 34 & I_{22} & \textrm{Inertia cross section (beam/truss)} \\ 35 & I_{12} & \textrm{Inertia cross section (beam/truss)} \\ 36 & J & \textrm{Polar inertia cross section (beam/truss)} \\ 37 & \kappa_1 & \textrm{Shear factor plate} \\ 38 & \kappa_2 & \textrm{Shear factor plate} \\ 39 & - & \textrm{Non-linear flag (beam/truss)} \\ 40 & - & \textrm{Inelastic material model type} \\ 41 & Y_0 & \textrm{Initial yield stress (Mises)} \\ 42 & Y_{\infty} & \textrm{Final yield stress (Mises)} \\ 43 & \beta & \textrm{Exponential hardening rate} \\ 44 & H_{iso} & \textrm{Isotropic hardening modulus (linear)} \\ 45 & H_{kin} & \textrm{Kinematic hardening modulus (linear)} \\ 46 & - & \textrm{Yield flag} \\ 47 & \beta_1 & \textrm{Orthotropic thermal stress} \\ 48 & \beta_2 & \textrm{Orthotropic thermal stress} \\ 49 & \beta_3 & \textrm{Orthotropic thermal stress} \\ 50 & - & \textrm{Error estimator parameter} \\ 51 & \nu_1 & \textrm{Viscoelastic shear parameter} \\ 52 & \tau_1 & \textrm{Viscoelastic relaxation time} \\ 53 & \nu_2 & \textrm{Viscoelastic shear parameter} \\ 54 & \tau_2 & \textrm{Viscoelastic relaxation time} \\ 55 & \nu_3 & \textrm{Viscoelastic shear parameter} \\ 56 & \tau_3 & \textrm{Viscoelastic relaxation time} \\ 57 & \texttt{nvis} & \textrm{Number of viscoelastic terms (1-3)} \\ 58 & - & \textrm{Damage limit} \\ 59 & - & \textrm{Damage rate} \\ 60 & k & \textrm{Penalty parameter} \\ 61 & K_1 & \textrm{Fourier thermal conductivity} \\ 62 & K_2 & \textrm{Fourier thermal conductivity} \\ 63 & K_3 & \textrm{Fourier thermal conductivity} \\ 64 & c & \textrm{Fourier specific heat} \\ 65 & \omega & \textrm{Angular velocity} \\ 66 & Q & \textrm{Body heat} \\ 67 & - & \textrm{Heat constitution added indicator} \\ 68 & - & \textrm{Follower loading indicator} \\ 69 & - & \textrm{Rotational mass factor} \\ 70 & - & \textrm{Damping factor} \\ 71 & g_1 & \textrm{Ground acceleration factor} \\ 72 & g_2 & \textrm{Ground acceleration factor} \\ 73 & g_3 & \textrm{Ground acceleration factor} \\ 74 & p_1 & \textrm{Ground acceleration proportional load number} \\ 75 & p_2 & \textrm{Ground acceleration proportional load number} \\ 76 & p_3 & \textrm{Ground acceleration proportional load number} \\ 77 & a_0 & \textrm{Rayleigh damping mass ratio} \\ 78 & a_1 & \textrm{Rayleigh damping stiffness ratio} \\ 79 & - & \textrm{Plate/Shell/Rod shear activation flag} \\ 80 & & \textrm{Method: Type 1} \\ 81 & & \textrm{Method: Type 2} \\ 82 & - & \textrm{Truss/Rod quadrature number} \\ 83 & - & \textrm{Axial loading value} \\ 84 & - & \textrm{Constitutive start indicator} \\ 85 & - & \textrm{Polar angle indicator} \\ \end{array}\)

\(\begin{array}{rll} 86 & - & \textrm{Polar angle coord 1} \\ 87 & - & \textrm{Polar angle coord 2} \\ 88 & - & \textrm{Polar angle coord 3} \\ 89 & - & \textrm{Constitution transient type} \\ 90 & d_{31} & \textrm{Plane stress recovery} \\ 91 & d_{32} & \textrm{Plane stress recovery} \\ 92 & \alpha_3 & \textrm{Plane stress recovery} \\ 93 & \texttt{sref} & \textrm{Shear center type} \\ 94 & y_1 & \textrm{Shear center coordinate} \\ 95 & y_2 & \textrm{Shear center coordinate} \\ 96 & \texttt{lref} & \textrm{Reference vector type} \\ 97 & n_1 & \textrm{Reference vector parameter} \\ 98 & n_2 & \textrm{Reference vector parameter} \\ 99 & n_3 & \textrm{Reference vector parameter} \\ 100 & - & \textrm{Cross section shape type: 1 - rectangles; 2 - tube; 3 - Wide flange;} \\ & & \textrm{4 - Channel; 5 - Angle; 6 - Circle} \\ 101-126 & - & \textrm{Shape data} \\ 127 & - & \textrm{Surface convection} (h) \\ 128 & - & \textrm{Free-stream temperature} (T_{\infty}) \\ 129 & - & \textrm{Reference absolute temperature} \\ 130 & \texttt{nseg} & \textrm{Number of hardening segments} \\ 131-148 & - & \textrm{Segment data sets} e_pY_{iso}H_{kin} \\ 149 & - & \textrm{Total variables on frame section} \\ 150 & - & \textrm{Piezoelectric flag} \\ 151-159 & - & \textrm{Piezoelectric data} \\ 160 & - & \textrm{Initial stress flag} \\ 161-166 & \sigma_{ij} & \textrm{Initial stresses (constant)} \\ 167 & - & \textrm{Tension/compression only indicator} \\ 168 & - & \textrm{Thermal activation indicator} \\ 169 & - & \textrm{Mechanical activation indicator} \\ 170 & - & \textrm{Volume model number (default 1)} \\ 171 & a_1 & \textrm{Fung model energy parameter} \\ 172 & a_2 & \textrm{Fung model energy parameter} \\ 173 & a_3 & \textrm{Fung model energy parameter} \\ 174 & a_4 & \textrm{Fung model energy parameter} \\ 175 & a_5 & \textrm{Fung model energy parameter} \\ 176 & a_6 & \textrm{Fung model energy parameter} \\ 177 & a_7 & \textrm{Fung model energy parameter} \\ 178 & a_8 & \textrm{Fung model energy parameter} \\ 179 & a_9 & \textrm{Fung model energy parameter} \\ 180-181 & - & \textrm{Viscoplastic rate parameters} \\ 182 & - & \textrm{Nodal quadrature parameters} \\ 183 & \beta_m & M_L - M_C \textrm{mass scaling factor} \\ 184 & c & \textrm{Estimate on maximum wave speed} \\ 185 & - & \textrm{Augmentation switch: <on/off>} \\ 186 & - & \textrm{Augmentation explicit indicator} \\ 187 & & \textrm{Implicit = 0; Explicit = 1 element integration} \\ 188 & - & \textrm{Number stress components in rod elements} \\ 189 & - & \textrm{Nurbs and VEM flag} \\ 190-192 & - & \textrm{Nurbs quadrature values/direction} \\ 193 & \texttt{tmat} & \textrm{Thermal material numbers} \\ 194 & \texttt{ietype} & \textrm{Element type} \\ 195 & T - frac & \textrm{Fraction of work to heat} \\ 196 & q - prop & \textrm{Proportional load factor for pressure loading} \\ 197-198 & - & \textrm{Body patch loading values} \\ 199 & - & \textrm{Axisymmetric 1-d: Plane stress in thickness} \\ 200 & \texttt{nsiz} & \textrm{Size of modulus or compliance array} \\ 201-236 & - & \textrm{Anisotropic Modulus or Compliance array} \\ 237 & - & \textrm{Number of element global equations} \texttt{nge} \\ 238 & - & \textrm{Partition of element global equations} \\ 239 & - & \textrm{Unused} \\ 240 & - & \textrm{0 - Element based; 1 - nodal based formulation} \\ 241 & - & \textrm{Number of active element degrees of freedom} \\ 242 & - & \textrm{Plastic Vector orientation indicator} \\ 243-245 & V_1 & \textrm{Three Components of vector} \ V_1 \\ 246-248 & V_2 & \textrm{Three Components of vector} \ V_2 \\ 249-255 & - & \textrm{Reference vector types and values} \\ 260-279 & \texttt{nstv} & \textrm{Number structure vectors/values} \\ 280-282 & g_i & \textrm{Thermal-elastic temperature function} \\ 283 & - & \textrm{Unused} \\ 283-286 & - & \textrm{Delete element data} \\ 287 & - & \textrm{Total energy computation switch} \\ 288 & - & \textrm{Shell thickness change flag} \\ 289 & - & \textrm{Rate switch (on=0,off=1)} \\ 290-293 & - & \textrm{Constitutive equation coordinate frame} \\ 294 & - & \textrm{Rotatory inertia on/off flag} \\ 295-296 & - & \textrm{Body force user parameters} \\ \hline \textrm{Parameter} & \textrm{Name} & \textrm{Description} \\ \hline \end{array}\)

d(16): Geometry types identifier stype

\(\begin{array}{rl} \hline \textrm{Value} & \textrm{Type} \\ \hline 1 & \textrm{Plane stress} \\ 2 & \textrm{Plane strain} \\ 3 & \textrm{Axisymmetric without torsion} \\ 4 & \textrm{Plate / Shell} \\ 5 & \textrm{Truss} \\ 6 & \textrm{Thermal} \\ 7 & \textrm{3-D Solid} \\ 8 & \textrm{Axisymmetric with torsion} \\ 9 & \textrm{Spherical symmetry} \\ \hline \end{array}\)

d(20): Non-linear elastic material type imat

\(\begin{array}{rl} \hline \textrm{Value} & \textrm{Type} \\ \hline 1 & \textrm{Regular compressible Neo-Hookean model} \\ 2 & \textrm{Modified compressible Neo-Hookean model} \\ 3 & \textrm{Ogden compressible model} \\ 4 & \textrm{Isotropic linear elastic model} \\ 5 & \textrm{Transversely isotropic/Orthotropic linear elastic model} \\ 6 & \textrm{Energy Conserving model} \\ 7 & \textrm{Fung type exponential model} \\ 8 & \textrm{Anisotropic linear elastic model} \\ 9 & \textrm{Regular compressible Mooney-Rivlin model} \\ 10 & \textrm{Modified compressible Mooney-Rivlin model} \\ 11 & \textrm{Compressible Arruda-Boyce hyperelastic model} \\ 12 & \textrm{Compressible Yeoh hyperelastic model} \\ 13 & \textrm{Solid Mechanics Representative Volume Material} \\ 14 & \textrm{Fiber models} \\ \hline \end{array}\)

Common Block Definitions

\(\begin{array}{lll} \hline \textrm{Block Name} & \textrm{Variable} & \textrm{Definition} \\ \hline \texttt{bdata} & \texttt{o} & \textrm{Page eject option} \\ & \texttt{head} & \textrm{Title record} \\ \hline \texttt{cdat1} & \texttt{ndd} & \textrm{Size of program material parameters} \ \texttt{d(ndd, nummat)} \\ & \texttt{nie} & \textrm{Size of element control array} \ \texttt{ie(nie, nummat)} \\ & \texttt{nud} & \textrm{Size of user material parameters} \ \texttt{ud(*)} \\ \hline \texttt{cdata} & \texttt{numnp} & \textrm{Number of mesh nodes} \\ & \texttt{numel} & \textrm{Number of mesh elements} \\ & \texttt{nummat} & \textrm{Number of material sets} \\ & \texttt{nen} & \textrm{Maximum nodes/element} \\ & \texttt{neq} & \textrm{Number active equations} \\ & \texttt{ipr} & \textrm{Real variable precision} \\ \hline \texttt{comblk} & \texttt{hr} & \textrm{Real array data} \\ & \texttt{mr} & \textrm{Integer array data} \\ \hline \texttt{elcoor} & \texttt{xref} & \textrm{Reference coordinates for the constitutive point} \\ & \texttt{xcur} & \textrm{Current coordinates for the constitutive point} \\ \hline \texttt{counts} & \texttt{nstep} & \textrm{Total number of time steps} \\ & \texttt{niter} & \textrm{Number of iterations current step} \\ & \texttt{naugm} & \textrm{Number of augments current step} \\ & \texttt{titer} & \textrm{Total iterations} \\ & \texttt{taubm} & \textrm{Total augments} \\ & \texttt{iaugm} & \textrm{Augmenting counter} \\ & \texttt{iform} & \textrm{Number residuals in line search} \\ \hline \texttt{eldata} & \texttt{dm} & \textrm{Element proportional load} \\ & \texttt{n_el} & \textrm{Current element number} \\ & \texttt{ma} & \textrm{Current element material set} \\ & \texttt{mct} & \textrm{Print counter} \\ & \texttt{iel} & \textrm{User element number} \\ & \texttt{nel} & \textrm{Number nodes on current element} \\ \hline \texttt{elplot} & \texttt{tt} & \textrm{Element stress values for} \ \texttt{TPLOt} \\ \hline \texttt{eltran} & \texttt{bpr} & \textrm{Principal stretch} \\ & \texttt{ctan} & \textrm{Element multipliers} \\ \hline \texttt{eluser} & \texttt{ut} & \textrm{Element user values for} \ \texttt{TPLOt} \\ \hline \texttt{hdata} & \texttt{nh1} & \textrm{Pointer to} \ t_n \ \textrm{history data} \\ & \texttt{nh2} & \textrm{Pointer to} \ t_{n+1} \ \textrm{history data} \\ & \texttt{nh3} & \textrm{Pointer to element history} \\ \hline \texttt{iofile} & \texttt{ior} & \textrm{Current input logical unit} \\ & \texttt{iow} & \textrm{Current output logical unit} \\ \hline \texttt{pointer} & \texttt{np} & \textrm{Pointer for standard program array} \\ & \texttt{up} & \textrm{Pointer for user defined array} \\ \hline \texttt{prstrs} & \texttt{nph} & \textrm{Pointer to global projection arrays} \\ & \texttt{ner} & \textrm{Pointer to global error indicator} \\ & \texttt{erav} & \textrm{Element error value} \\ & \texttt{j-int} & \textrm{J integral values} \\ \hline \texttt{qudshp} & \texttt{jac} & \textrm{Jacobian matrix determinant at points} \\ & \texttt{lint} & \textrm{Number of quadrature points} \\ & \texttt{sg1} & \textrm{Points natural coordinates and weights in 1D} \\ & \texttt{shp1} & \textrm{Shape functions and derivatives of points in 1D} \\ & \texttt{sg2} & \textrm{Points natural coordinates and weights in 2D} \\ & \texttt{el2} & \textrm{Points natural area coordinates and weights in 2D} \\ & \texttt{shp2} & \textrm{Shape functions and derivatives of points in 2D} \\ & \texttt{sg3} & \textrm{Points natural coordinates and weights in 3D} \\ & \texttt{el3} & \textrm{Points natural volume coordinates and weights in 3D} \\ & \texttt{shp3} & \textrm{Shape functions and derivatives of points in 3D} \\ \hline \texttt{sdata} & \texttt{ndf} & \textrm{Maximum dof/node} \\ & \texttt{ndm} & \textrm{Mesh space dimension} \\ & \texttt{nen1} & \textrm{Dimension 1 on IX array} \\ & \texttt{nst} & \textrm{Size of element matrix} \\ & \texttt{nneq} & \textrm{Total dof in problem} \\ \hline \texttt{tdata} & \texttt{ttim} & \textrm{Current time} \\ & \texttt{dt} & \textrm{Current time increment} \\ & \texttt{ci} & \textrm{Integration parameters} \\ \hline \end{array}\)

Orthotropic Material Array and Thermal Modulus Transformation

Rotation of material arrays from principal to local element directions

Constitutive equation(Using Voigt’s notation):

\(\boldsymbol{\sigma} = \textbf{D} \boldsymbol{\epsilon}\)

where

\(\boldsymbol{\sigma} = \begin{bmatrix} \sigma_{11} & \sigma_{22} & \sigma_{33} & \sigma_{12} & \sigma_{23} & \sigma_{31} \end{bmatrix}^{\textrm{T}}\)

\(\boldsymbol{\epsilon} = \begin{bmatrix} \epsilon_{11} & \epsilon_{22} & \epsilon_{33} & 2\epsilon_{12} & 2\epsilon_{23} & 2\epsilon_{31} \end{bmatrix}^{\textrm{T}}\)

Material array \(\textbf{D}\) defined based on principal axes \(x_1x_2x_3\):

\(\textbf{D} = \begin{bmatrix} d_{11} & d_{12} & d_{13} & & & \\ & d_{22} & d_{23} & & 0 & \\ & & d_{33} & & & \\ & & & g_{12} & & \\ & sym. & & & g_{23} & \\ & & & & & g_{31} \\ \end{bmatrix}\)

Coefficient of thermal exoansion \(\boldsymbol{\alpha}\) defined based on principal axes \(x_1x_2x_3\):

\(\boldsymbol{\alpha} = \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ 0 \\ 0 \\ 0 \end{bmatrix}\)

Thermal modulus \(\boldsymbol{\beta}\) defined based on principal axes \(x_1x_2x_3\):

\(\boldsymbol{\beta} = \begin{bmatrix} \beta_1 \\ \beta_2 \\ \beta_3 \\ 0 \\ 0 \\ 0 \end{bmatrix}\)

And they have the relationship:

\(\beta_i = d_{ij}\alpha_j \quad i,j = 1, 2, 3\)

Condition 1: \(\psi\)

\(\psi\) describes the angle in degrees which the principal material axis 1 (i.e. \(x_1\)) makes with the \(x\) axis.

N.B. It is assumed that the principal material axis 3 (i.e. \(x_3\)) coincides with the direction of the \(z\) axis.

Rotation matrix \(\textbf{R}\):

\(\textbf{R} = \begin{bmatrix} \cos^2\psi & \sin^2\psi & & \cos\psi\sin\psi & & \\ \sin^2\psi & \cos^2\psi & & -\cos\psi\sin\psi & & \\ & & 1 & & & \\ -2\cos\psi\sin\psi & 2\cos\psi\sin\psi & & \cos^2\psi - \sin^2\psi & & \\ & & & & \cos\psi & -\sin\psi \\ & & & & \sin\psi & \cos\psi \\ \end{bmatrix}\)

Then, Material array \(\textbf{D}_{\textrm{local}}\) based on local axes \(xyz\):

\(\textbf{D}_{\textrm{local}} = \textbf{R}^\textbf{T} \textbf{D} \textbf{R}\)

Thermal modulus \(\boldsymbol{\beta}_{\textrm{local}}\) based on local axes \(xyz\):

\(\boldsymbol{\beta}_{\textrm{local}} = \textbf{R}^\textbf{T} \boldsymbol{\beta}\)

Condition 2: \(\textbf{v}_1\), \(\textbf{v}_2\)

\(\textbf{v}_1\) and \(\textbf{v}_2\) describe the orientation of the principal material axes which lie in the plane of the principal material axes 1 and 2.

Vectors \(\bar{\textbf{v}}_1\), \(\bar{\textbf{v}}_2\) and \(\bar{\textbf{v}}_3\) form a orthonormal triad.

Matrix \(\textbf{R}_0\):

\(\textbf{R}_0 = \begin{bmatrix} \bar{v}_{11} & \bar{v}_{21} & \bar{v}_{31} \\ \bar{v}_{12} & \bar{v}_{22} & \bar{v}_{32} \\ \bar{v}_{13} & \bar{v}_{23} & \bar{v}_{33} \end{bmatrix}\)

where \(\bar{v}_{1i}\), \(\bar{v}_{2i}\), \(\bar{v}_{3i}\) are components of vectors \(\bar{\textbf{v}}_1\), \(\bar{\textbf{v}}_2\) and \(\bar{\textbf{v}}_3\).

Transformation matrix \(\textbf{R}\):

\(\textbf{R} = \begin{bmatrix} \bar{v}_{11}^2 & \bar{v}_{12}^2 & \bar{v}_{13}^2 & \bar{v}_{11}\bar{v}_{12} & \bar{v}_{12}\bar{v}_{13} & \bar{v}_{11}\bar{v}_{13} \\ \bar{v}_{21}^2 & \bar{v}_{22}^2 & \bar{v}_{23}^2 & \bar{v}_{21}\bar{v}_{22} & \bar{v}_{22}\bar{v}_{23} & \bar{v}_{21}\bar{v}_{23} \\ \bar{v}_{31}^2 & \bar{v}_{32}^2 & \bar{v}_{33}^2 & \bar{v}_{31}\bar{v}_{32} & \bar{v}_{32}\bar{v}_{33} & \bar{v}_{31}\bar{v}_{33} \\ 2\bar{v}_{11}\bar{v}_{21} & 2\bar{v}_{12}\bar{v}_{22} & 2\bar{v}_{13}\bar{v}_{23} & \bar{v}_{11}\bar{v}_{22}+\bar{v}_{13}\bar{v}_{21} & \bar{v}_{12}\bar{v}_{23}+\bar{v}_{13}\bar{v}_{22} & \bar{v}_{11}\bar{v}_{23}+\bar{v}_{13}\bar{v}_{21} \\ 2\bar{v}_{21}\bar{v}_{31} & 2\bar{v}_{22}\bar{v}_{32} & 2\bar{v}_{23}\bar{v}_{33} & \bar{v}_{21}\bar{v}_{32}+\bar{v}_{22}\bar{v}_{31} & \bar{v}_{22}\bar{v}_{33}+\bar{v}_{23}\bar{v}_{32} & \bar{v}_{21}\bar{v}_{33}+\bar{v}_{23}\bar{v}_{31} \\ 2\bar{v}_{11}\bar{v}_{31} & 2\bar{v}_{12}\bar{v}_{32} & 2\bar{v}_{13}\bar{v}_{33} & \bar{v}_{11}\bar{v}_{32}+\bar{v}_{12}\bar{v}_{31} & \bar{v}_{12}\bar{v}_{33}+\bar{v}_{13}\bar{v}_{32} & \bar{v}_{11}\bar{v}_{33}+\bar{v}_{13}\bar{v}_{31} \end{bmatrix}\)

Then, the transformation of geometrical points between the two coordinate systems are:

\(\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \textbf{R}_0 \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\)

Change the vector \(\boldsymbol{\beta}\) (defined based on principal axes \(x_1x_2x_3\) i.e. \(\beta_4 = \beta_5 = \beta_6 = 0\))

\(\boldsymbol{\beta} = \begin{bmatrix} \beta_1 & \beta_2 & \beta_3 & \beta_4 & \beta_5 & \beta_6 \end{bmatrix}^\textbf{T}\)

into matrix form \(\boldsymbol{\beta}\)

\(\boldsymbol{\beta} = \begin{bmatrix} \beta_1 & \beta_4 & \beta_6 \\ & \beta_2 & \beta_5 \\ sym. & & \beta_3 \end{bmatrix}\)

Thermal modulus \(\boldsymbol{\beta}_{\textrm{local}}\) based on local axes \(xyz\):

\(\boldsymbol{\beta}_{\textrm{local}} = \textbf{R}_0 \boldsymbol{\beta} \textbf{R}_0^\textbf{T}\)

The transformation of material arrays:

\(\textbf{D}_{\textrm{local}} = \textbf{R}^\textbf{T} \textbf{D} \textbf{R}\)

Reference: