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