TMM4175 Polymer Composites

Home About Python Links Table of Contents

Thermal effects in laminates¶

The Plane stress thermomechanical relation for a layer in the laminate coordinate system is

\begin{equation} \boldsymbol{\sigma}' = \mathbf{Q}'(\boldsymbol{\varepsilon}' - \boldsymbol{\alpha}'\Delta T ) \tag{1} \end{equation}

The strain (see Laminate deformation is

\begin{equation} \boldsymbol{\varepsilon}' = \boldsymbol{\varepsilon}^0 + z \boldsymbol{\kappa} \tag{2} \end{equation}

Substituting the strain (1) in (2):

\begin{equation} \boldsymbol{\sigma}' = \mathbf{Q}'(\boldsymbol{\varepsilon}^0 + z \boldsymbol{\kappa} - \boldsymbol{\alpha}'\Delta T ) \tag{3} \end{equation}

The force resultants (see Laminate loads and constitutive relations ) are now:

\begin{equation} \mathbf{N} = \sum_{k=1}^{n} \int_{h_{k-1}}^h \boldsymbol{\sigma}'dz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k(\boldsymbol{\varepsilon}^0 + z \boldsymbol{\kappa}-\boldsymbol{\alpha}'_k\Delta T)dz= \\ \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\varepsilon}^0 dz+ \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\kappa}zdz - \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\alpha}_k'\Delta Tdz= \\ \Big[ \sum_{k=1}^{n} \mathbf{Q}'_k (h_k-h_{k-1}) \Big] \boldsymbol{\varepsilon}^0 + \Big[\frac{1}{2}\sum_{k=1}^{n} \mathbf{Q}'_k(h_k^2-h_{k-1}^2 \Big] \boldsymbol{\kappa} + \Big[\Delta T \sum_{k=1}^{n} \mathbf{Q}'_k \boldsymbol{\alpha}'_k (h_k-h_{k-1}) \Big] \tag{4} \end{equation}

The equation above will be written as

\begin{equation} \mathbf{N} = \mathbf{A} \boldsymbol{\varepsilon}^0+\mathbf{B} \boldsymbol{\kappa} - \mathbf{N}_{th} \tag{5} \end{equation}

where $\mathbf{A}$ and $\mathbf{B}$ are the previously defined submatrices of the laminate stiffness matrix, and $\mathbf{N}_{th}$ is called thermal forces given by

\begin{equation} \mathbf{N}_{th}= \Big[\Delta T \sum_{k=1}^{n} \mathbf{Q}'_k \boldsymbol{\alpha}'_k (h_k-h_{k-1}) \Big] \tag{6} \end{equation}

Correspondingly, moment resultants are

\begin{equation} \mathbf{M} = \frac{1}{2}\Big[ \sum_{k=1}^{n} \mathbf{Q}'_k (h_k^2-h_{k-1}^2) \Big] \boldsymbol{\varepsilon}^0 + \Big[\frac{1}{3}\sum_{k=1}^{n} \mathbf{Q}'_k(h_k^3-h_{k-1}^3 \Big] \boldsymbol{\kappa} + \Big[\Delta T \frac{1}{2} \sum_{k=1}^{n} \mathbf{Q}'_k \boldsymbol{\alpha}'_k (h_k^2-h_{k-1}^2) \Big] \tag{7} \end{equation}

or

\begin{equation} \mathbf{M} = \mathbf{B} \boldsymbol{\varepsilon}^0+\mathbf{D} \boldsymbol{\kappa} - \mathbf{M}_{th} \tag{8} \end{equation}

where $\mathbf{M}_{th}$ are thermal moments given by

\begin{equation} \mathbf{M}_{th}=\Big[\Delta T \frac{1}{2} \sum_{k=1}^{n} \mathbf{Q}'_k \boldsymbol{\alpha}'_k (h_k^2-h_{k-1}^2) \Big] \tag{9} \end{equation}

The constitutive relations for a laminate subjected to both mechanical and thermal loading are summarized as

\begin{equation} \begin{bmatrix} \mathbf{N} \\ \mathbf{M} \end{bmatrix}+ \begin{bmatrix} \mathbf{N}_{th} \\ \mathbf{M}_{th} \end{bmatrix}= \begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B} & \mathbf{D} \end{bmatrix} \begin{bmatrix} \boldsymbol{\varepsilon}^0 \\ \boldsymbol{\kappa} \end{bmatrix} \tag{10} \end{equation}

Disclaimer:This site is about polymer composites, designed for educational purposes. Consumption and use of any sort & kind is solely at your own risk.
Fair use: I spent some time making all the pages, and even the figures and illustrations are my own creations. Obviously, you may steal whatever you find useful here, but please show decency and give some acknowledgment if or when copying. Thanks! Contact me: nils.p.vedvik@ntnu.no www.ntnu.edu/employees/nils.p.vedvik