TMM4175 Polymer Composites

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Thermal expansion¶

The thermally induced strains $\boldsymbol{\varepsilon}_ {th}$ divided by the change in temperature $\Delta T$ is called the material's coefficients of thermal expansion $\boldsymbol{\alpha}$. A general anisotropic material has six coefficients:

\begin{equation} \boldsymbol{\alpha}= \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \alpha_{23} \\ \alpha_{13} \\ \alpha_{12} \end{bmatrix}= \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \gamma_{23} \\ \gamma_{13} \\ \gamma_{12} \end{bmatrix}_{th} \frac{1}{\Delta T} \tag{1} \end{equation}

Hence, the thermally induced strain is

\begin{equation} \boldsymbol{\varepsilon}_{th}=\boldsymbol{\alpha}\Delta T \tag{2} \end{equation}

The total strain is the sum of thermal strain and mechanical strain:

\begin{equation} \boldsymbol{\varepsilon}=\mathbf{S}\boldsymbol{\sigma} + \boldsymbol{\alpha}\Delta T \tag{3} \end{equation}

Solving for stress:

\begin{equation} \boldsymbol{\sigma} = \mathbf{C}(\boldsymbol{\varepsilon} - \boldsymbol{\alpha}\Delta T ) \tag{4} \end{equation}

Hence, the relations for general anisotropic materials are:

\begin{equation} \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \tau_{23} \\ \tau_{13} \\ \tau_{12} \end{bmatrix}= \begin{bmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\ C_{16} & C_{26} & C_{36} & C_{45} & C_{56} & C_{66} \end{bmatrix} \begin{bmatrix} \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \gamma_{23} \\ \gamma_{13} \\ \gamma_{12} \end{bmatrix}- \Delta T \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \alpha_{23} \\ \alpha_{13} \\ \alpha_{12} \end{bmatrix} \end{bmatrix} \tag{5} \end{equation}

Due to the symmetry of orthotropic materials, the coefficients $\alpha_{23},\alpha_{13},\alpha_{12}$ are all zero and the stiffness matrix has only nine independent elastic constants:

\begin{equation} \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \tau_{23} \\ \tau_{13} \\ \tau_{12} \end{bmatrix}= \begin{bmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{bmatrix} \begin{bmatrix} \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \gamma_{23} \\ \gamma_{13} \\ \gamma_{12} \end{bmatrix}- \Delta T \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{bmatrix} \tag{6} \end{equation}

Example¶

The temperature is increased by 100$^\circ$C while the total strain is kept zero. Compute the stress for the unidirectional Kevlar-49/Epoxy material read from the Material library.

import matlib
m = matlib.get('Kevlar-49/Epoxy')
display(m)

{'name': 'Kevlar-49/Epoxy',
 'units': 'MPa-mm-Mg',
 'type': 'UD',
 'fiber': 'Kevlar-49',
 'Vf': 0.55,
 'rho': 1.4e-09,
 'description': 'Typical UD Kevlar-49/Epoxy from TMM4175',
 'E1': 73000,
 'E2': 5000,
 'E3': 5000,
 'v12': 0.35,
 'v13': 0.35,
 'v23': 0.45,
 'G12': 2200,
 'G13': 2200,
 'G23': 1700,
 'a1': -1e-06,
 'a2': 5e-05,
 'a3': 5e-05,
 'XT': 1400,
 'YT': 20,
 'ZT': 20,
 'XC': 300,
 'YC': 120,
 'ZC': 120,
 'S12': 40,
 'S13': 40,
 'S23': 20,
 'f12': -0.5,
 'f13': -0.5,
 'f23': -0.5}

import numpy as np
from compositelib import S3D, C3D

C=C3D(m)

dT = 100

strain = np.array([0,0,0,0,0,0])

CTE = np.array([ m['a1'], m['a2'], m['a3'], 0, 0, 0 ])

thermalstrain=dT*CTE

mechanicalstrain=strain-thermalstrain

stress=np.dot(C,mechanicalstrain)

print('Thermally induced stresses:')
print(np.array2string(stress, precision=3, suppress_small=True, separator='  ', floatmode='maxprec') )

Thermally induced stresses:
[-25.29   -46.557  -46.557    0.       0.       0.   ]

References and further readings¶

Herakovich, Carl T. Mechanics of Fibrous Composites. New York: Wiley, 1998.
Daniel, Isaac M., and Ori Ishai. Engineering Mechanics of Composite Materials. 2nd ed. New York: Oxford University Press, 2006.
Kollár, Lázló P., and George S. Springer. Mechanics of Composite Structures. Cambridge: Cambridge University Press, 2003.

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