Sheared woven fibers thermal properties

Thermal expansion

The in-plane thermal expansion properties of the virtual plies relate to the thermal expansion properties of the nominal woven fabric as:

k,{t_{0}},\begin{bmatrix}{{Q}{\mathit{warp}}}\end{bmatrix},\begin{bmatrix}{{\alpha}{1}}\{{\alpha}{2}}\0\end{bmatrix}\Delta T+\left( 1-k\right),{t{0}},\begin{bmatrix}{{Q}{\mathit{weft}}}\end{bmatrix},\begin{bmatrix}{{\alpha}{2}}\{{\alpha}{1}}\0\end{bmatrix}\Delta T={t{0}},\begin{bmatrix}{Q}\end{bmatrix},\begin{bmatrix}{{\alpha}{x}}\{{\alpha}{y}}\0\end{bmatrix}\Delta T

which yields to:

\alpha_1=\frac{\left( \left( \left( {{E}{2}}-{{E}{1}}\right) ,{{\nu}{12}}-{{E}{2}}+{{E}{1}}\right) ,{{k}^{2}}+\left( {{E}{2}},{{\nu}{12}^{2}}+\left( {{E}{1}}-{{E}{2}}\right) ,{{\nu}{12}}+{{E}{2}}-2{{E}{1}}\right) k-{{E}{2}},{{\nu}{12}^{2}}+{{E}{1}}\right) ,{{\alpha}{y}}+\left( \left( \left( {{E}{1}}-{{E}{2}}\right) ,{{\nu}{12}}+{{E}{2}}-{{E}{1}}\right) ,{{k}^{2}}+\left( {{E}{2}},{{\nu}{12}^{2}}+\left( {{E}{2}}-{{E}{1}}\right) ,{{\nu}{12}}-{{E}{2}}\right) k\right) ,{{\alpha}{x}}}{\left( 2{{E}{2}},{{\nu}{12}^{2}}-2{{E}{1}}\right) k-{{E}{2}},{{\nu}{12}^{2}}+{{E}{1}}}

\alpha_2=\frac{\left( \left( \left( {{E}{2}^{2}}-{{E}{1}},{{E}{2}}\right) ,{{\nu}{12}}-{{E}{1}},{{E}{2}}+{{E}{1}^{2}}\right) ,{{k}^{2}}+\left( {{E}{2}^{2}},{{\nu}{12}^{2}}+\left( {{E}{1}},{{E}{2}}-{{E}{2}^{2}}\right) ,{{\nu}{12}}-{{E}{1}^{2}}\right) k\right) ,{{\alpha}{y}}+\left( \left( \left( {{E}{1}},{{E}{2}}-{{E}{2}^{2}}\right) ,{{\nu}{12}}+{{E}{1}},{{E}{2}}-{{E}{1}^{2}}\right) ,{{k}^{2}}+\left( {{E}{2}^{2}},{{\nu}{12}^{2}}+\left( {{E}{2}^{2}}-{{E}{1}},{{E}{2}}\right) ,{{\nu}{12}}-2{{E}{1}},{{E}{2}}+{{E}{1}^{2}}\right) k-{{E}{2}^{2}},{{\nu}{12}^{2}}+{{E}{1}},{{E}{2}}\right) ,{{\alpha}{x}}}{\left( 2{{E}{2}^{2}},{{\nu}{12}^{2}}-2{{E}{1}},{{E}{2}}\right) k-{{E}{2}^{2}},{{\nu}{12}^{2}}+{{E}{1}},{{E}{2}}}

If the warp and weft modules of elasticity of the nominal material are identical (Ex=Ey) as well as the warp and weft coefficients of thermal expansion (ɑx=ɑy), then ɑ1=ɑ2=ɑx=ɑy.

The in-plane thermal expansion properties of the sheared fabric is expressed as:

\alpha_{2D}={{\left( k,{[{Q}{\mathit{warp}}]}+\left( 1-k\right) ,{[{T}{\sigma}]^{-1}},{[{Q}{\mathit{weft}}]},{[{T}{\epsilon}]}\right) }^{-1}},\left( k,{[{Q}{\mathit{warp}}]},\begin{bmatrix}{{\alpha}{1}}\{{\alpha}{2}}\0\end{bmatrix}+\left( 1-k\right) ,{[{T}{\sigma}]^{-1}},{[{Q}{\mathit{weft}}]},\begin{bmatrix}{{\alpha}{2}}\{{\alpha}_{1}}\0\end{bmatrix}\right)

The out-of-plane thermal expansion coefficients of the fabric are assumed to remain unchanged.

Thermal conductivity

The woven ply is virtually divided into unidirectional plies that are aligned with the warp and weft orientations. The following graphic iIlustrates the thermal resistance of the virtual plies:

The transverse thermal conductivity of these virtual plies is assumed to be the same as the thermal conductivity in the virtual matrix direction.

{{K}{\mathit{warp}}}=\begin{bmatrix}{{K}{1}} & 0 & 0\0 & {{K}{2}} & 0\0 & 0 & {{K}{2}}\end{bmatrix}

{{K}{\mathit{weft}}}=\begin{bmatrix}{{K}{2}} & 0 & 0\0 & {{K}{1}} & 0\0 & 0 & {{K}{2}}\end{bmatrix}

The following equations can be inferred:

\frac{{{t}{0}},{{K}{x}}}{b}=\frac{{{t}{0}},{{K}{1}}k}{b}+\frac{{{t}{0}},{{K}{2}},\left( 1-k\right) }{b}

\frac{{{t}{0}},{{K}{y}}}{b}=\frac{{{t}{0}},{{K}{2}}k}{b}+\frac{{{t}{0}},{{K}{1}},\left( 1-k\right) }{b}

\frac{{{t}{0}},{{K}{y}}}{b}=\frac{{{t}{0}},{{K}{2}}k}{b}+\frac{{{t}{0}},{{K}{1}},\left( 1-k\right) }{b}

KT is the relative importance of the warp virtual ply thermal conductivity to the woven fabric thermal conductivity. This yields to:

{{K}{1}}=-{{K}{z}}+{{K}{y}}+{{K}{x}}

{{K}{2}}={{K}{z}}

{{k}{T}}=\frac{{{K}{z}}-{{K}{x}}}{2{{K}{z}}-{{K}{y}}-{{K}{x}}}

The thermal conductivity of the weft is rotated according to the shear angle.

[K_{weft_sheared}]=\begin{bmatrix}\cos{\left( \beta\right) } & \sin{\left( \beta\right) } & 0\-\sin{\left( \beta\right) } & \cos{\left( \beta\right) } & 0\0 & 0 & 1\end{bmatrix}[K_{weft}]\begin{bmatrix}\cos{\left( \beta\right) } & -\sin{\left( \beta\right) } & 0\\sin{\left( \beta\right) } & \cos{\left( \beta\right) } & 0\0 & 0 & 1\end{bmatrix}

The thermal conductivity properties of the sheared fabric are calculated as:

\frac{b}{{{t}{0}},{{K}{\mathit{ij}}}}=\frac{1}{\frac{{{t}{0}},{{ {{K}{\mathit{warp}}} }{ij}},{{k}{T}}}{b}+\frac{{{t}{0}},{{ {{K}{\mathit{weft_ sheared}}} }{ij}}\left( 1-{{k}{T}}\right) }{b}};(i,j=1,2)

\frac{{{t}{0}}}{b,{{K}{\mathit{ij}}}}=\frac{{{t}{0}},{{k}{T}}}{{{ {{K}{\mathit{warp}}} }{33}}b}+\frac{{{t}{0}},\left( 1-{{k}{T}}\right) }{{{ {{K}{\mathit{weft_ sheared}}} }{33}}b}

{[{K}{\mathit{smeared}}]}=\begin{bmatrix}{{K}{x}}+\left( {{K}{y}}-{{K}{z}}\right) ,{{\sin{\left( \beta\right) }}^{2}} & \left( {{K}{y}}-{{K}{z}}\right) ,\cos{\left( \beta\right) },\sin{\left( \beta\right) } & 0\\left( {{K}{y}}-{{K}{z}}\right) ,\cos{\left( \beta\right) },\sin{\left( \beta\right) } & {{K}{y}},{{\cos{\left( \beta\right) }}^{2}}+{{K}{z}},{{\sin{\left( \beta\right) }}^{2}} & 0\0 & 0 & {{K}_{z}}\end{bmatrix}

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