User:EML4500.f08.JAMAMA/FE/ex13

= HW 2.1 =

Find
By taking infinitesimal slice of the bar (shown in red in Figure 1),$$ dx $$, develop expression for elastodynamic response in Heat Problem in 1-D

Solution
Free Body Diagram:

Heat is transferred in the form of Conduction, Convection and Thermal Radiation. In this problem we will do Heat Conduction only.

Here we have consider a small infinitesimal portion of wall of thickness dx as shown in figure.

Let,The Heat Flux is represented by "q".


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$$ \alpha_2=1, \alpha_1=0, \alpha_3=...=\alpha_n=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


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$$ \left \{ \alpha _i \right \}=\left \{ 0,1,0,...,0 \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


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$$ \underline w=\sum_{i}\alpha _i\underline b_i=\sum_{i}\left \{ 0,1,0,...,0 \right \}\underline b_i=\underline b_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


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$$ \underline w\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


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$$ \underline b_2\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


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$$ \alpha_n=1,\ \alpha_1=...=\alpha_{n-1}=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


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$$ \left \{ \alpha_i \right \}=\left \{ 0,..., 0,1 \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }


 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\alpha _i\underline b_i=\sum_{i}\left \{ 0,...,0,1 \right \}\underline b_i=\underline b_n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 *  $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline b_{n}\cdot \textrm{\underline P(\underline v)}=0\ \forall \ \left \{ \alpha_1,...,\alpha_n \right \}\in \ \mathbb{R}^n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\alpha_i\underline b_i $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline b_{i}\cdot \textrm{\underline P(\underline v)}=0\ \forall \ i=1,...,n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \left \{\underline a_{i},i=1,...,n \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
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$$ \underline a_{i}\cdot \underline a_{j}= \delta _{ij} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
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$$ \delta _{ij}=\left\{\begin{matrix} 1\ for\ i=j\\ 0\ for\ i\neq j \end{matrix}\right. $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \left \{\underline a_{i} \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \textrm{\underline v}=\sum_{i=1}^{n}\ \textrm{v}_{i}\underline a_{i} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \textrm{\underline v}=\sum_{j=1}^{n}\ \textrm{v}_{j}\underline a_{j} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
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$$ \textrm{\underline P (\underline v)}=\sum_{j=1}^{n}\ \textrm{v}_{j}\underline a_{j} - \textrm{\underline v}=\underline 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
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$$ \underline a_{i}\cdot\textrm{\underline P (\underline v)}=\underline a_{i}\cdot \sum_{j=1}^{n}\ \textrm{v}_{j}\underline a_{j} - \underline a_{i}\cdot \textrm{\underline v}=\underline 0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
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$$ \underline a_{i}\cdot \sum_{j=1}^{n}\ \textrm{v}_{j}\underline a_{j}=\underline a_{i}\cdot \textrm{\underline v}=\sum_{j=1}^{n}(\underline a_{i}\cdot \underline a_{j})\textrm{v}_{j}=\sum_{j=1}^{n}\ \delta_{ij}\cdot \textrm{v}_{j} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline a_{i}\cdot \textrm{\underline P(\underline v)}=0\ \forall \ i=1,...,n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w\cdot \underline P (\textrm{\underline v})=0 \ \forall \ \underline w\in \mathbb{R}^{n} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w = \sum_{i}\beta_i\underline a_i $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w \cdot \underline P(\textrm{\underline v})=0 \ \forall \ \left \{ \beta_1,..., \beta_n \right \}\ \in \ \mathbb{R}^n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\beta_i\underline a_i $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline a_{i}\cdot \textrm{\underline P(\underline v)}=0\ \forall \ i=1,...,n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \beta_1=1, \beta_2=...=\beta_n=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \left \{\beta _1, ..., \beta _n \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \forall \ \left \{\beta _1, ..., \beta _n \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\beta _i\underline a_i=\sum_{i}\left \{ 1,0,...,0 \right \}\underline a_i=\underline a_1 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline a_1\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \beta_2=1, \beta_1=0, \beta_3=...=\beta_n=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \left \{ \beta _i \right \}=\left \{ 0,1,0,...,0 \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\beta _i\underline a_i=\sum_{i}\left \{ 0,1,0,...,0 \right \}\underline a_i=\underline a_2 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline a_2\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \beta_n=1,\ \beta_1=...=\beta_{n-1}=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \left \{ \beta_i \right \}=\left \{ 0,..., 0,1 \right \} $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\beta _i\underline a_i=\sum_{i}\left \{ 0,...,0,1 \right \}\underline a_i=\underline a_n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w\cdot \underline P(\textrm{\underline v})=0 $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline a_{n}\cdot \textrm{\underline P(\underline v)}=0\ \forall \ \left \{ \beta_1,...,\beta_n \right \}\in \ \mathbb{R}^n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline w=\sum_{i}\beta_i\underline a_i $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }

<span id="(1)">
 * {| style="width:100%" border="0"

$$ \underline a_{i}\cdot \textrm{\underline P(\underline v)}=0\ \forall \ i=1,...,n $$ $$
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right"> $$ \displaystyle (Eq. x.x)
 * }