Linearly Elastic Solid

As the name says, linearly elastic solid denote the solid material which there its relation between stress and strain is linear. We give $T$ the stress tensor and $E$ the strain tensor $T$ and $E$ both are second-order tensor (9 members each). Therefore ${{T}_{ij}}$ and ${{E}_{kl}}$ denote the member of the tensor where i,j,k,l are $\in \left\{ 1,2,3 \right\}$ and independent of each other. For linearity relation between $T$ and $E$ ; \[{{T}_{11}}={{C}_{1111}}{{E}_{11}}+{{C}_{1112}}{{E}_{12}}+....................+{{C}_{1133}}{{E}_{33}}\] \[{{T}_{12}}={{C}_{1211}}{{E}_{11}}+{{C}_{1212}}{{E}_{12}}+....................+{{C}_{1233}}{{E}_{33}}\] \[.......................................................\] \[.......................................................\] \[{{T}_{33}}={{C}_{3311}}{{E}_{11}}+{{C}_{3312}}{{E}_{12}}+....................+{{C}_{3333}}{{E}_{33}}\] It involves total of 81 coefficients to do the linear job. This $C$ is the fourth-order tensor where ${{C}_{ijkl}}$ is the member. The above set of equations can be written as ${{T}_{ij}}={{C}_{ijkl}}{{E}_{kl}}$ However, $T$ and $E$ are proof to be symmetric tensors. The fact that $E$ is symmetric tensor, we can combine the sum of two term such as ${{C}_{1112}}{{E}_{12}}+{{C}_{1121}}{{E}_{21}}$ into $({{C}_{1112}}+{{C}_{1121}}){{E}_{12}}$ and make $({{C}_{1112}}+{{C}_{1121}})$ one coefficient. This combination of two term lead to reduction of coefficients from 81 to 54. The fact that $T$ is symmetric reduces the coefficient (by reducing number of equations from above set of equation) to 18. For the simplified set of linear equation, we write \[\left[ \begin{matrix} {{T}_{11}} \\ {{T}_{22}} \\ {{T}_{33}} \\ \begin{matrix} {{T}_{23}} \\ \begin{matrix} {{T}_{31}} \\ {{T}_{12}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right]=\left[ \begin{matrix} {{C}_{11}} \\ {{C}_{12}} \\ {{C}_{13}} \\ \begin{matrix} {{C}_{14}} \\ \begin{matrix} {{C}_{15}} \\ {{C}_{16}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{12}} \\ {{C}_{22}} \\ {{C}_{23}} \\ \begin{matrix} {{C}_{24}} \\ \begin{matrix} {{C}_{25}} \\ {{C}_{26}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{13}} \\ {{C}_{23}} \\ {{C}_{33}} \\ \begin{matrix} {{C}_{34}} \\ \begin{matrix} {{C}_{35}} \\ {{C}_{36}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{14}} \\ {{C}_{24}} \\ {{C}_{34}} \\ \begin{matrix} {{C}_{44}} \\ \begin{matrix} {{C}_{45}} \\ {{C}_{46}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{15}} \\ {{C}_{25}} \\ {{C}_{35}} \\ \begin{matrix} {{C}_{45}} \\ \begin{matrix} {{C}_{55}} \\ {{C}_{56}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{16}} \\ {{C}_{26}} \\ {{C}_{36}} \\ \begin{matrix} {{C}_{46}} \\ \begin{matrix} {{C}_{56}} \\ {{C}_{66}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right]\left[ \begin{matrix} {{E}_{11}} \\ {{E}_{22}} \\ {{E}_{33}} \\ \begin{matrix} 2{{E}_{23}} \\ \begin{matrix} 2{{E}_{31}} \\ 2{{E}_{12}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right]\] or \[\left[ \begin{matrix} {{T}_{1}} \\ {{T}_{2}} \\ {{T}_{3}} \\ \begin{matrix} {{T}_{4}} \\ \begin{matrix} {{T}_{5}} \\ {{T}_{6}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right]=\left[ \begin{matrix} {{C}_{11}} \\ {{C}_{12}} \\ {{C}_{13}} \\ \begin{matrix} {{C}_{14}} \\ \begin{matrix} {{C}_{15}} \\ {{C}_{16}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{12}} \\ {{C}_{22}} \\ {{C}_{23}} \\ \begin{matrix} {{C}_{24}} \\ \begin{matrix} {{C}_{25}} \\ {{C}_{26}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{13}} \\ {{C}_{23}} \\ {{C}_{33}} \\ \begin{matrix} {{C}_{34}} \\ \begin{matrix} {{C}_{35}} \\ {{C}_{36}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{14}} \\ {{C}_{24}} \\ {{C}_{34}} \\ \begin{matrix} {{C}_{44}} \\ \begin{matrix} {{C}_{45}} \\ {{C}_{46}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{15}} \\ {{C}_{25}} \\ {{C}_{35}} \\ \begin{matrix} {{C}_{45}} \\ \begin{matrix} {{C}_{55}} \\ {{C}_{56}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix}\begin{matrix} {{C}_{16}} \\ {{C}_{26}} \\ {{C}_{36}} \\ \begin{matrix} {{C}_{46}} \\ \begin{matrix} {{C}_{56}} \\ {{C}_{66}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right]\left[ \begin{matrix} {{E}_{1}} \\ {{E}_{2}} \\ {{E}_{3}} \\ \begin{matrix} {{E}_{4}} \\ \begin{matrix} {{E}_{5}} \\ {{E}_{6}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right]\] Where $C$ is symmetric tensor