Isotropic Linearly Elastic Solid

Isotropic Linearly Elastic Solid denotes the type of solid material which mechanical properties are the same regardless of directions. We had the linearity with respect to ${{e}_{i}}$ \[{{T}_{ij}}={{C}_{ijkl}}{{E}_{kl}}\] and with respect to ${{{e}'}_{i}}$ \[{{{T}'}_{ij}}={{{C}'}_{ijkl}}{{{E}'}_{kl}}\] For this material, it is obligated that \[{{C}_{ijkl}}={{{C}'}_{ijkl}}\] under all orthogonal transformations of basis (e.g. rotation). For example $I$, the identity matrix, can do this job in any catrtesian coordinate. Also its multiplication with scalar does the job. For isotropic fourth-order tensor, we use the form of identity sencond-order tensor, ${{\delta }_{ij}}$ , multipy to three new isotropic fourth-order tensors \[\begin{align} & {{A}_{ijkl}}={{\delta }_{ij}}{{\delta }_{kl}} \\ & {{B}_{ijkl}}={{\delta }_{ik}}{{\delta }_{jl}} \\ & {{H}_{ijkl}}={{\delta }_{il}}{{\delta }_{jk}} \\ & \\ \end{align}\] Therefore, the isotropic ${{C}_{ijkl}}$ could be defined as \[\begin{align} & {{C}_{ijkl}}=\lambda {{A}_{ijkl}}+\alpha {{B}_{ijkl}}+\beta {{H}_{ijkl}} \\ & \\ \end{align}\] where $\beta$, $\lambda$, and $\alpha$ are constants. Then it becomes \[{{T}_{ij}}={{C}_{ijkl}}{{E}_{kl}}=\lambda {{\delta }_{ij}}{{\delta }_{kl}}{{E}_{kl}}+\alpha {{\delta }_{ik}}{{\delta }_{jl}}{{E}_{kl}} +\beta {{\delta }_{il}}{{\delta }_{jk}}{{E}_{kl}}\] Thus, \[\begin{align} & {{T}_{ij}}=\lambda {{E}_{kk}}{{\delta }_{ij}}+(\alpha +\beta ){{E}_{kl}} \\ & \\ \end{align}\] denoting $(\alpha +\beta )$ by $2\mu $ , we have \[{{T}_{ij}}=\lambda e{{\delta }_{ij}}+2\mu {{E}_{kl}}\] where \[e\equiv {{E}_{kk}}\] denotes the dilation . Therefore, \[T=\lambda eI+2\mu E\] In detail, it says \[\begin{align} & {{T}_{11}}=\lambda ({{E}_{11}}+{{E}_{22}}+{{E}_{33}})+2\mu {{E}_{11}} \\ & {{T}_{22}}=\lambda ({{E}_{11}}+{{E}_{22}}+{{E}_{33}})+2\mu {{E}_{22}} \\ & {{T}_{33}}=\lambda ({{E}_{11}}+{{E}_{22}}+{{E}_{33}})+2\mu {{E}_{33}} \\ & {{T}_{12}}=2\mu {{E}_{12}} \\ & {{T}_{13}}=2\mu {{E}_{13}} \\ & {{T}_{23}}=2\mu {{E}_{23}} \\ & \\ \end{align}\] resource: ISBN: 978-0-7506-8560-3