5.4 Summary of the tensor strain-deformation equations

\[\begin{equation} \begin{split} {\varepsilon_{xx}}=& {u {}_{,x}} + \frac{1}{2} \left[({u {}_{,x}})^2 + ({v {}_{,x}})^2 + ({w {}_{,x}})^2 \right]\\ {\varepsilon_{yy}}=& {v {}_{,y}} + \frac{1}{2} \left[({u {}_{,y}})^2 + ({v {}_{,y}})^2 + ({w {}_{,y}})^2 \right]\\ {\varepsilon_{zz}}=& {w {}_{,z}} + \frac{1}{2} \left[({u {}_{,z}})^2 + ({v {}_{,z}})^2 + ({w {}_{,z}})^2 \right]\\ {\varepsilon_{xy}}=& \frac{1}{2} \left({u {}_{,y}} + {v {}_{,x}}\right) + \frac{1}{2} \left[{u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \right]\\ {\varepsilon_{xz}}=& \frac{1}{2} \left({u {}_{,z}} + {w {}_{,x}}\right) + \frac{1}{2} \left[{u {}_{,x}} {u {}_{,z}} + {v {}_{,x}} {v {}_{,z}} + {w {}_{,x}} {w {}_{,z}} \right]\\ {\varepsilon_{yz}}=& \frac{1}{2} \left({v {}_{,z}} + {w {}_{,y}}\right) + \frac{1}{2} \left[{u {}_{,y}} {u {}_{,z}} + {v {}_{,y}} {v {}_{,z}} + {w {}_{,y}} {w {}_{,z}} \right]\\ \end{split} \end{equation}\]

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• It is notable that these higher order strain equations are completely consistent with the linearized versions that are typically used in linear analysis.
  • There are many circumstances where the linear versions of these equations are perfectly adequate.
• However, in some circumstances such as stability analysis or calculations where the deformations are relatively large, it will be necessary to carry the higher-order terms.