5.5 The principal strains

\[ \left[ \begin{array}{ccc} {\varepsilon_{xx}}- {\varepsilon_{\mathrm{P}}}& {\varepsilon_{xy}}& {\varepsilon_{xz}}\\ {\varepsilon_{xy}}& {\varepsilon_{yy}}- {\varepsilon_{\mathrm{P}}}& {\varepsilon_{yz}}\\ {\varepsilon_{xz}}& {\varepsilon_{yz}}& {\varepsilon_{zz}}- {\varepsilon_{\mathrm{P}}}\\ \end{array} \right] \left\{ \begin{array}{c} l \\ m \\ n \end{array} \right\} = \left\{ \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right\} \]

• The strain tensor, like the stress tensor, has principal values and principal directions.
  • The computation is identical to the computation for stress.
  • The principal strains exist in a coordinate frame where the shear strain components are zero
• It is notable that the principal stress and strain directions are not necessarily the same.
  • In general, they will be the same for linear elastic materials but may not be so for anisotropic materials.

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