Tensor Components of Infinitesimal Strain, II

Abstract

In the last chapter it was claimed that

$$ \left[ \begin{gathered} {e_{{11}}}\quad {e_{{12\quad }}}{e_{{13}}} \hfill \\ {e_{{21}}}\quad {e_{{22}}}\quad {e_{{23}}} \hfill \\ {e_{{31}}}\quad {e_{{32}}}\quad {e_{{33}}} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} {\varepsilon_{{11}}}\quad \frac{1}{2}{\gamma_{{12}}}\quad \frac{1}{2}{\gamma_{{13}}} \hfill \\ \frac{1}{2}{\gamma_{{21}}}\quad {\varepsilon_{{22}}}\quad \frac{1}{2}{\gamma_{{23}}} \hfill \\ \frac{1}{2}{\gamma_{{31}}}\quad \frac{1}{2}{\gamma_{{32}}}\quad {\varepsilon_{{33}}} \hfill \\ \end{gathered} \right] $$

where the e _ij were defined in terms of displacement gradients and the ∊ _ij and γ _ij are the elongations and shear strains of lines initially parallel to the coordinate axes. We now demonstrate why this is so, first for a normal component (e₁₁) and then for a shear component (e₃₁).