(Solved): 5. A Lagrangian strain tensor is given as E=20000.750.2500.250.75 Find t ...

5. A Lagrangian strain tensor is given as $\mathbf{E}=?\begin{array}{ccc}2& 0& 0\\ 0& 0.75& 0.25\\ 0& 0.25& 0.75\end{array}?$ Find the principal strains and the corresponding principal directions. 6. The infinitesimal strain tensor is defined as ${\tilde{e}}_{kl}?\frac{1}{2}\left(u_{k,l}+u_{l?k}\right)$ Prove that the following compatibility conditions are identically satisfied. $\begin{array}{l}{\tilde{e}}_{xx,yz}+{\tilde{e}}_{yz,xx}?{\tilde{e}}_{xz,xy}?{\tilde{e}}_{xy,zx}=0\\ {\tilde{e}}_{yy,xz}+{\tilde{e}}_{xz,yy}?{\tilde{e}}_{xy,yz}?{\tilde{e}}_{yz,xy}=0\\ {\tilde{e}}_{zz,xy}+{\tilde{e}}_{xy,zz}?{\tilde{e}}_{yz,xz}?{\tilde{e}}_{xz,yz}=0\\ 2{\tilde{e}}_{xy,xy}?{\tilde{e}}_{xx,yy}?{\tilde{e}}_{yy,xx}=0\\ 2{\tilde{e}}_{yz,yz}?{\tilde{e}}_{zz,yy}?{\tilde{e}}_{yy,z}=0\\ 2{\tilde{e}}_{xz,xz}?{\tilde{e}}_{xx,zz}?{\tilde{e}}_{zz,xx}=0\end{array}$