8.27 Surfaces - Principal Strains

ε_1,+

Strain in the direction of the principal axis 1 on the positive side of the surface
(i.e. side in direction of positive surface axis z)

${\epsilon }_{1,+}=\frac{1}{2}\left({\epsilon }_{x,+}+{\epsilon }_{y,+}+\sqrt{({\epsilon }_{x,+}-{\epsilon }_{y,+}{)}^2+{\gamma }_{xy,+}^2}\right)$

ε_2,+

Strain in the direction of the principal axis 2 on the positive side of the surface
(i.e. side in direction of positive surface axis z)

${\epsilon }_{2,+}=\frac{1}{2}\left({\epsilon }_{x,+}+{\epsilon }_{y,+}-\sqrt{({\epsilon }_{x,+}-{\epsilon }_{y,+}{)}^2+{\gamma }_{xy,+}^2}\right)$

α₊

Angle between local axis x (or y) and principal axis 1 (or 2) for the strains on the positive side of the surface

${\alpha }_{+}=\frac{1}{2}\left(\text{arctan}\left(\frac{{\gamma }_{xy,+}}{{\epsilon }_{x,+}-{\epsilon }_{y,+}}\right)\right)$

ε_1,−

Strain in the direction of the principal axis 1 on the negative side of the surface

${\epsilon }_{1,-}=\frac{1}{2}\left({\epsilon }_{x,-}+{\epsilon }_{y,-}+\sqrt{({\epsilon }_{x,-}-{\epsilon }_{y,-}{)}^2+{\gamma }_{xy,-}^2}\right)$

ε_2,−

Strain in the direction of the principal axis 2 on the negative side of the surface

${\epsilon }_{2,-}=\frac{1}{2}\left({\epsilon }_{x,-}+{\epsilon }_{y,-}-\sqrt{({\epsilon }_{x,-}-{\epsilon }_{y,-}{)}^2+{\gamma }_{xy,-}^2}\right)$

α₋

Angle between local axis x (or y) and principal axis 1 (or 2) for the stresses on the negative side of the surface

${\alpha }_{-}=\frac{1}{2}\left(\text{arctan}\left(\frac{{\gamma }_{xy,-}}{{\epsilon }_{x,-}-{\epsilon }_{y,-}}\right)\right)$