8.26 Surfaces - Basic Strains

ε_x,+

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

${\epsilon }_{x,+}=\frac{\partial u}{\partial x}+\frac{d}{2}\frac{\partial {\phi }_y}{\partial x}$

where d: thickness of surface

ε_y,+

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

${\epsilon }_{y,+}=\frac{\partial v}{\partial y}+\frac{d}{2}\left(-\frac{\partial {\phi }_x}{\partial y}\right)$

γ_xy,+

Related rotation on the positive side of the surface

${\gamma }_{xy,+}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}+\frac{d}{2}\left(\frac{\partial {\phi }_y}{\partial y}-\frac{\partial {\phi }_x}{\partial x}\right)$

ε_x,−

Strain in the direction of the axis x on the negative side of the surface

${\epsilon }_{x,-}=\frac{\partial u}{\partial x}-\frac{d}{2}\frac{\partial {\phi }_y}{\partial x}$

ε_y,−

Strain in the direction of the axis y on the negative side of the surface

${\epsilon }_{y,-}=\frac{\partial v}{\partial y}-\frac{d}{2}(-\frac{\partial {\phi }_x}{\partial y})$

γ_xy,−

Related rotation on the negative side of the surface

${\gamma }_{xy,-}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}-\frac{d}{2}\left(\frac{\partial {\phi }_y}{\partial y}-\frac{\partial {\phi }_x}{\partial x}\right)$