# Two dimensional strain / bi-axial strain

Notations and units:

$\varepsilon$ = Normal strain, $\gamma$ = Shear strain, $\theta$ = Angle

Units of the notations are in SI units.

Substitute $\sigma=\varepsilon, \tau=\frac{\gamma}{2}$ in two dimensional stress system

Normal strain at an angle from the reference

$\varepsilon_{\theta}=\frac{\varepsilon_{x}+\varepsilon_{y}}{2}+\frac{\varepsilon_{x}-\varepsilon_{y}}{2}\cos2\theta+\frac{\gamma_{xy}\sin2\theta}{2}$

Shear strain at an angle from the reference

$\frac{\gamma_{\theta}}{2}=\frac{\varepsilon_{x}-\varepsilon_{y}}{2}\sin2\theta-\frac{\gamma_{xy}\cos2\theta}{2}$

Resultant strain

$\varepsilon_{R}=\sqrt{\varepsilon_\theta^{2}+\left( \frac{\gamma_{\theta}}{2} \right )^{2}}$

Angle between normal strain to one plane and resultant strain (angle of obliquity)

$\tan\phi=\frac{\gamma_{\theta}}{2\varepsilon_{\theta}}$

Maximum principal strain

$\varepsilon_{1}=\frac{\varepsilon_{x}+\varepsilon_{y}}{2}+\sqrt{\left ( \frac{\varepsilon_{x}-\varepsilon_{y}}{2} \right )^{2}+\left( \frac{\gamma_{xy}}{2} \right )^{2}}$

Minimum principal strain

$\varepsilon_{2}=\frac{\varepsilon_{x}+\varepsilon_{y}}{2}-\sqrt{\left ( \frac{\varepsilon_{x}-\varepsilon_{y}}{2} \right )^{2}+\left( \frac{\gamma_{xy}}{2} \right )^{2}}$

Angle of maximum principal strain from the reference plane

$\tan2\theta_{p}=\frac{\gamma_{xy}}{\varepsilon_{x}-\varepsilon_{y}}$

Angle of minimum principal strain from the reference plane

$\theta_{p} + 90^{\circ}$

Angles of plane where shear strain is maximum

$\theta_{s}, \theta_{s} + 90^{\circ}$

$\theta_{s} = \theta_{p} + 45^{\circ}$

$\tan2\theta_{s}=\frac{\varepsilon_{y}-\varepsilon_{x}}{\gamma_{xy}}$

Maximum shear strain

$\frac{\gamma_{max}}{2}=\sqrt{\left ( \frac{\varepsilon_{x}-\varepsilon_{y}}{2} \right )^{2}+\left( \frac{\gamma_{xy}}{2} \right )^{2}}$

$\gamma_{max}=\varepsilon_{1}-\varepsilon_{2}$

Maximum normal strain on maximum shear plane

$\varepsilon'=\frac{\varepsilon_{1}+\varepsilon_{2}}{2}$