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}

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