5.3 Definition of shear strain and its relation to linear strain

5.3.1 Shear strain for larger deformations

• Examiner initial and final configuration for two vectors of a strained body. In the initial configuration, the angle defined is 90 degrees.

Vector in a body

\[\begin{equation} \angle BOA = 90 {^\circ} \end{equation}\]

We will define the shear strain as: \[\begin{equation} {\gamma_{xy}}= 90 {^\circ}- \angle B'O'A' \end{equation}\]

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Manipulating our definition of \({\gamma_{xy}}\): \[\begin{equation} \begin{split} \sin {\gamma_{xy}}=& \sin \left(90 {^\circ}- \angle B'O'A' \right) \\ =& \cos \left( \angle B'O'A' \right) \end{split} \end{equation}\]

• The latter of these two can be attributed to the phase angle between sine and cosine.

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• Separately, we can take the scalar product of the two vectors: \[\begin{equation} \begin{split} \vec{O'B'} \cdot \vec{O'A'} =& ||O'B'|| \; ||O'A'|| \cos \left(\angle B'O'A'\right) \\ \cos \left(\angle B'O'A'\right) =& \frac{\vec{O'B'} \cdot \vec{O'A'}}{||O'B'|| \; ||O'A'||} \\ \end{split} \end{equation}\]

• Therefore, we have: \[\begin{equation} \sin {\gamma_{xy}}= \frac{\vec{O'B'} \cdot \vec{O'A'}}{||O'B'|| \; ||O'A'||} \end{equation}\]

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• We need calculate the scalar product: \[\begin{equation} \begin{split} \vec{O'A'} =& ({\delta x}+ {\delta u}) {\vec{i}}+ {\delta v}{\vec{j}}+ {\delta w}{\vec{k}}\\ \vec{O'B'} =& {\delta u}{\vec{i}}+ ({\delta y}+ {\delta v}) {\vec{j}}+ {\delta w}{\vec{k}}\\ \end{split} \end{equation}\]

\[\begin{equation} \vec{O'B'} \cdot \vec{O'A'} = ({\delta x}+ {\delta u}) {\delta u}+ ({\delta y}+ {\delta v}) {\delta v}+ {\delta w}{\delta w} \end{equation}\]

• Now, for the purposes of the change in angle, we can assume that the length of the vectors does not change significantly. Therefore: \[\begin{equation} \begin{split} O'A' \approx& \; {\delta x}\\ O'B' \approx& \; {\delta y}\\ \end{split} \end{equation}\]

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Therefore:

\[\begin{equation} \begin{split} \sin {\gamma_{xy}}=& \, \frac{({\delta x}+ {\delta u}) {\delta u}+ ({\delta y}+ {\delta v}) {\delta v}+ {\delta w}{\delta w}}{{\delta x}{\delta y}} \\ =& \, (1 + {u {}_{,x}}){u {}_{,y}} + (1 + {v {}_{,y}}){v {}_{,x}} + {w {}_{,x}} {w {}_{,y}}\\ =& \, {u {}_{,y}} + {v {}_{,x}} + {u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \\ \end{split} \end{equation}\]

Finally, we can also know that for small shear angles:

\[\begin{equation} \sin {\gamma_{xy}}\approx {\gamma_{xy}} \end{equation}\]

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Therefore: \[\begin{equation} {\gamma_{xy}}= {u {}_{,y}} + {v {}_{,x}} + {u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \end{equation}\]

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5.3.2 Summary of the engineering strain-deformation equations

\[\begin{equation} \begin{split} {\varepsilon_{xx}}=& {u {}_{,x}} + \frac{1}{2} \left[\left({u {}_{,x}}\right)^2 + \left({v {}_{,x}}\right)^2 + \left({w {}_{,x}}\right)^2 \right]\\ {\varepsilon_{yy}}=& {v {}_{,y}} + \frac{1}{2} \left[\left({u {}_{,y}}\right)^2 + \left({v {}_{,y}}\right)^2 + \left({w {}_{,y}}\right)^2 \right]\\ {\varepsilon_{zz}}=& {w {}_{,z}} + \frac{1}{2} \left[\left({u {}_{,z}}\right)^2 + \left({v {}_{,z}}\right)^2 + \left({w {}_{,z}}\right)^2 \right]\\ {\gamma_{xy}}=& {u {}_{,y}} + {v {}_{,x}} + {u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}}\\ {\gamma_{xz}}=& {u {}_{,z}} + {w {}_{,x}} + {u {}_{,x}} {u {}_{,z}} + {v {}_{,x}} {v {}_{,z}} + {w {}_{,x}} {w {}_{,z}}\\ {\gamma_{yz}}=& {v {}_{,z}} + {w {}_{,y}} + {u {}_{,y}} {u {}_{,z}} + {v {}_{,y}} {v {}_{,z}} + {w {}_{,y}} {w {}_{,z}}\\ \end{split} \end{equation}\]

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5.3.3 Engineering strain vs tensor strain

Lets compare the following equations: \[\begin{equation} \begin{split} {\varepsilon_{xx}}=& {u {}_{,x}} + \frac{1}{2} \left( ({u {}_{,x}})^2 + ({v {}_{,x}})^2 + ({w {}_{,x}})^2 \right) \\ {\gamma_{xy}}=& {u {}_{,y}} + {v {}_{,x}} + {u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \\ \end{split} \end{equation}\]

\[\begin{equation} \begin{split} 2 \, {\varepsilon_{xx}}=&{u {}_{,x}} + {u {}_{,x}} + {u {}_{,x}} {u {}_{,x}} + {v {}_{,x}} {v {}_{,x}} + {w {}_{,x}} {w {}_{,x}} \\ {\gamma_{xy}}= 2 \, {\varepsilon_{xy}}=&{u {}_{,y}} + {v {}_{,x}} + {u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \end{split} \end{equation}\]

From the above equations we observe: \[\begin{equation} {\varepsilon_{xy}}= \frac{1}{2} {\gamma_{xy}} \end{equation}\]

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Similarly: \[\begin{equation} \begin{split} {\gamma_{xy}}= 2 \, {\varepsilon_{xy}}=&{u {}_{,z}} + {w {}_{,x}} + {u {}_{,x}} {u {}_{,y}} + {v {}_{,x}} {v {}_{,y}} + {w {}_{,x}} {w {}_{,y}} \\ {\gamma_{xz}}= 2 \, {\varepsilon_{xz}}=&{u {}_{,z}} + {w {}_{,x}} + {u {}_{,x}} {u {}_{,z}} + {v {}_{,x}} {v {}_{,z}} + {w {}_{,x}} {w {}_{,z}} \\ {\gamma_{yz}}= 2 \, {\varepsilon_{yz}}=&{v {}_{,z}} + {w {}_{,y}} + {u {}_{,y}} {u {}_{,z}} + {v {}_{,y}} {v {}_{,z}} + {w {}_{,y}} {w {}_{,z}} \end{split} \end{equation}\]