5.2 Normal strain for larger deformations

Strain is derived from deformation. Therefore we must take a closer look at deformation in order to fully understand strain.

Examine a deformation vector in a body under strain.

Vector in a body

In this derivation, the vector is assumed to be originally aligned with the \(x\) direction, and the normal strain (\({\varepsilon_{xx}}\)) is computed.

(This assumption is not necessary, however it simplifies the calculation.)

Vector in a body

Assume the following: \[\begin{equation} \begin{split} {\varepsilon_{xx}}=& \lim_{OA \rightarrow 0} \frac{(O'A')^2-(OA)^2}{2 (OA)^2} \\ =& \lim_{OA \rightarrow 0} \left(\frac{1}{2}\right)\left[ \frac{(O'A')^2}{(OA)^2} -1 \right] \end{split} \end{equation}\]

It is noteworthy that we are looking a quadratic equation with respect to vector length, i.e., this definition examines a higher order strain equation than the linear strain encountered in many undergraduate textbooks.

Examine the displacement vector:

Vector in a body \[\begin{equation} \begin{split} OA =& \, {\delta x}{\vec{i}}+ 0 {\vec{j}}+ 0 {\vec{k}}\\ OO' =& \, u {\vec{i}}+ v {\vec{j}}+ w {\vec{k}}\\ AA' =& \, (u + {\delta u}) {\vec{i}}+ (v +{\delta v}) {\vec{j}}+ (w +{\delta w}) {\vec{k}}\\ \end{split} \end{equation}\]

\[\begin{equation} \begin{split} OA' =& \, OA + AA' \\ =& \, OO' + O'A' \\ \end{split} \end{equation}\]

\[\begin{equation} \begin{split} O'A' =& \, OA+AA'-OO' \\ \end{split} \end{equation}\]

\[\begin{equation} \begin{split} O'A' =& [{\delta x}{\vec{i}}] + [(u + {\delta u}){\vec{i}}+ (v+{\delta v}){\vec{j}}+(w+{\delta w}){\vec{k}}] - [u {\vec{i}}+ v {\vec{j}}+w {\vec{k}}]\\ =& ({\delta x}+{\delta u}) {\vec{i}}+ {\delta v}{\vec{j}}+ {\delta w}{\vec{k}}\\ (O'A')^2 =& ({\delta x}+{\delta u})^2 + {\delta v}^2 + {\delta w}^2 \\ =& {\delta x}^2 +2 {\delta x}{\delta u}+ {\delta u}^2 + {\delta v}^2 + {\delta w}^2 \\ \end{split} \end{equation}\]

\[\begin{equation} \begin{split} \frac{(O'A')^2}{(OA)^2}=& 1 +2 \frac{{\delta u}}{{\delta x}} + \left(\frac{{\delta u}}{{\delta x}}\right)^2 + \left(\frac{{\delta v}}{{\delta x}}\right)^2 + \left(\frac{{\delta w}}{{\delta x}}\right)^2 \end{split} \end{equation}\]

\[\begin{equation} \begin{split} \frac{(O'A')^2}{(OA)^2}-1 =& 2 \frac{{\delta u}}{{\delta x}} + \left[\left(\frac{{\delta u}}{{\delta x}}\right)^2 + \left(\frac{{\delta v}}{{\delta x}}\right)^2 + \left(\frac{{\delta w}}{{\delta x}}\right)^2 \right] \end{split} \end{equation}\]

Recall our quadratic definition of strain.

\[\begin{equation} \begin{split} {\varepsilon_{xx}}=& \lim_{OA \rightarrow 0} \left(\frac{1}{2}\right)\left[ \frac{(O'A')^2}{(OA)^2} -1 \right] \\ \end{split} \end{equation}\]

Therefore: \[\begin{equation} \begin{split} {\varepsilon_{xx}}=& {\frac{\partial u}{\partial x}} + \frac{1}{2} \left[\left({\frac{\partial u}{\partial x}}\right)^2 + \left({\frac{\partial v}{\partial x}}\right)^2 + \left({\frac{\partial w}{\partial x}}\right)^2 \right]\\ \approx & {\frac{\partial u}{\partial x}} \end{split} \end{equation}\]

Similarly, we can derive:

\[\begin{equation} \begin{split} {\varepsilon_{yy}}=& {\frac{\partial v}{\partial y}} + \frac{1}{2} \left[\left({\frac{\partial u}{\partial y}}\right)^2 + \left({\frac{\partial v}{\partial y}}\right)^2 + \left({\frac{\partial w}{\partial y}}\right)^2 \right]\\ \approx & {\frac{\partial v}{\partial y}} \end{split} \end{equation}\]

\[\begin{equation} \begin{split} {\varepsilon_{zz}}=& {\frac{\partial w}{\partial z}} + \frac{1}{2} \left[\left({\frac{\partial u}{\partial z}}\right)^2 + \left({\frac{\partial v}{\partial z}}\right)^2 + \left({\frac{\partial w}{\partial z}}\right)^2 \right]\\ \approx & {\frac{\partial w}{\partial z}} \end{split} \end{equation}\]