Strain rate and velocity fields

Normal strain is deformation which changes the length of a material element. It is defined as the fractional change in length in a particular direction. For example, if the length of a piece of ice is \(L\) in the \(x\) direction, then the strain in the \(x\) direction is

\[ \epsilon_{xx} = \frac{\Delta L}{L}. \]

The rate of change of strain (the strain rate) is denoted \(\dot{\epsilon}\) and this is related to gradients in the velocity field as follows

\[ \dot{\epsilon}_{xx} = \frac{\partial u}{\partial x}. \]

Proof of this strain rate - velocity relationship

Consider a segment of ice within a glacier that is \(L\) long in the \(x\) direction. It sits in a velocity field \(\underline{u}(x) = u(x)\) which only varies in the \(x\) direction. Let’s define \(x=0\) at the left side of the block. Arbitrarily, we say that the velocity at the left side of the block is \(u\). Because the block is in a velocity field, the ends of the block move at different speeds.

After a time \(\Delta t\), the left side of the block has moved from \(x=0\) to \(x = u\Delta t\).

The right side of the block moves at a velocity of \(u + L\frac{\partial u}{\partial x}\). To understand this, consider that the spatial gradient inthe velocity (\(\partial u/\partial x\)) expresses how much \(u\) increases for every meter you shift in the \(x\) direction. So the difference in the velocity between the left of the block and the right is simply this gradient times the length of the block \(L\). This approach is valid as long as we consider the distance \(L\) small enough that \(u\) varies linearly.

In \(\Delta t\) the right side of the block has moved \(\Delta t(u + L\frac{\partial u}{\partial x})\). It started at \(x = L\), so its new position is \(L + \Delta t(u + L\frac{\partial u}{\partial x})\)

Now, if the left side is at \(x = u\Delta t\) and the right side of the block is at \(x = L+ \Delta t(u + L\frac{\partial u}{\partial x})\), the new length of the block is $\( L + \Delta t(u + L\frac{\partial u}{\partial x}) - \Delta t u = L + \Delta t L\frac{\partial u}{\partial x} \)$

and the change in length is

\[ \Delta L = L + \Delta t L\frac{\partial u}{\partial x} - L = \Delta t L\frac{\partial u}{\partial x}. \]

Rearranging shows that the strain in a time \(\Delta t\) is

\[ \epsilon_{xx} = \Delta t \frac{\partial u}{\partial x}. \]

Dividing through by the time gives us the relationship between the strain rate and gradient in velocity,

\[ \dot{\epsilon}_{xx} = \frac{\partial u}{\partial x}. \]

A general relationship

Doing the same exercise for shear strain shows that a general equation for this relationship is

\[ \dot{\epsilon}_{ij} = \frac{1}{2}\left(\frac{\partial \underline{u}_i}{\partial \underline{x}_j} + \frac{\partial \underline{u}_j}{\partial \underline{x}_i}\right), \]

where \(\underline{u}\) is velocity field \((u, v, w)\) components and \(\underline{x}\) are the three coordinates \(x\), \(y\), and \(z\).