1.8. Extensional and Shear Strain Rates

The dilatation or extensional strain is defined as the ratio of the rate of the change in length to the original length. Along the x direction, the original length was dx, and the increased length at time t + dt is given by $\frac{\partial u}{\partial x} \cdot dx \cdot dt$. Hence, the change in length is $\frac{\partial u}{\partial x} \cdot dx \cdot dt$. This implies the rate of linear strain is given by $\frac{\partial u}{\partial x}$.

Extensional strain measures how much an object stretches or compresses. It is defined as the change in length per unit of original length. In this case, if you have a segment of material that has an original length (dx), and when you measure it later, it has changed due to forces acting on it. The rate of change in the length divided by the original length gives you a measure of how much the material is experiencing extensional strain. The specific formula indicates how the displacement (u) varies with respect to position (x).

Imagine a rubber band. When you pull on it, it stretches. The original length of the rubber band represents dx. The change in its length after you pull on it can be thought of as the increase in length, given the formula in the chunk. The further you stretch the band, the greater the extensional strain; we can relate that back to the friction or drag caused by whatever is pulling on the band.

Similarly, along y and z direction, $\epsilon_{yy}$ can be written as $\frac{\partial v}{\partial y}$ and $\epsilon_{zz}$ can be written as $\frac{\partial w}{\partial z}$.

Shear strain measures how much a shape deforms due to forces that cause it to slide past itself. In the y and z directions, we define shear strain components as $\epsilon_{yy}$ and $\epsilon_{zz}$ corresponding to displacements (v and w) based on how they vary with respect to the y and z positions. This means that just like we measure extensional strain in one dimension, we can also measure how much the material is twisting or shearing in other dimensions.

Think of a deck of cards. If you hold one end of the deck stationary and push the other end, the cards will slide over each other - this is similar to shear strain. The amount that one card slides on top of another is akin to how we define $\epsilon_{yy}$ and $\epsilon_{zz}$ as the shear strain values.

The extensional and shear strain rates, taken as together, form a second-order symmetric tensor $\epsilon_{ij}$ given by; so, $\epsilon_{ij}$ captures both extensional and shear components.

In fluid mechanics, strain rates reflect how the velocity of different layers of the fluid change relative to each other. When combined, extensional strains (which stretch) and shear strains (which twist) can create a detailed picture of material behavior as a tensor. A tensor is a mathematical object that generalizes the concept of scalars and vectors to higher dimensions, allowing us to represent multiple strain components simultaneously in a compact form.

Consider a loaf of bread being squashed. When you press down on the loaf, the top layer is compressed while the underlying layers are pushed sideways - both extensional (compression) and shear (sliding past) are happening at once. A tensor helps summarize the effects of these different deformations happening simultaneously, much like how a recipe succinctly outlines many steps of baking bread.

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