1.7. Dilatation or Extensional Strain

The dilatation or extensional strain is defined as the ratio of the rate of the change in length to the original length.

Dilatation or extensional strain measures how much a material stretches or shrinks compared to its original size. This is represented as a ratio, showing the change in size as a proportion of the original size. For instance, if a rod of length 'L' stretches to 'L + ΔL', the change in length (ΔL) is compared to the original length (L). The formula is:

$\text{Extensional Strain} = \frac{\Delta L}{L}$

This concept is similar to how students often calculate grades in percentages, where the change in their scores is compared to the total possible scores.

Imagine stretching a rubber band. If you pull it and it goes from 5 cm to 7 cm, the change in length is 2 cm. The original length was 5 cm, so the extensional strain would be 2 cm / 5 cm = 0.4 or 40%. This analogy helps visualize how materials stretch over a baseline measurement.

Along the x direction, the original length was dx. The increased length at time t + dt is represented as del u del x into dx dt.

When considering the changes in length in relation to time, we can say that the original length in the x direction is 'dx'. After a small interval of time 'dt', the new length changes based on how much the speed or velocity (u) affects it. Hence, we use partial derivatives to describe these changes. The expression del u / del x (represented by the symbol ‘∂u/∂x’) helps us understand how velocity varies with position, allowing us to determine how much the length has changed during that short time frame.

Think of a car moving forward. If you denote the initial distance from a stop sign to a point as 'dx', and the car travels at a velocity that changes over time, then after a few seconds (dt), the car will have moved forward a certain distance depending on how fast it's going. This analogy illustrates the relationship between velocity and change in length over time.

Similarly, along y and z direction, epsilon yy can be written as del v by del y and epsilon zz can be written as del w by del z.

Extensional strain is not limited to one direction; it applies to multiple dimensions. In the y direction, the extensional strain is calculated by taking the derivative of velocity (v) with respect to position (y), while in the z direction, it involves the derivative of another velocity (w) concerning position (z). Thus, we represent these strains as epsilon yy (for y direction) and epsilon zz (for z direction), which reflect how much the material changes in respect to its original dimensions in those directions.

Imagine a balloon being inflated. As you blow air into it, each part of the balloon (length, width, height) expands. The way the balloon stretches in different directions mimics the idea of extensional strain across y and z directions. This illustrates how materials can deform in multiple dimensions simultaneously.

The extensional and shear strain rates taken together form a second-order symmetric tensor epsilon ij given by epsilon ij.

In fluid mechanics, to comprehensively analyze how a fluid deforms under stress, we use mathematical structures called tensors. Specifically, a second-order symmetric tensor captures how both extensional and shear strains interact. For example, epsilon ij combines different strain rates like epsilon xx, epsilon yy, and epsilon zz, which summarize the changes across different dimensions. This tensor is important for engineers and scientists as it helps predict how materials respond to forces.

Think of a soccer ball in a game. As players kick and press the ball, it deforms. Each direction the ball morphs can be compared to different components of our tensor. Just as each indentation or stretch contributes to the overall shape of the ball, each element in the tensor contributes to understanding the material's behavior under stress.

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