1. Longitudinal Strain (contd.)

We finally need to reduce the length of the line element in the reference configuration to zero in order to obtain the local value of stretch, i.e., we have to take the limit of ||∆X|| → 0. The O(||∆X||) then vanishes.

In this step, we are focusing on determining the local stretch of a material. To find the stretch accurately, we consider a very small segment of the material, denoted as ∆X. By taking the limit as the length of this segment approaches zero, represented mathematically as ||∆X|| → 0, we can eliminate higher-order infinitesimals from our calculations, resulting in a more precise understanding of the stretch at a particular point.

Think of trying to measure the stretch of a rubber band. If you pull at one end, you might look at a very tiny segment of the band. By narrowing your focus down to an infinitely small length, you can better understand the true nature of the stretch occurring in that spot.

We can simplify this even further. In this course, we will only be working with displacements such that their gradients are very small.

We simplify our expressions by assuming that the displacements, which indicate how much a point in a material moves, have very small gradients. This means that changes in displacement are minor, allowing us to ignore certain higher-order terms in our calculations, thereby simplifying the mathematical model we are working with.

Consider a person walking very slowly on a straight path. The small changes in their position can be neglected when discussing larger movements (like jumping). This allows us to ignore minor fluctuations and focus on the overall distance traveled.

The term ∇uT∇u in (2), when written in matrix form, has two matrices multiplied and each of them contains derivatives of u. So, the product matrix will have components which are quadratic combinations of displacement gradients. As the displacement gradients themselves are very small, their quadratic combinations will be even smaller.

When we express the mathematical relationships in matrix form, we see that certain terms (like ∇uT∇u) end up representing combinations of displacement derivatives. As we established earlier, these derivatives are quite small, meaning the combinations of these small values will result in even smaller values, which can be considered negligible in our calculations.

Imagine trying to measure the tiny deviations in a tree's growth over time. If the tree grows very slightly each year, any multiple low values representing those changes will yield even smaller differences. This allows you to say that the tree's growth was consistent without needing to measure every tiny change precisely.

But, in the definition for strain, it will turn out to be the most significant term (see equation (6) below). The idea is that we keep the most significant term while neglecting the other less significant terms.

In the context of strain, we acknowledge that while some terms can be disregarded because they yield negligible values, one specific term remains important and must be retained for accurate analysis. This process of distinguishing between significant and insignificant terms allows us to simplify our equations without losing critical information.

Consider budgeting your expenses. If you have many minor costs (like small daily coffee purchases), you can often ignore those in your big-picture budget planning. However, substantial expenses (like rent) cannot be overlooked, and they must be accurately accounted for to maintain financial stability.

Having obtained longitudinal stretch, the longitudinal strain then becomes. ... Here again, we have dropped quadratic and higher order terms from binomial expansion for the same reason as earlier.

After calculating the longitudinal stretch, we derive an expression for longitudinal strain using established mathematical principles like binomial expansion. In this context, we again disregard smaller order terms that do not significantly affect the result, focusing on the primary contributing factors to ensure our calculations reflect meaningful outcomes.

Think about boiling water to make pasta. When adding salt, a pinch might seem negligible, but the right amount significantly enhances the flavor. You focus on adding a sufficient quantity rather than minuscule adjustments that would have little impact.

If the line element direction (n) is taken to be e1, the matrix form of (6) in (e1,e2,e3) coordinate system will be similar analysis can be done for n=e2,...

Here, we examine how the longitudinal strain behaves when measured along different coordinate axes. By changing the direction of our line element, we can apply similar mathematical formulations to derive results that adhere to the same principles we previously established. This flexibility is crucial for analyzing strain in three-dimensional systems.

Think of using a ruler to measure the height of people standing in different orientations. Just like you can measure their height regardless of facing north or south, you can analyze strain in any direction by simply adjusting your reference frame.

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