2.1. Formulation

Our goal is to find a mathematical expression for shear strain just as we found one for longitudinal strain. Consider a body before and after deformation as shown in Figure 2. At the point of interest X in the reference configuration, we identify two perpendicular line elements ∆X and ∆Y. After deformation of the body, ∆X becomes ∆x and ∆Y becomes ∆y. Let the unit vectors along line elements ∆X and ∆Y be denoted by n and m respectively.

In this section, we begin by establishing the goal of finding a mathematical expression for shear strain, similar to what we previously derived for longitudinal strain. We start by examining a deformation scenario where two initially perpendicular line elements (denoted as ∆X and ∆Y) change to new orientations post-deformation. The important aspect here is that after deformation, these line elements are no longer perpendicular, and their change will help us calculate the shear strain. The vectors n and m represent the directions of these line elements.

Imagine a rubber band stretched in two directions at right angles. Before stretching, they are at 90 degrees to each other, but when you twist the rubber band, that angle changes. This concept helps in visualizing how the original perpendicular line elements become distorted under stress, similar to how the rubber band’s sections become slanted.

In general, the two line elements undergo stretching, and the angle between them also changes. We have already derived the expression for deformed line element in terms of the deformation gradient tensor (F) according to which The angle between these two line elements in deformed configuration (denoted by β) will be given by the following relationship.

Here, we delve into the mathematical derivation of shear strain. After the line elements have undergone stretching and rotation, we introduce the deformation gradient tensor, which relates to how these elements transform during deformation. The angle β is introduced to show how this newly transformed angle differs from its original, indicating the shift caused by deformation.

Think of a paper plane that you fold. Initially, the wings are at right angles to each other. When you twist the wing slightly while flying, the angle between them changes, which is akin to the change in angle of line elements under shear strain. This change can be mathematically represented, showing precisely how much the wings have altered their angle.

Similar to the case of longitudinal strain, we keep the higher order terms because we don’t know if they are significant or not. Let us substitute equation (13) in the numerator above and express the terms in denominator using stretch.

This section emphasizes the importance of retaining higher order terms in our mathematical expressions for shear strain. Unlike before, where we neglected small terms, here we carefully consider whether they could influence the results. As we substitute derived equations, we will use the relationship between different terms while expressing everything concerning stretch to ensure all relevant factors are included in our calculations.

Imagine you are measuring the height of a tall building with a tape measure. If you were to overlook small bumps or dips along the way, your measurement could be off. Similarly, in our calculations, ignoring higher order terms might lead to inaccuracies; we must account for all details to ensure precision, just as you would walk smoothly up to the building to get an accurate height.

As m and n are perpendicular, their dot product will be zero. Also, as longitudinal strains are much less than 1, the terms (1-ϵ ) and (1-ϵ ) can be approximated to be 1. Thus, we finally have using basic trigonometry and equation (11), the shear strain is given by the change in angle between the two elements.

Finally, we arrive at the simplified expression for shear strain. By recognizing that the vectors n and m are perpendicular, we can simplify calculation by understanding that their mutual relationship yields specific results. Consequently, we confirm that shear strain can be defined through basic trigonometry by incorporating these angles into our final formula. The understanding gained here is critical for applying these concepts in complex situations.

Consider a famous example - the act of stretching a piece of dough. While pulling the dough, you can visibly see that the angles between the lengths change, indicating shear strain. The simple yet effective representation of shear strain allows bakers to understand how their kneading affects the dough's structure in many practical scenarios, much like we do in our calculations.

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