3.1. Geometric interpretation of shear strain formula

Let us visualize the above expression for shear strain. Consider the body shown in Figure 3 and try to obtain shear strain at point X. We choose the line elements along e₁ and e₂ in the reference configuration having their lengths ∆X₁ and ∆X₂ respectively. After deformation, the point X as well as the line vectors will shift to new positions as shown in the figure.

In this part of the explanation, we are invited to visualize how shear strain can be calculated at a specific point in a material. Here, we focus on point X and consider two line elements e₁ and e₂ originating from it. Initially, these lines are at specific lengths (∆X₁ and ∆X₂). As the body deforms, these line elements change orientation and position, leading to the observation of the shear strain. This configuration is crucial in understanding how materials behave when subjected to forces that cause distortion rather than uniform stretching.

Imagine a book being pushed sideways while it is leaning against a wall. The corners of the cover represent the line elements. Initially, the corners are at 90 degrees relative to each other. As you push the book, you notice that those angles change. The amount the angles deviate from the original 90 degrees illustrates the shear strain experienced by the book cover due to the applied force.

We have also drawn horizontal and vertical axes (shown by dotted lines) at the deformed position x. These lines will help us to evaluate the angle change. The total change in angle of the two line elements will be the sum of the angles shown in blue and green. We call the angles shown in blue as αₐ and the angles shown in green as αᵍ. Thus α = αₐ + αᵍ.

In this part, horizontal and vertical axes are added to the visualization to better analyze how the angles between the line elements change through deformation. The total change in angle between the two line elements is represented as the sum of two smaller angles, αₐ (blue) and αᵍ (green), formed due to the deformation. This geometric interpretation allows us to quantify how much the original configuration is distorted by representing the resultant shear strain as a combination of individual angular changes.

Consider a square piece of paper that is pushed diagonally from one corner. The angles at two adjacent corners of the square will shift from 90 degrees to something less than that. The angle you initially had divided into two parts before the deformation can help visualize how distorted the original shape has become due to the push, yielding a clear calculation for the shear strain.

Let’s find αₐ first. Consider the right-angled triangle containing angle αₐ. The red distance there is the difference in y-displacement of the tip and vertex of the initially horizontal line element. The length of the base of this right-angled triangle will be approximately ∆X₁ since the longitudinal strain is much smaller than 1. Thus, the angle αₐ will be given by (29). Finally, to find this angle at point X itself, ∆X₁ should tend to zero.

To calculate αₐ, we use a right triangle formed by the original configuration and its deformation. The length of the base of the triangle corresponds to the original line element ∆X₁. Since the change in angle, represented as αₐ, is related to the vertical displacement difference of the tip of the line element, we can set it up using the small-angle approximations and triangle properties. By letting ∆X₁ approach zero, we can analyze the limit case more precisely, which then gives us the actual expression for αₐ.

Think of a tightrope walker. As the tightrope wiggles slightly, the walker adjusts angles with tiny movements. When examining those angles from the perspective of small deviations, you'll see that even a minor shift at the rope can cause visible changes at the wearer's position, similar to how minute variations in angles lead to the overall shear strain of materials.

Using equation (28), we then get α = αₐ + αᵍ. Thus we have realized the physical significance of this expression for shear strain. It is the sum of the change in angles of the two line elements from their initial orientation.

At this point, we summarize that the total shear strain at point X is represented as the composite angle change from the original configuration to the new configuration after deformation. This total angle change gives a clear indication of how the material has responded to applied forces, illustrating the nature of shear strain in a geometric context.

Consider an artist stretching a canvas on a frame. As she pulls the canvas, it alters the angles formed at the corners of the frame. The total change in these angles from their original state directly represents how much the canvas has been pulled and deformed, akin to how we assess shear strain based on angle distortions in engineering.

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