2. An alternate physical meaning of shear strain

We know already that shearstrain denotes change in angle between two perpendicular line elements. There is another physical interpretation of shear strain. Let us consider the following displacement function:

u = αX , u = 0, u = 0

and understand its underlying deformation. After deformation, the position vector of a typical point in the body changes as follows:

x = X + αX , x = X , x = X .

Shear strain refers to how much something twists or deforms. It can also be thought of as the change in angle between two lines that were originally at a 90-degree angle. To illustrate an alternate way of understanding shear strain, we can look at a specific mathematical function which shows how a point's position changes as deformation occurs. The expression provided describes how the displacement (the movement of a point from its original position) depends on a parameter (α) multiplied by the original position (X). This means as α changes, it affects how much the point moves or shifts.

Imagine holding a rectangular piece of paper flat on a table. If you push one side of the paper while keeping the opposite side fixed, the paper will no longer be a rectangle but rather will become a parallelogram. This process of pushing distorts the shape, similar to how shear strain describes the change in angles of line elements in materials.

Think of a rectangular slice in the reference configuration of the body in e − e plane. This slice deforms to a parallelogram according to (6) as shown in the figure inducing change in angle between its two perpendicular edges.

To visualize how shear strain works, consider a rectangular piece (like a slice of bread) lying flat. When one part of the slice is pushed while the other is held still, the shape turns into a parallelogram. The angle between the edges changes as a result of this push, much like how the displacement function calculated before gives an idea of how much the points have shifted due to strain.

Think about playing with a deck of cards. If you hold one side of the deck and push the other side, the cards slide against each other without bending. This sliding motion resembles shear strain, where layers move parallel to each other, changing the angle between the cards (or the slices) slightly.

Therefore, the displacement prescribed by equation (6) is also called shear displacement. To measure the amount of shear, we can directly notice from Figure 4 that (for small α).

The amount of shear displacement can be quantified using the deformation function we've introduced. In simple terms, 'shear displacement' describes how far one side of an object has moved in relation to the other side due to an applied force. For small values of α, this change can be viewed and measured as linear, allowing for easier calculations.

If you are sliding a stack of papers on your desk, the top sheet moves while the sheets below remain in place. The distance the top sheet slides over the stack is like shear displacement. The greater the push (or value of α), the further the top sheet goes, similar to our earlier calculations.

Alternatively, using the formula for shear, we can also see. Let us now imagine a plane section of the body in (e − e ) plane. These planes have constant X coordinate. According to (6), all points in this plane displace by the same amount in e direction.

In a different view of shear strain, consider a flat plane within the rectangular slice. The deformation along the e-axis (for shear displacement) indicates that every point on that plane moves uniformly, as if the entire plane is simply shifting without changing shape. This rigid translation helps visualize how layers within a material might shift under strain.

Imagine a stack of sticky notes. If you push the top note sideways, every note below it moves the same distance horizontally, just shifting as a group. This is analogous to the rigid translation under shear strain, where the plane shifts without deformation, highlighting the shear effect.

This is an alternate physical interpretation of shearing: sliding of parallel planes in a direction perpendicular to the plane normal. Here, sliding is along e direction which is perpendicular to e (the plane normal). We can also let the planes slide in an arbitrary direction perpendicular to e and that will also be shear.

We can understand shear strain as layers (or planes) of materials sliding over each other. The motion occurs in a direction that's perpendicular to the normal of the plane. This sliding can happen in various orientations, just as long as it remains along a line that's at 90 degrees to the 'normal' direction of each plane.

Think of ice skaters on ice. When skaters push off each other laterally, they glide parallel to the ice surface without the surface itself lifting. This sliding closely resembles how planes might move against each other under shear stress, creating strain in the material structure.

The angle β in Figure 4 and 5 is the measure of intensity of sliding of parallel planes which also equals the shear strain value.

The intensity of the sliding motion between parallel planes can be quantified by measuring the angle (β) created due to the shift. Notably, the larger this angle, the greater the shear strain. This angle directly connects the spatial orientation of planes with the shear strain experienced by the material.

Imagine turning a deck of cards where the top card shifts more dramatically at its edges. The angle created at the edge where the cards slide past each other represents how significantly the shear strain is present. Larger angles indicate more deformation, akin to measuring how much a material has been strained.

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