3. Significance of [ (∇ u + ∇ uT)]

Let us write the matrix form of (∇u + ∇uT) in (e1, e2, e3) coordinate system, i.e.,

(25)

In this section, we are looking at the mathematical representation of the expression (∇u + ∇uT) in a specific coordinate system. The notation e1, e2, and e3 represents the standard basis vectors in a three-dimensional space. This matrix form is essential for understanding how the tensor behaves under various transformations. Since (∇u + ∇uT) is an asymmetric tensor, its matrix form is symmetric, meaning that the elements of the matrix are mirrored across the main diagonal.

Imagine a rubber sheet marked with a grid. When the sheet is stretched, the grid changes shape, but the symmetry of the grid's layout remains intact. Similarly, the symmetry in (∇u + ∇uT) reflects how changes in shape do not break certain underlying relations, just as the grid maintains its pattern despite being deformed.

We can see upon comparing from equation (8) that the diagonal elements of the above matrix give us longitudinal strains along e1, e2, and e3 directions.

This chunk explains the relationship between the diagonal elements of the matrix form of (∇u + ∇uT) and the longitudinal strains in the material. Longitudinal strains are the changes in length along a particular direction due to external forces. This means that if we analyze the diagonal entries of the matrix, they directly correspond to how much the material stretches or compresses along the x, y, and z axes, which is crucial for understanding material behavior under stress.

Think of pulling a rubber band in a straight line. The amount it stretches along that line represents the longitudinal strain in that direction. If you were to graph the rubber band's stretch, the values you record at various points would form a diagonal line, representing the diagonal entries of the strain matrix in this context.

To understand the significance of the off-diagonal elements, let us choose the two line element directions n and m in equation (24) to be e1 and e2 respectively. We denote the shear strain for this case as α because of the directions chosen which equals (26)

This part discusses the off-diagonal elements of the matrix, which relates to shear strain. Off-diagonal elements represent the interactions between different axes in the strain tensor. By selecting specific directions (like e1 and e2), we can calculate shear strain, which is the change in angle between these directions due to deformation. The shear strain is crucial for understanding how materials twist or distort under load, rather than just stretching.

Imagine a deck of cards. When you slide the top half of the deck sideways while keeping the bottom half in place, the angle between the cards changes. This change in angle is similar to shear strain, and it is reflected in the off-diagonal elements of our strain matrix.

When worked out using the matrix form, it turns out to be twice of the off-diagonal term in the second row and first column of (25), i.e.,

(27)

We can thus conclude that in matrix (25), the off-diagonal elements represent half the shear strains.

Here, we conclude that the off-diagonal elements of the strain matrix provide an essential relationship to shear strain measurements. By deriving the relationship to be twice the off-diagonal elements, we illustrate how these values encapsulate significant information about the material's response to shearing forces. Thus, these elements have direct applications in engineering and materials science, enabling accurate predictions of material behavior during deformation.

Consider how a classroom can be arranged. If the teacher moves some desks around, the angle between certain rows may change. In this analogy, the rearranged desks represent the material undergoing shear strain. The amount of rearrangement corresponds to the values represented in the off-diagonal elements of the strain tensor—essentially detailing 'how much' the classroom layout has changed due to 'forces' applied to the desks.

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 ∆X and ∆X respectively. After deformation, the point X as well as the line vectors will shift to new positions as shown in the figure. 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 αb and the angles shown in green as αg. Thus

α = αb + αg (28)

In this section, we visualize shear strain geometrically. By representing the original and deformed states of a body, we can see how the angles between line elements change. This sum of angle changes (αb and αg) provides a holistic view of shear strain at the specified point, showing how physical deformation can be tracked and quantified. Understanding shear strain through geometrical interpretation aids in visual learning and real-world applications.

Imagine folding a sheet of paper. When you fold it, the angle between the two halves changes. Just as you can measure how much the paper has deviated from its original flat state by the angles created during the fold, we measure shear strain through the angle changes represented geometrically in this body model. This tactile experience of bending the paper parallels our theoretical understanding of shear strain.

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