3.2.1. An example

Let us consider an example where our coordinate system is (e₁, e₂, e₃) and the displacement components are given as follows:

u₁ = u₁(X₁, X₂), u₂ = u₂(X₁, X₂), u₃ = 0.

In this example, we focus on a coordinate system defined by three axes: e₁, e₂, and e₃. The displacement components represent how points in the material move from their original positions under deformation. Here, u₁ and u₂ depend only on the first two coordinates (X₁ and X₂), indicating that the movement in the x₃ direction is zero, or non-existent. This scenario is particularly typical in plane strain conditions where we consider motion in a two-dimensional plane (the e₁-e₂ plane), ignoring any movement along the e₃ axis.

Think of a thin sheet of paper being crumpled. The movement (displacement) occurs primarily in two dimensions on the surface of the paper (like e₁ and e₂), while the height (the third dimension, e₃) remains unchanged. This simplification makes it easier to analyze the strains and rotations occurring in the plane of the paper.

We have assumed u₁ and u₂ components to be independent of the third coordinate and u₃ is assumed to be zero. So, the strain matrix in this case according to (18) will be:

$\epsilon = \begin{pmatrix} \epsilon_{11} & \epsilon_{12} \\ \epsilon_{12} & \epsilon_{22} \end{pmatrix}$

Given the plane strain condition, the strain matrix becomes a 2x2 matrix that only contains components related to the first two dimensions (e₁ and e₂). Here, $\epsilon_{11}$ and $\epsilon_{22}$ are normal strains, while $\epsilon_{12}$ represents shear strain. This formulation simplifies calculations by focusing only on the relevant directions involved in deformation, allowing us to neglect the e₃ component which does not contribute to strain changes within the plane.

Imagine stretching a rubber band horizontally. The strain caused by this stretching can be captured in a two-dimensional matrix, where one side is the length of the rubber band and the other side is the width of it as it stretches. The width may change due to the tension, and this is represented by the shear strain in our matrix.

We can then obtain the axial vector corresponding to local rotation using equation (30) to be:

$w = \begin{pmatrix} w_{1} \\ w_{2} \\ 0 \end{pmatrix}$

This axial vector $w$ represents the direction and magnitude of any local rotation that occurs due to the deformation. Since we are in a plane strain state, the third component is zero, indicating rotation is confined to the e₁-e₂ plane. The components $w_{1}$ and $w_{2}$ essentially help us quantify how much and in which direction the material elements in this plane rotate as a result of the underlying deformation.

Think about trying to twist a paper towel roll while holding it at its ends. As you twist, the roll rotates about its axis. The axial vector gives us a way to describe how much and which direction the towel roll has rotated, just like how the local rotation of the materials in the e₁-e₂ plane can be described using this vector.

Let us now see how the above strain and rotation matrices act on the system. As the displacement is confined in the e₁ - e₂ plane, we take two line elements in this plane along e₁ and e₂ directions with magnitudes ∆X₁ and ∆X₂ respectively.

In analyzing deformation, we consider how two small line segments—along the e₁ and e₂ axes—change in length and orientation after applying strains and rotations. First, we observe the effects of strain on these line segments, which indicates how they deform under the applied stress. Next, we assess the impact of local rigid rotation, which describes how these segments additionally rotate due to the material's response to displacement.

Imagine holding two flexible straws right next to each other. When you pull on one end, the straws can elongate (strain) and also rotate slightly due to the pull. The initial stretching changes their lengths and their angles relative to each other. This illustrates how both strain causes deformation and the dynamics of rigid rotation come into play simultaneously.

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