Problem 17 - An Introduction to the Mechanics of Solids in SI Units

Question

In a case of plane strain in which each point displaces radially in a rotationally symmetric fashion about the origin \(O\), the displacement can be expressed by a single displacement component \(u\) in the radial direction. Show that the strain components referred to the radial, tangential \((r, \theta)\) set of axes are $$ \epsilon_{r}=\frac{d u}{d r} \quad \epsilon_{\theta}=\frac{u}{r} \quad \gamma_{r_{0}}=0 $$

Step-by-Step Solution

Answer

The strain components in the radial, tangential (\(r\), \(\theta\)) set of axes are \(\epsilon_{r} = \frac{d u}{d r}\), \(\epsilon_{\theta} = \frac{u}{r}\) and \(\gamma_{r_{0}} = 0\)

Step 1: Define strain components

Strain is defined as the change in length per unit length. Thus, for a radial displacement \(u\), \(\epsilon_{r}\) is the change in length in the radial direction per unit length which is equivalent to \(\frac{d u}{d r}\). Similarly, \(\epsilon_{\theta}\) is the change in length in the tangential or circular direction per unit length. Since the displacement is radial, \(u\) is the change in length and \(r\) is the original length along the tangential direction or circle circumference, thus \(\epsilon_{\theta} = \frac{u}{r}\). \(\gamma\) characterizes the change in angle between two lines originally perpendicular. In this scenario, the displacement \(u\) is radial, meaning every point is moving in a line that is always perpendicular to the circle circumference, hence \(\gamma_{r_{0}}=0\)

Step 2: Write the formulas

Insert the formulas for the strain components: \(\epsilon_{r} = \frac{d u}{d r}\), \(\epsilon_{\theta} = \frac{u}{r}\), and \(\gamma_{r_{0}}=0\)

Step 3: Confirm they satisfy the problem statement

Check these strain components satisfy the description of the problem which is each point displaces radially in a rotationally symmetric fashion. As described, due to radial displacement, \(\frac{d u}{d r}\) accounts for radial stretch, \(\frac{u}{r}\) accounts for tangential shrink, and the angle between radial and tangential direction does not change which justifies \(\gamma_{r_{0}}=0\).

Key Concepts

Strain Components

Understanding strain components is crucial when dealing with materials under load. In the context of plane strain, where deformation is restricted to a single plane, we look at how a material deforms in response to an applied force in a two-dimensional perspective. The strain components consist of the normal strains, \(\epsilon_{r}\) and \(\epsilon_{\theta}\), and the shear strain, \(\gamma_{r_{0}}\). The radial strain, \(\epsilon_{r}\), represents the change in length in the radial direction per unit length. This is associated with how much the material stretches or compresses along a line emanating from the origin outward. The mathematical expression to describe this is \(\epsilon_{r} = \frac{d u}{d r}\), where \(u\) is the radial displacement and \(r\) stands for the radial distance from the origin. The tangential strain, \(\epsilon_{\theta}\), on the other hand, refers to the change in length in the circumferential direction, which is equivalent to how much the material stretches or compresses around the circle. It is defined by the ratio \(\epsilon_{\theta} = \frac{u}{r}\), signifying that the displacement \(u\) around a circular path is normalized by the radial distance \(r\).Lastly, the shear strain \(\gamma_{r_{0}}\) characterizes the deformation in shape without changing the area. However, under plane strain conditions with rotationally symmetric deformation, this component is zero, \(\gamma_{r_{0}} = 0\), indicating no angular deformation occurs between elements originally perpendicular to one another.

Radial Displacement

Radial displacement is a term used to describe the movement of points in a body towards or away from a central point or axis. In the context of our exercise, all points in the material are moving radially with reference to the origin, which means their motion can be described with respect to a single axis, the radius \(r\).The radial displacement \(u\) is the distance a point moves along this radius. The displacement is considered positive when the point moves away from the origin, indicating tension, and negative when it moves towards the origin, indicating compression. To establish the relationship between radial displacement and radial strain, we derive the former with respect to \(r\), resulting in \(\epsilon_{r} = \frac{d u}{d r}\). This represents how the displacement varies along the radius - whether the material is uniformly stretching or compressing, or if there's a gradient to this deformation.

Rotationally Symmetric Deformation

Rotationally symmetric deformation refers to the movement of a body where points at the same distance from the center of rotation undergo the same displacement. This kind of symmetry is also known as axisymmetric deformation, where the displacement does not depend on the angle around the axis.In our example, each point of the material displaces radially, which means points along a circular path centered at the origin move the same amount. This uniformity simplifies analyses and calculations, as the behavior in one radial direction can be considered to represent all radial paths. The significance of \(\epsilon_{\theta} = \frac{u}{r}\) becomes evident here, as this formula confirms that for a given radius, the tangential strain is consistent regardless of the angle around the center.Rotationally symmetric deformation plays a key role in the analysis of cylindrical structures and components, such as pipes and pressure vessels, where the load is often symmetrically distributed around an axis. Understanding this principle helps in predicting the behavior and failure of materials under such conditions.