Problem 18 - An Introduction to the Mechanics of Solids in SI Units

Question

A gcneral defomation in plane strain can be described in polar coordinates by expressing the displacement of each point as the vector sum of a radial component \(u\) and a tangential component \(v\). Show that in such a case the strain components referred to the \(r, \theta\) set of axes are $$ \begin{aligned} &\epsilon_{r}=\frac{\partial u}{\partial r} \\ &\epsilon_{\theta}=\frac{1}{r} \frac{\partial v}{\partial \theta}+\frac{u}{r} \\\ &\gamma_{r_{0}}=\frac{\partial v}{\partial r}+\frac{1}{r} \frac{\partial u}{\partial \theta}-\frac{v}{r} \end{aligned} $$

Step-by-Step Solution

Answer

The strain components \(\epsilon_{r}\), \(\epsilon_{\theta}\), and \(\gamma_{r0}\) in polar coordinates can be expressed with radial and tangential components of a deformation. They are represented as: \(\epsilon_{r}=\frac{\partial u}{\partial r}\), \(\epsilon_{\theta}=\frac{1}{r} \frac{\partial v}{\partial \theta}+\frac{u}{r}\), and \(\gamma_{r_{0}}=\frac{\partial v}{\partial r}+\frac{1}{r} \frac{\partial u}{\partial \theta}-\frac{v}{r}\) respectively. These expressions are derived from the definitions of each strain component - radial, tangential, and shear strain - in relation to radial and tangential displacements.

Step 1: Define displacement vector

Firstly, understand that the displacement of each point in the plane can be represented in polar coordinates as a vector sum of a radial component \(u\) and a tangential component \(v\). This is expressed as \(\vec{r} = u\hat{r} + v\hat{\theta}\).

Step 2: Define strain components

Then, define the strain components \(\epsilon_{r}\), \(\epsilon_{\theta}\), and \(\gamma_{r_{0}}\) in terms of \(u\) and \(v\). The radial strain \(\epsilon_{r}\) is the change in length per unit length in the radial direction, the circumferential or tangential strain \(\epsilon_{\theta}\) is the change in length per unit length in the tangential or circumferential direction, and the shear strain \(\gamma_{r_{0}}\) represents the change in angle.

Step 3: Show the expression for \(\epsilon_{r}\)

Using the definition of radial strain, we can derive its expression: \(\epsilon_{r}=\frac{\partial u}{\partial r}\). This is implying that radial strain is the rate of change of the radial displacement \(u\) with respect to the radial distance \(r\).

Step 4: Show the expression for \(\epsilon_{\theta}\)

The tangential strain \(\epsilon_{\theta}\) can be expressed as: \(\epsilon_{\theta}=\frac{1}{r} \frac{\partial v}{\partial \theta}+\frac{u}{r}\). This expression is narrower than \(\epsilon_{r}\), reflecting the fact that radial displacement contributes to the tangential strain.

Step 5: Show the expression for \(\gamma_{r_{0}}\)

Finally, the shear strain \(\gamma_{r_{0}}\) is given by the equation: \(\gamma_{r_{0}}=\frac{\partial v}{\partial r}+\frac{1}{r} \frac{\partial u}{\partial \theta}-\frac{v}{r}\). This equation reflects how the shear strain accounts for changes in both radial and tangential displacements, as well as the inherent relationship between these two variables due to the circular (polar) coordinate system.

Key Concepts

Plane Strain

In mechanical engineering, plane strain is a type of deformation that occurs in structures when the strain across one of its dimensions is negligible. It's a simplifying assumption often used for materials that are long in one dimension but are thin in others, like sheets of metal or hollow cylinders. The assumption of plane strain helps engineers analyze problems in two dimensions instead of three, thereby making complex calculations more manageable.

• This condition assumes that axial strain or out-of-plane movement is zero, meaning all the deformation takes place in the remaining two dimensions.
• It is particularly useful for problems where the object is subjected to uniform strain in the direction perpendicular to the length, such as earth dams, tunnels, or long walls.

Plane strain problems are solved using mathematical tools suited for two-dimensional analysis, while polar coordinates often serve as the best choice in problems involving cylindrical symmetry.

Polar Coordinates

Polar coordinates provide a way to describe the position of a point using angles and distances, which can be particularly beneficial in problems involving circular or rotating systems. In this system, any point is expressed in terms of its distance from a reference point known as the pole (similar to the origin in Cartesian coordinates) and the angle from a reference direction, usually the positive x-axis.

• The position is given by two values: the radial distance (r) from the pole and the angular coordinate (θ), which indicates the point's direction relative to the reference axis.
• Polar coordinates are especially advantageous for circular objects or phenomena, as the properties are naturally described as functions of radius and angle, which are more intuitive than Cartesian coordinates.

Polar coordinates allow for a more natural description of problems where symmetry about a point is present, such as circular plates, tires, or pipes.

Strain Components

Strain is a measure of deformation representing the displacement between particles in the material body. In the context of polar coordinates, strain components help quantify how much an object is deformed or the change in size or shape due to external forces. The main strain components in this context are:

• Radial Strain (\(\epsilon_{r}\)): Indicates how much a material element stretches or compresses in the radial direction.
• Tangential or Circumferential Strain (\(\epsilon_{\theta}\)): Describes the deformation along the angular section of the material.
• Shear Strain (\(\gamma_{r_{0}}\)): Represents the change in the angle between two line segments originally perpendicular.

Understanding these components is crucial for analyzing material behavior under load, predicting potential failures, and ensuring stability and safety in designs.

Radial and Tangential Displacement

Radial and tangential displacement are key concepts when using polar coordinates to analyze deformation. In mechanical engineering, these displacements can provide insights into how an object or material might behave under certain conditions.
Radial Displacement (\(u\))
- Radial displacement refers to how much a point moves in the direction pointing away from or towards the center of the system, often resulting from forces applied outward or inward.
- This type of displacement is crucial when assessing structures like pressure vessels or pipes, where forces commonly act in a radial manner.
Tangential Displacement (\(v\))
- Tangential displacement, on the other hand, measures movement along a circular path around the axis or center.- It's relevant in systems that involve rotational dynamics, such as gears or turbines, where the material has to endure angular displacement.
By considering both radial and tangential displacements, engineers can develop a comprehensive understanding of the object's behavior, optimizing design and safe performance.