A uniform bar of rectangular section is elastically bent into a circular arc by applying a pair of equal and opposite couples \(M_{0}\) to its two ends. By using the fact that each "fiber" under tension (or compression) shrinks (or expands) sideways an amount given by Poisson's ratio \(\sigma\) times the fiber elongation, show that the neutral surface does not remain plane but has a lateral radius of curvature \(R^{\prime}\) given by $$ R^{\prime}=\frac{R}{\sigma} $$ where \(R\) is the longitudinal radius of curvature given as usual by \(R=Y I / M_{0}\). If the bar is of thickness \(t\), then on the top surface the lateral and longitudinal curvatures are \(\rho^{\prime}=R^{\prime}+\frac{1}{2} t\), \(\rho=R-\frac{1}{2} t\), and $$ \sigma=\frac{R}{R^{\prime}}=\frac{\rho+\frac{1}{2} t}{\rho^{\prime}-\frac{1}{2} t} $$ (By polishing the surface of the undistorted bar to optical flatness, it is possible to use interference fringes, made with the help of an optically flat glass plate, to measure \(\rho\) and \(\rho^{\prime}\), from which an accurate value of \(\sigma\) can be computed.)

Step by step solution

01

Define the fibers

Fibers are the elongated or compressed sections of the bar that contribute to its deformation. When a bar is bent, the fibers on the concave side experience compression, whereas the fibers on the convex side are subject to tension. The neutral surface is the surface within the bar that experiences zero strain.

02

Express the fiber elongation and compression in terms of Poisson's ratio

The amount of lateral compression or elongation of fibers is given by \(\sigma\) times the axial elongation. Using Poisson's ratio, we can write the elongation or compression of fibers as: $$ L_{f}=\sigma \times L_{a} $$ where \(L_{f}\) is the lateral elongation or compression, and \(L_{a}\) is the axial elongation or compression.

03

Find the relationship between R and R'

On the neutral surface, we have \(R' = \frac{R}{\sigma}\), which means the fibers at the neutral surface are not inclined but bent during the deformation process. We can now use the information from the compression or elongation of fibers to derive the relationship between \(R'\) and \(R\) as follows: $$ R'=\frac{R}{\sigma} $$

04

Determine the lateral and longitudinal curvatures on the top surface

As given, the lateral and longitudinal curvatures on the top surface can be defined as: $$ \rho^{\prime}=R^{\prime}+\frac{1}{2} t $$ $$ \rho=R-\frac{1}{2} t $$ where \(t\) is the thickness of the bar.

05

Compute the Poisson's ratio using the curvatures

Finally, we can use the expression for Poisson's ratio to find its value from the curvatures we defined earlier. The expression is: $$ \sigma=\frac{R}{R^{\prime}}=\frac{\rho+\frac{1}{2}t}{\rho^{\prime}-\frac{1}{2}t} $$ By following these steps and using the information given, you will be able to show that the neutral surface does not remain plane during the deformation process and derive the expressions for \(R'\), lateral and longitudinal curvatures on the top surface, and Poisson's ratio.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Neutral Surface Deformation

When we bend a bar or a beam, it undergoes something quite fascinating called neutral surface deformation. This concept is at the heart of understanding how materials flex without breaking. Imagine a uniform bar; when it is bent into an arc, the central layer, known as the neutral surface, is the one that does not experience any elongation or compression. It theoretically remains undeformed.

However, in reality, during bending, the neutral surface also deforms due to the material's Poisson's ratio, which describes how much a material contracts or expands in a direction perpendicular to the applied force. We can notice that the neutral surface takes on a curved shape, and this curvature has a particular radius that can be mathematically related to the Poisson's ratio, and this is pivotal for engineers and designers to understand material behavior under bending stresses.

Fiber Elongation in Bending

To get a better grasp of the bending process, let's dive into the concept of fiber elongation in bending. Each 'fiber' in the bar we're discussing can be thought of as a small, parallel strip running along the length of the bar. These fibers deform differently depending on their location: fibers on the outer side elongate while those on the inner side contract.

The amount by which these fibers shrink or stretch sideways is directly tied to Poisson's ratio (\(\sigma\)) and the original length. The resulting elongation is not just an interesting physical phenomenon but is crucial for predicting the materials behavior during bending and avoiding structural failures in practical applications.

Radius of Curvature Explained

The radius of curvature is an essential concept in material bending, and it tells us a lot about the shape and stress distribution within a bent bar. For a bending beam, there's a longitudinal radius of curvature, represented by\(R\), and a lateral radius of curvature,\(R^{prime}\), which corresponds to the deformation on the neutral surface caused by Poisson's ratio. Essentially, the radius of curvature indicates how 'tight' the bend is — the smaller the radius, the sharper the bend.

Understanding the radius of curvature isn't just academic; it's a critical component for safety and functionality in the construction of bridges, automobiles, and even in aerospace engineering. Correctly calculating this radius ensures that structural integrity is maintained under various loads and bends.

Measuring Poisson's Ratio

Measuring Poisson's ratio is a fundamental aspect of material science. It helps us to predict how materials will behave under various loads. This ratio is not just a fixed value; it can change depending on the material and the conditions of the test. To measure it, experts often use the method described in the exercise, involving optical interference fringes.

By polishing the surface of the bar to a flatness comparable to optical wavelengths, using an optically flat glass plate to create a pattern of light and dark 'fringes', professionals can observe deformations at a very small scale. By carefully measuring the lateral and longitudinal curvatures before and after applying force (\(\rho^{prime}\) and\(\rho\)), they can determine Poisson's ratio with precision. This is more than a clever trick — it's a window into the inner workings of materials, essential for creating safer and more efficient designs in engineering.