Physics Bootcamp

Subsection 16.2.1 Elasticity

Real materials react to external forces, as we have seen in the development of the Normal force. If the deformation is small compared to the physical dimensions of the object, many materials often recover the original shape when the external force is removed. This property is called elasticity.

You can see the elastic behavior clearly when you hang weights by a wire as shown in Figure 16.2.1. The reaction force here is the tension in the wire caused by the weight. Initially, the stretch and tension are linearly related, \(F = k \Delta L\text{,}\) much like a spring. This part is the linear elastic region. If the tension increases beyond the linear limit, we get nonlinear behavior behavir, \(F \ne k \Delta L\text{,}\) but if you remove the tension, the wire still returns to the original length.

Beyond the elastic limit, there is a region, where wire still stretches, but when you remove the tension-causing weights, the wire is pemanently elongated. This is called plastic region.

If you continue to add more weight to the wire, eventually the wire will break at some point, called the breaking point. The breaking point gives us information about the ultimate strength of the material.

We will be concerned with the elastic range here since we want a predictable linear restoring force to help analyze the equilibrium behavior of structures so as to prevent permanent deformation or fracture.

In the linear elastic regime, the elongation \(\Delta L\) is proportional to the force \(F\) applied to the wire. Not only that, but longer wires stretch more, i.e., \(\Delta L\) is proportional to the length \(L\) of the wire.

We can rewrite this as

where \(\alpha\) is a constant. By analogy with spring, we can identify \(\alpha/L\) as “spring constant” of the wire.

This observation was first made by Robert Hooke (1635-1703), a contemporary of Isaac Newton, and is often called Hooke's law, which states that “Ut tensio sic uis,” in Latin, meaning “As the extension, so the force”.

The reason for a material “to act like” springs can be traced to the inter-atomic bonds in materials as illustrated in Figure 16.2.2, which have a spring-like behavior for small displacement from equilibrium.

Subsection 16.2.2 Strain

From Eq. (16.2.1), we note that force is proportional to the fractional change of the wire. We call the fractional change in length the strain or relative deformation of length. We will denote strain by the Greek letter epsilon, \(\epsilon\text{.}\)

that is,

Clearly, strain is a dimensionless quantity since it is a ratio of two lengths.

There are other types of strain when we have deformation of the area or volume. The relative change in volume is called bulk strain, which we can denote by \(\epsilon_V\text{.}\)

where \(V\) is volume and \(\Delta V\) the change in volume. We will discuss other types of strain in another section.

Subsection 16.2.3 Stress

When you hang different masses to a wire the strain in the wire and the force exerted on the wire are linearly related, as seen above by plotting \(F\) vs \(\epsilon=\Delta L/L\text{.}\)

When you perform the same same experiment on wires of different thicknesses, you get different plots. However, when you plot \(F/A\) vs \(\epsilon=\Delta L/L\text{,}\) wires of all thicknesses of the same material, fall in one line as shown schematically in Figure 16.2.3. That means \(F/A\) captures more universal physics.

Force per unit area is called stress. We will denote stress by the Greek symbol sigma, \(\sigma\text{.}\)

that is,

Note that stress will have direction since force has direction. Even the area of cross-section has an orientation. For instance, the six sides of a cube have different orientations in space. Therefore, in general, the direction of stress is dependent on the direction of the force and the orientation of the area upon which the force acts.

The dimension of stress will come from the ratio of the dimensions of force and area.

This gives the unit of stress to be \(\text{N/m}^2\text{,}\) which is also called Pascal denoted by \(\text{Pa}\text{.}\)

Subsection 16.2.4 General Hooke's Law

The right side part of Figure 16.2.3 says that stress in the material is directly proportional to its strain. This observation is called Hooke's law, named after Robert Hooke (1635 - 1703).

The proportionality constant \(Y\) is called Young's modulus. Young's modulus is a characteristic of the material, independent of the geometrical aspects of a sample.

The unit of Young's modulus is the same as that of stress, i.e., \(\text{N/m}^2\) or Pascal, since strain in unitless.

A more flexible material has smaller \(Y\text{.}\) From wikipedia, \(Y_\text{rubber} = 0.01\text{-}0.1\text{ GPa}\) and that of diamond is \(Y_\text{diamond} = 1,050\text{-}1,210\text{ GPa}\text{.}\) Here \(G\) stands for Giga, i.e., a factor of \(10^9\text{.}\)