Introduction, Stress, and Strain

Why Do Solids Bend, Stretch, and Compress?

Think of a steel bridge supporting heavy traffic, or a rubber band stretched between your fingers. In earlier chapters on mechanics, we treated objects as perfectly rigid bodies that never change shape. In reality, no solid is perfectly rigid. Every solid deforms at least a little when a force acts on it, and the way it deforms tells us a great deal about the material it is made of.

This is the starting point of a fascinating area of physics: understanding how solids respond to external forces. The answers have enormous practical importance. Engineers need these concepts to design buildings that will not collapse, bridges that can carry heavy loads, aircraft that are light yet strong, and artificial limbs that can withstand daily use. Even everyday questions, like why railway tracks are shaped like the letter “I” or why glass shatters while brass bends, trace back to the mechanical properties of solids.

Elasticity and Plasticity: Two Ways Solids Respond

When you pull the ends of a helical spring gently, the spring stretches. Release it, and it snaps right back to its original length. The body “remembers” its original shape and works to restore it. This tendency of a body to regain its original size and shape after the deforming force is removed is called elasticity (the ability of a material to bounce back after deformation). The deformation that disappears once the force is gone is called elastic deformation.

Not every material behaves this way. Press a lump of putty or mud into a new shape, and it stays that way permanently. There is no tendency to spring back. Materials that retain the deformed shape permanently are called plastic (tending to stay deformed), and the property itself is called plasticity. Putty and mud are close to ideal plastics.

Most real materials fall somewhere between these two extremes. A rubber band is highly elastic. A piece of chewing gum is largely plastic. Metals like steel are elastic up to a point, but if you push them too far, they deform permanently.

Stress: The Internal Response to External Force

When forces act on a body and the body remains in static equilibrium (not accelerating), it deforms. The deformation might be too small to see with the naked eye, but it is always present. Inside the body, a restoring force develops. This restoring force is equal in magnitude to the applied force but acts in the opposite direction, trying to push the body back to its original configuration.

Stress is defined as the restoring force per unit area of cross-section. If a force $F$ is applied normal to a cross-sectional area $A$, then:

$\text{Stress} = \frac{F}{A} \qquad \text{(8.1)}$

SI unit: $N\,m^{-2}$, also called the pascal (Pa).

Dimensional formula: $[ML^{-1}T^{-2}]$

This is the same dimensional formula as pressure, and indeed stress and pressure share the same units. The key difference is conceptual: stress refers to the internal restoring force per unit area, while pressure typically refers to an externally applied force per unit area.

Three Types of Stress and Their Corresponding Strains

A solid can change its dimensions in three distinct ways depending on how the external force is applied. Each way produces a different type of stress and a matching type of strain.

1. Longitudinal Stress and Longitudinal Strain

Imagine a cylinder being pulled from both ends by equal and opposite forces applied perpendicular to its cross-sectional area. The cylinder stretches. The internal restoring force per unit area that resists this stretching is called tensile stress.

Now imagine the same cylinder being squeezed from both ends instead. The restoring force per unit area that resists the compression is called compressive stress.

Both tensile and compressive stress are grouped under the single name longitudinal stress, because in both cases the force acts along the length of the body (normal to the cross-section).

Fig 8.1(a): A cylindrical body under tensile stress elongates by $\Delta L$

The deformation that accompanies longitudinal stress is a change in the body’s length. The measure of this change is longitudinal strain, defined as the ratio of the change in length to the original length:

$\text{Longitudinal strain} = \frac{\Delta L}{L} \qquad \text{(8.2)}$

Here, $\Delta L$ is how much the length changes (increase for tension, decrease for compression) and $L$ is the original length.

2. Shearing Stress and Shearing Strain

What if, instead of pulling or pushing along the length, you apply two equal and opposite forces parallel to the cross-sectional area? Picture placing your hand flat on top of a thick book and sliding it sideways while the bottom stays fixed. The top face shifts horizontally relative to the bottom face.

Fig 8.1(b): Shearing stress on a cylinder deforming it by angle $\theta$

Fig 8.1(c): A body subjected to shearing stress

The restoring force per unit area that develops due to this tangential (sideways) force is called shearing stress or tangential stress.

The deformation it produces is measured by shearing strain. If the top face displaces by $\Delta x$ relative to the bottom face, and the height of the body is $L$, then:

$\text{Shearing strain} = \frac{\Delta x}{L} = \tan \theta \qquad \text{(8.3)}$

Here, $\theta$ is the angle by which the vertical edge of the body tilts from its original upright position. This angle is called the angular displacement.

For small deformations, which is the usual case in practice, $\theta$ is very small. When $\theta$ is measured in radians and is small, $\tan \theta$ is very nearly equal to $\theta$ itself. To give a sense of how good this approximation is: even at $\theta = 10°$, the difference between $\tan \theta$ and $\theta$ is only about 1%.

So for small deformations:

$\text{Shearing strain} = \tan \theta \approx \theta \qquad \text{(8.4)}$

3. Hydraulic Stress and Volume Strain

Now consider a completely different scenario. A solid sphere is placed inside a fluid that is under very high pressure. The fluid pushes inward on the sphere from every direction, with force acting perpendicular to the surface at each point. The sphere is being compressed uniformly from all sides. This is called hydraulic compression.

Fig 8.1(d): A solid body under hydraulic stress. The volume decreases by $\Delta V$, but the shape does not change

The body responds by developing internal restoring forces equal and opposite to the fluid’s forces. Once taken out of the fluid, the body would spring back to its original shape and size. The restoring force per unit area in this case is called hydraulic stress, and its magnitude equals the hydraulic pressure (the applied force per unit area from the fluid).

The important feature of hydraulic stress is that the body’s volume decreases, but its geometric shape stays the same. A sphere remains a sphere, just a smaller one.

The resulting deformation is measured by volume strain, defined as the ratio of the change in volume to the original volume:

$\text{Volume strain} = \frac{\Delta V}{V} \qquad \text{(8.5)}$

where $\Delta V$ is the change in volume and $V$ is the original volume.

Strain Is Always Dimensionless

Notice something that all three types of strain share: each one is a ratio of two quantities that have the same dimensions. Longitudinal strain is length divided by length. Shearing strain is length divided by length. Volume strain is volume divided by volume. In every case, the units cancel out. Strain is therefore a pure number with no units and no dimensional formula.

Quick Comparison: Types of Stress and Strain

Type	How the force is applied	What changes	Stress name	Strain name	Strain formula
Longitudinal	Normal to cross-section, along the length	Length	Tensile (stretching) or Compressive (squeezing)	Longitudinal strain	$\Delta L / L$
Shearing	Parallel to cross-section (tangential)	Shape (lateral displacement)	Shearing (tangential) stress	Shearing strain	$\Delta x / L = \tan \theta$
Hydraulic	Perpendicular to surface at every point (uniform pressure)	Volume (shape unchanged)	Hydraulic stress	Volume strain	$\Delta V / V$