8.2. STRESS AND STRAIN

When forces are applied on a body in such a manner that the body is still in static equilibrium, it is deformed to a small or large extent depending upon the nature of the material of the body and the magnitude of the deforming force. The restoring force per unit area is known as stress.

When a solid object is subjected to an external force, it undergoes deformation. This deformation is countered by an internal restoring force exerted by the material, which tries to return the object to its original shape. Stress is defined as this restoring force per unit area. Thus, if we have a force (F) acting on a surface with an area (A), the stress (σ) can be calculated using the formula σ = F/A. The SI unit of stress is pascal (Pa), which is equivalent to N/m².

Think of stress as how hard you press down with your finger on a soft sponge. The harder you press (increased force), the more the surface of the sponge deforms (more stress applied). If you stop pressing, the sponge wants to return to its original shape, similar to how materials behave under applied stress.

There are three ways in which a solid may change its dimensions when an external force acts on it. These are: tensile stress, compressive stress, and shear stress.

When an external force is applied to a solid, it can experience different types of stress based on the direction of that force. Tensile stress occurs when forces are applied to stretch the material (like pulling a rubber band). Compressive stress happens when forces push the material, causing it to compress (like pressing down on a sponge). Shear stress is observed when forces are applied parallel to the surface, causing layers of the material to slide over each other (imagine sliding one deck of cards over another).

Visualize a piece of clay. If you pull on it from opposite ends, you're applying tensile stress. If you push the clay down, you're applying compressive stress. If you press down on one side of the clay while keeping the other side steady, you're applying shear stress, causing the clay to distort.

The change in length ΔL to the original length L of the body is called longitudinal strain. Longitudinal strain is defined as ΔL/L.

Strain is a measure of how much deformation a material undergoes in response to an applied stress. For longitudinal strain, it quantifies the change in length (ΔL) of a material compared to its original length (L). The formula for longitudinal strain is expressed as ε = ΔL / L, which is a dimensionless quantity since it’s a ratio of lengths.

Consider stretching a rubber band. If you pull it from its original length of 10 cm to a new length of 12 cm, the change in length (ΔL) is 2 cm. The strain would be 2 cm / 10 cm = 0.2 or 20%. This indicates how much the band has deformed relative to its original size.

When two equal and opposite deforming forces are applied parallel to the cross-sectional area of the cylinder, there is a relative displacement ∆x between opposite faces of the cylinder. The strain is known as shearing strain and is defined as the ratio of relative displacement of the faces ∆x to the length of the cylinder L.

When forces are applied parallel to a surface, the material experiences shear stress that causes layers to slide over one another. The resulting displacement (∆x) between two faces of the object divided by the length (L) gives us shearing strain. This can be expressed as γ = ∆x / L, which typically involves small angles where the tangent can be approximated as equal to the angle in radians.

Imagine a thick book lying flat on a table. If you push the top of the book sideways while keeping the bottom fixed, the top slides over the bottom, creating an angle with respect to the vertical. This sliding motion is similar to shear deformation, where the book's layers experience shearing stress.

In this scenario, a solid sphere placed in the fluid under high pressure is compressed uniformly on all sides. The force applied by the fluid acts in a perpendicular direction at each point of the surface, leading to a decrease in its volume without any change of its geometrical shape.

Hydraulic compression occurs when a fluid exerts pressure evenly across the surface of an object. This results in volume strain expressed as the ratio of the change in volume (ΔV) to the original volume (V). The internal restoring forces counteract this pressure, allowing the material to potentially return to its original state when the pressure is released.

Think about a sponge submerged in water under pressure. The water compresses the sponge equally from all sides. Even though the sponge gets smaller in size, its shape remains consistent until the pressure is released, at which point the sponge can restore itself.

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