Elasticity

1. Introduction

When an external force is applied to a body, it can change the body's shape, size, or both. The applied force that tends to change the configuration (shape or size) of a body is called a deforming force. Whether the body returns to its original form after the force is removed depends on a property called elasticity.

This chapter deals with elasticity and plasticity, the ideas of stress and strain, Hooke's law, the three moduli of elasticity (Young's modulus, bulk modulus, and shear modulus (also called modulus of rigidity)), Poisson's ratio, and the energy stored in a stretched wire.

2. Elasticity and Plasticity

Elasticity is the property of a body by virtue of which it regains its original configuration (shape and size) after the deforming force is removed.

• A body that completely recovers its original configuration is said to be perfectly elastic. In practice quartz and phosphor bronze come very close to being perfectly elastic.

Plasticity is the property of a body by virtue of which it does not regain its original configuration after the deforming force is removed, but retains the deformed shape.

• A body that shows no tendency to recover at all is said to be perfectly plastic. Paraffin wax and wet clay are close to being perfectly plastic.

No real material is perfectly elastic or perfectly plastic; every solid lies somewhere between these two ideals.

2.1 Deforming force and restoring force

• Deforming force: the external force that changes the configuration (shape or size) of a body.
• Restoring force: the internal force developed inside the body that opposes the deformation and tends to bring the body back to its original configuration. Within the elastic limit the restoring force is equal in magnitude and opposite in direction to the deforming force.

3. Stress

Stress is the internal restoring force set up per unit area of cross-section of a deformed body. Since within the elastic limit the restoring force balances the applied deforming force $F$, stress is measured as the deforming force per unit area $A$:

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

• SI unit: $\text{N m}^{-2}$ (pascal, $\text{Pa}$).
• Dimensional formula: $[\mathrm{M\,L^{-1}\,T^{-2}}]$.
• Stress is not a simple scalar or vector; it is a tensor quantity, because its value depends on both the direction of the force and the orientation of the area on which the force acts.

3.1 Types of stress

1. Normal stress: the deforming force acts per unit area perpendicular (normal) to the surface. If it stretches the body it is tensile stress; if it compresses the body it is compressive stress.

2. Bulk stress (volume stress): when a body is immersed in a fluid, the fluid pushes on the surface equally from all directions. This force per unit area acting normally over the whole surface produces a change in volume and is called bulk stress. It is numerically equal to the pressure applied.

3. Shear stress (tangential stress): the deforming force acts tangentially (parallel) to a surface, with an equal and opposite force on the opposite face. It changes the shape of the body without changing its volume.

4. Strain

Strain is the ratio of the change in configuration to the original configuration of a body:

$\text{Strain} = \frac{\text{change in configuration}}{\text{original configuration}}$

Strain is a ratio of two like quantities, so it is a pure number with no units and no dimensions.