Solids have definite shapes and volumes, but when forces are applied to them, they deform. A bridge bends under the weight of vehicles, a spring compresses when pressed, and a wire stretches when a load is hung from it. The study of how solids respond to applied forces is called elasticity, and it is the subject of this chapter.
When a deforming force is removed, some solids return to their original shape and size. This property is called elasticity, and solids that possess it are called elastic bodies. Bodies that do not return to their original shape are called plastic bodies. The chapter introduces stress and strain, the measures of the force per unit area and the relative deformation, and the moduli that connect them through Hooke's law.
Elasticity is essential to engineering. Knowing the elastic constants allows engineers to choose materials for bridges, buildings, and aircraft. This chapter also introduces Poisson's ratio and the elastic potential energy stored in a deformed body, and explains the stress-strain curve for a metallic wire.
When a deforming force is applied to a body, the body offers an internal restoring force to regain its original shape. Stress is defined as the restoring force per unit area of the surface on which the force acts:
Stress = F / A
Stress has the same units as pressure, the pascal (Pa), and is a vector quantity. The three types of stress are longitudinal stress (for length changes), volume stress (for volume changes under pressure), and shearing stress (for shape changes).
Strain is the fractional change in a dimension of the body. It is defined as the ratio of the change in the dimension to the original dimension:
Strain = change in dimension / original dimension
Since strain is a ratio of two similar quantities, it has no units and is dimensionless. The three types of strain are longitudinal strain (change in length divided by original length), volume strain (change in volume divided by original volume), and shearing strain (the angle of deformation).
Hooke's law states that within the elastic limit, stress is directly proportional to strain:
Stress proportional to Strain
The constant of proportionality is called the modulus of elasticity. There are three moduli, one for each type of stress and strain. Young's modulus is defined for longitudinal stress and longitudinal strain:
Y = longitudinal stress / longitudinal strain = (F/A) / (delta L / L)
The bulk modulus applies to volume stress and volume strain under pressure:
B = - (delta p) / (delta V / V)
where the negative sign indicates that an increase in pressure decreases the volume. The modulus of rigidity (shear modulus) applies to shearing stress and shearing strain:
G = shearing stress / shearing strain
The SI unit of all moduli is the pascal (Pa) or N/m^2. The modulus of a material is a measure of its stiffness; a material with a large Young's modulus is difficult to stretch.
When a wire is loaded gradually, the stress-strain curve reveals the behaviour of the material. Up to point A, the proportionality limit, stress is proportional to strain and Hooke's law holds. Beyond A, the curve departs from a straight line, but up to point B, the elastic limit, the wire returns to its original length when the load is removed.
Between B and C, called the yield point, a small increase in stress produces a large increase in strain; the wire is said to yield. Beyond C, the wire undergoes plastic flow and continues to elongate without much increase in load. At point D, the breaking point, the wire breaks. Between C and D the strain increases faster than the stress, and the material becomes thinner at one point, a phenomenon called necking.
A ductile material like copper shows a large plastic region and can be drawn into wires, while a brittle material like glass breaks soon after the elastic limit with very little plastic deformation. Materials like rubber show large strains with a small Young's modulus.
Consider a wire of original length L and cross-sectional area A stretched by a force F, producing an elongation delta L. Young's modulus is:
Y = (F/A) / (delta L / L) = F L / (A delta L)
The maximum load the wire can bear without breaking determines its breaking stress, also called tensile strength. The breaking stress depends on the material and not on the dimensions of the wire. For a given material, a thicker wire can bear a larger total load because the stress (force per unit area) at breaking is fixed.
If the same material is used for wires of different lengths, the elongation is proportional to the original length for the same stress. The elastic behaviour of a material can be altered by temperature, by impurities, and by processes like hammering and annealing.
Work is done in deforming an elastic body, and this work is stored as elastic potential energy. For a wire stretched within the elastic limit, the elastic potential energy stored per unit volume is:
u = (1/2) x stress x strain
Since stress = Y x strain, this can also be written as:
u = (1/2) x Y x (strain)^2
The total energy stored in a stretched wire is therefore:
U = (1/2) x Y x (strain)^2 x A x L
The area under the stress-strain curve represents the energy stored per unit volume of the material. For a spring of force constant k stretched by x, the stored energy is (1/2)kx^2, which matches the general formula.
When a body is stretched along one direction, it contracts in the perpendicular directions. Poisson's ratio is the ratio of the lateral strain to the longitudinal strain:
sigma = lateral strain / longitudinal strain = - (delta d / d) / (delta L / L)
The negative sign shows that the lateral strain and longitudinal strain have opposite signs. Poisson's ratio is dimensionless and lies between -1 and 0.5 for isotropic materials. For most metals, it is around 0.3. Poisson's ratio connects Young's modulus, the bulk modulus, and the shear modulus through relations used in advanced elasticity.
| Quantity | Formula | Unit |
|---|---|---|
| Stress | F / A | Pa |
| Strain | Change / original | dimensionless |
| Young's modulus | Y = (F/A)/(delta L/L) | Pa |
| Bulk modulus | B = -delta p / (delta V/V) | Pa |
| Shear modulus | G = shear stress / shear strain | Pa |
| Elastic energy density | u = (1/2) stress x strain | J/m^3 |
| Type of Stress | Type of Strain | Modulus |
|---|---|---|
| Longitudinal | Longitudinal | Young's modulus Y |
| Volume | Volume | Bulk modulus B |
| Shearing | Shearing | Shear modulus G |
In this chapter we studied how solids deform under applied forces. Stress, the restoring force per unit area, and strain, the fractional deformation, are the fundamental measures of elastic behaviour. Hooke's law states that within the elastic limit, stress is proportional to strain, and the three moduli - Young's modulus for longitudinal stress, bulk modulus for volume stress, and shear modulus for shearing stress - quantify this proportionality for different types of deformation. The stress-strain curve revealed the elastic limit, yield point, and breaking point of materials, and we derived the elastic potential energy stored in a deformed body. These concepts of elasticity are applied directly in engineering and are related to the study of fluids, where we examine pressure and flow.