Young's Modulus
This article discusses the mathematical method used to calculate the elongation of a material under tensile force and elasticity of a material.
Hooke's Law
If a metal is lightly stressed, a temporary deformation, presumably permitted by an elastic displacement of the atoms in the space lattice, takes place. Removal of the stress results in a gradual return of the metal to its original shape and dimensions. In 1678 an English scientist named Robert Hooke ran experiments that provided data that showed that in the elastic range of a material, strain is proportional to stress. The elongation of the bar is directly proportional to the tensile force and the length of the bar and inversely proportional to the cross-sectional area and the modulus of elasticity.
Hooke's experimental law may be given by equation below.
This simple linear relationship between the force (stress) and the elongation (strain) was formulated using the following notation.
P = force producing extension of bar (lbf imperial, Newton’s metric)
= length of bar (inches imperial, or millimetre, centimetre, or metre for metric)
A = cross-sectional area of bar (inches squared imperial, or millimetre, centimetre, or metre squared metric)
δ = total elongation of bar (inches imperial, or millimetre, centimetre, or metre for metric)
E = elastic constant of the material, called the Modulus of Elasticity, or Young's Modulus (lbf/in.2 imperial, or Pascals (Pa) metric)
Note that one pascal is equal to one Newton per square metre.
The quantity E, the ratio of the unit stress to the unit strain, is a material’s modulus of elasticity when in tension or compression and is often called Young's Modulus.
Previously, we learned that tensile stress, or simply stress, was equated to the load per unit area, or force applied per cross-sectional area perpendicular to the force.
We also learned that tensile strain, or the elongation of a bar per unit length, is determined by:
Thus, the conditions of the experiment described above are adequately expressed by Hooke's Law for elastic materials. For materials under tension, strain (ε) is proportional to applied stress (σ).
Where
E = Young's Modulus (lbf/in.2 imperial, Pascals metric)
σ = stress (psi imperial, N/m2 metric)
ε = strain (any unit of length measurement comparing the change in length to the original length e.g. inch/inch for imperial, or centimetre/centimetre for metric etc.)
Young's Modulus (Elastic Modulus)
Young's Modulus (sometimes referred to as Modulus of Elasticity, meaning "measure" of elasticity) is an extremely important characteristic of a material. It is the numerical evaluation of Hooke's Law, namely the ratio of stress to strain (the measure of resistance to elastic deformation). To calculate Young's Modulus, stress (at any point) below the proportional limit is divided by corresponding strain. It can also be calculated as the slope of the straight-line portion of the stress-strain curve (the positioning on a stress-strain curve will be discussed later).
or
Note that in SI units, E is expressed as E=N/m2/(cm/cm). Strain is dimensionless, so any units of measurement may be used e.g. centimetre, metre etc.
We can now see that Young's Modulus may be easily calculated, provided that the stress and corresponding unit elongation or strain have been determined by a tensile test as described previously. Strain (ε) is a number representing a ratio of two lengths; therefore, we can conclude that the Young's Modulus is measured in the same units as stress (σ), that is, in pounds per square inch imperial, or Newtons per square metre metric. Table 1 gives average values of the Modulus E for several common metals and alloys. Yield strength and ultimate strength will be discussed in more detail later.
TABLE 1: Properties of Common Structural Materials
Example: What is the elongation of 200 in. of aluminium wire with a 0.01 square in. area if it supports a weight of 100 lb?
Solution:
Summary
The important information in this section is summarised below.
Young's Modulus Summary
Hooke's Law states that in the elastic range of a material strain is proportional to stress. It is measured by using the following equation:
Young's Modulus (Elastic Modulus) is the ratio of stress to strain, or the gradient of the stress-strain graph. It is measured using the following equation:
