P Pallav, PhD

Stress vs. strain diagrams are very important in materials science, because many mechanical properties can be read directly or derived readily from these.

The figure shows how materials are tested in a tensilometer. With such an experiment the force and the displacement are recorded continuously. With this data a so-called stress – strain diagram may be produced, when the force values are converted to stress and the displacement values (∆) to relative strain (ε).

Elastic Modulus

With many structures, up to a certain limit there is a linear relationship between the force (F) applied to it and how much (D) it deforms. This is Hook’s law

∆ / F = C (equation 1)

C is a constant, the compliance in for example millimeters per Newton. If a string needs 1256 Newtons (F) to be extended by 20 millimeters (∆), C is equal to about 0.016 millimeters per Newton (20/1256), or 16 micrometers per Newton.

On the level of the material this linear relationship is expressed as the first straight part of the curve in the stress-strain diagram. The angle of this part (tanφ) represents the stiffness of the material, i.e. the modulus of elasticity or Young’s modulus, E, because it is defined as the stress (σ) divided by the strain (ε) in the elastic part.

E = σ / ε (equation 2)

Rigid materials have a great elastic modulus, whereas softer materials such as rubbers have a low elastic module. As ε is dimensionless, E is expressed in units of stress, Pascals, just like σ. Because ε is normally much closer to 0.001 than to 1, E is often expressed in GigaPascals (1GPa=10³MPa).

In the case of a steel string (E=200GPa), which is extended (ε) by 0.2%, the stress (σ) will be 400MPa (Exε).

Proportional Limit

This is the point, up to which the plot is a straight line, i.e. where the strain is directly proportional to the stress and equation 2 applies.

With many materials but especially with most metals, the proportional limit coincides accurately with the elastic limit, the point up to which all deformations are reversible. 

Because plastic (=irreversible) deformation occurs only beyond this point, it is also called the plastic limit.
The elastic limit of ordinary construction steel is about 500MPa. A load of 400MPa is therefore within the elastic range.

Ultimate Strength

The strength (S) or ultimate strength (US) is the stress required to failure, the highest point of the curve. Depending on the type of test, it may be named UTS, UCS, UFS, etc., referring to the ultimate tensile, compressive, flexural, etc. strength respectively.

Strain

Up to the elastic limit all deformation is elastic (reversible). Beyond this point the deformation is mainly plastic, but the elastic deformation still increases in proportion to the stress.

The amount of elastic and plastic strain at any point can be inferred by drawing a vertical line and a line parallel to the elastic part. The elastic strain is the part between the lines and the plastic strain is from zero to the slant line.

A large horizontal distance to the right of the elastic limit indicates a ductile material. With brittle materials the stress-strain diagram of a simple tensile experiment is generally just a straight line without any sign of plastic deformation, i.e the proportional limit coincides with the ultimate strength. 

Plastic deformability may lead to necking shortly before failure. In such a case a drop is visible at the end of the curve and the point of fracture doesn’t coincide with the ultimate strength.

Resiliency

The simultaneous action of stress and strain requires strain energy. 
The amount of elastic strain energy, for example in the form of impact, that can be stored without permanent deformation is called the resiliency.

Because it is the integral (∫σdε), it is the area under the curve from O to the elastic limit. It is expressed in Pascals, units of pressure. This may seem strange, but units of energy per unit volume, such as Joules per cubic meter (J/m3), Watt seconds per cubic meter (Ws/m3), or Newton meters per cubic meter (Nm/m3) are all the same as Pascals (N/m2).

Toughness

This is the maximum amount of strain energy up to fracture, the area under the complete curve. Toughness should not be confused with fracture toughness, which is an entirely different property.