Material science and mechanics are branches of physics which are devoted to the study of chemical and physical properties of every single material that finds applications in engineering and pure science. While the reductionist approach of studying the properties of materials at atomic or molecular level may help us understand certain physical properties, it cannot predict how matter will behave in bulk, when the intermolecular forces come into play. In extreme engineering applications, materials need to be tested for their strength, to determine their yield point or tensile strength limit. The macroscopic forces come into play here. One of the associated physical properties, which is measured when conducting stress tests, is the proportional limit.
About Stress and Strain
The strength of a material is measured by how it responds to stress and strain. Stress is a measure of the internal force per unit area acting on a body, when it is subjected to deforming forces. The unit of stress is same as that of pressure and it's equal to Newton per square meter.
In engineering terms, strain is a ratio of change in the dimensions of any body, with respect to its original dimensions. It is distortion of the body that occurs due to the application of external forces on it. It is the shifting of layers of the material, which occurs under application of deforming force.
Hooke's law is a relation between stress and strain felt by a deforming body. This approximate law states that the strain felt by a material is directly proportional to stress, within the elastic limit. The elastic limit is the point beyond which, if an object is stretched, it's permanently deformed. The proportionality factor between stress and strain is known as the modulus of elasticity of the material.
Proportional limit is the maximum amount of stress that an object can handle, while still obeying the Hooke's law. In other words, it is the highest limit of stress that a material may be subjected to, while still maintaining a linear relation with strain.
If you plot a curve of stress against strain, there will be a linear relation between them until the yield point is reached. After this point, the strain goes on increasing without increase in stress. The highest value of stress, just before the yield point is reached, is known as the proportional limit.
If you apply a force, beyond this limit, the proportionality relation between stress and strain is no longer maintained. Apply more stress beyond this point and you may deform the material permanently. There is no proportional limit formula per se, as the exact value of maximum stress that a material can take, is entirely dependent on its internal molecular or crystalline structure.
If you pursue production or mechanical engineering, you will know that studying the properties of materials like their proportional limit and grading them according to their yield strength is one of the most important tasks of a designer and production manager.