Stress-Strain Relationship
Stress-Strain Relationship for elastic materials
Solids have fixed shape and size but can be modified upon application of force. The change in shape and size of a solid on application of force is deformation. The relation between application of force and consequent deformation depends on its elasticity.
What is Elasticity ?
Elasticity is the property of a matter by virtue of which a body regains its shape and size on removal of the force that deforms it. Such materials showing the property of elasticity are called as elastic materials. e. g. steel, copper etc. Deformation produced in such materials within elastic limit is called as elastic deformation.
What is Plasticity?
Plasticity is the property of a material by virtue of which a body is permanently deformed on the application of a force. It does not restore to its original dimensions even after removal of applied force. such materials showing the property of plasticity are plastic materials. e. g. mud, putty etc. Deformation produced in plastic materials are plastic deformation.
Stress-Strain Relationship
What is Stress?
A deforming force acting on a body results in development of a restoring force that is equal in magnitude but opposite in direction to the applied force. This restoring force acting per unit cross sectional area is the stress. The force and the area vectors are either parallel or perpendicular.
\(\textbf{Stress}=\dfrac{F_{r}}{A}\)
SI unit of stress is pascal same as that of pressure. 1 Pa = 1 \(N.m^{-2}\)
Stress is a scalar quantity.
Types of Stress
On the basis of deformation produced by the applied force, stresses can be classified as
- Longitudinal stress
- Shear stress
- Hydraulic stress
Longitudinal stress, \(\sigma\)
If the application of a force (tension or compression) on a body changes its length, the stress developed in the body is longitudinal stress. In this case, the applied force and hence the restoring force acts perpendicular to the cross sectional area of the matter. We represent it by \(\sigma\) (Sigma).
\(\sigma = \frac{\text{Longitudinal restoring force}}{\text{cross sectional area}}\)
Longitudinal strain
The measure of longitudinal deformation, elongation or shortening, caused to a material by the deforming force is the longitudinal strain. It is the ratio of elongation/shortening of the material to the original length of the body.
Longitudinal strain is a fraction.
However, we can express it in percentage too. We represent it by the symbol \(\epsilon\), epsilon.
If the original length of the material is \(L\) and the deforming force cause a deformation \(\Delta L\), then \[\epsilon = \frac{\Delta L}{L} = \frac{\Delta L}{L} \times 100 \%\]
Also, the longitudinal strain may be positive or negative. If the acting force is tension, the strain is positive. If it is compression, the strain is negative.
Longitudinal strain is unit less as well as dimensionless.
Shear Stress and Shear Strain
When two forces parallel to cross sectional area of a body act on it in opposing directions, the restoring force developed in the body is shear stress. Another name for it is tangential stress.
The strain developed due to shear stress is shear strain. It is also unit less as well as dimensionless.
\(\textbf{Shear Strain}=\tan \theta = \frac{\Delta{L}}{L}\)
Hydraulic Stress and Volume Strain
Due to the pressure by a fluid over a body, a restoring stress develops. We call it hydraulic stress. The stress is always perpendicular to the surface of the body.
The strain developed due to hydraulic stress is volume strain. It has no unit and no dimension.
\(\textbf{Volume Strain}= \frac{\Delta{V}}{V}\)
Hooke’s Law
Hooke’s law states that the stress \(\sigma\) developed in a material due to small deformation in it is directly proportional to the strain \(\epsilon\) developed. The law establishes a linear relationship between stress and strain.
\[\sigma \propto \epsilon\]
\[\sigma = k \cdot \epsilon \], where \(k\) is proportionality constant
We call this proportionality constant \(\textbf{Modulus of Elasticity}\).
However,
Hooke’s law is an empirical law. Though the law is not universal in nature, it is valid for most of the materials. There are also some materials that do not obey this law.
Stress-Strain Relationship (The Curve)
Stress-strain curve for a material like steel under stress is an experimentally observed relation between stress and strain developed in it.
Region OA
The curve in this region is linear and obeys Hooke’s law. Point A refers to proportionality limit. The body behaves as an elastic material.
Region AB
Stress strain curve is not linear but the body still behaves as an elastic material. Point B refers to yield strength, \(\sigma_y\).
Region BC
This is the strain hardening region. In this region, there is a rapid change in the strain even for small change in the load applied over the material. The matter turns plastic and never gains its original dimensions even if all the applied load is removed. The body develops a permanent strain, called as permanent set which is less than 1%.
Point C refers to ultimate strength, \(\sigma_u\).
Region CD
This is the region of necking. Steel quickly grows thinner and finally breaks at point D. This point is the fracture point.
Define yield strength.
Yield strength is the maximum stress developed in a material up to which it behaves as an elastic body. Refer here for diagram.
Ultimate strength
Ultimate strength of a material is the maximum load it can withstand before failure. Once a load equal to it is applied on it, the matter starts developing additional strain that continues to grow even if load is reduced. Refer here for diagram.
Brittle and ductile materials
We can identify a material as brittle or ductile on the basis of Stress-Strain Relationship. A material is brittle if the points of ultimate strength and point of fracture are close. If they are far away, the material is ductile. More these points are away, more ductile the matter is.
