Define Stress and Strain
First and foremost, we need to understand what is stress and strain and their basic equations. Basic stress and strain definition along with their formula and some general information has been described below.
Stress: When a force is applied to an object, it tries to deform the object. Internal resistance is generated in the object that resists this deformation. This internal resistance is referred to as stress. Stress can be defined as the resistance force per unit area. Stress is explained with the help of axial force P acting at the end of a prismatic bar as shown below.
It is generally denoted by σ and is expressed mathematically as -
P = Force acting on the object
A = Area of the cross-section of the object
Unit = N/m2
Above we have assumed that the force P is uniformly distributed throughout the cross-section. If the force is not uniformly distributed over the cross-section than we consider a fraction P of the total force is P applied over a small area A. Stress can hence be expressed as
As stress generally holds true only at a point, therefore mathematical expression for stress will be
Strain: An object when subjected to a force, undergoes a change in length. It will either elongate or shorten depending on the nature of the force. The elongation or shortening per unit length of the object is termed as strain. The change in length due to the force is shown in the figure below
It is generally denoted by ε and mathematically expressed as
ε=Change in length/Original length=ΔL/L (Unitless)
Types of Stress
Different types of stress and strain need to be understood to analyse whether a material can withstand the loading condition. In this section, we will look at different types of stress. Stress is basically of two types: normal stress and shear stress.
Normal Stress: When the applied force is perpendicular to the cross-section area of the object, then the stress generated is defined as normal stress. Normal stress can either be tensile or compressive depending on the nature of the force. Tensile stress is taken as positive and compressive stress is taken as negative. It is generally denoted by σ. Normal stress is of 2 types discussed below.
- Direct stress: Stress generated when an axial force is acting at the centre of gravity of the cross-section is termed direct stress. For a prismatic body, direct stress is uniformly distributed over the cross-section.
- Bending stress: This type of normal stress is indirectly generated when a body is subjected to a bending moment. It varies linearly across the cross-section with the highest bending stress at the extreme fibre and zero bending stress at the neutral axis.
Shear Stress: When a body is subjected to forces which are coplanar to the cross-section, then the stresses produced is defined as shear stress. It is shown in the figure below.
It is generally denoted by and it is mathematically expressed as-
P = total force acting parallel to the concerned area
A = area of the concerned section
Shear stress can also be of two types: Direct shear stress which is produced by direct shear force and torsional shear stress which is produced when a body is subjected to twisting. Shear stress always has balancing shear stress of equal magnitude in a perpendicular direction. This is shown in the figure given below along with the sign convention.
Types of Strain
As mentioned earlier, it's important to understand different types of stress and strain. In this section, we will look into different types strain. Strain can be of four types: longitudinal strain, lateral strain, volumetric strain and shear strain.
- Longitudinal strain: Strain due to deformation of the body along the direction of applied force is defined as longitudinal strain. It is the ratio of change in longitudinal dimension to original longitudinal dimension.
Longitudinal strain for example in the above diagram is given as,
Longitudinal Strain, ε = Change in longitudinal dimension/Original longitudinal dimension =ΔL/L
- Lateral Strain: Strain due to deformation of the body in directions perpendicular to the direction of applied force is defined as lateral strain. It is the ratio of change of lateral dimension to the original lateral dimension.
Lateral strain for example in the above diagram is given as,
Lateral strain, εL=Change in lateral dimension/Original lateral dimension=ΔB/B
- Volumetric Strain: Volumetric strain is defined as the ratio of change in volume of a body to its original volume under the application of stress.The volumetric strain can be expressed mathematically as-
εv=Change in volume/Original volume=ΔV/V
- Shear Strain: Strain due to angular deformation caused by shear forces is defined as shear strain. It is denoted by Φ. Shear strain can be expressed mathematically as
Relation Between Stress and Strain
Hooke’s law provides a relation between stress and strain for elastic materials for small deformation. According to Hooke’s Law, within elastic limit, strain in a body is directly proportional to applied stress. Hooke’s law holds true only for small deformation i.e., till the proportionality limit of the material.
Stress (σ) ∝Strain (ε)
Where, E is the proportionality constant which is called Modulus of Elasticity.
Stress and Strain Curve
By applying a steadily increasing force on a particular material, the stress vs. strain graph is generated by plotting the applied stress against the corresponding strain. The stress and strain curve for mild steel under tension is shown below.
In the above graph, point A represents the proportionality limit, B represents the elastic limit, C and C’ are the upper and lower yield points, respectively, E is the point of ultimate stress and F is the failure point. Hooke’s law is valid only for OA (till proportionality limit). C’D represents a perfectly plastic region, DE represents strain hardening and EF represents the necking region.
Stress and Strain Behaviour of Different Materials
For most materials, it is impossible to define the entire stress and strain curve. So, for most material, we use idealized stress and strain curves to analyze their behaviour. Stress and strain behaviour for a different types of materials are given below -
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