Stress at any point in a material is defined as -
Resisting force per unit area
Understanding Stress in Materials
Stress is a fundamental concept in mechanics of materials, representing the internal forces that neighboring particles of a continuous material exert on each other. When an external force or load is applied to an object, the material within the object resists this deformation. This internal resistance force is distributed over the area of the material.
Defining Stress
Stress ($\sigma$) is formally defined as the internal resisting force per unit cross-sectional area upon which the force acts. When an external force is applied to a body, internal resistance forces are developed within the body to oppose the deformation caused by the external force. Stress is a measure of the intensity of these internal forces.
The formula for stress is typically given by:
$$\sigma = \frac{F_{\text{resisting}}}{A}$$
Where:
- $\sigma$ is the stress
- $F_{\text{resisting}}$ is the internal resisting force
- $A$ is the cross-sectional area over which the force is distributed
In many practical situations, especially when dealing with elastic materials under static loads, the internal resisting force is assumed to be equal to the applied external force under equilibrium conditions.
Analyzing the Options for Stress Definition
Let's examine the given options based on the definition of stress:
- Option 1: Load per unit time
Load per unit time is related to concepts like power or flow rate of load application, not stress. Stress is defined per unit area, not per unit time. This option is incorrect. - Option 2: Young's modulus of elasticity per unit strain
Young's modulus ($E$) is the ratio of stress to strain in the elastic region ($E = \sigma / \epsilon$). Therefore, Young's modulus per unit strain would be $E/\epsilon = (\sigma/\epsilon)/\epsilon = \sigma/\epsilon^2$. This is not the definition of stress. This option is incorrect. - Option 3: Modulus of rigidity
Modulus of rigidity ($G$ or $\tau/\gamma$) is the ratio of shear stress ($\tau$) to shear strain ($\gamma$). It is a material property related to shear deformation, but it is not the definition of stress itself. This option is incorrect. - Option 4: Resisting force per unit area
This directly matches the fundamental definition of stress as the internal resisting force distributed over the cross-sectional area. This option correctly defines stress.
Units of Stress
Since stress is defined as force per unit area, its units in the SI system are Newtons per square meter ($\text{N/m}^2$), which is also known as Pascal ($\text{Pa}$). Commonly used larger units include megapascals ($\text{MPa}$) and gigapascals ($\text{GPa}$). In the imperial system, units like pounds per square inch ($\text{psi}$) or kilopounds per square inch ($\text{ksi}$) are used.
| Option | Description | Matches Stress Definition? |
|---|---|---|
| Load per unit time | Rate of load application | No |
| Young's modulus of elasticity per unit strain | $(\sigma/\epsilon)/\epsilon = \sigma/\epsilon^2$ | No |
| Modulus of rigidity | Ratio of shear stress to shear strain | No |
| Resisting force per unit area | Internal force divided by area | Yes |
Conclusion on Stress Definition
Based on the analysis, stress is correctly defined as the internal resisting force per unit area. This definition quantifies the intensity of the internal forces within a material when subjected to external loads.
Revision Table: Key Concepts in Material Stress
| Term | Definition | Formula/Relationship |
|---|---|---|
| Stress ($\sigma$) | Internal resisting force per unit area | $\sigma = F/A$ |
| Strain ($\epsilon$) | Deformation per unit original length | $\epsilon = \Delta L / L_0$ |
| Young's Modulus ($E$) | Ratio of normal stress to normal strain (in elastic region) | $E = \sigma / \epsilon$ |
| Shear Stress ($\tau$) | Tangential resisting force per unit area | $\tau = F_{\text{tangential}}/A$ |
| Shear Strain ($\gamma$) | Angular deformation | $\gamma \approx \Delta x / h$ |
| Modulus of Rigidity ($G$) | Ratio of shear stress to shear strain (in elastic region) | $G = \tau / \gamma$ |
Additional Information on Material Stress and Strain
Stress is a tensor quantity, although for simple uniaxial loading, it can often be treated as a scalar or vector quantity representing the stress component normal to the surface or parallel to the surface (shear stress). There are different types of stress:
- Normal Stress: Acts perpendicular to the surface area. It can be tensile (pulling) or compressive (pushing).
- Shear Stress: Acts parallel to the surface area. It is caused by forces that tend to slide one layer of material over another.
Strain is the measure of deformation resulting from stress. Like stress, there are different types of strain, such as normal strain (change in length) and shear strain (change in angle).
The relationship between stress and strain within the elastic limit of a material is described by Hooke's Law, which states that stress is directly proportional to strain, with the constant of proportionality being the material's modulus of elasticity (like Young's modulus for normal stress/strain or Modulus of Rigidity for shear stress/strain).
Important Questions from Stress and Strain
Dimensional formula for stress is
Unit of stress in SI unit is
A hollow steel column has to carry an axial load of 2,00,000 kg and the ultimate stress for the steel column is 4800 kg/cm 2and allows a load factor of 4. What is the sectional area of the column?
When a body is subjected to two equal and opposite pulls, as a result of which the body tends to extend its length, the stress and strain induced are
The failure of a material under varying load after a number of cycles of such load is known as
