Graph between stress & strain
Graph between Stress and Strain
The relationship between stress and strain is fundamental in the study of materials science and mechanical engineering. When a material is subjected to an external force, it experiences a deformation. The measure of this deformation is known as strain, while the force applied per unit area is called stress. The graph that represents the relationship between stress (σ) and strain (ε) is crucial for understanding the behavior of materials under load.
Stress (σ)
Stress is defined as the force applied per unit area of a material and is given by the formula:
$$ \sigma = \frac{F}{A} $$
where:
- ( \sigma ) is the stress,
- ( F ) is the force applied, and
- ( A ) is the cross-sectional area perpendicular to the force.
Stress is typically measured in pascals (Pa) or newtons per square meter (N/m²).
Strain (ε)
Strain is the measure of deformation experienced by the material and is defined as the change in dimension divided by the original dimension. There are two types of strain: normal strain and shear strain. For normal strain (which is dimensionless), the formula is:
$$ \epsilon = \frac{\Delta L}{L_0} $$
where:
- ( \epsilon ) is the strain,
- ( \Delta L ) is the change in length, and
- ( L_0 ) is the original length.
Stress-Strain Curve
The graph between stress and strain typically results in a curve that can be divided into several distinct regions:
- Proportional Limit
- Elastic Limit
- Yield Point
- Plastic Region
- Necking
- Fracture
Proportional Limit
In the initial part of the curve, stress is directly proportional to strain. This means that the material obeys Hooke's Law, which is given by:
$$ \sigma = E \cdot \epsilon $$
where ( E ) is the modulus of elasticity or Young's modulus.
Elastic Limit
The elastic limit is the maximum stress that a material can withstand without undergoing permanent deformation. Beyond this point, the material will not return to its original shape when the load is removed.
Yield Point
The yield point is where the material begins to deform plastically. For some materials, there is a distinct yield point where the curve levels off.
Plastic Region
Beyond the yield point, the material will undergo plastic deformation, meaning it will not return to its original shape even when the stress is removed.
Necking
As the material is stretched further, it may begin to neck, which is a localized reduction in cross-sectional area. The stress calculated by the original area continues to increase, but the true stress based on the actual area is higher.
Fracture
The fracture point is where the material ultimately fails and breaks apart.
Table of Differences and Important Points
| Property | Stress (σ) | Strain (ε) |
|---|---|---|
| Definition | Force per unit area | Deformation per unit dimension |
| Formula | ( \sigma = \frac{F}{A} ) | ( \epsilon = \frac{\Delta L}{L_0} ) |
| Units | Pascals (Pa) or N/m² | Dimensionless (no units) |
| Types | Normal stress, Shear stress | Normal strain, Shear strain |
| Measurement | Requires force and area | Requires original and changed dimensions |
| Dependency | Depends on external force | Depends on material deformation |
| Hooke's Law | ( \sigma = E \cdot \epsilon ) | Applicable in the proportional limit |
Examples
Example 1: Elastic Behavior
A steel rod with a cross-sectional area of 0.01 m² is subjected to a force of 10,000 N. Calculate the stress and strain if the rod stretches by 1 mm and the Young's modulus of steel is 200 GPa.
Stress: $$ \sigma = \frac{F}{A} = \frac{10,000 \text{ N}}{0.01 \text{ m}^2} = 1,000,000 \text{ Pa} $$
Strain: $$ \epsilon = \frac{\Delta L}{L_0} = \frac{1 \text{ mm}}{L_0} $$
Since we don't have the original length ( L_0 ), we can't calculate the strain without it. However, if we assume the original length is 1 m, then:
$$ \epsilon = \frac{1 \text{ mm}}{1000 \text{ mm}} = 0.001 $$
Using Hooke's Law, we can verify the stress:
$$ \sigma = E \cdot \epsilon = 200 \times 10^9 \text{ Pa} \cdot 0.001 = 200 \times 10^6 \text{ Pa} = 200 \text{ MPa} $$
Example 2: Plastic Deformation
Consider a plastic material that does not have a distinct yield point. When a stress of 50 MPa is applied, it undergoes a permanent strain of 0.05. This indicates that the material has entered the plastic region and will not return to its original shape after the load is removed.
In conclusion, the graph between stress and strain is a powerful tool for understanding material behavior under various loads. It helps engineers and scientists predict how materials will react to different forces, which is essential for designing safe and efficient structures and components.