Strain measurement
Strain measurement
What is Stress?
In mechanics, stress is defined as a force applied per unit area. It is given by the formula
where,
F is the force applied
A is the area of force application
The unit of stress is
Stress applied to a material can be of two types. They are:
● Tensile Stress: It is the force applied per unit area which results in the increase in length (or area) of a body. Objects under tensile stress become thinner and longer.
● Compressive Stress: It is the force applied per unit area which results in the decrease in length (or area) of a body. The object under compressive stress becomes thicker and shorter.
What is Strain?
The strain is the amount of deformation experienced by the body in the direction of force applied, divided by initial dimensions of the body. The relation for deformation in terms of length of a solid is given below.
where,
L is the original length of the material.
The strain is a dimensionless quantity as it just defines the relative change in shape.
Depending on stress application, strain experienced in a body can be of two types. They are:
● Tensile Strain: It is the change in length (or area) of a body due to the application of tensile stress.
● Compressive Strain: It is the change in length (or area) of a body due to the application of compressive strain
When we study solids and their mechanical properties, information regarding their elastic properties is most important. These can be obtained by studying the stress-strain relationships, under different loads, in these materials
Stress-Strain Curve
Explaining Stress-Strain Graph
The stress-strain graph has different points or regions as follows:
- Proportional limit
- Elastic limit
- Yield point
- Ultimate stress point
- Fracture or breaking point
(i) Proportional Limit
It is the region in the stress-strain curve that obeys Hooke’s Law. In this limit, the ratio of stress with strain gives us proportionality constant known as young’s modulus. The point OA in the graph is called the proportional limit.
(ii) Elastic Limit
It is the point in the graph up to which the material returns to its original position when the load acting on it is completely removed. Beyond this limit, the material doesn’t return to its original position and a plastic deformation starts to appear in it.
(iii) Yield Point
The yield point is defined as the point at which the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs. There are two yield points (i) upper yield point (ii) lower yield point.
(iv) Ultimate Stress Point
It is a point that represents the maximum stress that a material can endure before failure. Beyond this point, failure occurs.
(v) Fracture or Breaking Point
It is the point in the stress-strain curve at which the failure of the material takes place.
Hooke’s Law
In the 19th-century, while studying springs and elasticity, English scientist Robert Hooke noticed that many materials exhibited a similar property when the stress-strain relationship was studied. There was a linear region where the force required to stretch the material was proportional to the extension of the material. This is known as Hooke’s Law.
Hooke’s Law states that the strain of the material is proportional to the applied stress within the elastic limit of that material.
Mathematically, Hooke’s law is commonly expressed as:
- F is the force
- x is the extension length
- k is the constant of proportionality known as spring constant in N/m
What is Young’s Modulus?
Young’s modulus is also known as modulus of elasticity and is defined as:
The mechanical property of a material to withstand the compression or the elongation with respect to its length.
Young’s Modulus (also referred to as the Elastic Modulus or Tensile Modulus), is a measure of mechanical properties of linear elastic solids like rods, wires, and such. There are other numbers that give us a measure of elastic properties of a material, like Bulk modulus and shear modulus, but the value of Young’s Modulus is most commonly used. This is because it gives us information about the tensile elasticity of a material (ability to deform along an axis).
Young’s modulus describes the relationship between stress (force per unit area) and strain (proportional deformation in an object. The Young’s modulus is named after the British scientist Thomas Young. A solid object deforms when a particular load is applied to it. If the object is elastic, the body regains its original shape when the pressure is removed. Many materials are not linear and elastic beyond a small amount of deformation. The constant Young’s modulus applies only to linear elastic substances.
Young’s Modulus Formula
Young’s Modulus Formula From Other Quantities
Notations Used In The Young’s Modulus Formula
- E is Young’s modulus in Pa
- 𝞂 is the uniaxial stress in Pa
- ε is the strain or proportional deformation
- F is the force exerted by the object under tension
- A is the actual cross-sectional area
- ΔL is the change in the length
- L0 is the actual length
Elastic Moduli of Materials
Following is the table with Young’s modulus, shear modulus, and bulk modulus for the common materials that we use every day in the life:
| Material | Young’s modulus (E) in GPa | Shear modulus (G) in GPa | Bulk modulus (K) in GPA |
| Glass | 55 | 23 | 37 |
| Steel | 200 | 84 | 160 |
| Iron | 91 | 70 | 100 |
| Lead | 16 | 5.6 | 7.7 |
| Aluminium | 70 | 24 | 70 |
Note: GPa is gigapascal and 1 GPa = 1,00,00,00,000 Pa.
What is Poisson’s Ratio?
Poisson’s ratio is “the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here,
- Compressive deformation is considered negative
- Tensile deformation is considered positive.
| Symbol | Greek letter ‘nu’,ν |
| Formula | Poisson’s ratio = – Lateral strain / Longitudinal strain |
| Range | -1.0 to +0.5 |
| Units | Unitless quantity |
| Scalar / Vector | Scalar quantity |
Poisson’s Ratio Formula
Imagine a piece of rubber, in the usual shape of a cuboid. Then imagine pulling it along the sides. What happens now?
It will compress in the middle. If the original length and breadth of the rubber are taken as L and B respectively, then when pulled longitudinally, it tends to get compressed laterally. In simple words, length has increased by an amount dL and the breadth has increased by an amount dB.
In this case,
The formula for Poisson’s ratio is,
where,
εt is the Lateral or Transverse Strain
εl is the Longitudinal or Axial Strain
The strain on its own is defined as the change in dimension (length, breadth, area…) divided by the original dimension.
For most materials, the value of Poisson’s ratio lies in the range, 0 to 0.5.
A few examples of Poisson ratio is given below for different materials.
| Material | Values |
| Concrete | 0.1 – 0.2 |
| Cast iron | 0.21 – 0.26 |
| Steel | 0.27 – 0.30 |
| Rubber | 0.4999 |
| Gold | 0.42 – 0.44 |
| Glass | 0.18 – 0.3 |
| Cork | 0.0 |
| Copper | 0.33 |
| Clay | 0.30 – 0.45 |
| Stainless steel | 0.30 – 0.31 |
| Foam | 0.10 – 0.50 |
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