# Young’s Modulus and it’s applications with Example

Young’s Modulus

Young’s modulus, also known as the elastic modulus, is a measure of the stiffness of a material. It quantifies the relationship between stress (force per unit area) and strain (proportional deformation) in a material undergoing tension or compression. The modulus is named after Thomas Young, a 19th-century British scientist who first introduced the concept.

Definition and Formula

Young’s modulus is defined as the ratio of stress to strain within the elastic limit of a material. Mathematically, it is represented by the formula:

The SI unit of Young’s modulus is pascals (Pa) or newtons per square meter (N/m²).

Importance and Applications

Young’s modulus is a crucial property for materials used in engineering and construction. It helps engineers and designers understand how materials will behave under different loading conditions. Materials with high Young’s moduli are stiffer and less deformable under load, while those with lower moduli are more flexible.

This property is essential in designing structures such as buildings, bridges, and aircraft components. It also plays a significant role in material selection for various applications, including automotive, aerospace, and civil engineering.

The basic principle is that a material undergoes elastic deformation when it is compressed or extended, returning to its original shape when the load is removed. More deformation occurs in a flexible material compared to that of a stiff material.

- A low Young’s modulus value means a solid is elastic.
- A high Young’s modulus value means a solid is inelastic or stiff.

The behavior of a rubber band illustrates Young’s modulus. A rubber band stretches, but when you release the force it returns to its original shape and is not deformed. However, pulling too hard on the rubber band causes deformation and eventually breaks it.

### Young’s Modulus Formula

Young’s modulus compares tensile or compressive stress to axial strain. The formula for Young’s modulus is:

E = σ / ε = (F/A) / (ΔL/L_{0}) = FL_{0} / AΔL = mgL_{0}/ *π*r^{2}ΔL

Where:

- E is Young’s modulus
- σ is the uniaxial stress (tensile or compressive), which is force per cross sectional area
- ε is the strain, which is the change in length per original length
- F is the force of compression or extension
- A is the cross-sectional surface area or the cross-section perpendicular to the applied force
- ΔL is the change in length (negative under compression; positive when stretched)
- L
_{0}is the original length - g is the acceleration due to gravity
- r is the radius of a cylindrical wire

### Young’s Modulus Units

While the SI unit for Young’s modulus is the pascal (Pa). However, the pascal is a small unit of pressure, so megapascals (MPa) and gigapascals (GPa) are more common. Other units include newtons per square meter (N/m^{2}), newtons per square millimeter (N/mm^{2}), kilonewtons per square millimeter (kN/mm^{2}), pounds per square inch (PSI), mega pounds per square inch (Mpsi).

## Applications of Young’s Modulus

Young’s modulus has numerous applications in various fields, including civil engineering, materials science, and structural analysis. Some of the key applications of Young’s modulus are:

**Design and analysis of structures:**Engineers use Young’s modulus to design and analyze the structural behavior of various components, such as beams, columns, and trusses. It helps in determining the load-carrying capacity and the deformation of a structure under different loads.**Selection of materials:**Engineers and scientists use Young’s modulus to compare the stiffness and strength of different materials. This information is crucial in selecting the most appropriate material for a specific application, considering factors such as weight, cost, and durability.**Fatigue testing:**Young’s modulus is also used in fatigue testing, which involves subjecting a material to repeated cycles of stress to determine its ability to withstand such conditions without failing.**Vibrations and dynamic analysis:**Young’s modulus plays a significant role in the analysis of vibrations and dynamic behavior of structures, as it helps in understanding the natural frequencies and modes of vibration.**Biomechanics**: In the field of biomechanics, Young’s modulus is used to study the mechanical properties of biological tissues, such as bones, muscles, and tendons, to understand their behavior under various loads and conditions.

### Example Problem

For example, find the Young’s modulus for a wire that is 2 m long and 2 mm in diameter if its length increases 0.24 mm when stretched by an 8 kg mass. Assume g is 9.8 m/s^{2}.

First, write down what you know:

- L = 2 m
- Δ L = 0.24 mm = 0.00024 m
- r = diameter/2 = 2 mm/2 = 1 mm = 0.001 m
- m = 8 kg
- g = 9.8 m/s
^{2}

Based on the information, you know the best formula for solving the problem.

E = mgL_{0}/ *π*r^{2}ΔL = 8 x 9.8 x 2 / 3.142 x (0.001)^{2} x 0.00024 = 2.08 x 10^{11} N/m^{2}

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