Young’s Modulus of Elasticity

What is Young’s Modulus

Young’s modulus, denoted by the letter (E), is a mechanical property that measures the tensile or compressive stiffness of a material when a longitudinal force is acting on it. In other words, it measures a material’s resistance to elastic (non-permanent) longitudinal deformation when it’s under tensile or compressive stress.

Diagram showing longitudinal strains in tensile and compression — Longitudinal strains in tensile and compression

When a material is under tensile or compressive stress, it undergoes longitudinal deformation. Up to a specific stress limit, known as the elastic limit, that deformation is recoverable, meaning that the material will return to its original shape and dimensions once the applied force is removed. The ratio between the stress and strain within the elastic limit is young’s modulus.

Young’s modulus, Shear modulus, and Bulk modulus are the three main elastic moduli used to describe the elasticity / stiffness of materials. Young’s modulus is the modulus of elasticity we use when dealing with longitudinal stress and strain.

Young’s modulus, same as all the other elastic moduli, is only practical while the deformation is elastic, meaning that deformation is recoverable once the load/force is removed, and it is not permanent.

Formula & Units

Diagram shwowing the slope of the elastic region or young’s modulus on a stress-strain curve — Young’s Modulus (E) = Slope of the elastic region line = Δσ / Δϵ

Young’s modulus = normal stress / axial strain

E = \frac{𝜎}{𝜀}

Where:

E is the young’s modulus
σ is the normal stress (F / A)
ε is the longitudinal strain (ΔL / L_o)

The SI unit of young’s modulus is pascal (Pa), which is equal to 1 Newton per square meter (N/m²).

The US customary unit of young’s modulus is pounds per square inch (psi).

Using Young’s Modulus Example

Suppose you needed to figure out how much stress is required to stretch a steel rod elastically by 0.8%. Well, you would simply take the young’s modulus of steel, which is approximately (160 GPa), and multiply it by 0.8%.

So the stress required to stretch the steel rod by 0.8% is equal to:

0.008 x 160 = 1.28 GPa

Young’s Modulus Calculation

Find the young’s modulus of a cylindrical sample that is under a tensile force of 8 N and a longitudinal deformation / stretching of 5 mm. The dimensions of the sample are given as: length = 300 mm, and radius = 60 mm.

Solution

Step 1) We write down the given parameters:

F = 8 N L = 300 mm ΔL = 5 mm r = 60 mm

Step 2) We find the tensile stress caused by the force:

𝜎 = \frac{F}{A} = \frac{8}{𝜋 (0.06)^{2}} = 707.4 N / m^{2}

Step 3) We find the tensile strain:

𝜀 = \frac{Δ L}{L_{o}} = \frac{5}{300} = 0.017

Step 4) Now, we can find the young’s modulus of the cylindrical sample:

E = \frac{𝜎}{𝜀} = \frac{707.4}{0.017} = 41611.8 N / m^{2}

Factors Affecting Young’s Modulus

There are several factors that can affect the young’s modulus of a material, including:

Temperature: Changes in temperature can affect the atomic structure of a material and how its atoms vibrate, which can affect its stiffness and Young’s Modulus. Generally, young’s modulus decreases with increasing temperature.
Strain rate: The rate at which a material is subjected to stress can also affect its young’s modulus. In general, young’s modulus increases as the strain rate increases.
Microstructure: The internal structure of a material, including the size and distribution of its grains or crystals, can affect its young’s modulus. Generally, materials with larger grains or crystals have higher young’s moduli.
Moisture content: Some materials, such as wood and paper, can absorb moisture from their environment, which can affect their young’s modulus. Generally, young’s modulus decreases as the moisture content increases.
Porosity: The presence of pores or voids in a material can affect its young’s modulus. Generally, materials with higher porosity have lower young’s modulus.

These factors can interact with each other in complex ways, and understanding their effects on Young’s Modulus is an active area of research in materials science and engineering.

Applications of Young’s Modulus

Young’s Modulus has many practical applications in various fields. Some of the most common applications of young’s modulus include:

Structural engineering: Young’s modulus is used extensively in structural engineering to design and analyze structures such as bridges, buildings, and towers. Engineers use young’s modulus to determine the strength and stiffness of different materials and to ensure that structures are safe and stable.
Manufacturing: Young’s modulus is used in manufacturing to select the appropriate materials for different applications. For example, a material with a high young’s modulus may be used in the construction of airplane wings, where stiffness is essential for flight performance.
Medical implants: Young’s modulus is used in the design of medical implants, such as hip replacements and dental implants. The implants need to have the right stiffness to withstand the forces of everyday use while also being compatible with the human body.
Geology: Young’s modulus is used in geology to understand the properties of rocks and minerals. For example, geologists use young’s modulus to determine the strength of rocks and to understand how they may deform under stress.

These are just a few examples of the many practical applications of Young’s Modulus. Its importance in materials science and engineering cannot be overstated, as it provides a fundamental understanding of how materials behave under stress and strain.

Young’s Modulus Summary
Definition	The material’s resistance to longitudinal - elastic deformation when under a uniaxial stress
Symbol	$E$
Formula	$E = \frac{𝜎}{𝜀}$
Units	Si unit (Pa)	US unit (psi)

Is young’s modulus the same as the elastic modulus?: Yes, young’s modulus is sometimes referred to as the elastic modulus, tensile modulus, or the modulus of elasticity in tensile and compression.

Is young’s modulus stiffness?: Young’s modulus is a measure of the stiffness of a material. It represents the uniaxial stiffness of a material or how much it resists stretching or compression.

What happens when young’s modulus increases?: Young’s modulus is a measure of the uniaxial stiffness of a material, so a higher young’s modulus means it would be harder to stretch or compress a material.

Is young’s modulus constant?: For isotropic materials (materials that exhibit the same mechanical properties in all orientations) and at constant temperature and pressure, young’s modulus is constant. However, for anisotropic materials, young’s modulus value depends on the direction of the force.

Does young’s modulus change with temperature?: Yes, in general, an increase in temperature would decrease young’s modulus and vice versa.

How is Young’s Modulus determined experimentally?: Young’s modulus can be determined experimentally through a variety of methods, including tensile testing, bending tests, and ultrasonic measurements.

What materials have high Young’s Modulus values?: Materials that have strong, rigid, and tightly bonded atomic structures tend to have high Young’s Modulus values. Examples of such materials include metals such as steel, titanium, and tungsten, as well as ceramics like diamond, and silicon carbide.