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ResourcesMaterialsA Step-by-Step Guide on How to Calculate Young's Modulus
Elasticity. Image Credit: Shutterstock.com/Anita Silalahi

A Step-by-Step Guide on How to Calculate Young's Modulus

Xomety X
By Team Xometry
August 18, 2023
 10 min read
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Young's modulus, also referred to as the modulus of elasticity, is a basic mechanical characteristic that measures a material's capacity to deform elastically under stress. It is often employed in engineering and materials science to evaluate a material’s stiffness and resistance to external forces. It’s a critical factor to understand if you wish to predict the material behavior in structures you design. 

Figure 1 depicts a generalized stress-strain curve. Young’s modulus can be determined by calculating the linear region's slope:

stress-strain curve

Stress-strain curve.

Image Credit: Shutterstock.com/Anshuman Rath

This detailed tutorial will go over Young's modulus and how to calculate it. We will examine the underlying ideas behind this important parameter and give precise instructions for completing the calculations.

1. Determine the Initial Length of the Substance Using a Micrometer

In the first step of the process, the initial length of the material or item must be measured using a micrometer. This crucial measurement serves as a baseline reference point for all subsequent calculations. Only when we have an accurate measurement of the original configuration can we properly assess its deformation and behavior under applied forces. This initial measurement is fundamental when calculating Young's modulus. Engineers and scientists can make well-informed decisions regarding the materials and structural design if they have a precise starting point that lays the groundwork for trustworthy and meaningful results.

2. Use the Same Micrometer to Determine the Material's Cross-Sectional Area

While the material's length marks the first measurement, the same micrometer should be used to assess its cross-sectional area. Carefully measure both its width and height so you can accurately calculate all subsequent changes. Since the cross-sectional area directly impacts the material's response to applied forces, it plays a vital role in determining Young's modulus. This precise assessment allows for a comprehensive evaluation of the material's stiffness and its ability to withstand deformation. Engineers can design structurally sound and efficient systems based on this valuable information.

3. Measure the Different Diameters of the Substance Using the Micrometer

The next step is to use the micrometer to gauge the item’s various diameters, making sure to take readings at multiple locations. This meticulous approach creates an accurate baseline for subsequent calculations by accounting for any irregularities in the material's dimensions. Precise diameter measurements are essential to calculate Young’s modulus because they tell you the original cross-sectional area of the substance.

4. Apply Forces on the Substance by Using Various Slotted Masses

Engineers next exert forces on the material using a variety of slotted masses to create controlled tension and deformation. They can see and examine how the material responds to various loads during this process. This step evaluates the material's elasticity and ability to effectively withstand external forces, which is important for calculating Young's modulus. Engineers can create dependable and long-lasting structures because the data gathered from these experiments tell them whether a given material will meet the structure’s needs. 

5. Use a Vernier Scale To Measure the Length As the Substance's Components Extend to Different Lengths

Use a Vernier scale to ascertain the substance's elongation during tensile testing precisely. Align the scale parallel to the specimen and take precise readings at regular intervals. The Vernier scale will let you measure deformation accurately enough to calculate Young's modulus. That precision guarantees more dependable Young's modulus values, enhancing the validity of your findings.

6. Graph the Length Measurements Against the Applied Forces

This visual representation of the relationship between stress and strain is crucial for determining Young's modulus. The graph helps you verify and analyze experimental data. The whole process gives you a quantitative measure of the material's stiffness, making it an essential aspect of the calculation process.

7. Calculate Young's Modulus Using the Equation E = Tensile Stress / Tensile Strain = (FL) / (A * change in L)

Young's modulus (E) is the material's stiffness. It’s determined by dividing the tensile stress (F/A) by the tensile strain (ΔL/L₀). This quantification of the material's resistance to deformation is essential for engineers and materials scientists. It aids in material selection, part design, and prediction of behavior. 

8. Analyze the Graph To Determine Whether the Material Exhibits Elasticity

Examine the graph of your experimental data. Identify the linear region — it indicates elastic behavior, the stress range from which the material will return to its original shape after the applied stress is removed. Young's modulus only applies to the elastic behavior of materials. It cannot be applied to the material’s performance after its elastic limit is exceeded. 

What Is Young's Modulus?

Young's modulus, also known as elastic modulus (E), is a numerical constant that quantifies a solid material's ability to withstand tensile or compressive deformation in a single direction. It represents the ratio of stress (force per unit area) applied to the material to the resulting strain (change in length per original length). A higher Young's modulus indicates a stiffer material, as it can withstand larger stresses with minimal elongation or compression. 

To learn more, see our guide on Using Young's Modulus.

How Does Young's Modulus Work?

Young's modulus is a material property that measures the stiffness and elasticity of a material. When a metal bar is subjected to a force "F" at each end, it deforms, resulting in a change from its original length "L0" to a new length "Ln." Stress is the force "F" divided by the cross-sectional area "A," while a strain is the relative deformation, (Ln - L0) / L0. Young's modulus is then given by (F × L0) / (A × (Ln - L0)), representing the material's resistance to deformation.

Why Is Young's Modulus Important?

Young's modulus is important because it allows engineers to predict when a structural implant will deform. They can thus design the implant to withstand any reasonable stresses it might encounter. Additionally, Young's modulus enables the calculation of dimensional changes in isotropic elastic materials under tensile or compressive loads. This property is valuable in materials science and engineering, ensuring that materials and structures can withstand stress and maintain their integrity.

What Is an Example of Young's Modulus Calculation?

Here are three examples of Young's modulus calculations for different materials:

1. Example 1: Steel

Suppose we have a steel rod with an original length (L0) of 2 meters, a final length (Ln) of 2.04 meters (the elongation is Ln - L0 = 4 centimeters), and a cross-sectional area (A) of 1 square centimeter (0.0001 square meters). The force applied (F) is 1,000 Newtons.

Young's modulus = (F × L0) / (A × (Ln - L0))

Young's modulus = (1000 N × 2 m) / (0.0001 m2 × 0.04 m)

Young's modulus ≈ 5 × 108 Pascal (Pa)

2. Example 2: Aluminum

For an aluminum rod with the same dimensions but a force of 800 Newtons:

Young's modulus = (800 N × 2 m) / (0.0001 m2 × 0.04 m)

Young's modulus ≈ 4 × 108 Pascal (Pa)

3. Example 3: Rubber

Now, consider a rubber band with an original length of 10 centimeters, a final length of 15 centimeters (an elongation of 5 centimeters), and a cross-sectional area of 1.0 square centimeters (0.0001 square meters). The force applied is 20 Newtons.

Young's modulus = (20 N × 0.1 m) / (0.0001 m2 × 0.05 m)

Young's modulus = 4 × 105 Pascal (Pa)

The comparison shows that steel and aluminum have Young's moduli that are several orders of magnitude higher, indicating they are stiffer and less elastic than rubber. This means steel and aluminum are better suited for load-bearing applications, while rubber is more flexible and deformable.

What Is Young's Modulus for Steel?

The Young's modulus of steel usually falls between 190 and 215 GPa. This value represents the material's stiffness or ability to resist deformation when subjected to tensile or compressive forces. It is a measure of the material's elasticity, indicating how much it will stretch or compress under stress. Factors such as the: composition, temperature, microstructure, and strain rate of the steel will affect its overall Young’s modulus.

What Is Young's Modulus for Aluminum?

Young's modulus of aluminum material is approximately 69 gigapascals (GPa). This value is well established and has been extensively studied in the field of materials science and engineering. Experimental measurements, theoretical calculations, and simulations have consistently confirmed this value. Factors that may affect Young's modulus in aluminum include: temperature, alloy composition, crystal structure, and manufacturing processes. For instance, alloying elements can alter the modulus by affecting the material's intermolecular lattice arrangement and bonding, leading to variations in its mechanical properties.

What Is Young's Modulus for Polyester?

Pure polyester exhibits Young's modulus of around 920 MPa, denoting its stiffness and resistance to deformation under tensile stress. Several factors can impact Young's modulus in polyester, including: its molecular structure, crystallinity, temperature, and manufacturing processes used to create it. Alterations in these aspects may influence the material's elasticity and overall mechanical properties, resulting in fluctuations in the modulus value.

To learn more, read our guide on Polyester Plastic.

What Are the Benefits of Calculating Young's Modulus?

The benefits of calculating Young's modulus (modulus of elasticity) are as follows:

  1. Material Characterization: Many engineers rely on Young's modulus to gain insights into the behavior of materials when exposed to external forces, enabling them to make accurate predictions about how the materials will respond to stress and strain.
  2. Design Considerations: Young's modulus helps engineers design and analyze structures to meet safety and performance standards.
  3. Stress Calculation: This parameter lets engineers calculate stress in a material, which is crucial for designing components to withstand external forces without failure.
  4. Predicting Material Failure: Young’s modulus helps designers predict the point at which a material will deform permanently or fail under stress.

Are There Downsides in Calculating Young's Modulus?

No, there are no downsides in calculating Young's modulus other than the destruction of test articles. It is a fundamental and essential property used in engineering and materials sciences. The calculation involves simple stress-strain testing, which is widely accepted and practiced. Moreover, understanding a material's stiffness and deformation behavior is vital for designing safe and reliable structures and predicting material performance under different conditions.

Are Young's Modulus and Elasticity the Same Thing?

No, Young's modulus and elasticity are not the same. Young's modulus measures a material's stiffness within the elastic limit, while elasticity is a broader concept encompassing a material's ability to deform and return to its original shape. Young's modulus is just one measure of elasticity, which includes other properties like shear and bulk modulus.

Summary

This article presented how to calculate Young's modulus, explained each step, and discussed the steps in detail. To learn more about Young's modulus, contact a Xometry representative.

Xometry provides a wide range of manufacturing capabilities and other value-added services for all of your prototyping and production needs. Visit our website to learn more or to request a free, no-obligation quote.

Disclaimer

The content appearing on this webpage is for informational purposes only. Xometry makes no representation or warranty of any kind, be it expressed or implied, as to the accuracy, completeness, or validity of the information. Any performance parameters, geometric tolerances, specific design features, quality and types of materials, or processes should not be inferred to represent what will be delivered by third-party suppliers or manufacturers through Xometry’s network. Buyers seeking quotes for parts are responsible for defining the specific requirements for those parts. Please refer to our terms and conditions for more information.

Xomety X
Team Xometry
This article was written by various Xometry contributors. Xometry is a leading resource on manufacturing with CNC machining, sheet metal fabrication, 3D printing, injection molding, urethane casting, and more.

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