Stress and strain are two critical concepts in materials science and engineering, determining how materials respond under different forces. From the tension in a bridge cable to the compression in a concrete column, understanding stress and strain is essential for designing structures that can withstand various loads.
Although stress measures the force per unit area within a material, strain quantifies the deformation or displacement that occurs due to this force. These concepts help engineers predict how materials will behave under different conditions, whether it’s a stretching rubber band or a bending beam.
In this blog, we will explore the definitions and types of stress and strain, the mathematical formulas behind them, and the differences between related concepts like plane stress and plane strain. By the end, you’ll have a clear understanding of these fundamental principles and their applications in engineering. You will also learn about 2D and 3D stress tensors. Uniaxial tensile test and strain stress diagram and its different parts will be examined
1. What is stress?
Stress is the force or pressure that a material feels when something pushes or pulls on it. Imagine squeezing a rubber ball or stretching a rubber band both are under stress. The more you push, pull, or twist something, the more stress it feels. If the stress becomes too much, the material can change shape or even break. Stress helps engineers understand how much weight or pressure a material can handle before it starts to get damaged.
Stress in the context of the mechanics of materials is defined as the internal force per unit area within a material.
1.1. Real Examples of Stress
- Tensile Stress: The tension in a cable supporting a suspended bridge
- Compressive Stress: The compression in a concrete column supporting a building
- Shear Stress: When a material is subjected to a force that causes sliding, such as in scissors cutting paper.
- Torsion Stress: The torsion in a drive shaft of a car
- Bending Stress: The bending stress in an arch bridge
1.2. Stress Formula
The formula for stress (
) is:
Where:
is the stress,- F is the applied force,
- A is the original cross-sectional area over which the force is applied.
Example: If a rod with a cross-sectional area of 0.01 m² is subjected to a tensile force of 1000 N, the stress is:
1.3. Stress Unit
The unit of stress in the International System of Units (SI) is the Pascal (Pa), which is equivalent to one Newton per square meter (N/m²). Other units of stress include:
- Kilopascal (kPa): 1 kPa=1000 Pa
- Megapascal (MPa): 1 MPa=1,000,000 Pa
- Pounds per square inch (psi): Commonly used in the United States, where 1 psi=6894.76 Pa
Figure 1: Stress concept [reference]
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In this Abaqus training package for beginners, which is designed for FEM Simulation students in mechanical engineering, various examples in the most widely used fields are presented:
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2. What is strain?
Strain is a measure of deformation representing the displacement between particles in the material body. It quantifies how much a material reshape under forces. Imagine a rod getting stretched under tension. Strain tells you by what fraction its length increased compared to its initial length. Strain is a dimensionless quantity as it represents the ratio of the change in length to the original length.
2.1. Real Examples of Strain
- Tensile Strain: When a rubber band is stretched, the length increases compared to its original length.
- Compressive Strain: When a sponge is squeezed, its length decreases compared to its original length.
- Shear Strain: When a stack of paper is pushed sideways, the layers shift relative to each other.
2.2. Strain Formula
The formula for linear strain (
) is:
Where:
is the linear strain,
is the change in length,
is the original length.
is the change in length,
is the original length.Example: If a steel bar originally 2 meters long stretches by 0.01 meters under a load, the strain is:
2.3. Strain Unit
Strain is a dimensionless quantity because it is a ratio of two lengths. Therefore, it does not have any units. It is often expressed as a percentage or a fraction.
Figure 2: Strain concept [reference]
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3. What are engineering and true stress and strain?
Engineering Stress and Strain and True Stress and Strain are two different ways of measuring the stress and strain in a material.
3.1. Engineering Stress and Strain
- Engineering Stress (
): Engineering stress is calculated by dividing the applied load (F) by the original cross-sectional area (
) of the material:
): Engineering stress is calculated by dividing the applied load (F) by the original cross-sectional area (
) of the material:
- Engineering Strain (
) Engineering strain is calculated by dividing the change in length (
) by the original length (
):
) Engineering strain is calculated by dividing the change in length (
) by the original length (
):
3.2. True Stress and Strain
- True Stress (): True stress is calculated by dividing the applied load (F) by the instantaneous cross-sectional area (A) of the material:
): True stress is calculated by dividing the applied load (F) by the instantaneous cross-sectional area (A) of the material:
True Strain ( 
): True strain is calculated by integrating the incremental change in length over the original length:
3.3. Differences between Engineering and True Stress and Strain
Engineering stress and strain are sufficient for many engineering problems, especially in the initial linear elastic region. True stress and strain become more important for highly deformable materials (plastics, metals undergoing large strains) where necking (reduction in area) occurs.
- Engineering Stress and Strain are simpler to calculate and are often used in the initial design and analysis of materials. They are based on the original dimensions of the material and do not account for the changes in dimensions that occur during deformation.
- True Stress and Strain provide a more accurate representation of the material’s behavior, especially at higher strains. They account for the actual changes in the dimensions of the material as it deforms.
3.4. Example: Tensile Test
Imagine a round metal bar being stretched in a tensile test. Initially, the engineering and true stress and strain will be very close. However, as the bar necks (becomes thinner in a specific region), the true stress will increase due to the reduced area, while the engineering stress will decrease based on the constant original area.
- Engineering Stress and Strain: Suppose a metal rod with an original cross-sectional area of 10 mm2 and an original length of 100 mm is stretched by 5 mm under a load of 5000:
- True Stress and Strain: Assume the cross-sectional area at the stretched length is 9.5 mm2 due to necking.
In the third lesson of Abaqus for beginners package, there is a simple example of a beam that would help you to simulate concentrated load on it and do a stress analysis to have the stress distribution one the beam.
4. What are types of stress and strain?
Stress and strain can be categorized into several types based on the nature of the forces and deformations involved. Below are detailed explanations of different types of stress and strain, along with examples and formulations.
4.1. Normal Stress and Strain
- Normal Stress is the stress that occurs perpendicular to the surface of the material. It can be tensile (stretching) or compressive (squeezing).
- Normal Strain is the deformation per unit length in the direction of the applied load.
In the previous sections, formulas and examples for this type of stress and strain were presented.
Figure 3: Normal stress and strain [reference]
Learn the normal stress with a practical example: 3D truss modeling in Abaqus.
4.2. Bending Stress and Strain
- Bending Stress occurs when a moment or bending force is applied to a member of structure like beam, causing it to bend.
- Bending Strain is the deformation resulting from the bending moment.
Formulation:
Where:
- M is the bending moment,
- y is the distance from the neutral axis,
- I is the moment of inertia,
- E is the modulus of elasticity.
Example: Let’s consider a beam (like a metal rod) with a length of 2 meters, supported at both ends. If you apply a force of 500 Newtons directly in the middle of the beam, it will bend downwards. To calculate the bending stress, let’s assume the beam has a rectangular cross-section with a width of 0.1 meters and a height of 0.05 meters.
- Bending Moment:
- Distance from the neutral axis (half the height): y=0.025 m
- Moment of inertia:
- Bending Stress:
Figure 4: Example of bending [reference]
4.3. Shear Stress and Strain
- Shear Stress (
) occurs when forces are applied parallel to the surface of the material. - Shear Strain (
) is the angular deformation resulting from the applied shear stress.
) occurs when forces are applied parallel to the surface of the material.
) is the angular deformation resulting from the applied shear stress.Formulation:
Where:
is the angle of deformation.
is the angle of deformation.Example: A bolt with a cross-sectional area of 20 mm2 is subjected to a shear force of 1000 N. The angle of deformation is also equal to 
.
- Shear Stress:
- Shear Strain: base on formula, shear strain is equal to
.
Figure 5: Shearing [reference]
4.4. Torsional Stress and Strain
- Torsional Stress () occurs when a material is subjected to a twisting moment or torque.
- Torsional Strain () is the angular deformation resulting from the applied torque.
Formulation:
Where:
- T is the applied torque,
- r is the radius of the shaft,
- J is the polar moment of inertia,
- G is the modulus of rigidity,
- θ is the angle of twist.
Example: Let’s say you have a solid metal rod with a length of 1 meter and a circular cross-section with a diameter of 0.05 meters. You twist one end of the rod while the other end is fixed. The twisting force you apply is called torque. If you apply a torque of 100 Nm to the rod, you can calculate the torsional stress using the formula (The modulus of rigidity for the material is equal to 
):
- Torsional Stress:
- Angle of Twist:
Figure 6: Torsion [refernce]
4.5. Thermal Stress and Strain
- Thermal Stress (
) occurs when a material is subjected to a change in temperature, causing it to expand or contract. - Thermal Strain () is the deformation resulting from the change in temperature.
Formulation:
Where:
- E is the modulus of elasticity,
- α is the coefficient of thermal expansion,
- ΔT is the change in temperature.
Example: A steel rod with a coefficient of thermal expansion of 
and a modulus of elasticity of 200 GPa is heated from 20
to 100.
- Thermal Strain:
- Thermal Stress (assuming no deformation is allowed):
Figure 7: All types of stress[reference]
Now you have learned types of stress and strain, right? But I’m sure you have heard of Principal stress and strain, and also Mohr’s circle. What are these? what is the difference between them? You can learn all of it in this article: “What is Von Mises Stress? | Mohr’s Circle and Principal Stress and Strain“
5. What is the stress-strain curve?
The stress-strain curve is a graphical representation of the relationship between stress and strain for a material under uniaxial loading. It provides critical insights into the mechanical properties of the material, such as its strength, yield stress, and ultimate stress.
5.1. Uniaxial Test
A uniaxial tensile test involves stretching a material sample along a single axis until it fractures. During the test, the applied force and the corresponding elongation of the specimen are measured. From these measurements, stress and strain can be calculated. The most important data of this test is the stress-strain curve, which we will examine.
5.2. Stress-Strain Curve Regions
The stress-strain curve typically includes several distinct regions, each representing different mechanical behaviors of the material:
- Elastic Region:
- In this region, the material deforms elastically, meaning the deformation is reversible. The material returns to its original shape when the applied stress is removed.
- The slope of the curve in the elastic region is the modulus of elasticity or Young’s modulus (E).
- Hooke’s Law: This law represents the relation between strain and stress in elastic region. (
)
)- Yield Point and Yield Stress:
- The yield point marks the transition from elastic to plastic deformation. The yield stress (
) is the stress at which this transition occurs. - Beyond the yield point, the material undergoes permanent deformation.
) is the stress at which this transition occurs.- Plastic Region:
- In this region, the material deforms plastically, meaning the deformation is permanent and non-reversible.
- The stress continues to increase with strain but at a decreasing rate.
- Ultimate Stress and Ultimate Strength:
- The ultimate stress or ultimate tensile strength (UTS) is the maximum stress the material can withstand.
- Beyond this point, the material begins to neck, and the cross-sectional area reduces significantly.
- Necking:
- Beyond the UTS, the material’s cross-sectional area starts to localize and reduce in a specific region (necking). This leads to a decrease in the calculated engineering stress even though the true stress continues to increase until fracture.
- Fracture Point:
- The fracture point is where the material ultimately fails and breaks apart.
- The strain at this point is known as the fracture strain.
Figure 4: Stress-Strain curve[reference]
6. Plane stress vs. plane strain
In mechanical engineering, plane stress and plane strain are two fundamental concepts used to simplify the analysis of stress and deformation in materials. Plane stress applies to thin objects where stress in one dimension (like thickness) is negligible, often seen in thin plates under load. Plane strain, on the other hand, is used for long structures where strain in one direction is minimal, such as in long walls or dams. Understanding these concepts helps engineers predict material behavior under various loading conditions effectively.
Learn the plain stress and strain with practice in Abaqus with this workshop of the beginners package.
6.1. Stress and Strain Tensors
The stress and strain tensors are mathematical representations used to describe the state of stress or strain at a point in a material. Here are the key points about stress and strain tensors:
- The stress and strain tensor are a 3×3 matrix that fully describe the state of stress or strain at a point.
- The diagonal elements represent normal stresses or strains.
- The off-diagonal elements represent shear stresses or strains.
- For 2D plane stress or plane strain conditions, these are reduced to 2×2 matrices.
6.2. Plane Stress
Plane stress occurs when the stress across one dimension (usually the thickness) is negligible compared to the stresses in the other two dimensions. This is often seen in thin plates subjected to forces perpendicular to the plane. For example, a thin metal sheet is subjected to tension in its plane. The stresses in the thickness direction are negligible compared to the in-plane stresses.
- Stress Tensor for Plane Stress:
6.3. Plane Strain
Plane strain refers to a condition in which deformation occurs only in two dimensions while the strain in the third dimension is negligible or zero. This is common in long structures where the length dimension is much greater than the other two. For example, a long concrete dam is subjected to loading along its height and width. In these cases, the strain across the length of the structure is assumed to be zero, and only the strains in the other two dimensions (height and width) are considered.
- Strain Tensor for Plane Strain:
6.4. 3D Stress and Strain
3D Stress and Strain involve all three dimensions, accounting for stresses and strains in every direction. For example, a cube of material subjected to forces on all faces, creating a complex stress state.
- Stress Tensor (3D):
- Strain Tensor (3D):
Figure 5: 3D and 2D stresses graphical element [reference]
Question: What are the units of stress and strain in Abaqus? Get the answer in: “Used Units in Abaqus | Abaqus units“
6. Conclusion
This article discussed the concepts of stress and strain in materials under deformation. Stress, defined as internal force per unit area, was explored through examples and its formula and units. Strain, a measure of deformation, was explained along with real-world examples and its calculation method. The article also covered the differences between engineering and true stress and strain, and their applications. Various types of stress and strain, including normal, bending, shear, torsional, and thermal, were examined with relevant formulas and examples. The stress-strain curve and its regions were outlined, highlighting its importance in understanding material behavior. Finally, the distinctions between plane stress and plane strain, and their application in engineering analysis were discussed. This comprehensive overview provided insights into the fundamental mechanics of stress and strain, crucial for material engineering and analysis.
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