Stress and strain are fundamental concepts in engineering used to describe the mechanical behavior of materials under various loading conditions. Understanding stress and strain is crucial in designing and analyzing structures, as it allows engineers to predict how materials will behave and it ensures that these materials can withstand their real-world applications.

Strain, the underlying core of any FEA solver, measures the deformation of an object. Objects that are stretched, compressed, bent, or twisted undergo a change in shape. Strain is defined as this change in length divided by its original length. Although it is commonly presented as a percentage, strain itself is the unitless measure of deformation. A simple example of uniaxial strain is presented below:

The strain in the bar is defined by the change of length (in this case due to tension) divided by its original length.

Stress, on the other hand, is defined by the amount of pressure or tension applied over the cross-sectional area of the object.

The tensile stress in this bar can be calculated by taking the amount of force applied and dividing by the cross-sectional area of the member.

## The Stress-Strain Curve

The relationship between stress and strain can be graphed using what is called the Stress-Strain Curve. Each material has its own unique stress-strain curve. Materials that have similar characteristics, like ductile metals or elastomeric foams, will all share a similar looking Stress-Strain Curve. A stress-strain curve representative of most common metals is shown below for reference.

## The Elastic Region

The graph starts with a linear segment (left of the dashed line) referred to as the elastic region. This region is characterized by the linear relationship between stress and strain: as strain increases, stress increases by a proportional amount. The Proportional Limit marks the point when the stress-strain relationship moves away from being linear.

As implied by its name, any strain that occurs in the elastic region of the graph is considered recoverable. This means an object that is loaded short of the proportional limit will be able to return to its original shape. After loading exceeds the proportional limit, permanent deformation begins to occur.

Young’s modulus (E) describes the linear relationship between stress (σ) and strain (ε) of a material. It can be calculated by taking the slope from the elastic region of the stress-strain curve.

Young’s modulus is a convenient way to quantify the stiffness of various materials. A large Young’s modulus means that high amounts of stress (load) is required to achieve small amounts of strain. Stiff materials such as steel will have a large Young’s modulus (E~200,000 MPa), whereas more compliant materials such as nylon will have a much lower value (E~2,000 MPa).

### Yield Point

The yield point (marked on the graph above) is described as the point when “severe” permanent deformation begins to occur. Technically speaking, permanent deformation begins immediately after exceeding the proportional limit. The yield point, however, is defined when 0.2% plastic strain is achieved. This value is an industry standard for defining the elastic limit of a material because it is too difficult to experimentally record the exact point when plastic deformation begins to occur. The 0.2% offset yield ensures that empirical data remains reliable and repeatable.

## The Plastic Region

The non-linear segment of the stress-strain curve (right of the dashed line) is referred to as the plastic region. In this region, the stress-strain relationship described by Young’s Modulus no longer applies. Any small amounts of additional stress (load) cause increasingly large amounts of strain. Strain occurring within the plastic region of the graph induces permanent deformation and is no longer recoverable when unloaded. Note that the elastic strain is always recoverable, even after undergoing plasticity.

Say a given material is exposed to the stress (load) level marked on this graph. Since it entered the plastic region of the graph, there will be permanent deformation upon unloading. As previously stated, the elastic portion of strain is still recoverable. Thus, the total permanent deformation (or plastic strain) can be calculated by subtracting the elastic strain from the total strain. These areas are marked on the graph above.

### Ultimate Tensile Strength

The maximum stress level marked on this graph is defined as the ultimate tensile strength for the given material. At this point, no additional load can be supported and is typically used as the “failure” stress value when performing FEA. Note that this is not the point at which fracture occurs (unless it’s a very brittle material). Ductile materials will continue to stretch well beyond the ultimate tensile strength point despite strain occurring from no additional force. This is a process called necking (right of the dashed line) where the cross-section becomes overloaded, and the area starts decreasing by a greater proportion than the remaining material. Think when you pull a piece of playdough apart. The material continues to stretch until there’s only a thin filament formed between the two solid ends.

### Necking

During necking, it appears that the stress-strain curve undergoes decreasing amounts of stress despite the increasing strain, but this is misleading. Mathematically speaking, stress is defined as force over area. Since the area is decreasing during necking, the true stress would be rapidly increasing. In engineering, however, stress is calculated with respect to the original area since that is much easier to measure. For more details on this concept of True vs Engineering stress values, check out our other blog!

### Fracture

The final stop on our trip through the stress-strain curve is the fracture point where the material literally ruptures (breaks). This is commonly referred to as “Elongation at Break.” As previously discussed, materials that are more brittle (like glass) will fail at or very close to the ultimate tensile strength value. Whereas ductile materials (like stainless steel) are more representative of the behavior shown in this graph.

## Final Thoughts

The stress-strain curve is a fundamental concept in material science and engineering. When performing FEA, it’s very important to ensure the material properties in your model are accurately defined. You now should understand the defining points needed to create the graph.

If you need additional assistance about material properties or anything else in FEA, don’t hesitate to **reach out** to our team!