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# Types Of Nonlinearity In Finite Element Analysis – And Why You Might Want To Consider Them

Finite element analysis (FEA) solvers such as Abaqus are powerful tools for predicting the behavior of structures under load, but it's important to remember that the assumptions of linearity underlying FEA can break down under certain conditions. When that happens, the results of the analysis may not accurately represent the real-world behavior of the structure.

Nonlinearity can take many forms, but the three most common types are geometric, material, and contact nonlinearity. In this blog post, we'll take a closer look at each of these types of nonlinearity and explain why they matter when it comes to FEA.

## Geometric Nonlinearity

Geometric nonlinearity occurs when the deformation of a structure is so large that it causes significant changes in the shape. This can happen in structures that are subjected to large loads, where strain is above a typical threshold of around 5%. When deformation is significant enough, it can lead to changes in the stiffness and mass of the structure. This can have an important impact on the results of a linear analysis, as the assumptions of linearity are no longer valid.

Geometric nonlinearity can occur in a variety of structures, and one example is a washing line under tension. In a linear analysis, the washing line is assumed to be perfectly straight and the tension force is assumed to be proportional to the displacement. However, as the tension force increases, the washing line will begin to stretch and its shape (and the stiffness matrix in the FEA) will change, leading to a nonlinear relationship between force and displacement.

Mathematically, the nonlinear equation for the deformation of a washing line under tension can be represented as follows:

where F is the tension force, L is the length of the line under tension, L0 is the initial length of the line, and k is a constant representing the line's stiffness.

As we can see, the relationship between force and displacement is no longer linear, hence a traditional linear analysis would not be able to capture this behavior. In this case, a nonlinear analysis must be performed to accurately predict the behavior of the washing line under tension. This nonlinear analysis would take into account the change in shape of the line and the change in stiffness due to the stretching, which is often labeled ‘stress stiffening’.

For more information on nonlinear geometry, check out this blog post! To accurately analyze this type of behavior, a nonlinear analysis must be performed.

## Material Nonlinearity

Material nonlinearity occurs when the material properties of a structure change with deformation. This is relevant to most materials, but can become very important when simulating highly nonlinear materials such as rubber and certain types of metals. Material nonlinearity can also occur in materials that are subjected to high temperatures. The stress-strain relationship of the material is changing as it deforms. This can lead to changes in the stiffness and strength of the material and can result in a significant departure from the results that would be found in a linear analysis.

The stress-strain relationship of metals is perhaps the best example of material nonlinearity, where the stiffness of the material reduces as the strain increases beyond the yield point.

## Contact Nonlinearity

Finally, contact nonlinearity occurs when two or more parts of a structure come into contact and interact with each other. This can happen in structures that have moving parts, such as gears and bearings, or in structures that have multiple components that are connected together like a car body and chassis. The main cause of contact nonlinearity is the interaction between the two parts and, importantly, the change in the contact conditions as the parts move during the analysis. This can lead to changes in the stiffness and strength of the structure as a whole, and can have a significant impact on the results of a linear analysis.

## Final Thoughts

Nonlinearity is an important aspect of FEA that can greatly impact the accuracy and reliability of analysis results. It is important to take into account geometric nonlinearity, material nonlinearity, and contact nonlinearity when analyzing structures and systems, in order to ensure that the analysis results are accurate and reliable.

We’re always here to help, so if you have questions about nonlinearities in your models, or just FEA in general, don’t hesitate to reach out!

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