Linear FEA calculations are the most common type of static analysis done with finite elements. There are a lot of benefits to the method itself, but of course, there are risks involved as well. Today you will learn when it is safe to ignore material nonlinearity. I will also tell you what things you should watch out for!

#### Linear FEA approach to static design

When your task is to assess a capacity of any given element/structure, you will automatically think about linear static. The name already suggests several very important things about calculations you are about to make.

Firstly “static” indicates that you don’t consider dynamic effects in the analysis. This is, however, a topic of its own and I won’t discuss it here.

What we will focus on is the “linear” approach:

Depending on whom you ask there are several “aspects” that can be treated as linear in the analysis:

• Material – an obvious choice. I think that most engineers automatically think about material nonlinearity when they are asked what nonlinearity is. This is a big thing, as there are many different material models avaiable and a lot of settings needed. It is good to know when you can ignore this effect and simply use linear material.
• Geometry – second obvious aspect. Nonlinear geometry can help you with buckling design or second order effects in the analysis. Unfortunately defining a case usually takes some time, also computing alone is much longer. This make it seems like a great idea to wonder when you can avoid all this trouble safely!
• Contact – this is a tricky one. Depending on the source you may have issues to determine if contact is always nonlinear, or can it be linear as well. I won’t take part in the discussion about definition – I hate argues about semantics! Whatever side of the fence you will take, contact can be nonlinear – so we will try to answer when ignoring it makes sense.
• Follower forces – this is a relatively small thing. If this is a “nonlinearity” at all again is a discussion I would say. If we will “clear” geometrical nonlinearity we are certain that deformations in the model are small. In such cases it doesn’t really matter if the loads follow the shape of the geometry or not. This would play a role in geometrical nonlinear analysis, but we are staying in linear zone today.

Since you already learned what problems you must consider let’s discuss how to deal with them. Today I will describe material nonlinearity, and when it is ok to ignore it.

#### When can you ignore material nonlinearity?

First of all, it would be great to understand what material nonlinearity do. In short, FEA calculates deformations of the model first. Then it calculates strain and based on this strain it calculates stress. If you follow Hooke’s law, the relation between stress and strain is linear. However, most materials show a nonlinear relation between stress and strain after the initial linear part of this relationship. This means that initial material is “linear” but when strain gets bigger material starts to be “nonlinear”: This means that when you define the linear case, you basically assume that you will always have “small strain”. Small here means that strain will never get high enough in your model to “reach” the nonlinear part. In essence, instead of a “real” material (marked in green below), you model a “fake” material (broken line below). In the small strain region they have identical properties – so everything is well! Unfortunately, there is no safety net here. If your model will get higher strains there are no “warning messages” – you will simply get unrealistically high stress. Mechanism of how solver does those calculations is simple. Look at the schematic below. Assume that analyzed point in your structure has a certain strain, higher than the “linear limit”. In reality, the nonlinear material would display stress marked in green. Since we are using linear material solver will “blindly” believe that the relation between strain and stress is linear and will produce much higher value! This is why you often get stresses in GPa instead of MPa in linear FEA. They are simply calculated with the assumption that stress always depends linearly on the strain.

#### Material nonlinearity – summary

You can safely ignore material nonlinearity when strains in the model (and resulting stresses) do not go out of the “linear zone”. The further away you get from the limit, the worse outcomes you get.

Just be aware that such conditions aren’t met in many models. You need to be aware that there are stress concentrations in some places. In those regions, stress will reach higher values. Of course, it is best to use nonlinear analysis in such cases. There are however code rules and “best practices” that allow you to estimate if the stress you have obtained is “dangerous or not” even if it is higher than yield (or even higher than ultimate stress). Such analysis requires vast post-processing time (where you analyze outcomes by hand) but is doable and quite popular.

In essence, when you use linear material and the strains are high you will have to spend a lot of time on post-processing to check (according to various standards) if the stress is “acceptable”. Code rules are based on experience in this regard as there is no good way to build a mathematical model for such checks. In my personal opinion, if post-processing takes a lot of time it is more effective to use a nonlinear approach where all you need to do is to check capacity and maximal plastic strain.