Dig even a little bit into the different fea packages out there and you will notice a distinction between linear and nonlinear analysis. You might then wonder under what circumstances to use linear analysis and what circumstances to use nonlinear analysis?
First off, let me say that mechanically speaking, there is no such thing as a linear situation. That being said, there are circumstances under which a linear approximation is close enough. It’s just like when you neglected friction back in college, your answer isn’t going to be right, but maybe (just maybe) it’s right enough.
As an aside, one of my favorite ah-ha moments in college was when one of my professors, in explaining ‘accuracy’ vs. ‘precision’ said (i paraphrase)”even if you could measure things to an infinite precision, uncertainty prevents you from measuring with infinite accuracySo let’s get into it. The major circumstances that will dictate whether you should be using a nonlinear analysis are material stiffness, displacement, and contact.
material stiffness
Specifically, I’m talking about the stiffness (Young’s modulus). If it’s been a while, Young’s modulus (or tensile modulus) is the slope of the elastic region (the far left) on a stress-strain diagram.
As you can see, the graph is linear up to the proportional limit. As you might imagine, the proportional limit marks the end of linearity. It is also around this point that the yield strength and elastic limit occur. Everything below this limit can be accurately approximated using a linear material model. This is often suitable for analyses because most designs typically try to stay well below the yield strength in order to maintain a healthy factor of safety.
But…
The stress-strain diagram I mentioned before (this one) is typical of metals. Plastics are a different story (and anisotropic materials are another story).
Just look at this set of stress-strain curves from Azom (one of the best materials sites). Some plastics like ABS are linear for a stretch, others (like polypropylene and polyethylene) are not. To perform an accurate (not precise, FEA will give you precision regardless) analysis, you would need to crank the analysis up to nonlinear.
It’s worth noting that it can be difficult to get stress-strain curves for specific material. It is usually not found on any materials sites, and involves getting in contact with Sabic, DuPont, etc. At that point you are at the mercy of the customer service of the resin supplier.large deflection
Every mechanical system has a stiffness. A given force applied to a system will yield a displacement (or if you prefer Hooke’s Law, F=kx).
A linear analysis assumes that the stiffness of the stays constant regardless of deflection. Strictly speaking, this is never true; as a system deflects, its change in shape results in a change in stiffness. It is useful however because the small change in stiffness resulting from a small deflection can be neglected. It’s the same idea (from trigonometry) as when we say it is ok to assume that sin(x) = 0 for all small x. It makes the math easier (which is to say it results in less equations and therefore less computing power required) and is often accurate enough.
Where this linear assumption falls on it’s face is when deflections are large. Consider a 24″x1″x1″ cantilever beam: if 1lbf were to give you .005″ deflection (making those numbers up), it’s a safe bet that 2 lbf would give you about .010″ deflection. A linear analysis would then tell you that 10,000 lbf would give you a 50″ deflection (impossible because the beam itself is only 24″ long ). That is a gross exaggeration, but it proves the point that displacements aren’t linear.
Put another way, the ‘sin(x) = 0 for all small x’ theory that is foundational to linear analysis likewise fails for large deflection. sin(45°) certainly does not equal zero.
contact
Contact can also impact stiffness. As a system deflects, if faces come into (or come out of) contact with each other, the effective stiffness of the system can change. Depending on the system geometry that change can be small (in which case a linear approximation may be acceptable) or large. I find the magnitude of the stiffness delta to be difficult to estimate, so if given the option I would recommend using a nonlinear analysis for any kind of complicated contact setup.
thin wall parts
The idea with thin wall parts is that they are more likely to deflect from stretching as well as bending, whereas stretching can be neglected in solid body parts
rapidly applied loads
Generally speaking, linear solvers assume that a load is gently applied, which is to say that inertial effects are negligible. If this is not the case, you would do well to turn to nonlinear analysis.
There you have it; the main reasons to use nonlinear analysis. Is there anything I missed? When do you most often turn to nonlinear analysis? How far do you try to push linear analysis?
Let me know in the comments.
Related posts:
- some notes on nonlinear fea with contact
- when in doubt…
- statistical tolerance analysis basics: worst-case tolerance analysis
Article by chris loughnane
Chris is Design Engineer + Sustainability Advocate at a Boston area design consultancy and has written 134 sweet posts for the Product Development Notebook.
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