When To Go Non-linear with Finite Element Analysis (FEA)

The first question an experienced structural analyst often asks themselves when confronted with a simulation problem is, “Do I need to go nonlinear with this one?” Years ago the answer would typically be, “Hopefully not.” The reason being was that this method requires more expertise, pre- and post-processing time, and computational time.

Historically, the computational time may have in fact made ‘going nonlinear’ impractical, given whatever product development schedule was in play. Nowadays, taking things nonlinear is more practical. Computers are blazing fast and software more amenable to setting up the analyses. Today the answer is often “Yes, I must capture the nonlinearity to get a proper assessment of the product performance.” Of course, there are still times when the answer is “No,” for example, if the nonlinearity is mild, or the product is significantly overdesigned. But the truth is we live in a nonlinear world, and our simulations should try to capture as close as possible the actual behavior of the assembly/component.

The nonlinear effects we’re talking about are things like, "Is the material stressed beyond its yield point?" The stress-strain curve is no longer a straight line at that point, so the nonlinearity is relatively obvious. There are less obvious situations though. Nonlinearity is generally broken up into 3 categories: Material, Geometric, and Contact.

1. Material nonlinearities consist of plasticity (yielding), creep, viscoelasticity, superelasticity, etc. FEA codes provide access to a myriad of complex nonlinear material constitutive models.
2. Geometric nonlinearities occur due to changes in the shape or location of the object that in turn cause changes in the loading or stiffness. For example, a guitar string gets stiffer and hits a higher note when it’s stretched, much like a turbine blade vibrates at higher frequency when it’s spinning. Linear analyses assume small deflections and rotations. If that’s not the case for your model, then the results can be very wrong. Rigid body rotations in particular invalidate the linear stiffness matrix and require it to be updated as it rotates.
3. When multiple bodies come into contact, or if a body contacts itself, then there is an abrupt change in behavior, hence the contact nonlinearity. The friction between the two would be an additional nonlinearity. Frictional effects are path dependent, another characteristic of nonlinear behavior.

In the video below, the expanding stent simulation incorporates all three types of nonlinearity. The stent plastically deforms as it is expanded such that when the balloon deflates, the stent remains in its expanded state holding the artery open. As it is deforming, local sections of the stent undergo large displacements and rotations. Also, once it’s expanded enough, it contacts the interior artery wall with frictional contact. Because of this extensive nonlinear behavior, the analysis has to be broken up into small steps. At each step the stiffness matrix has to be updated to account for the accumulated plasticity and large rotations. Eventually, it will also account for the abrupt stiffness change when the stent contacts the artery.

Stay tuned for future blog entries which will look at the various nonlinearities in more detail as well as cover other practical finite element modeling techniques.