Tuesday, June 9, 2009

Application of Bi-Linear Material Models

Based on the comment from Burhop on my post about bi-linear material models, I'm expanding on the applications of bi-linear models. This deserves a topic on its own because I need to first clarify a few implied assumptions, mainly on how I would like to see this in "first-past solvers." I made the statement that the math is easy because I'm only switching E1 with E2 in the stiffness matrix after I reach the yield point and therefore should be able to be included in first-pass solvers. Well, it's not really that easy and here's where I need to clarify. I'll define first-pass solvers and typical thresholds of those solvers in terms of linear, nonlinear, static, and dynamic analysis.

Static vs. Dynamic
Most first-pass solvers, or those built into CAD packages, only run static solvers. Static solvers assume that the load is applied slowly and remains constant, or static. But along with that is the assumption that the deflections of the material are small. So what is a small deflection? Well, small deflection can be defined the same way that sin(theta) = theta for small angles. Basically, it is as small or large as you need it to be while still producing acceptable levels of error in the result.

Dynamic solvers not only solve simulations that require movement or changing loads over time, but they also have less error when solving problems with large deflections. I'll get back to this concept in a minute, so hold this thought.

Linear vs. Nonlinear
In my prior post I commented on linear material models. Linear models assume stresses and strains only within the elastic, or linear, range of the material (E1) and project the same elasticity of the material if the loads exceed yield. Nonlinear models define equations of state for the entire stress-strain curve. Bi-linear material models estimate the plastic range of the material with a linear curve following the slope of the line from yield to ultimate stress/strain points (E2) as shown in this repost of the stress strain curve of a typical steel material.
Where these definitions start to get confusing is that many materials that have large deflections, like rubbers, are defined with non-linear material models and linear material models are assumed to have small deflections. Therefore, FEA analysis tend to fall into two groups: 1)linear-static and 2)nonlinear-dynamic. Why would anyone want to do a non-linear static analysis?

Application of Bi-linear Material Model
That's just it, I'm doing a bi-linear material model in a static analysis. That's my typical application. It crosses the border between static load but with large deflections. If I were to assume linear materials, an increased load does not give large deflections as I'm still following E1, even above the yield point. By switching to E2, a small load causes a large deflection - as typically seen during necking of a metal material - while stress barely rises. In other words, I'm not going to get a strange stress riser in my post processor that I have to explain away. Instead, I'm going to get large deflections that will either a)crash my simulation or b)more accurately represent the system including interactions (interferences) with other components within the assembly.

What are some specific examples?
• When I'm designing sacrificial parts. My actual design my not ever exceed yield, but if there is an overload condition due to handling or unforeseen use of the product, then I will create a simulation that overloads the assembly. I need to make sure I design in a specific - and safe - failure mode. Shear pins are cheap to replace if it saves the motor!
• When I'm designing for deflection. Usually when designing for deflection, the parts are so overbuilt that I don't have to worry about stress failure. But as mentioned in my previous post, if I'm doing FEA it is often because the complexity of the part precludes me from being able to visualize load paths or failure modes. If I happen to yield a part, I need to know that the deflection in my simulation accurately represent deflections beyond yield with just a slight increase in load.
• Always. Yeah, I (almost) always use a bi-linear material model. The overhead to run the stiffness matrix with a bi-linear material model (static analysis) is so low that there is hardly an increase in run-times compared to a linear static analysis. I use a bi-linear material model just for the increased fidelity of the FEA model.