4 main nonlinear material models – general version

Recently a friend challenged me to do something in Nastran. At a certain point, I realized that I know too little about nonlinear material types. So as typical me, I stopped a challenge and decided to learn more about material nonlinearity, starting from the simplest models to the most advanced ones when it comes to steel 🙂

This article will be general in nature. If you want to learn more specific things for NX Nastran (and Femap to a certain degree) – read the full “geek version” here.

Most common nonlinear material models:

• Nonlinear elastic material
• Bilinear elastoplastic material
• Multilinear plastic material
• Rigid-plastic material

1. Nonlinear elastic material

This is the first type of material nonlinearity. It is dedicated only to isotropic materials. A nonlinear elastic material will not yield, which means that however high the load will be, after taking that load away the material will return to its initial state without any permanent deformations. It also does not show strain hardening (after several times loading-unloading cycles it acts the same).

To define it you should make stress vs strain function (in Femap it is called Function Dependence). This function can be defined for the first and third quadrant (positive stress + positive strain and negative stress + negative strain) – this takes into account that material may exhibit different properties in compression. If you define the function only in the first quadrant the first point must be at 0.0 point. It will be assumed that the relation in tension and compression are the same, but some codes may actually require you to define both even if they are symmetric. If you are using Femap please note that you should use a chart 4.. vs Stress when you are defining the nonlinear property (there is also stress vs strain function).

2. Bilinear elastoplastic material

I’m tempted to write this is a “default” nonlinear material for steel. It is not as robust as the third option, but really easy to set up instead. This material can include strain hardening (also kinematic+isotropic hardening is an option). You can define this material in several ways. First of all, you need to define which yield criterion will be used – 4 possibilities are usually present (this can be a different list in your FEA package, but for steel von Mises is used and I cannot imagine a code that does not have this one!):

Yield criterion:

• von Mises
• Tresca
• Mohr – Coulomb
• Drucker – Prager

When you decide on the proper yield criterion (for steel von Misses is used as I mentioned), you need to input initial yield stress (for von Mises and Tresca) or 2*cohesion and angle of internal friction (for Mohr-Coulomb and Drucker–Prager). Then the work hardening slope has to be defined unless you want your material to be perfectly plastic (which is a default setting in most codes I think). Work hardening slope H (in units of stress) is a slope of stress vs plastic strain, as shown below:

If you will have cyclic loads in your analysis, defining the hardening rule is a good idea. The most common possibilities are:

• Isotropic (usually default)
• Kinematic
• Isotropic + Kinematic

3. Multi-linear plastic material

This is the more advanced option than the bilinear material described previously. A lot of settings remain the same (yield criterion, hardening rule, initial yield point). Instead of work hardening slope H, you have to define a curve showing the multilinear relation between stress and strain.

The stress-strain curve for this type of material must be defined starting in (ε=0; σ=0) point. The second point on the curve should be at initial yield (ε1; σy) for von Mises and Tresca. For Mohr – Coulomb and Drucker – Prager models this second point should be (ε1; 2c). The slope of the line connecting those first 2 data points must equal the value of E (different codes react differently if the data are mismatched). Work hardening slope Hk for each following step is calculated as follows:

where  is a plastic strain in point k.

4. Rigid plastic material

I don’t think this material is used all too often, but since it is a possibility I decided to write something about it as well. Most of the settings are identical as for the previous 2 examples. Selecting this type of material leads to a situation where we define only the plastic part of a stress-strain curve – the solver will assume that material is rigid in the elastic regime. I think that the graph below clearly shows everything:

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