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21 February 2017

# 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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If you enjoyed the post you can share it with friends – that would be a great help! If you have a spare 15 seconds write a comment with your thoughts on the matter or any questions you might have. I have a good history of replying to each and every comment.

#### Author:Łukasz Skotny Ph.D.

I have over 10 years of practical FEA experience (I'm running my own Engineering Consultancy), and I've been an academic teacher for a decade. Here, I gladly share my engineering knowledge through courses, and on the blog!

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### Join the discussion

Gustavo - 2021-05-21 15:18:02

Hi, Łukasz.

On the multi-linear hardening model. Could you suggest to me any material (web page, paper, book) which would explain it in more detail? I am still getting used to this topic.

Gustavo

Łukasz Skotny Ph.D. - 2021-05-30 16:19:11

Hey Gustavo!

For now, I don't really recall any good materials. I'm working on this topic for my nonlinear FEA course, but it's months before the course will be ready...

All the best!
Ł

Farha - 2021-04-23 08:32:26

Hi Łukasz,

Thank you for the article, It's really informative.

If I were to define the material behavior using the Ramberg-Osgood equation, how is the curve different from the bi-linear and multi-linear curve?

Łukasz Skotny Ph.D. - 2021-04-29 10:30:25

Hey Farha!

There is no way I could answer your question in a single comment - I'm literally working on lessons for my nonlinear FEA course right now, and it's a lot to cover.

In super short: Ramberg-Osgood equation would lead to a "smooth curve" - you can approximate that with a reasonable multi-linear material model, although there are some limitations for that I would say (like when the yielding can start, etc.).

All the best!
Ł

Thomas - 2021-03-26 19:00:38

Hi Łukasz,

Thanks for this great article. Really clarifies these categories in a straight forward way. One question- would there be any issue with using a nonlinear elastic model for stainless steel using actual stress strain data? This would be for a single loading cycle.

Łukasz Skotny Ph.D. - 2021-03-26 20:58:11

Hey Thomas!

I usually use a nonlinear plastic model for stainless steel, since I like to know what the plastic strain is. I haven't tested how this works with the nonlinear elastic model, but apart from the obvious (you get elastic strain, so you have to figure out how much of it is actually plastic strain), I think that this should work. I'm not sure though, as I never tried - so please test that before using it in actual design :)

All the best!
Ł

Martha - 2020-12-01 00:44:17

Hi Lukasz,
Great explanation! In my FEA model in ABAQUS, I have used bilinear material properties, and the graph is similar to what you have shown in no. 2, (of course). After the yield point, however, my axial deformation (U1) is increasing but U3 values (displacement along the lateral direction) are decreasing. Can you help me understand what might be happening?

Łukasz Skotny Ph.D. - 2020-12-01 15:13:50

Hey Martha!

Sure, I think it's a classical Poisson thing. Ar you stretch something, it wishes to maintain the same volume (at least to some degree). So as you stretch something along it's axis - the other two dimensions are getting smaller. This is what the Poisson Ration tells you about given material (how much of this "thinning" will happen). You can google for this - information shouldn't be hard to find. Sadly I never did a post about this subject.

But also, please note that on the charts I used the horizontal axis is strains, not deformations. While they will "look similar" in some cases, this is not precisely the same thing :)

All the best!
Ł

Alexander Karachun - 2019-10-29 14:44:19

Put my 2c. Look like picture that represent rigid-plastic material is actually related to multilinear plastic material, it represents only plastic part.
Solver calculate yield point from E and sigma_y and then use this curve for plasticity. With full curve user should care that E in linear material property and E calculated from curve should be same. With only plastic part user have one thing less to worry about.

Łukasz Skotny Ph.D. - 2019-10-30 07:36:31

Hey Alex!
This is a really good point, thank you for mentioning it. I really appreciate this!

Have a great day!
Ł

Miroslaw - 2019-04-19 07:02:11

Bart,
Here you go, handy info on materials. Some mat props to be find there too. https://www.varmintal.com/aengr.htm
Mirek

Bart - 2017-05-25 15:46:32

Hi Lukasz,

I am trying to set-up bi-linear elastic-plastic material in my software (MIDAS FEA) and in this particular piece of software, it is require to define graph of Plastic Strains / Yield Stress for Von Misses criterion (so I guess, depending on how many points on graph I define this can be bi-linear or multi-linear model).

There is no option to change to hardening slope or at least total strain.

I was looking through net but cannot find this data for any steel grade (I mean plastic strain-stress function). Could you maybe point me to right direction where I can get this kind of data?

Bart

Cormac - 2017-05-04 15:15:32

Firstly, many thanks for the interesting articles. It's great to see a practicing FE user's take on how to build trustworthy models. There's very little out there to bridge to gap between FEA classes and real world use.

I think there's something up with the notation in the Multi-linear plastic material section as there is no (ε1; 2c) point in the diagram.

Is there a case where the rigid plastic material model is the best choice? I haven't seen this model before.

Łukasz Skotny Ph.D. - 2017-05-04 18:31:40

They Cormac!

Thank you for the kind words, I'm really glad that you like the blog :)

The diagram is made for von Misses yield criterion, if you would use a Mohr – Coulomb or Drucker – Prager models there is no "yeild stress" in those. Instead a 2c (twice the cohesion) is used.

Have a great day
Ł