When we run a structural Abaqus model, it might be fine to assume linear-elastic material behavior. This means that all of the deformation caused by loading is fully recoverable and no permanent damage or 'set' is incurred by the material. This is the case only when the analysis does not generate stresses exceeding the material yield point at any location within the structure.
However, if the loading is expected or known to cause stresses beyond yield and, as a result, the effect of plasticity needs to be captured, then we have a decision to make… which Abaqus hardening model should we use?
We’re focusing on metals here, since hydrostatic stresses in metals cause very minimal deformation, all of which is recoverable. Other materials, such as foams and soils, might require very little hydrostatic pressure to permanently deform, and we’ll leave them for another blog topic.
Although there are a number of options in Abaqus to define plastic hardening behavior, we’re typically bound to think about it in the familiar terms; isotropic or kinematic. There is also the option to combine these models, which we’ll expand on later in the article. To understand what these mean, we first need some background into yield criteria and what a material’s ‘yield surface’ looks like in 3-dimensional stress space. Fortunately, we published a post a few weeks ago that does exactly that. You can find it here, and we strongly recommend that you take a look if you’re not intimately familiar with deviatoric stress and the yield criteria.
Plasticity occurs in metals when we introduce enough strain energy that dislocations in the crystal lattice begin to move around, and, once that happens, the material will be deformed permanently. We’re all familiar with the stress-strain curve that looks like this:
This plot is representative of a typical tensile test that is loaded to its yield stress, σy, and then beyond to an arbitrary stress level, σ1, that produces plastic strain of εp. Seems pretty straightforward, right? Until we begin to consider what happens if we unload the material, and then reload (either in tension or compression). We know the material has deformed permanently, but how does that affect the stress-strain behavior upon loading for a second time?
The phenomenon of ‘strain-hardening’ might also be familiar, where upon reloading in tension the apparent yield strength is equivalent to the maximum stress seen in the previous loading sequence. To capture this constitutively, we must go back to the yield surface and think about what has happened to it after the yield criterion was met. Not getting too much into the mathematics of this, because there are a few different options for mathematical representation and material definition, but, from a conceptual standpoint, the isotropic model sees the yield surface simply grow in size as a response to material flow.
That gives us a stress-strain response that looks like this during fully-reversing cyclic loading.
As can be seen from the image above, if we were to encounter one cycle of fully-reversing tension-compression loading, it would result in a completely linear-elastic response after the very first cycle. Hence, Isotropic models are fine for representing material that is only to be loaded once - or is only to be reloaded in the same direction cyclically (i.e. tension-zero-tension) - but the expansion of the yield surface does not adequately capture behavior when loading is tension-compression-tension in nature. A term that might be a little less familiar is the ‘Bauschinger effect’. This is the terminology used to describe the combination of the permanent strain hardening and backstresses that the material will acquire as it flows. Kinematic hardening models capture this more appropriately, because instead of growing, the yield surface is shifting or translating, as shown in the image below.
The resulting stress-strain behavior much more closely reflects reality, as we can see here.
Mixed-Mode or Combined Hardening
In reality, we are always trying to fit mathematical models to mimic microscopic material deformation phenomena, and as such, there are always tradeoffs. Some materials have a tendency to soften after cycling while others harden. This means that, rather than the tension-compression stress-strain curve simply looping around the same path as is shown in the image above, the path of the curve can shift with number of cycles. To capture this, we can employ a model whereby the yield surface shifts and changes in size concurrently. This would be known as a mixed-mode or combined hardening model and is often considered the most accurate way to predict material behavior.
The stress-strain behavior seen below represents the response that might be seen over the first four cycles of a cyclically hardening material, which requires a mixed-mode hardening model to be accurately represented in FEA.
Which One Should You Choose?
Hopefully, if you’ve read to here, you now understand the differences between the available hardening models within Abaqus and many other FEA tools. To summarize, if you plan on one-off or pulsating loading cases, isotropic hardening is acceptable. It is also often the easiest to implement into the model. Kinematic hardening models allow for fully cyclic behavior, but miss the effects of cyclic hardening and softening. If these factors are important, mixed-mode or combined models allow their inclusion in the constitutive material behavior and are typically considered to offer the most 'complete' representation of the physics.
When you need to model loading that is above yield and cyclic in nature, it is important to select the most appropriate material model for the use case. Here, we’ve addressed the three main options – conceptually at least - and we hope this article can help you in identifying which that is.