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Which phenomenological flow stress model is the best?

A couple of years ago a colleague who wanted to simulate high-speed machining asked me: " Which is the best phenomenological flow stress model for metals?" I wasn't able to give an answer right away and decided to look in the literature.

What I found was, every ten years or so, a new model appears in the literature that tries to solve some of the problems of older models. However, a clear ranking of models has not been established yet.

The word "best" may mean very different things to different people. Modern flow stress models are usually used in computational settings. So a general rule of thumb is that the "best" model should be both fast and accurate over a large range of conditions. Thus, the model should be able to represent strain hardening, strain rate effects, and temperature effects. The model should be amenable to a Newton type root search. Also an evaluation of the flow stress should take a minimal number of arithmetic operations. There are of course several other considerations depending on the application.

The plastic flow properties of metals can be extremely variable. In fact, even for a well characterized material such as OFHC (oxygen free high conductivity) copper, the flow stress measured by two different research labs may vary by 50% or more. Hence a model from one lab might match experimental data from that lab but be off the charts when compared with experimental data from another lab. There are many reasons for this variability, for instance texture, inadequate annealing, etc.

However, engineers who use these models do not usually have the resources to recalibrate model parameters for every material they simulate. The best bet is to go with a model that is at least partially based on the underlying physics and also computationally inexpensive. Which is a tall order!

Given these constraints, I have found that:

  • The Zerilli-Armstrong model is a computationally cheap option. However, this model has to be recalibrated for high strain-rate and high temperature applications.
  • The Mechanical Threshold Stress (MTS) model is more physically based and can be used over a larger range of conditions and is also easy to calibrate. However, the MTS model is computationally expensive.
  • The Preston-Tonks-Wallace model works for the largest range of temperatures and strain rates. But this model is also quite expensive and its calibration is more involved than that of the MTS model.

    For the non-expert, I would recommend the Zerilli-Armstrong model with the caveat that the model should be recalibrated if you are interested in high strain rates or high temperatures.

    I have attached an unpublished paper reporting some of my investigations and findings on models of OFHC copper. The paper also contains a discussion of validation metrics that I have found useful for comparing Taylor impact tests. Some of these metrics may also be used to compare time-dependent profiles of velocity or strain.

  • AttachmentSize
    PDF icon CuPlasticCompare.pdf2.2 MB


    Henry Tan's picture

    Flow stress is a function of F(texture, annealing, temperature, strain rate, ...). Make sure that the function variables set is complete, then can we compare the experimental data.

    Ah!  There's the catch!  If we knew all the prior processing history, the distribution of dislocations and their density, the geometric organization of crystals and their size, etc. of a particular piece of material then life would be easier.  However, the typical engineer or analyst will not have that information.  In fact, it can be argued that we don't have a methodology yet of coarse graining dislocation dynamics results to dynamic crystal plasticity models to macroscale models.  Clearly, multiscale models as they exist today cannot be used to simulate a car crash.  Does that mean that we should stop simulating plastic deformation until we have models that are good enough to represent reality?

    Henry Tan's picture

    We can still do the modeling and simultion even though the complete microscopic description of the system is inaccessible.

    The uncertainty can be treated as the randomness, as Amit wrote (

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