# Plotting the Johnson-Cook strength model

I'm trying to plot the stress-strain curve described by the Johnson-Cook strength (and eventually damage) models. The strength model is defined as:

σ=[A+Bεn][1+C ln(ε_dot*)][1-T*m]

where A, B, C, n, and m are material constants, ε_dot* is the non-dimensionalized strain rate, and T* is the homologous temperature where T*=(T-T0)/(Tmelt-T0)

To calculate the thermal softening (term in the last bracket of the J-C model), I need to determine the increase in temperature related to an increase in stress (and strain). I'm using the following equation:

ΔT=∫ Χ (σ/(ρ*cp)) dε

where Χ is the Taylor-Quinney coefficient (i've set it to 0.9), ρ is the density, and cp is the specific heat.

So my problem is that to calculate the thermal softening, I need to work out the increase in termperature - but that is dependant on stress! Can anybody help me with plotting this mode? The only way that I can think to do it is rearrange the ΔT equation in terms of σ, and then set up some kind of minimization function where ΔT or T* is the variable. I've tried doing this in MATLAB using the fminsearch command, but it's not working.

Any help would be really appreciated!!

AttachmentSize Johnson-Cook plasticity curves.xls270.5 KB

## Comments

### Re: Plotting the Johnson-Cook strength model

The answer depends on what you're trying to achieve and how you got the parameters in the first place.

The J-C parameters are usually determined by curve fitting a number of experimental true-stress vs true-strain plots.   Some of these experiments are under nominally isothermal conditions while others are under adiabatic conditions.  Temperature measurements under these conditions are difficult, if not close to impossible, to get.   In the absence of knowledge of termperature as a function of time, thermal softening effects cannot be subtracted out from the stress-strain curves.

As a result, the J-C parameters already have temperature effects built into them (including thermal softening) for a given plastic strain, strain rate, and temperature.

However, when you try to simulate something complex, say a Taylor impact test,  the plastic strain and strain rate at a material point can be quite different from an adjacent point.  Also, the strain rate will rarely be constant and in many situations the temperature will increase as the energy is dissipated via plastic deformation.

The usual way to deal with variable strain rates is to solve the momentum equation and estimate the strain rate from local velocity gradients.

To deal with  variable temperatures is to start with a reference temperature and do all the calculations at that temperature.  The energy equation is solved next (the one for delta T) and an increment of temperature is calculated keeping stress constant and using the increment of plastic strain.  This increment in T is used to update the temperature before going to the next time step.  This process is equivalent to splitting the coupled moementum and energy equations into two parts.  A significant body of research can be found that discusses numerical issues related to this type of "operator splitting".

-- Biswajit

### Reproducing curves

What i'd like to do is reproduce the 4340 steel curves from the Johnson-Cook 1985 publication in Engineering Fracture Mechanics. The following parameters are defined for the material:

A=792 MPa, B=510 MPa, n=0.26, C=0.014, m=1.03 and the reference strain rate ε_dot* is 1.0 1/s. I'm using a room temperature of 293K, and a melting temp of 1793K. The article assumes adiabatic compression, so i've set the Taylor-Quinney coefficient, Χ, equal to 1.0.

The curves are drawn for three strain rates, 1.0, 10.0, and 100.0. I've tried two different approaches, but neither agree with the curves in the article.

In the first approach, i've tried to apply what I understood from the last paragraph in your post. That is...calculate the stress at a constant temperature (i.e. using just the first two terms in the J-C equation), then determine the associated increase in temperature and use that to update the stress at the next increment of plastic strain. This differs to your solution as my calculation is not time-dependent like it would be in a code, so there is some variance there. The curve shape looks similar to that in the publication, but it fails to reach the maximum stress values.

In the second approach I rearranged the equation for deltaT in terms of stress, and wrote a minimization script to find the increase in temperature associated with an increase in plastic strain that would give the same stress value (in both the J-C model and the deltaT equation). Again, the curve shape is reproduced, but it fails to reach the maximum stress values that are plotted in the J-C journal article.

l'll try to attach my Excel working sheet.

Thanks again for you help!

### Re: Reproducing curves

Your approach seems to be OK.  Congratulations on one of the rare attempts to reproduce the original JC curves.

I recall having to use very small steps in the time integration for the temperature but got reasonably good results for a constant strain rate test.   A deltaT less than 10^-6 should give you very accurate results.  The discrepancy is puzzling.

-- Biswajit

### Drop Test of PP at minus 30oC

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Dear
All,

I would like to study the drop behaviour of Polypropylene at -30oC from a
height of 1.2 m in Altair Radioss. I have two questions in my mind:

Q 1: Is the Johnson/Cook model is the right choice for my kind of application?
I have seen many people using this model in their publications. But my concern
is that is it good model for Polypropylene as it is thermoplastics? I have
checked the material models from Abacus and in their matreial data base they
say it is good for metals only. I can see in this model that it takes into
consideration effects of strain rate, temperature and hardening.

Q2: What kind of tests I have to perfrom to get material data for my model:
tensile, compression etc and at what temperature these tests have to be
conducted? Is it possible that I perform my test at room temperature and at a
particular strain rate and then enter this data and my model will take into
consideration the variation of temperature and strain rates based upon my
boundary conditions etc. I reallly donot understand how it works?

I would be thankful if anyone could write me about my matter.

Thanks very much

Best Regards

Shahid

### JC Model: Parameter C determination

I have Stress Vs. Strain data for high speed tensile test

Test conducted at 1 m/s,5 m/s, 10 m/s and 15 m/s

(Corresponding Strain rate are 31.25,156.25,312.5 and 468.75 /S )

I don't know how to determine the parameter C, from the test results

Can you suggest me how to find the parameter C in JC model

Arun 