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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:
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!!