# Establishment of GJV450 Constitutive equation of vermicular Graphite cast Iron

There are many commonly used constitutive equations of metallic materials, including Power-Law model, Zerilli-Arstrong model, Johnson-Cook model and so on. The lohnson-Cook model is suitable for describing most metals at high strain rates and high temperatures. In this chapter, the Johnson-Cook model is used to fit the stress-strain curve.

The basic form of Johnson-Cook model is.

In the formula, δ is the flow stress, the first term on the right indicates the effect of strain on stress, An is yield stress, B is hardening coefficient, n is strain strengthening index; the second item on the right indicates the effect of strain rate on stress, and C is the influence factor of strain rate; the third term on the right represents the effect of temperature on stress, and m represents the sensitivity of the material to temperature, that is, the sensitive factor of thermal softening. It can be seen that the items in the formula are independent of each other, but from the stress-strain curve, the temperature softening term is independent and is not affected by the strain rate, but the effect of the strain rate on the flow stress is affected by temperature, so it is necessary to modify the second item in the formula.

The linear regression method is used to fit the stress-strain curve. The premise of linear regression is that the relationship between independent variable and dependent variable is linear. JC constitutive equation has three terms, two of which are constant, then it is transformed into a linear problem. First fit the strain strengthening term, then fit the thermal softening term, and finally fit the strain rate strengthening term.

(1) strain strengthening term fitting
If ε = ε 0, T = 0.001s-1, T = 20 °C, the constitutive equation of JC is changed to.

An is the yield strength, which can be obtained directly from the static tensile test, and the logarithm on both sides of the formula can be obtained.

The formula is changed into a straight line with a slope of n and an intercept of InB. The formula is fitted in MATLAB to get Bao 287 and n = 0.244.

(2) fitting of thermal softening term.
Keep the strain ε and strain rate ε constant, and divide the stress and temperature when T is 7.

Take the logarithm on both sides of the formula.

When the strain rate is kept constant, Tr=20 °C ·C ~ (- 1) T = 200 °C, 400 °C and 600 °C are substituted into the formula and fitted in MATLAB. The values of m are all about 0.985, indicating that the thermal softening term is independent and is not affected by the strain rate, which is consistent with the previous analysis results.

(3) solution of strain rate influence factor.
The nonlinear regression is carried out by using “Levenberg-Marquardt+Universal Global Optimization” in the mathematical optimization toolkit of IstOpt, and the C value under each parameter is obtained. The C value under each parameter fluctuates greatly, indicating that the strain rate strengthening term is affected by temperature and strain rate, which is consistent with the previous analysis results.

In the process of high-speed machining, the strain rate can reach 10000s-1. In order to give priority to ensuring the accuracy at high temperature and high strain rate, the average value of C under 10000s-1 is 0.0345. Therefore.

Under various experimental conditions, the error between the fitting value and the actual value is generally considered to be less than 15%. It can be seen from the table that except for the large error under 3000W, the other parameters are in a reasonable range, so the constitutive equation is effective.

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