Estimation of johnson-cook constitutive model constants from sheppard and jockson model

This work introduces a novel approach for determining the material constants of the Johnson Cook constitutive model, which is used to estimate flow stress when a material is open to higher strain rates, stresses, and temperatures. In this study the material characteristics/constants of the Johnson-Cook constitutive model were derived by using another flow stress model which was introduced by Sheppard and Jackson. The input data was collected from published hot compression test results. This method can potentially reduce the experimental efforts required for evaluating the data of the Johnson-Cook model. In this work aluminum alloys 7075-T65I and Ti- 6Al-4V have been considered for analysis. These materials have their mechanical properties at high temperatures and are of excellent significance for new manufacturing procedures that use deformation to produce a bonding in the solid state, such as friction stir welding and magnetic impulse welding. The flow stress variation of the two models and similar characteristics have been observed at some points over the temperature range. The strain rate constant (c ) and temperature softening coefficient (m) for each material have been determined and results were compared with experimental data which were found in the literature. By adopting this method it is possible to eliminate torsion test to know the strain rate effect on a material. These data could also be used to simulate possibly the best processes as forging, rolling at higher temperatures, and creep.


Introduction
A typical model consisting of the plastic behavior in the current generation manufacturing processes, such as friction stir welding (FSW, magnetic impulse welding and other well recognized processes, such as hot rolling, forging to estimate reliable material properties, strains, and strain rates at high temperatures are needed. There is always a need for the newer techniques as the current techniques have lack of experimental measurements at high temperatures, high strain rates, and high strains. The problems involved in measuring mechanical properties at high temperatures are also high. In the case of friction stir welding (FSW), reproducing the conditions anticipated in the shear layer experimentally is exceptionally hard, and the community agrees that one of the most difficult challenges to bypass the creation of effective and reliable FSW simulations is a shortage of available mechanical properties data. Atomistic simulations can be used to obtain mechanical behaviour of monolithic metals and alloys, however simulations of alloys with grain sizes higher than 1 µm are extremely difficult.
In the absence of real measurements at high temperatures, strain rates, and strains, realistic extrapolations based on trustworthy prepared models are the next best choice. For hot deformation of materials, the Johnson-Cook material model and the constitutive model presented by Sheppard and Jackson are the most extensively used.
For materials subjected to huge strains, high strain rates, and high temperatures, Johnson and Cook (1983) established a constitutive model . This model requires data from torsion testing over a wide range of strain rates, static tensile tests, and dynamic Hopkinson bar tests at elevated temperatures for the material constants. Maheswari et al., (2013) established a dimensionless constitutive modelling containing the relationship between stress, strain rate and temperature is the prominent input for assessing thermo mechanical deformation processing results. 350-500 • C at strain rates ranging from 0.01-10/s up to a real strain of 0.9. The flow stress rapidly climbed to a maximum value. With increase in deformation temperature and decreasing strain rate, the peak stress was decreased. A Zener-Hollomon parameter with an Arrhenius term can be used to model the effects of strain rate and temperature on hot deformation behaviour. Ning et al. (2018) [11] predicted machining forces under various cutting conditions using the identified J-C constants and then compared them to the corresponding experimental forces. There was a high level of agreement between predicted and experimental forces. Song et al. (2019) [12] adapted the Johnson-Cook constitutive model to define the deformation behavior of the material when it was exposed to high temperatures and strain rates. Ballistic impact testing was used to verify the correctness of the improved model. The hot deformation behaviour of aluminum matrix composites was investigated by Neelima et.al., (2010) [13] developing constitutive equations using Johnson-Cook (JC), modified JC (m-JC), Arrhenius, and modified Zerilli and Armstrong (m-ZA) models, and the capacity of these models was estimated and compared using average absolute error.
In this work a comparison of both Johnson-Cook constitutive model and Sheppard-Jackson model was carried out by using MATLAB software for two materials Al7075-T651 and Ti-4Al-6V. With these observations, the material constants strain rate constant (c ) and temperature softening coefficient (m) of Johnson-Cook constitutive model have been determined and the results were compared with known values which were found in the literature.

Johnson-Cook Constitutive Model
Gordon R. Johnson and William H. Cook proposed this model in 1983 for materials subjected to significant strains, high strain rates, and high temperatures. The model's basic form is as shown (1) Where is the flow stres is the equivalent plastic strain is the dimensionless plastic strain rate for = 1.0s -1 is the homologous temperature.
A, B, n, C, m are the material constants.
In this model the data for the material constants is collected from torsion tests over a broad spectrum of strain rates, static tensile tests, dynamic Hopkinson bar tensile tests and Hopkinson bar tests at elevated temperatures. The following sections provide detailed information about these tests.

Torsion Test
Torsion tests across a wide variety of strain rates are the primary source of Johnson-Cook material data. The state of the test specimen is well defined, high strains may be reached without forming geometric instabilities, and small to big range of strain can be obtained with the same testing technique, are some important features.

Dynamic Hopkinson Bar Tensile Test
This is also a important test in getting the data of the Johnson-Cook material model. This is done over a range of temperatures. The higher temperatures can be achieved by surrounding the in-place test specimen in an oven such that the temperatures are gradually given for several minutes before to testing. It is also possible to test materials to larger strains; the Hopkinson bar tensile test cannot be accurately evaluated after necking begins to occur in the tensile specimens. In addition, the effects of adiabatic heating at significant strains might make the results even more complicated. The materials are softening as a result of the higher temperatures.

Tensile Test
Static tensile tests should also be used in order to obtain test data of the Johnson-Cook model. The stress for the tension test data is based on the neck current area, and the strain is referred as ln(A 0 /A), where A 0 and A represent the initial and areas of the neck. Static tensile testing should also be used to get test results for the Johnson-Cook model. Because of the presence of hydrostatic tension, this is the case. The Bridgman correction factor is used to approximate the equivalent tensile flow stress obtained from the tension measurements. Usually both tensile and torsion tests have to be conducted such that the disagreement between two modes of deformation can be recognized and rewarded.

Sheppard-Jackson Constitutive Model
Constitutive equation introduced by sheppard and Jackson Where Z is the Zener-Hollomon parameter: Where T is temperature is the strain rate Q is the activation energy R is gas constant A,n, are the material constants.
The Zener-Hollomon parameter is used to relate the flow stress to the deformation strain-rate and temperature. It can be used to describe the material properties during solid state joining, such as friction stir welding, cold spray and magnetic impulse welding.

Material Constants
Material constants are used to describe the constitutive model for a particular material. Experimental work has been done by many researchers to derive the data of the constitutive models. Material constants for few materials was given in Table 1.

Flow Stress Variation
When a material is subjected to large strains, high strain rates and high temperatures its flow stress varies. Many models were proposed for this flow stress variation with temperature and strain rates.
Johnson-Cook constitutive model and Sheppard-Jackson model were the most useful in estimating this flow at particular temperature and strain rate. The flow stress variation for Al7075-T651 using both models at different strain rates is given in Figure 1.

Comparison of Flow Stress
From the above graphs (Figure 3.) it can be say that the value flow stress is increasing with strain rate at a temperature in both models. We have increased strain rate value from 1s -1 to 100s -1 and the corresponding increment in flow stress in both models has been drawn in flowing figures.

Constitutive equation given by sheppard and Jackson is (4)
First derivative of flow stress with respect to temperature is given as  The variation of the its first derivative over temperature is given in the following Figure 5   The change in flow stress value with increasing strain rate from 1s -1 to 100s -1 in both models was shown in above Figures (Figure 6 (a-f)). By using these observations the detailed procedure to determine C and m was explained in the following section.

Determination of Material Constants
By observing above graphs, at a particular temperature, as we increase the strain rate, flow stress is also increasing. And this increment in flow stress is proportionally equal. This characteristic of these two models can be used to predict the value of strain rate constant of the Johnson-Cook constitutive model by using data of the sheppard-Jackson model.

Determination of Strian Rate Constant C
At a particular temperature, as we increase the strain rate, flow stress is also increasing. And this increment in flow stress is proportionally equal in both the models from Fig. 3.6. with these observations the increment in flow stress at room temperature considered equal for both models and the material constants strain rate constant C and temperature softening coefficient of Johnson-Cook constitutive model have been determined as follows.

5.
The change in value of strain rate constant C effects flow stress variation with temperature at a strain rate. The flow stress variation and error in flow stress from experimental value is shown in the below Figure 7.  Al7075-T651 0.017 0.021 23.5 4

Conclusions
From the above results and observations it is concluded that  The Johnson-Cook constitutive model always gives more flow stress than Sheppard-Jackson model at a particular temperature and strain rate for any material over temperature range. From the graphs of first derivative of the flow stress  Flow stress of the material using Johnson-Cook model is decreasing with temperature rapidly.
But it was decreasing gradually if use Sheppard and Jackson model. Flow stress is increasing with strain rate at a particular temperature in both models and it was observed that the increment in flow stress is proportionally equal in the both models.  The values of C and m for the materials have been calculated from the Sheppard and Jackson model. The results were well approximated with the know values which were found in the literature. The value of C for Ti-4Al-6V is found to be 0.0091 and for Al7075-T651 is 0.021and the error from the experimental value is found to be low.