Elsevier

Mechanics of Materials

Volume 71, April 2014, Pages 52-61
Mechanics of Materials

A comparative study on the capability of Johnson–Cook and Arrhenius-type constitutive equations to describe the flow behavior of Mg–6Al–1Zn alloy

https://doi.org/10.1016/j.mechmat.2013.12.001Get rights and content

Highlights

  • Modeling the flow behavior of Mg–6Al–Zn alloy using two phenomenological equations.

  • High accuracy of Arrhenius-type model comparing with Johnson–Cook one.

  • JC model cannot describe the work-hardening and softening behaviors at one time.

Abstract

In the current study, the predictability of two phenomenological constitutive equations, Johnson–Cook (JC) and Arrhenius-type ones, for describing the flow behavior of a magnesium alloy (Mg–6Al–1Zn) under hot deformation conditions has been evaluated. Towards this end, a series of hot compression tests were performed over a temperature range of 250–450 °C, under strain rates of 0.001, 0.01 and 0.1 s−1. Using the experimental results obtained through implementing the predetermined compression tests, the related parameters and material constants in the constitutive equations were calculated. In order to compare the performance of the models, the statistical parameters of correlation coefficient and absolute mean error were employed. The results imply that the predictability of the Arrhenius-type equations is much stronger in estimating the flow behavior compared to that of the JC model; although more constants are needed to be calculated when using the former equation. It is concluded that the JC model, in contrast to the Arrhenius-type equations, is not reliable for the materials possessing tangible softening stage in their stress–strain curves such as magnesium AZ series.

Introduction

Magnesium alloys, in general, and the Mg-AZ series, in particular, have recently received great attention due to their potential applications in diversified industries such as transportation ones (Mordike and Ebert, 2001). These materials, however, suffer from an inadequate mechanical performance especially ductility because of their hexagonal crystal structure with only two independent operative basal slip systems (Abedi et al., 2010). As a promising method to refine the microstructure, and, in turn, enhance the mechanical characteristics of such alloys, thermomechanical processing (TMP) has been put into practice by a vast number of researchers worldwide. It is taken as a given that to successfully optimize and control the desired thermomechanical processing cycles, the identification of the material response to the external loading conditions i.e., strain, strain-rate and temperature is highly necessitated (Quan et al., 2011).

In this context, the constitutive equations which are mathematical relations between the aforementioned parameters have been extensively and successfully used to numerically analyze and model the flow behavior during thermomechanical procedures (Lin and Chen, 2011). Such numerical descriptions have been also widely introduced into the finite element codes in order to computationally simulate metal forming processes. In general, the constitutive equations can be divided to 3 categories (Lin and Chen, 2011): Empirical, phenomenological, and physical based models. Although physical based models give a more precise estimate of flow behavior, most of them involve large number of material constants and need quite a few accurate experiments in order to extract the material constants. On the contrary, the phenomenological constitutive models predict the flow stress using predetermined limited experiments; meanwhile they often have a sustaining predictability. The main advantage of these models is that they need fewer constants than physical based models, and the required experiments can be run more easily compared to the ones for physical based models.

To date, many efforts have been put towards developing several phenomenological models. Among these models, the Johnson–Cook (JC) model (Johnson and Cook, 1983) and Arrhenius type equation (Sellars and McTegart, 1966) have been widely employed for different metallic materials. Up till now, the JC model which provides a function with few material constants has been employed in many researches on Al alloys (Clausen et al., 2004, Lin et al., 2012, Roy et al., 2012, Vural and Caro, 2009) and steels (Li et al., 2013, Lin and Chen, 2010, Lin et al., 2010a, Prawoto et al., 2012, Wang et al., 2010) in a wide range of strain rates and temperature. However, there are limited studies surveying capability of this model for AZ series of Mg alloys. Accordingly, it would be of high interest to devote a part of the present paper to establish a JC model for a typical Mg-AZ alloy (e.g., Mg–6Al–1Zn).

When it comes to the Arrhenius type (Garofalo) constitutive equation, it is to be mentioned that there are many works in which the applicability of this equation has been approved for different materials (i.e., Al alloys (Haghdadi et al., 2012, Lin et al., 2010b, Wu et al., 2012a), Mg alloys (Changizian et al., 2012, Qin et al., 2010, Slooff et al., 2007), Steels (Ji et al., 2011, Krishnan et al., 2011, Li et al., 2011, Li et al., 2012a, Li et al., 2012b, Mirzaei et al., 2014, Phaniraj et al., 2011, Samantaray et al., 2010, Samantaray et al., 2011, Yin et al., 2013), and Ti alloys (Cai et al., 2011, Pilehva et al., 2013, Shafaat et al., 2011, Zeng et al., 2009)). It has been recently (Haghdadi et al., 2012, Li et al., 2011, Lin et al., 2010b) shown that considering the effect of strain may result in a very good agreement between the flow stresses estimated by the equations with those measured by experimental results. Although there are several works on steels (He et al., 2013, Li et al., 2013, Samantaray et al., 2009) in which the capability of strain-compensated Arrhenius type constitutive equation has been compared with the JC model, there is not such a comparative report for Mg-AZ alloys in the literature.

Considering all the above into account, the present work was conducted to analyze and predict the flow behavior of Mg–6Al–1Zn alloy under different hot compression conditions using two phenomenological constitutive equations. Towards this end, the experimental results obtained from the tests were used to develop the Arrhenius type and Johnson–Cook equations. Finally, the predictability of the models was assessed using standard statistical parameters.

Section snippets

Experimental procedure

The experimental alloy was received in as-extruded condition in the form of rod with the chemical composition of Mg–6.53Al–0.74Zn(wt.%). As is seen in Fig. 1, the initial microstructure of the alloy consists of grains with the mean size of about 13 μm. The cylindrical compression specimens of 12 mm in height and 8 mm in diameter (according to ASTM E209) were machined from the rod. The deformation direction was selected parallel to the extrusion direction. The hot compression tests were then

Flow behavior

The typical true stress–true strain curves which were obtained through hot compression tests are illustrated in Fig. 2. At the initial stage, the stress increases rapidly due to the work hardening resulted from the continuous accumulation of dislocations. Following to the increase of strain, stress reaches to a peak value (corresponding to εP) and then decreases gradually into a steady state stress by increasing the strain. This form of the curve is more distinguished at higher strain rates and

Summary

In this paper a comparative study has been conducted to predict the flow behavior of Mg–6Al–1Zn alloy under hot deformation condition using Johnson–Cook and Arrhenius type constitutive equations. For this purpose, the hot compression tests were performed in the temperature range of 250–450 °C under the strain rates of 0.001, 0.01 and 0.1 s−1. The main obtained results are as follows.

  • 1.

    The JC model is inappropriate to make a good description for predicting the flow behavior in the applied range of

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