Technical ReportDevelopment of constitutive models for dynamic strain aging regime in Austenitic stainless steel 304
Highlights
► Tensile testing of Austenitic stainless steel 304 at elevated temperatures. ► Prediction of flow stress in Dynamic Strain Aging (DSA) regime. ► Constitutive modeling using JC, modified ZA, modified Arrhenius type, and ANN models.
Introduction
Austenitic Stainless Steel (ASS) 304 has found various applications in the field of defense and nuclear science because of its excellent corrosion resistance in seawater environment due to having addition of molybdenum which prevents chloride corrosion. It has low carbon content due to which it has improved wear and friction properties, less carbide precipitation in the heat-affected zone during welding and a lower susceptibility to intergranular corrosion [1]. This steel is particularly useful in nuclear reactors as it is used for the purpose of cladding of fuel rods. As the temperature attained in the reactors is very high, it becomes essential to study the properties of the material at elevated temperatures. Generally, these steels exhibit a peculiar phenomenon at elevated temperatures with low strain rates in which the mobility of solute atoms can become large enough that they can follow a dislocation during its motion and segregate to its core while it has to wait in front of an obstacle. The repeated segregation and detachment process causes the flow stress to oscillate. In such cases, serrations are observed, i.e. wavy pattern like saw tooth in the stress–strain plot. This effect is called Dynamic Strain Aging (DSA) or Portevin-Le Chatelier effect. Also, DSA is manifested by a negative strain rate sensitivity, which results in unstable, jerky flow [2], [3].
Constitutive equations give a mathematical representation of the flow stress behavior of materials and it involves a particular number of material constants which are evaluated using limited number of experimental data [4]. These equations are used in finite element software to simulate the material’s response under specified loading conditions [5]. Therefore, the accuracy of the numerical simulation largely depends on how accurately the deformation behavior of the material is being represented by the constitutive equation. Recently, several constitutive models have been developed which can be broadly classified into three categories, namely phenomenological constitutive models, physical based constitutive models and artificial neural networks [6]. In this work, Johnson Cook (JC), modified Zerilli–Armstrong (m-ZA), modified Arrhenius type equations (m-Arr) and Artificial Neural Networks (ANNs) model are employed on the flow stress data of ASS304. Out of these models, JC and m-Arr models are phenomenological constitutive models, while m-ZA model is a physical based constitutive model.
The JC model as put forth by Johnson and Cook [7] has been successfully used for a variety of materials for different ranges of temperature and strain rate as it requires fewer material constants and also few experiments to evaluate these constants. A number of modifications to this model are also available in literature. To represent the behavior of the materials with different strain rate sensitivity and to incorporate adiabatic temperature rise during material deformation, the model has been revised [8], [9].
On the other hand, the m-ZA model formulated by Zerilli and Armstrong [10] model has been used for different FCC and BCC materials over different strain rates and at temperatures between room temperature and 0.6Tm, where Tm is the melting point of the material [11]. The m-ZA model is preferred over JC model as it considers the coupled effects of strain rate and temperature. However, it is particularly not suitable to predict the flow behavior of material at temperatures above 0.6Tm and at lower strain rates by this model [8], [9].
The sine-hyperbolic law in an Arrhenius-type equation has been successfully applied for prediction of high-temperature flow behavior of materials. The relation between the flow stress, temperature and strain rate, particularly at elevated temperatures, can be expressed by an Arrhenius-type equation [12]. The original model has been revised several times to suitably represent the high-temperature flow behavior of various grades of materials. In this model, an exponential strain-dependent parameter was introduced in the sine-hyperbolic constitutive equation to predict the flow stress. The combined effect of the temperature and strain rate on the deformation behavior is represented by the Zener–Hollomon parameter (Z) in an exponent-type equation [13].
Recently, ANN has been applied for describing the hot deformation processes [14], [15]. ANN is an artificial intelligence technique that emulates the behavior of biological neural systems in digital software or hardware and this approach need not to have a well-defined process for algorithmically converting an input to an output. ANN methods are very valuable in processes where a complete understanding of the physical mechanisms is very difficult, or even impossible to acquire and where no satisfactory analytical model exists. The greatest advantage of ANNs is their ability to be used as an arbitrary function approximation mechanism that ‘learns’ from a collection of representative data of the desired mapping. The ANN then adapts itself to reproduce the desired output when presented with training sample input. ANN is ideally suited for the problem of estimating the flow stress from the available experimental data because of its inherently high parallelism. This model does not require explicit mathematical and physical knowledge of deformation mechanism and has good generalization performance. As compared to modeling by mathematical equations, the understanding of flow stress behavior in DSA regime becomes easier by using ANN modeling. [16].
The objective of this paper is to make a comparative study on JC, m-ZA, m-Arr and ANN models on their capability to represent the flow behavior of ASS304 at elevated temperatures and also to capture the effect of DSA phenomenon. Experimental data from isothermal uniaxial tensile tests at elevated temperatures were employed to determine the material constants of these models. Subsequently, the suitability of these models were evaluated by comparing the correlation coefficient, the number of material constants involved and the ability to track the deformation behavior.
Section snippets
Experimental details
The composition of ASS304 is given in Table 1. Flat tensile test specimens of thickness 0.9 mm were used to perform the experiments. The dimensions of the specimen were as per the Defence Metallurgical Research Laboratory, Hyderabad (DMRL) standards as shown in Fig. 1, which is a reduced version of ASTM: E8/E8M-11 standards. The samples were machined out of the raw sheet material by wire-cutting electro-discharge machining process for high accuracy and finish. Isothermal tensile tests were
Development of constitutive models
In this work, four-different constitutive models are considered: JC, m-ZA, m-Arr and ANN model. For flow behavior at elevated temperatures, JC model considers isotropic strain hardening, strain rate hardening and thermal softening as three independent phenomena, while m-ZA model also considers the coupled effects of temperature and strain rate on the flow stress. In m-Arr model, the effects of deformation temperature and strain rate are represented by the Zener–Holloman parameter and also the
Results and discussions
The equation for JC model is given by:
Fig. 5a–c shows the predicted values of flow stress for JC model. As it can be seen from Table 3, the value of C obtained is negative. This indicates the negative strain rate sensitivity, which is responsible for the occurrence of DSA. However, the R value for the model came out to be 0.7975 as shown in Fig. 6 and value of average absolute error was 33.0266%. In addition, influence of the coupled effect of
Conclusion
A comparative study has been made to compare the capability of JC model, m-ZA model, m-Arr model and ANN model to predict the flow stress behavior of ASS304 in DSA regime. Based on the analysis done, the following conclusions can be made:
- (1)
ASS 304 exhibits phenomenon of DSA at elevated temperatures and lower strain rates as presented in Table 2.
- (2)
JC model is not suitable to provide flow stress predictions of ASS 304 in DSA regime as it does not consider coupled effects of strain, strain rate and
Acknowledgements
The financial support received for this research work from Department of Atomic Energy (DAE), Government of India, through Young Scientist Research Award 2009/36/45-BRNS/1751 is gratefully acknowledged.
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