Estimation of damage in high strength steels
Graphical abstract
Introduction
Ductile fracture progresses through void nucleation, growth and either coalescence under the influence of favourable plastic strain and hydrostatic stress, which is well established [1]. The voids which are the basic source of ductile fracture process are nucleated heterogeneously at sites where compatibility of deformation is quite difficult/complex. The preferred sites for void nucleation are: inclusions, second phase particles, while in high purity metals, voids can nucleate at grain boundary triple points also. Under tension, void nucleates prior to necking but after a neck is formed and hydrostatic tensile stresses develop, the void formation becomes more prominent. The frequency of occurrence of nucleating particles is having strong influence on ductile fracture processes [2], [3]. It has been demonstrated in references [2], [3] that the true strain to fracture decreases rapidly with increasing volume fraction of second phase particles. Present authors has clearly demonstrated that the dimple geometries and its distributions on the tensile fracture surfaces in systematically varied microstructures of copper strengthened high strength low alloy steels (HSLA 100) and metastable austenitic stainless steel (AISI 304LN) are having strong influence on their corresponding mechanical properties [4], [5], [6], [7], [8].
The strain-induced growth of voids within the cluster is accelerated when the voids are closely spaced in a low strain-hardening material subjected to high level of stress triaxiality [9]. It is then logical to assume that initially void nucleating strains are quite smaller than in fracture strain. In fully densed materials, voids are generally nucleated during straining, usually by the decohesion or fracture of large inclusions or precipitates [2], [3]. The nature of void formation results in cavities being nucleated over a range of strains, void nucleation in most of the alloys begins early in the deformation process, and as a result the fracture process is controlled by the void growth and coalescence. Predicting the ductile fracture micro-mechanisms of structural materials subjected to complex loading conditions (i.e., multiaxial stress-states) requires modelling and simulations. However, the prediction of fracture of structures manufactured of high-toughness steels, such as are used in submarine hulls, pipe line components, etc. cannot be made solely by employing either linear or nonlinear fracture mechanics. The tensile fracture behaviour of quenched and tempered HY 100 and HSLA 100 steels have been explored in detail as a function of stress-state elsewhere [10], [11], [12], [13], [14], [15]. The result of these studies show that, for loading parallel to the long transverse direction of the plate, failure at high stress triaxilities occurs as a result of localised shear deformation (i.e., shear bands, dislocation piled up, etc.), as void links up due to void sheeting micro-mechanism [9], [10], [11], [12], [13], [14], [15]. Such failure has been shown to depend on the presence of elongated MnS inclusions distribution, which are oriented normal to the tensile axis and which nucleate large, elongated, cylindrically shaped voids [9], [10], [11], [12], [13], [14], [15]. Material separation subsequently occurs at shear localisation develops between the elongated, primary voids, resulting in the formation of secondary voids (Fig. 1). Secondary void sites are active in initiating voids only after large local strains are induced, due to deformation localization in the ligament between the primary voids (shown in Fig. 1). The detail of the ductile fracture micro-mechanisms (i.e., void nucleation, growth and coalescence) is schematically represented in Fig. 1.
From this figure, it is clearly noted that void nucleates at initial inclusions present in the material (i.e., image a, and corresponding point a on the curve) at UTS/just before UTS (i.e., point b on the curve), where there is favourable amount of plastic strain and hydrostatic stress. Beyond UTS (i.e., at point c, d and e on the curve and their corresponding images), void growth and coalescence take place under the influence of more strain (i.e., more hydrostatic stress due to necking) and fractured at point f (i.e., corresponding fracture surface showing dimples). Hence, the void accumulation inside the material strongly depends upon the parameter mentioned in the figure. These parameters are also interacting each other while deformation takes place.
A large number of constitutive and micro-mechanical models are available in the published domain to explain the ductile fracture process since last 40 years. Some elegant studies are reviewed and briefly discussed in Table 1 year wise. Recently Gorkunov et al. [28] has proposed an approach for estimating the accumulated damage level in the steel under tension or torsion by magnetic measurements. A large body of published literatures is also available on the application of neural network technique for damage estimation and continuum damage mechanics (CDM) in engineering components and structures. In the present study, damage is defined as per the concept propounded by Gurson is [1]: void nucleation, growth and coalescence and hence fracture, which means the extent of void area fraction. The literatures related to the application of neural network technique to predict the damage accumulation with its influencing parameters is very limited. Hence, it will be worthy now to investigate the role of each individual variable on damage accumulation in high strength low alloy steels under tension. The complex correlation between damage accumulations with its influencing parameters is extremely important to design a new structural component.
Neural networks are having enormous usefulness in these circumstances, not only to estimate the mechanical properties of the materials but wherever the complexity of the problem is overwhelming from a fundamental perspective and where simplification is unacceptable [29]. Accordingly, the modelling of damage estimation has to cover a range of conditions, and it is not easy to predict the extent of damage of an unknown material. In this study, in view of the complexity of the phenomena, neural network techniques are applied in place of the usual regression analysis or physical models.
The problem of damage accumulation inside a material clearly involves many variables and considerable complexity. The purpose of this study is not only to identify the parameters which control the deformation and fracture of porous or cavitating alloys but also to correlate the complex relationship between the extent of damage with its influencing parameters. In this present study, we shall distinguish the influence of each individual parameter on damage accumulations in high strength low alloy steels under tension. In the present context, the optimization of damage accumulation needs access to a quantitative relationship between the stress, strain, temperature, strain rate, stress triaxility and the ultimate the extent of damage accumulations. A neural network method has been developed to correlate those and applied extensively for applications within a Bayesian framework [30], [31], [32], [33], [34], [35], [36].
Section snippets
Organisation
In order to understand the basic philosophy of the whole content of the research work, the organisation of the paper has been schematically represented in Fig. 2. It has been designed in such a way that it would be an easier task for any readers to understand the whole philosophy of the present study.
Technique
Neural network is a simple regression method in which a flexible non-linear function is fitted with the experimentally measured data, the details of which have been reviewed extensively elsewhere [29], [30], [31], [32]. It is nevertheless worth emphasising some of the features of the particular method employed here, which is referred by MacKay in his pioneer studies [33], [34], [35], [36].
The Bayesian framework of the network used in the present context is able to indicate two uncertainties. A
Variables and database
We have extensively carried out literature study to understand the ductile fracture micro-mechanisms and their interpretations while explaining the mechanical behaviour of high strength low alloy steels under various operating conditions. Fig. 1 clearly defines the basic philosophy of the problem. It is indicated that the input parameters are interactive each other during damage accumulation. The analysis is based on the published data and is, therefore, limited to quantities that are readily
Neural network analysis
In this present study, both the input and output variables were first normalised within the range ±0.5 as follows in Eq. (2):where xN is the normalised value of x; xmin and xmax are respectively the minimum and maximum values of x in the entire dataset (Table 2). The normalisation is not necessary for the analysis but facilitates the subsequent comparison of the significance of each of the variables. The normalisation is straightforward for all the quantitative variables.
Application of the model
The neural network can capture interactions between the input variables because the functions involved are non-linear in nature. The nature of these interactions is implicit in the values of the weights, but the weights are not always easy to interpret. For example, there may exist more than just pair wise interactions, in which case the problem becomes difficult to visualise from an examination of the weights. A better method is to actually use the network to make predictions and to see how
Prediction of the model
The optimised committee model has been used to predict the influence of all individual input variables on damage accumulation in high strength low alloy steels and they have been discussed in the following subsections. Fig. 15(a–e) shows the prediction according to the example shown in Table 2 (i.e., last column). When we observe the effect of stress, other parameters were kept constant to investigate how the void area fraction is varying with stress. The isolated influence of all individual
Conclusions
A neural network model has been created to enable the estimation of the fraction of void area in high strength low alloy steels as a function of stress, strain, stress triaxility ratio, strain rate and temperature of deformation. Nevertheless, it would have been better model if we could have included initial inclusions fraction, its shape factor and distribution and initial microstructural quantities. The model successfully reproduces experimentally observed trends. It can be exploited in two
Acknowledgements
The authors are grateful to Professor H.K.D.H. Bhadeshia, University of Cambridge, UK for the provision of Neuromat Neural Network software for the present analysis. The authors would also like to thank to all the respected reviewers for their positive and constructive recommendations which helped a lot to prepare this manuscript.
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