Microcrack propagation under monotonic and cyclic loading conditions using generalised phase-field formulation
Introduction
Advancements in the computer technology gave rise to the numerical methods combining the field of mechanics of materials with the field of materials science. Conventional macrohomogeneity in modern materials is no longer valid and microheterogeneity influence must be considered. Combination of non-destructive technology (NDT) methods for material inspection, experimental analysis and numerical simulations give rise to different computational methods. Perhaps the best example of NDT method is computed microtomography [1]. It is able to inspect different sized samples and create 2D and 3D computer models for numerical simulation. Such generated models often use direct numerical simulations (DNS) [2] which, although computationally very costly, allows accurate modelling of microstructure and results close to the experimentally observed ones [3]. The time-consumption problem can be alleviated by employing the multiscale methods [4] which separate the scales by transferring the information obtained at one scale to the next one. Detailed overview of multiscale methods, their development and application, can be found in [5], [6], [7]. The most popular method within the class of multi-scale methods is the computational homogenization method. It is based on the averaging of certain mechanical properties over a representative volume element (RVE), assumed to be a statistical representative of the macroscopic material behaviour, as first explained by Hill [8]. These methods have been proven successful in general when no material softening is expected [9], [10], and while they give promise to multiscale modelling of material fracture, there still seems to be a long way before they can be consistently and reliably applied in actual structural problems. When it comes to modelling the material failure, which is essentially a multiscale phenomenon [11], [12], there are still major problems and open questions with computational homogenization methods, as reported in Budarapu et al. [13].
Macroscopic cracks leading to a component failure are a direct result of the cascade of complex fracture events at the microscale. Since fracture processes at the microstructural level of highly heterogeneous materials are often accompanied by complex crack configurations, there is an ongoing tendency to use the phase-field fracture (P-F) theory in the frame of the computational multiscale methods, to reliably describe these processes including the crack nucleation, propagation, merging, kinking or branching without the introduction of any ad hoc criteria. The phase-field approach to fracture has become an area of major interest in the last decade, owing its success to its thermodynamical consistency, ease of implementation and resolution of different complex fracture processes [14], [15].
A considerable number of various phase-field fracture formulations have been developed for modelling of brittle fracture [16], [17], [18], [19], [20], [21] and extended to ductile behaviour [22], [23], [24], [25], [26]. The scope of applicability of these formulations is huge. Different multi-physics problems were analysed, e.g., related to the thermomechanical fracture [27], [28], electromechanical fracture [29] and hydraulic fracture [30]. Moreover, different phase-field formulations for crack description in compressive stress state were investigated [16], [31], [32]. Recently, the phase-field approach has emerged as a powerful computational tool to simulate and predict the fracture processes occurring at microlevel of heterogeneous materials [21], [33], [34]. This shows the great potential of this method.
The phase-field approach has recently been extended to the fatigue fracture problems, too. The phase-field models for fatigue are usually able to reproduce the main features of fatigue failure with fracture-based input parameters. Phase-field models with Ginzburg-Landau formalism were adopted in [35], while [36] presented the phase-field fracture model coupled with thermal and fatigue behaviour. More recently, the fatigue behaviour within the phase-field fracture models is introduced via fracture energy degradation on the account of strain or stress history in [37], [38], under the assumption of elastic material behaviour. The effects of plastic material behaviour inherent to low-cyclic analysis are introduced in [39] via Ramberg-Osgood relation, while complex cyclic plasticity effects like ratcheting are introduced in [40]. Very recently, a general phase-field framework for low- and high-cycle fatigue, which is able to accurately reproduce the features of monotonic fracture without the influence of the fatigue extension is proposed in authors’ work [41]. The potential of the phase-field models is thus enormous and not yet fully researched, especially when dealing with fatigue failure. The presented models show the ability to reproduce the main stages of fatigue failure naturally, i.e., the crack nucleation, stable and unstable propagation and other features like Paris law, Wöhler curve or Miner’s rule.
In this work, the capabilities of the generalised phase-field model for brittle, ductile and fatigue fracture [41] are further investigated by studying the fracture processes in highly heterogeneous microstructural geometries. The numerical examples are conducted on the nodular cast iron microstructure obtained by metallography imaging from authors’ previous work [42]. Different sized samples are selected, respecting the volume fraction of graphite nodules, to present the well-known size-effect. Moreover, three nodule modelling options are investigated: perfect spherical porosities (i.e. circles in 2D settings) instead of graphite inclusions, porosities closely following the graphite inclusion geometries, and finally the inclusions with graphite material properties. The latter option is introduced via contact modelling options, showing the great advantage of the proposed framework implemented within the commercial software ABAQUS by the author [43]. The clear brittle and ductile fracture patterns are observed in the monotonic loading examples, as well as the transition between brittle and ductile fracture behaviour through a parametric investigation of the influence of material critical energy release rate. Moreover, the specimens are subjected to cyclic loading conditions clearly showing the transition between low- and high-cyclic fatigue regime in terms of fracture patterns and Wöhler-type curve.
The paper is structured as follows. First, the concepts of the proposed generalised phase-field model are presented in Section 2. The extension of the brittle model to ductile and fatigue behaviour is explained. The elastoplastic model with combined nonlinear kinematic and isotropic hardening is briefly discussed. Section 3 covers the numerical implementation within the ABAQUS finite element software with the additional features, like contact problems. Section 4 presents the numerical examples under monotonic and cyclic loading scenarios on different specimen sizes and graphite inclusions modelling options. The thorough discussion on the obtained results is given. Finally, concluding remarks are drawn in Section 5.
Section snippets
Phase-field approach to fracture
A theoretical overview of the general phase-field model for monotonic and cyclic fracture in brittle and ductile solid deformable bodies is outlined here, closely following the authors’ recent work [41]. The phase-field model for brittle fracture in monotonic loading conditions is first presented as a foundation, and then extended to ductile and fatigue problems. The model is derived under isothermal and small-strain settings assumptions.
Numerical examples
In this section, the presented phase-field framework for brittle, ductile and fatigue fracture is employed to the analysis of heterogeneous microstructural specimens. The metallographic images of nodular cast iron grade EN-GJS-400–18-LT, investigated in authors’ previous work [42], are used to create three different sized specimens (S, M, L) with dimensions shown in Fig. 6. legend. The selected specimens satisfy the global average graphite nodules content of 7%. The metallographic image is
Conclusion
The microstructural geometries of nodular cast iron grade EN-GJS-400–18-LT were numerically analysed with the recently developed phase-field model for monotonic and cyclic fracture. Three different specimen sizes were selected from the experimental metallographic image satisfying the global average nodule content. Different nodule modelling options, ascending in the level of detail, were analysed. Great advantage of the proposed model implementation into ABAQUS FE software is presented here
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work has been fully supported by Croatian Science Foundation under the project “Multiscale Numerical Modelling and Experimental Investigation of Aging Processes in Sintered Structural Components” (MultiSintAge, PZS-1 2019-02-4177).
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