Statistical modelling of fracture using cellular atomata finite element
Introduction
The integrity of materials with limited plasticity such as ferritic steels at their transition temperature and lower shelf or tungsten are commonly assessed by probabilistic fracture mechanics. Since the plastic deformation of these materials are limited, they are prone to sudden fracture which can pose a threat to the integrity of components made from them (e.g. first wall in fusion reactor or a reactor pressure vessel). To further complicate their safe engineering design their toughness is not a unique value but can only be presented as a distribution [1]. That is, even if the tests to evaluate their toughness are conducted at identical conditions, the measured values would be non-unique. This could be because fracture initiates predominantly from the pre-existing flaws such as micro-cracks [2] and/or inclusions [3] present in the material. This means that the cleavage stress required to initiate fracture in the material is variable and is generally regarded as a function of distribution of these flaws present in the micro-structure and the stress applied to them which is deponent on the grain orientation chousing them [4]. Catastrophic failure can occur in as manufactured components made of materials with limited plasticity such as tungsten in the first wall in a fusion reactor but also in initially ductile materials, such as ferritic steels of reactor pressure vessels, that embrittle due to ageing or irradiation [5]. If lower-bound deterministic fracture mechanics is used in the assessment of such components, the outcome may not be economically feasible to operate [6]. To mitigate this issue, it is more realistic to predict the scatter in the material properties and quantify the uncertainty associated with the variation in the fracture toughness and employ them within a probabilistic procedure. Probabilistic design methods can be exploited, such as those used in the aerospace industry, to give a good account of failure rate allowing for an appropriate choice of component safe lifetime [7].
Damage and fracture scatter have been modelled using probabilistic approaches since the 80′s. They provides a better understanding of fracture behaviour and variability in its micro-mechanism [8] compared to deterministic methods. They hypothesise that the occurrence of fracture is motivated by micro-mechanical response of the material thus requiring their application to include some micromechanical information. For example, in Beremin model [9] failure in ferritic steels is assumed to initiate from an inclusion when the stresses in the fracture process zone containing the inclusion reaches a critical value. Various micro-structural factors affecting the fracture are encoded in the models to describe realistically the fracture event. One of the main advantages associated with this approach is that the model parameters depend only on the material rather than the geometry. Therefore, this approach finds applicability from laboratory specimens to engineering structures. The probabilistic approach was introduced initially by [10] and was later expanded for example [11] effectively used the probabilistic approach to estimate the risk of fracture of nuclear pressure vessels. They derived an analytical expression for fracture critical stress that governs the cleavage micro-crack nucleation and propagation as a function of plastic deformation. They enabled the formulation to address various loading histories and the results were verified experimentally. Recent advances in the local approach considers the role played by the inclusions in determining the fracture behaviour. [12] considered the influence of inclusions size distribution in their micro-structural model and investigated the temperature dependency of Beremin parameters and surface energy required for crack propagation.
While treating the fracture behaviour of in a continuum framework, as the probabilistic approaches do, is mathematically elegant, they can neglect the intricacies caused by the microstructural sensitivities. For example, such approaches present a formulation for estimating the probably of the presence of a fracture initiator within the fracture process zone and may even assume a distributing for the critical stress applied on them. However, it not possible for a continuum mechanics model to account for the orientation of the grain and its cleavage planes that houses the inclusions although the orientation is a deciding factor in the magnitude of the stress applied normal to the cleavage plane which causes fracture. It is therefore becoming more desirable to combine the probabilistic approaches with microstructurally informed models. For example, MIBF, micro-structurally informed brittle fracture [13] captures the heterogeneities of the microstructure by combining the probabilistic approach and crystal plasticity finite element. The originality of MIBF model is that it introduces a second source of scatter in the form of stress distribution created by the representative aggregate of bainitic micro-structures within a crystal plasticity finite element framework to simulate the microstructural effect on stress distribution.
In this paper, we aim to combine probabilistic approach with cellular automata finite element model (CAFE). Cellular automata (CA) is a mathematical, discrete time and discrete space framework that can be employed to model a wide range of physical phenomena. It consists of a series of cells arranged regularly in a uniform fashion in a three-dimensional grid, with each of the cells corresponding to a particular phenomenon to be analysed. It uses an iterative procedure where the state of each cell is determined by the state of the cells in the previous iteration and that of the cells in the nearest neighbourhood. When used independently, the CA framework can be used to simulate a range of physical processes like lattice gas diffusion, phase transitions, wave propagation, multi-phase fluids etc. However, in most cases, these models are used in tandem with a finite element model and continuum fields like stress, strain, temperature etc., which are mapped over the CA space to drive certain physical phenomena. This integrated approach captures more intricate details at the micro-structure level, where a conventional continuum model could only describe the properties at scale pertinent to the macro model. Within the fracture simulation context, CAFE is realized as a multi-scale model that integrates effectively a macro model, such as a finite element model, and the micro-structure, represented by a cellular automata model, enabling the simulation of mechanical response with specific input from the material’s inherent heterogeneity in the lower length scales.
Modelling by CAFE started in the late 1990s for simulating solidification [14] and has since been used by many researchers to simulate a variety of engineering problems. Pioneering work on CAFE modelling for fracture was performed by [15] where they modelled fracture at the ductile–brittle transition regime in a thermo-mechanically controlled rolled steel. Innovations in high performance computing such as Co-array Fortran [16]coupled with the message passing interface [17] have since been used to reduce runtimes for the modelling of cleavage fracture using CAFE, enabling computationally intensive processes such as dynamic loading to be tackled [18]. Several other researchers have focused their efforts not only on fracture related topics, but also on various other subjects like oxide cracking [19], dynamic re-crystallization [20] and friction stir welding [21].A multi-scale modelling technique allows us to combine two or more different length and time scales which are often dissimilar in terms of their characterization due to the change in the scale [22]. CAFE framework is one such approach where the macroscopic model, e.g. finite element model, is combined with the micro-structural model to bring in an accurate description of the structural integrity of the component at a global level. Limited work has been carried out to integrate CAFE with large scale modelling outside the high-performance computing community. They include, for example[23] extended CAFE model to include the distribution of four different cleavage fracture nucleation sites found in Grade A ship plate steel. They used the fracture stress derived from Griffith’s energy balance approach [24] to account for the strength of micro-structural features considered in their material interface. However, in spite recent computing advancement, CAFE is still not computationally efficient enough to allow its wide application in industrial contexts.
The novelty of our work is that we adopt a probabilistic approach that not only considers the distribution of fracture initiators but also the effects of the microstructural features such as grain orientation on stresses that are applied don the fracture initiators. We use these as inputs to create a microstructurally representative CAFE model which effectively simulates the distribution of fracture toughness due to inclusion size variation and grain orientation distribution in the material. We use Weibull’s distribution to quantify the scatter predicted by the CAFE model. The predicted fracture distribution is later compared with experimental results previously obtained for validation.
Section snippets
Background
A generic representation of the methodology is shown in Fig. 1. To model the failure of a cracked component/specimen, its continuum representation is created to calculate the stress field in response to external loads and boundary conditions. The stresses are then mapped to the CA model which simulates the fracture and adjust the elastic modulus at local level according to perceived damage. The elastic modulus is fed back to the continuum model and the process is repeated until convergence is
Background
To calibrate and to validate the proposed CAFE method two sets of intendent experiments were required; these were two sets of previously carried out tests on as-received and Warm Pre-Stress specimens [39].Warm pre-stressing is a term used in describing the effects of a pressure test of a reactor pressure vessel at the upper shelf on the behaviour of the pressure vessel in the lower shelf [40]. This is to simulate the deformation of the vessel at its as-received ductile stage (upper shelf)
Experimental verification of inclusion distribution
To elucidate the rationale behind the selection of critical stress distribution shown in Fig. 7a, an experimental observation of inclusions in the tested steel (BS1501-224 28B – see its composition in Table 2) was carried out. The chemical composition of the steel is given in Table 2 [46].
Conclusions
The paper models the distribution of brittle fracture using an established statistical method, that is, Weibull distribution. The modelling framework CAFE provides an ideal setting for this work, modelling fracture at different length scales and supplying the modelling foundation required for the stochastic simulation. Fracture was induced at a micro level through the random allocation of seeds and introduction of inclusions in the material. These inclusions are initially characterized by
CRediT authorship contribution statement
A. Balasubramanian: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft. L. Margetts: Methodology, Software, Data curation, Writing - original draft, Supervision. V.D. Vijayanand: Methodology, Validation, Formal analysis, Investigation, Writing - original draft. M. Mostafavi: Conceptualization, Data curation, Writing - review & editing, Supervision, Funding acquisition.
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.
Acknowledgements
The authors gratefully acknowledge the support of EPSRC grants EP/R013047/1 (AB, MM), EP/R020108/1 (MM) and EP/N026136/1 (LM). Professor Paul Mummery, University of Manchester, is thanked for the useful discussion. MM would like to thank Royal Academy of Engineering for the financial support provided through a Senior Research Fellowship.
References (52)
- et al.
Microstructural parameters governing cleavage fracture behaviors in the ductile–brittle transition region in reactor pressure vessel steels
Mater. Sci. Eng., A
(2004) - et al.
Brittle fracture of nuclear pressure vessel steels—I. Local criterion for cleavage fracture
Int. J. Press. Vessels Pip.
(1997) - et al.
A micromechanical interpretation of the temperature dependence of Beremin model parameters for French RPV steel
J. Nucl. Mater.
(2010) - et al.
Application of local approach to fracture of an RPV steel: effect of the crystal plasticity on the critical carbide size
Procedia Struct. Integrity
(2016) - et al.
A massively parallel multiscale CAFE framework for the modelling of fracture in heterogeneous materials under dynamic loading
Adv. Eng. Softw.
(2020) - et al.
Modelling fracture in heterogeneous materials on HPC systems using a hybrid MPI/Fortran coarray multi-scale CAFE framework
Adv. Eng. Softw.
(2018) - et al.
A statistical approach for transferring fracture events across different sample shapes
Eng. Fract. Mech.
(2011) - et al.
The statistical modelling of brittle fracture in homogeneous and heterogeneous steel microstructures
Acta Mater.
(2000) - et al.
Statistical analysis of the effects of prior load on fracture
Eng. Fract. Mech.
(2007) - et al.
Effects of microstructure on cleavage fracture in pressure vessel steel
Acta Metall.
(1986)