Fracture modeling in dual-phase steel grades based on the random cellular automata finite element approach

Abstract

The development of a parallel version of the fracture model dedicated for multi-phase materials based on a combination of the finite element model and random cellular automata approach is the overall goal of this study. Dual-phase (DP) steel, commonly used in the automotive industry, is selected as a case study for the present investigation. Firstly, various fracture modes that can occur during deformation in DP steel grade microstructures are presented from an experimental point of view. To consider explicitly microstructure features that play a significant role during initiation and subsequent failure propagation, the digital material representation concept is used. Then, details of the developed random cellular automata model, fully embedded within the finite element framework, are discussed. The cellular automata space definition, internal variables, state variables and transition rules replicating investigated fracture modes are presented in detail and discussed. The concept of data transfer and parallelization based on the Message Passing Interface methodology in such an innovative hybrid numerical model is also clearly presented. The final section of the paper is devoted to examples of obtained results highlighting model predictive capabilities.

Keywords

cellular automata finite element random cellular automata finite element parallelization fracture

1. Introduction

Both the automotive and aerospace industries are in constant exploration for new metallic materials that can meet strict requirements regarding the weight/property ratio. This has recently been one of the major driving forces for fast development of modern innovative steel grades, which are rapidly increasing in number.¹ A series of innovative steel materials (transformation-induced plasticity [TRIP], twinning-induced plasticity [TWIP], dual-phase [DP], Bainitic, nano-Bainitic, etc.), as well as other non-ferrous metals, for example aluminum, magnesium, titanium or copper alloys, are being developed in various research laboratories around the world.^2–5 They are usually characterized by composite-type microstructures. Complex thermomechanical operations are applied to obtain the mentioned, highly sophisticated microstructures with combinations of, for example, large grains, small grains, inclusions, precipitates, multi-phase structures, etc. It is believed that these microstructure features and interactions between them at the lower length scales during manufacturing or exploitation stages result in highly elevated material properties at the macro scale. As a consequence, a significant increase in the application of modern steels to auto body components production has been observed.^6,7

One of these new steel grades is a group of advanced high-strength steels (AHSSs). They provide the possibility of reducing automobile weight (increasing fuel efficiency), while maintaining or even increasing their safety (crash-worthiness). A good representative of such a material is DP steels, with the tensile strength of 400–1200 MPa.

Unfortunately, due to the complex character of the DP microstructure with features characterized by significantly different properties, there is a problem of material failure during the manufacturing stages. Thus, zones where fracture can initiate during the production stages should be identified and the manufacturing cycle should be redesigned to avoid such behavior before industrial trials. Moreover, the extended service life of modern products requires determination of failure probability during exploitation conditions. Experimental analysis can provide all the required information; however it is time consuming and also expensive. Thus, to reduce the development costs of manufacturing technology for new products, computer-aided design is used more often. The finite element (FE) method is the main tool often used in many research facilities to simulate various deformation processes, and it gives satisfactory results.⁸ This method is usually applied to describe material behavior as a continuum, and it is based on general relationships between strains and stresses despite the presence of various phases in DP steels. However, when the problem of failure initiation is closely associated with morphological features of the microstructure, the classical FE approach cannot provide satisfactory results and has to be supported by methods taking into account microstructure features explicitly during simulations.^9–11 Literature findings suggest that failure in DP steels generally have a ductile character; however, the influence of brittle failure mechanisms can also be clearly visible. When DP fracture is considered, five mechanisms can be distinguished: cracking of martensite; decohesion at ferrite/martensite interfaces; void nucleation at non-metallic inclusions; decohesion between ferrite/ferrite interfaces; and finally voids coalescence. One of the solutions is to use the digital material representation (DMR) approach, which enables a precise description of the real material morphology, where different micro-scale features are included, that is, precipitates, inclusions, large and small grains, grain boundaries, crystallographic grains orientations, phases boundaries, etc. DMR enables detailed virtual analysis of real material behavior, while calculation errors are minimized. Numerical models based on DMR give more detailed results than those based on conventional approaches. Application of the DMR approach is important, as it provides a possibility to take the complex microstructure into account in an explicit manner.

It can be stated that presently the fracture modeling in DP steel is mainly simulated only at the macroscopic scale level because the morphology of the microstructure is usually neglected. At the same time, experimental research has proved that the size, shape or position of the hard martensitic phase directly influences failure initiation and propagation.^12,13

For this reason, the main aim of the approach proposed in this study is to create a fast and efficient model of failure for multi-phase materials, in particular DP steels, based on a modern numerical approach that take microstructure explicitly into account during simulation of deformation at room temperature during exploitation. The DMR concept in fracture modeling of DP steels was presented in an earlier work, and the obtained results were very encouraging.¹⁴ Thus, the concept was applied in the present work to simulate fracture, based on the developed hybrid random cellular automata finite element (RCAFE) approach.

Conventional fracture models do not give possibilities to take into account all fracture initiation modes that operate in DP steels (i.e., martensite phases fracture, delamination between martensite–ferrite phases, ferrite phases fracture and delamination between ferrite–ferrite grain boundaries). This is why these fracture modes were incorporated into the developed random cellular automata model and then combined with FE model using the VUMAT subroutine. Details of the model, including state variables, transition rules, data transfer protocols and most important implementation aspects are clearly described within this work. The relationship between transition rules and physical phenomena is particularly highlighted. To increase the computational efficiency, a parallelization algorithm based on the Message Passing Interface (MPI) technique was realized. The proposed RCAFE model was identified on the basis of the in situ tensile tests and used to simulate fracture during a tensile test case study.

2. Fracture phenomenain dual-phase steel

Failure in DP steels has in general a ductile character; however, the influence of brittle failure mechanisms is also clearly visible. Experimental observations clearly show that there are four major fracture modes, namely brittle cracking of martensite, decohesion at the ferrite/martensite interface, decohesion between ferrite/ferrite interfaces and voids coalescence.

Cracking of martensite: it has been observed during tensile test that cracks in the brittle martensite phase already occur at very low local strain levels¹⁵ (Figure 1).

Decohesion at the ferrite/martensite interface: the work of Avramovic-Cingara et al.¹² suggests that a dominant percentage of voids within the material nucleate at higher strain levels at the ferrite/martensite interface by the decohesion (Figure 2).

Ferrite grain boundary fracture: in addition, due to differences in hardness between ferrite and martensite, stress concentration perpendicular to the loading direction occurs most often, leading to the delamination of the ferrite grains¹⁵ (Figure 3).

Ferrite transcrystalline fracture: finally, during large deformation a mechanism of the transcrystalline fracture across ferrite grains starts to operate. Usually this modes starts within the vicinity of earlier cracks that developed in the material¹⁵ (Figure 4).

Figure 1.

Cracking of martensite (red = martensite, other colors = ferrite grains).

Figure 2.

Decohesion at ferrite/martensite interface (red = martensite, other colors = ferrite grains).

Figure 3.

Decohesion between ferrite/ferrite interfaces (red = martensite, other colors = ferrite grains).

Figure 4.

Ferrite transcrystalline fracture (red = martensite, other colors = ferrite grains).

As pointed out in the introduction, conventional numerical approaches, which are used at the industrial scale to model fracture in these steel grades, fail to take into account all those mechanisms that are directly related to micro features of the microstructure. This is why more complex numerical models based on the DMR concept are more often used.^11,16,17 However, they are still based only on the FE approach, which also has limitations in taking into account all four fracture mechanisms. This is why a discrete cellular automata model based on the DMR concept was developed to deal with this issue.

3. Digital material representation

To obtain a reliable representation of real microstructure morphology, an image-processing algorithm was developed that consisted of two supplementary approaches: binarization and cellular automata grain growth. Both are required, especially in the case of optical microscopy and scanning electron microscopy with electron back scattered diffraction (SEM/EBSD) images. In the present work the DMR models were created on the basis of real microstructure images obtained from light optical microscopy (LOM). The image-processing methodology was widely described in a previous study.¹⁸ Results of complex DP microstructure morphology where both martensite islands as well as subsequent ferrite grains are clearly visible are shown in Figure 5.

Figure 5.

(a) Real microstructure image and (b) digital representation received after the image-processing approach.

The generated DMR is then used as input for the developed RCAFE method.

4. Random cellular automatafinite element method

The RCAFE model developed here is a unique fully coupled solution combining two computational approaches: RCA and FE methods. The proposed model is designed to have the capability to take into account all main fracture mechanisms observed in DP steels.

The innovative nature of the proposed RCAFE method lies in the assumption that the cellular automata cells correspond to nodes of the FE mesh, as seen in Figure 6. Thus, the same point in each time step is considered as the FE node and then is considered as the CA cell.

Figure 6.

Idea of the random cellular automata finite element (RCAFE) method.

The proposed RCAFE model takes the advantages provided by the automata modeling and solves one of the major disadvantages: the reliable description of the cellular automata space deformation under plastic loading conditions.

Due to the irregular nature of the cellular automata space in the random CA method, classical definitions of neighborhoods cannot be used. Thus, a specified radius neighborhood is used to evaluate CA cells contributing to the cell changes via transition rules. The RCA model holds information on microstructure morphology and, by combination with the FE model, also information on microstructure deformation during processing. As an outcome, the influence of morphological features on fracture initiation and propagation can be obtained.

To develop the hybrid RCAFE model, a standard elasto-plastic flow rule is introduced into the ABAQUS code. The damage model is combined within the random cellular automata framework. Both models are implemented in the VUMAT subroutine available in ABAQUS/Explicit solver using the Fortran and C++ programming languages. The initial concept of the model with particular attention put on development of digital representation approach is described in a previous study,¹⁴ while a detailed description of its fully developed components, including transition rules, data transfer model, application of periodic boundary condition concept and identification of model coefficients, is presented below.

When the cellular automata model is being developed much attention has to be paid to the definition of the CA space, internal variables, cells states and transition rules. In the case of CA space, due to the hybrid character of the model, it exactly matches the FE mesh computational domain. To reflect the underlying microstructure of the DP steel, four kinds of cells internal variables were introduced: ferrite, martensite, ferrite–ferrite neighbor and ferrite–martensite neighbor. In addition, two types of cells states were defined: not fractured and fractured. The detailed idea of the developed RCAFE approach matching the DP steel microstructure morphology is presented in Figure 7.

Figure 7.

Concept of the proposed random cellular automata finite element (RCAFE) approach matching the dual-phase (DP) steel microstructure morphology.

Changes of the cell states are controlled by the defined transition rules, when fracture criteria are reached and the local neighborhood gives opportunities for fracture initiation. In the random cellular automata model, each cell has to define its CA neighbors at the beginning of each time step based on the defined neighborhood radius $R$ . All cells that are inside the prescribed circle belong to the neighborhood of a particular cell, as seen Figure 8.

Figure 8.

Cellular automata random neighborhood communication method (a) before neighbors selection and (b) after neighbors selection.

To increase computational efficiency, subsequent steps are realized in the ABAQUS/Explicit solver with the parallel MPI mode. In this case the computational mesh is divided into several domains (Figure 9). Each processor receives a similar number of FEs for the computations. In the ABAQUS approach, one master processor is responsible for handling connections between all existing elements. To use a RCA model implemented via the user subroutine in parallel mode a series of modifications were introduced to the developed code, for example, a parameter called separate index number for each FE in the FE mesh was defined.

Figure 9.

Schematic picture representing parallelization of the discretized microstructure.

In subsequent time steps the CA model requests data from FE integration points on equivalent plastic strain and stress values. The information is used within defined transition rules that are designed to replicate all the above-mentioned fracture mechanisms operating in DP steels. Transition rules then change the state of the analyzed CA cell from not fractured to fractured. They are defined on the basis of laboratory investigations^11,12 and their mathematical form is described as follows.

Martensite grains brittle fracture initiation and propagation rule:

Y_{m (m a r t e n s i t e)} (t_{i + 1}) = {\begin{matrix} f r a c t u r e \Leftrightarrow A \land M P I_{c o n n e c t} \\ Y_{m (m a r t e n s i t e)} (t_{i}) \end{matrix}

(1)

where $Y_{m (martensite)} (t_{i + 1})$ is the state of the mth cell from the martensite phase at the $t_{i + 1}$ time step, $MP I_{connect}$ is the parallel CPU’s connection check and $A$ is the logical function formulated on the basis of available knowledge on fracture initiation:

A = (ε_{i} \geq ε_{c}^{mI})

(2)

where $ε_{i}$ is the equivalent plastic strain and $ε_{c}^{mI}$ is the critical equivalent plastic strain for fracture initiation within martensite phase. A similar logical function was formulated for fracture propagation:

A = (ε_{i} \geq ε_{c}^{mP}) \land (σ_{i} \geq σ_{c}^{mP}) \land Y_{m (martensite)}^{l} (t_{i}) = fracture

(3)

where $ε_{i}$ is the equivalent plastic strain, $ε_{c}^{mP}$ is the critical equivalent plastic strain for fracture propagation in martensite, $σ_{i}$ is the equivalent plastic stress, $σ_{i}^{mP}$ is the critical equivalent plastic stress for fracture propagation in martensite, $Y_{m (martensite)}^{l} (t_{i})$ is the state of the lth neighbor of the mth cell from the martensite phase.

Ferrite–martensite interface ductile fracture initiation and propagation rule:

Y_{m (ferrite - martensite)} (t_{i + 1}) = {\begin{matrix} fracture \Leftrightarrow A \land MP I_{connect} \\ Y_{m (ferrite - martensite)} (t_{i}) \end{matrix}

(4)

where $Y_{m (ferrite - martensite)} (t_{i + 1})$ is the state of the mth cell from the boundary between ferrite and martensite at the $t_{i + 1}$ time step and $MP I_{connect}$ is the parallel CPU’s connection check. Logical functions for nucleation and propagation are described as follows:

A = (ε_{i} \geq ε_{c}^{fmI}) \land Y_{m (martensite)}^{l} (t_{i}) = fracture

(5)

where $ε_{i}$ is the equivalent plastic strain in the considered cell and $ε_{c}^{fmI}$ is the critical equivalent plastic strain for fracture initiation between ferrite–martensite.

A = (ε_{i} \geq ε_{c}^{fmP}) \land Y_{m (ferrite - martensite)}^{l} (t_{i}) = fracture

(6)

where $ε_{c}^{fmP}$ is the critical equivalent plastic strain for fracture propagation between ferrite–martensite and $Y_{m (ferrite - martensite)}^{l} (t_{i})$ is the state of the lth neighbor of the mth cell from the ferrite/martensite boundary.

Ferrite–ferrite boundary ductile fracture initiation and propagation rule:

Y_{m (ferrite - ferrite)} (t_{i + 1}) = {\begin{matrix} fracture \Leftrightarrow A \land MP I_{connect} \\ Y_{m (ferrite - ferrite)} (t_{i}) \end{matrix}

(7)

where $Y_{m (ferrite - ferrite)} (t_{i + 1})$ is the state of the mth cell from the boundary between ferrite–ferrite grains at the $t_{i + 1}$ time step and $MP I_{connect}$ is the parallel CPU’s connection check. Logical functions for nucleation and propagation are described as follows:

A = (ε_{i} \geq ε_{c}^{ffI}) \land Y_{m (ferrite - martensite)}^{l} (t_{i}) = fracture

(8)

where $ε_{i}$ is the equivalent plastic strain in the considered cell and $ε_{c}^{ffI}$ is the critical equivalent plastic strain for fracture initiation between ferrite grains.

A = (ε_{i} \geq ε_{c}^{ffP}) \land Y_{m (ferrite - ferrite)}^{l} (t_{i}) = fracture

(9)

where $ε_{c}^{ffP}$ is the critical equivalent plastic strain for fracture propagation between ferrite–ferrite grains and $Y_{m (ferrite - ferrite)}^{l} (t_{i})$ is the state of the lth neighbor of the mth cell from the boundary between ferrite grains.

Ferrite grains ductile fracture initiation and propagation rule:

Y_{m (ferrite)} (t_{i + 1}) = {\begin{matrix} fracture \Leftrightarrow A \land MP I_{connect} \\ Y_{m (ferrite)} (t_{i}) \end{matrix}

(10)

where $Y_{m (ferrite)} (t_{i + 1})$ is the state of the mth cell from the ferrite phase at the $t_{i + 1}$ time step and $MP I_{connect}$ is the parallel CPU’s connection check.

A = (ε_{i} \geq ε_{c}^{fIP})

(11)

where $ε_{c}^{fIP}$ is the critical equivalent plastic strain for fracture initiation and propagation in the ferrite phase.

The developed model performs a series of subsequent operations realized in each time increment for each FE and the corresponding CA cell of the RCAFE model. The schematic diagram of the proposed algorithm is presented in Figure 10, while a detailed description of the following steps is presented below.

Figure 10.

Cellular automata (CA) algorithm flow chart.

As mentioned, the RCAFE model is implemented within the VUMAT subroutine of the ABAQUS/Explicit FE code. The typical CPE3 (reduced integration 3 – node plane linear triangle) FEs are used during simulation. These types of elements have a single Gaussian integration point, which is why each FE can communicate with a single CA cell. Communication between FE and RCA models is schematically represented in Figure 11.

Figure 11.

Flow of information between finite element (FE) and random cellular automata (RCA) models.

5. Assigning material properties to the digital material representation

The possibility of assigning material properties to particular ferrite or martensite grains is the main advantage of the developed DMR model. Due to the fact that only cold deformation processes are investigated in the present work, a simple isotropic elasto-plastic nonlinear hardening rule was used:

\begin{matrix} σ_{i} = E ε_{i} for 0 < σ_{i} < σ_{Y} \\ σ_{i} = C {ε_{i}}^{n} for σ_{i} > σ_{Y} \end{matrix}

(12)

where $E$ is the Young’s modulus, $σ_{i}$ is equivalent stress, $σ_{Y}$ is yield stress, $ε_{i}$ is equivalent strain and $C, n$ are constants ( $C_{martensite} = 3500, C_{ferrite} = 700, n_{martensite} = 0.22, n_{ferrite} = 0.08)$ .

In the approach the size of the yield space is fixed in the stress space, while only the radius expands in subsequent steps. Elastic mechanical properties were set for martensite and ferrite phases by the values: Young’s modulus for ferrite ( $E = 202 GPa$ ), Young’s modulus for martensite ( $E = 210 GPa$ ) and the Poisson ratio was set the same for both phases ( $ν = 0.3$ ). In the DMR model there is a possibility to capture differences in the grain flow due to various crystallographic orientations. It was done through diversification in the flow curves for each particular grain. The diversification was introduced using the random Gauss distribution (Figure 12).

Figure 12.

Schematic representation of curves generated with the Gaussian distribution function.

To obtain such a diversification, the range in which the flow stress parameters can change was specified, and the obtained values were then automatically assigned to subsequent grains. The Gaussian distribution was used to generate material properties for all ferrite grains existing in the microstructure. Plastic hardening rules were generated on the basis of the laboratory microcompression test as described in a previous work.¹⁹ Mechanical properties for martensite phase were not differentiated in the present approach, as martensite phase deformation is significantly smaller than the ferrite phase.

To facilitate the process of properties assignment to subsequent grains in the model, a procedure using Python scripting language that automatically modifies the calculation input file in the ABAQUS .inp format was implemented. Examples of obtained results are presented in the following section.

6. Results

The random cellular automata fracture model parameters were identified on the basis of the literature data presented by Madej et al.¹⁹ and Ghadbeigi et al.^20,21 Based on the laboratory experiments presented in those works, critical parameters in the RCA transition rules were defined (Table 1).

Table 1.

Critical values for initiation of different fracture mechanisms in the developed random cellular automata finite element model.

Fracture mechanisms	Critical values for initiation
Brittle crack initiation in martensite phase.	$ε_{c}^{mI} = 0.18$
Brittle crack propagation in martensite phase.	$ε_{c}^{mP} = 0.18$ $σ_{i}^{mP} = 2300 MPa$
Ductile crack initiation and propagation at the grain boundaries between ferrite and martensite phases.	$ε_{c}^{fmI}, ε_{c}^{ffP} = 2.044$
Ductile crack initiation and propagation along the ferrite grain boundaries.	$ε_{c}^{ffI} = 2.31$
Ductile crack initiation and propagation within the ferrite grains.	$ε_{c}^{fIP} = 4.62$

As a case study, a simple tension test was selected to show model capabilities. The initial microstructure morphology was obtained on the basis of the described image-processing algorithm and then it was discretized using 128,000 FEs. The applied specific mesh was obtained with developed FE mesh generation software DMRmesh²² (Figure 13). Because fractures occur mainly in the borders between different material phases, mesh density is high along grain boundaries to capture local inhomogeneities.

Figure 13.

Density of finite element discretization.

To ensure the continuity of the solutions space, left–right periodic boundary conditions were employed.²³ To enforce periodicity to the DMR, a specific buffer zone to create an investigated DMR was added. Based on the simple rule of mixture, different material definitions were adopted to the surrounding buffer zone for the investigated cases (Figure 14).

Figure 14.

Buffer zone applied to the digital material representation (DMR) sample.

In the simulation DMR was fixed at the bottom border and deformation was set at the upper border (Figure 15).

Figure 15.

Digital material representation before the tension test.

Examples of obtained results from the RCAFE model after deformation to 0.10, 0.14, 0.20 and 0.25 of engineering strain are presented in Figures 16 –19.

Figure 16.

Distribution of (a) equivalent stress, (b) equivalent strain and (c) fractures after 0.10 of engineering strain.

Figure 17.

Distribution of (a) equivalent stress, (b) equivalent strain and (c) fractures after 0.14 of engineering strain.

Figure 18.

Distribution of (a) equivalent, (b) equivalent strain and (c) fractures after 0.20 of engineering strain.

Figure 19.

Distribution of (a) equivalent stress, (b) equivalent strain and (c) fractures after 0.25 of engineering strain.

It can be found that brittle fractures start to initiate from the 0.10 engineering strain. Full brittle fracture propagation ends when strain is equal to 0.20. In the same time decohesion at the boundaries between ferrite–martensite phases starts to initiate and propagate. When the strain value increases to 0.25, fractures start to grow along the ferrite–ferrite borders and finally coalesce with other cracks leading to complete material failure. As presented by the developed RCAFE model, different failure mechanisms can be modeled, which is the major advantage in comparison to other fracture models developed for DP steels.

7. Conclusions

The presented research clearly demonstrated that implementation of the RCA method with the FE approach in a concurrent fully coupled mode for parallel computing is possible and can be used to develop efficient fracture models of different multi-phase materials. Application of the MPI methodology with the master node assembling results from subsequent slave nodes seems to be a good solution to the matter. The random cellular automata model additionally combined with the DMR concept can be effectively used to predict complex failure system of multi-phase materials. The developed RCAFE method applied to the DP steels was able to qualitatively predict all major failure mechanisms observed in these steels: brittle fracture in martensite phases; decohesion between martensite and ferrite phases; fracture between ferrite grains; and transcrystalline ferrite grains fracture. This is the major advantage of the model in comparison to conventional fracture models, which are limited only to brittle and ductile fracture modeling across martensite and ferrite, respectively.

Footnotes

Acknowledgements

Numerical calculations were performed at the ACK Cyfronet: MNiSW/IBM_BC_HS21/AGH/076/2010.

Funding

This work was supported by the National Science Centre (project 2014/14/E/ST8/00332).

Author biographies

Konrad Perzynski defended his master’s thesis in applied information technology at the Faculty of Metallurgy and Materials Science in 2008. The research was on the development process of spinning copper tubes for the electrical industry. From 2008 he continued his education with PhD studies at the Department of Applied Computer Science and Modelling in Metallurgy. His PhD thesis was related to the development of a hybrid RCAFE model for fracture modelling in multiphase materials. He is a co-author 49 articles in reviewed journals (10 publications in journals ranked in the Philadelphia list) and international conference proceedings.

Lukasz Madej received his engineering degree in solid-state physics at the Department of Physics and Nuclear Techniques, AGH University, in 2003. The research was on analytical and numerical solutions of differential equations describing the behavior of the hot-deformed metals. In 2004 he defended his master’s thesis at the Faculty of Metallurgy and Materials Science in the field of information technology in materials science. The research was on inverse analysis and cellular automata method applied to determine the rheological parameters and modeling the development of microstructure of steel with different carbon contents. In 2007 he obtained a PhD at the Faculty of Metals Engineering and Industrial Computer Science in the field of applied informatics; the PhD thesis was awarded first prize for Best PhD Thesis in Europe by the ECCOMAS in 2007. He received the ESAFORM Scientific Prize 2015 for outstanding contributions in the field of materials forming. He is also the author and co-author of 2 books, 4 chapters in books, 45 publications in journals ranked in the Philadelphia list, 59 articles in other peer-reviewed journals, and 70 publications in national and international conference proceedings.

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