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(1)AGH University of Science and Technology Faculty of Metals Engineering and Industrial Computer Science. Department of Applied Computer Science and Modelling. PhD Thesis. Hybrid RCAFE model for fracture modelling in multi–phase materials by. MSc. Konrad Perzyński. Supervisor: Łukasz Madej, PhD. DSc.. Kraków 2014.

(2) Akademia Górniczo–Hutnicza Wydział Inżynierii Metali i Informatyki Przemysłowej. Katedra Informatyki Stosowanej i Modelowania. Rozprawa Doktorska. Hybrydowy model RCAFE w zastosowaniu do symulacji pękania materiałów wielofazowych. mgr inż. Konrad Perzyński. Promotor: dr hab. inż. Łukasz Madej, prof. n. AGH. Kraków 2014.

(3) Acknowledgment I am deeply grateful to my supervisor Professor Łukasz Madej for his guidance and encouragement during last six years. His insightful supervision and generous support were invaluable to the completion of the thesis. Moreover the friendly atmosphere that he has been creating was always a motivation in everyday hard work. I am grateful to Professor Maciej Pietrzyk for his valuable and constructive comments to my thesis. His support, especially at the beginning of my scientific work, was irreplaceable. I would like to say thank you to my family for their support during my work and comments on grammatical aspects of the text. Special thanks goes to my sister Ola for her time and valuable text amendments to the English version and to my father Janusz for his text amendments to the Polish version. My mother Mira for her general support. I would also like to thank my colleagues from the Faculty and in particular Joanna Szyndler, Rafał Gołąb, Mateusz Sitko and Mateusz Kopyściański for their help in completion of the thesis. Last but not least, special thanks goes to my beloved wife Magda for her never–ending support and most of all understanding. She organized our life so I could work and finalized the thesis. Without her, this work could never be realized.. The work is dedicated to Magda and Kamil.. 1.

(4) Content Content ....................................................................................................................................... 2 List of symbols ........................................................................................................................... 4 1.. Introduction ......................................................................................................................... 6. 2.. Dual phase steels ................................................................................................................. 8. 2.1. Microstructure formation ................................................................................................ 8 2.2. Failure mechanisms in DP steels .................................................................................. 11 2.2.1. Cracking of martensite .................................................................................................. 11 2.2.2. Decohesion at ferrite/martensite interface .................................................................... 12 2.2.3. Void nucleation at non–metallic inclusions .................................................................. 12 2.2.4. Decohesion between ferrite/ferrite interfaces ............................................................... 13 2.2.5. Voids coalescence ......................................................................................................... 14 3.. Fracture .............................................................................................................................. 15. 3.1. Basic concept ................................................................................................................ 15 3.2. Empirical fracture criteria ............................................................................................. 16 3.3. Fracture mechanics ....................................................................................................... 18 3.3.1. Linear fracture mechanics............................................................................................. 18 3.3.2. Nonlinear fracture mechanics ....................................................................................... 21 3.4. Continuum damage mechanics (CDM) ........................................................................ 23 3.5. DP fracture models based on the FEM approach ......................................................... 23 3.6. Fracture models based on the XFEM, cellular automata, Monte Carlo and molecular dynamic approaches ............................................................................................................... 27 3.6.1. Extended Finite Element Method (XFEM) .................................................................. 27 3.6.2. Cellular automata approach .......................................................................................... 27 3.7. Monte Carlo (MC) approach ........................................................................................ 30 3.8. Molecular dynamics (MD) approach ............................................................................ 31 4.. Digital material representation .......................................................................................... 33. 4.1. Basic information.......................................................................................................... 33 4.2. Incorporation of the DMR into the finite element code ............................................... 37 5.. Aim of the work ................................................................................................................ 39. 6.. Generation of digital material representation of dual phase steels .................................... 41. 6.1. Modified image processing algorithm for 2D DMR .................................................... 41 6.2. Modified version of the Monte Carlo grain growth algorithm for 3D DMR ............... 44 6.3. Assigning material properties to the DMR ................................................................... 46 2.

(5) 7.. Determination of material properties of digital material representation of the DP steel... 48. 8.. FE based model for ductile and brittle failure prediction in dual phase steels .................. 52. 8.1. Brittle failure with XFEM method ............................................................................... 53 8.2. Ductile failure with Johnson–Cook model ................................................................... 55 8.3. Combined (brittle–ductile) failure model ..................................................................... 58 9.. Selection of the representative volume element of dual phase steels ................................ 62. 9.1. Periodic boundary conditions ....................................................................................... 62 9.2. Buffer zone ................................................................................................................... 63 9.3. Mesh sensitivity analysis .............................................................................................. 66 9.4. Influence of DMR model size....................................................................................... 68 10. Model validation................................................................................................................ 71 10.1. Hole expansion (HE) test .............................................................................................. 71 10.2. Numerical model of the hole expansion test................................................................. 74 10.2.1. Macroscopic model of the hole expansion test .......................................................... 75 10.2.2. First level submodel ................................................................................................... 77 10.2.3. Second level submodel ............................................................................................... 77 10.2.4. Microscale model based on the digital material representation approach .................. 78 11. Hybrid RCAFE model for brittle and ductile failure prediction in dual phase steels ....... 83 11.1. FE model ....................................................................................................................... 83 11.2. RCAFE model .............................................................................................................. 84 11.2.1. Identification of critical fracture parameters of the martensite phase ........................ 90 11.2.2. Identification of critical fracture parameters of the ferrite phase ............................... 93 11.2.3. Modelling fracture in the three point bending test ..................................................... 96 12. Conclusions ..................................................................................................................... 101 12.1. General conclusions .................................................................................................... 101 12.2. Specific conclusions ................................................................................................... 101 12.3. Plans for future ........................................................................................................... 103 References .............................................................................................................................. 105 APPENDIX 1 – Gaussian distribution function for ABAQUS input file .............................. 115. 3.

(6) List of symbols 𝜀𝑖 – equivalent strain 𝜀𝑖𝑗 , (𝑖, 𝑗 = 1,2,3) – strain tensor 𝜀1 , 𝜀2 , 𝜀3 – principal strain components 𝜀 – true strain 𝜀𝐸 – engineering strain 𝑝 𝜀𝑖𝐷 – equivalent plastic strain at the onset of damage 𝑝 𝜀̇𝑖 – equivalent plastic strain rate 𝑝 𝑑𝜀𝑖𝑗 – plastic strain increment tensor 𝜎𝑖𝑗 , (𝑖, 𝑗 = 1,2,3) – stress tensor 𝜎1 , 𝜎2 , 𝜎3 – principal stresses components 𝜎 𝑎 – apparent stress 𝜎𝑖𝑗𝐼 – deviatoric stress tensor 𝜎𝑖 – equivalent tensile stress or von Mises stress 𝜎̃ – effective stress 𝜎𝑚 – mean stress 𝜎𝑌 – yield stress 𝜎𝑚𝑎𝑥 – maximal principal stress 𝜎𝑐 – critical stress 𝜎 – true stress 𝜎𝐸 – engineering stress 𝑑𝜆 – nonnegative scalar multiplayer 𝑑𝑓 – plastic potential 𝜂 – stress triaxiality 𝜏 – shear stress 𝜏𝑚𝑎𝑥 – maximum shear stress 𝐸 – Young modulus 𝜈 – Poisson ratio 𝑇 – thickness of the sample 𝛾𝑠 – surface specific energy 𝑈𝐸 – elastic strain energy 𝑈𝑆 – surface energy 𝐾𝐼 , 𝐾𝐼𝐼 , 𝐾𝐼𝐼𝐼 – stress intensity factors 𝐾𝐼𝐶 – critical stress intensity factor 𝑟 – distance between crack tip and single point Θ – angle between radius 𝑟 and axis situated at the surface where crack propagates 𝐽𝑐 – Rice’s integral value 𝑌𝑚 (𝑡𝑖+1 ) – state of the 𝑚𝑡ℎ cell at the 𝑡𝑖+1 time step 𝑌𝑙𝑚 – state of the 𝑙 𝑡ℎ neighbour of the 𝑚𝑡ℎ cell Δ𝐸 – energy difference of the selected CA cell prior and after random change of the CA state 𝑘 – Boltzmann constant 4.

(7) 𝑇 – temperature 𝑃 – loading force 𝐴 – cross–sectional area of pillar base 𝑌𝑚(𝑚𝑎𝑟𝑡𝑒𝑛𝑠𝑖𝑡𝑒) (𝑡𝑖+1 ) – state of the 𝑚𝑡ℎ cell from the martensite phase at the 𝑡𝑖+1 time step 𝜀𝑐𝑚𝐼 – critical equivalent plastic strain for fracture initiation in martensite phase 𝜀𝑐𝑚𝑃 – critical equivalent plastic strain for fracture propagation in martensite phase 𝜎𝑖𝑚𝑃 – critical Mises stress for fracture propagation in martensite phase 𝑙 (𝑡𝑖 ) – state of the 𝑙 𝑡ℎ neighbor of the 𝑚𝑡ℎ cell from the martensite phase 𝑌𝑚(𝑚𝑎𝑟𝑡𝑒𝑛𝑠𝑖𝑡𝑒) 𝑌𝑚(𝑓𝑒𝑟𝑟𝑖𝑡𝑒−𝑚𝑎𝑟𝑡𝑒𝑛𝑠𝑖𝑡𝑒) (𝑡𝑖+1 ) – state of the 𝑚𝑡ℎ cell from the boundary between ferrite and martensite phases at the 𝑡𝑖+1 time step 𝑓𝑚𝐼. 𝜀𝑐. – critical equivalent plastic strain for fracture initiation between ferrite–martensite phase. 𝑓𝑚𝑃 𝜀𝑐. – critical equivalent plastic strain for fracture propagation between ferrite–martensite. phase 𝑙 (𝑡𝑖 ) – state of the 𝑙 𝑡ℎ neighbor of the 𝑚𝑡ℎ cell from the boundary 𝑌𝑚(𝑓𝑒𝑟𝑟𝑖𝑡𝑒−𝑚𝑎𝑟𝑡𝑒𝑛𝑠𝑖𝑡𝑒) between ferrite and martensite phases 𝑌𝑚(𝑓𝑒𝑟𝑟𝑖𝑡𝑒−𝑓𝑒𝑟𝑟𝑖𝑡𝑒) (𝑡𝑖+1 ) – state of the 𝑚𝑡ℎ cell from the boundary between ferrite and martensite phases at the 𝑡𝑖+1 time step 𝑓𝑓𝐼. 𝜀𝑐. – critical equivalent plastic strain for fracture initiation between ferrite phases. 𝑓𝑓𝑃 𝜀𝑐 – critical equivalent plastic strain for fracture initiation between ferrite–ferrite 𝑙 (𝑡𝑖 ) – state of the 𝑙 𝑡ℎ neighbor of the 𝑚𝑡ℎ cell from the boundary 𝑌𝑚(𝑓𝑒𝑟𝑟𝑖𝑡𝑒−𝑓𝑒𝑟𝑟𝑖𝑡𝑒). phase between. ferrite phases 𝑌𝑚(𝑓𝑒𝑟𝑟𝑖𝑡𝑒) (𝑡𝑖+1 ) – state of the 𝑚𝑡ℎ cell from the ferrite phase at the 𝑡𝑖+1 time step 𝑓𝐼𝑃. 𝜀𝑐 – critical equivalent plastic strain for fracture initiation and propagation in ferrite phase 𝐽𝑔𝑏 – coefficient used to determine the grain boundary energy 𝛿𝑄𝑖 𝑄𝑗 –Kronecker delta (𝑄𝑖 𝑄𝑗 – state of the investigated pair (𝑖, 𝑗) of MC cells) 𝜀𝑐𝑟 – critical strain for fracture initiation. 5.

(8) Chapter 1. Introduction. 1. Introduction Significant need presented by automotive and aerospace industries for new metallic materials that can meet strict requirements regarding weight/property ratio has recently been observed. This necessity is a driving force for fast development of modern innovative steel grades, which number of is rapidly increasing (Jaroni, et al., 2010). A series of innovative steels (TRIP, TWIP, DP, Bainitic, nano–Bainitic etc.) as well as other metallic materials e.g. aluminium, magnesium, titanium or copper alloys are being developed in various research laboratories around the world (Beladi, et al., 2009; Robertson, et al., 2008; Sabirov, et al., 2008; Sun, 2011). They are usually characterized by a composite type microstructures. Complex thermomechanical operations are applied to obtain mentioned, highly sophisticated microstructures with combination of e.g. large grains, small grains, inclusions, precipitates, multi–phase structures etc. It is commonly believed that these microstructure features and interactions between them at the micro scale level during manufacturing or exploitation stages result in highly elevated material properties at the macro scale. As a consequence, significant increase in the application of modern steels to auto body components production has been observed (Liedl, et al., 2002; Saleh & Priestner, 2001). One of those new steel grades is a group of advanced high strength steels (AHSS). They provide possibility of reducing the automobile weight (increase of the fuel efficiency), while maintaining or even increasing their safety (crash worthiness). The most widely used example of those advances steels are Dual Phase (DP) steels with the tensile strength of 400–1200 MPa. The name “dual–phase” was first reported by Hayami and Furukawa in (Hayami & Furukawa, 1977). Unfortunately, due to the complex character of DP microstructure with features characterized by significantly different properties there is a problem of material failure during manufacturing stages. Thus, zones where fracture can initiate during production stages should be identified and manufacturing cycle should be redesigned to avoid such behaviour before industrial trials. Moreover 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 expensive at the same time. Thus, to reduce costs of development of manufacturing technology for new products, advantages of computer aided design are more often used. 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 (Malinowski, et al., 2004). This method is usually applied to describe material behaviour as a continuum and it is based on general relationships between strains and stresses despite the presence of various phases in a DP steel. However, when 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 a method taking into account microstructure features explicitly during simulations. One of the solutions is to use the Digital Material Representation (DMR) approach that enables a precise description of the real material morphology, where different micro scale features are included, i.e. precipitates, inclusions, large and small grains, grain boundaries, crystallographic grains orientations, phases boundaries etc. Due to that, the detailed virtual analysis of real material behaviour can be performed, while errors of calculations are minimized. Numerical models based on the DMR. 6.

(9) Chapter 1. Introduction. give more detailed results, than these, which are 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 modelling 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 (Avramovic–Cingara, et al., 2009; Park, et al., 2014). That is why, the main aim of the approach proposed by the Author is to create a robust model of failure for DP steels based on a modern numerical approaches that take microstructure explicitly into account during simulation of deformation at room temperature during exploitation. In this work the two different approaches for addressing the above problem were developed. The first, is based on conventional numerical damage models, however combined with the DMR approach. Author developed a combined model of ductile and brittle fracture that occur in ferrite and martensite, respectively. Ductile fracture is modelled by the Ductile Fracture criterion implemented within conventional FE model, while brittle one is predicted by more sophisticated eXtended Finite Element Method (XFEM). Proper data transfer protocols between these two methods were proposed to create a hybrid numerical model. Finally, the developed micro scale solution became the basis for the multiscale model, where micro model provided an information about failure initiation to the macro scale simulation. The model was validated with the experimental hole expansion test and proved its good predictive capabilities. However, some drawback of the model were also highlighted and to solve them a more complex model based on combination of discrete random cellular automata model with the finite element method was developed. The conventional fracture models do not give possibilities to take into account all fracture initiation modes that operate in DP steels namely: martensite phases fracture, delamination between martensite–ferrite phases, ferrite phases fracture and delamination between ferrite– ferrite grain boundaries. That is why these fracture modes were incorporated into the developed within the work random cellular automata model and then combined with finite element model using VUMAT subroutine. The proposed hybrid random cellular automata finite element (RCAFE) model was identified on the basis of the in–situ tensile tests. Finally, it can be summarized, that the first developed model is designed for more practical applications, while the second has more scientific character and can be used for detailed investigation of fracture behaviour in mentioned modern steel grades.. 7.

(10) Chapter 2. Dual phase steel. 2. Dual phase steels The DP steels have been successfully applied in the production of the automobile structural parts (Figure 1) because they are characterized by a combination of a good formability, high bake hardenability and crash worthiness.. Kg per vehicle. 250 200 150. DP. 100. Boron. 50. Martensite. 0. TRIP, CP, RA 2000. 2005. 2007. 2009. 2020. AHSS (other). Figure 1 Prediction of development of application of parts made of DP steels in car body. These elevated properties are the results of the properly designed microstructure morphologies, which consists mainly of ferrite matrix (around 70–90%) and a hard martensitic phase (around 10–30%) as seen in Figure 2. However, it has to be mentioned that small amounts of bainite, perlite or retained austenite may also be present in the DP microstructure. Properties of DP steels are affected by many factors, including: volume fraction of martensite, average carbon content and carbon distribution in martensite, ductility of martensite, distribution of martensite, ferrite grain size, alloying elements content in ferrite etc. (Calcagnotto, et al., 2011).. Figure 2 Dual phase steel containing 27% of martensite; white phase–ferrite, dark phase– martensite; LOM (Light Optical Microscopy). 2.1. Microstructure formation The most usual approach to obtain a dual phase microstructure is intercritical annealing of a ferritic–perlitic microstructure in the 𝛼 + 𝛾 two–phase region, which is then followed by a controlled cooling to enable the austenite/martensite phase transformation. The procedure is adopted in industrial conditions in two processes: controlled cooling after hot rolling or continuous annealing of cold rolled sheets. The former approach is usually used to produce thick DP steel strips, while the latter is used for manufacturing thin sheets often applied in automotive applications.. 8.

(11) Chapter 2. Dual phase steel. a) Hot rolling and laminar cooling (Figure 3) After hot rolling in two phase region, ferritic–perlitic structure is obtained. This is followed by controlled cooling, enabling the austenite to transform into martensite. This technology requires very precise control of the austenite/ferrite phase transformation. The determination of precise continuous cooling transformation (CCT) diagrams taking into account influence of alloying elements, heat treatment conditions and required properties is of great importance to ensure the proper outcome of this process.. Figure 3 Hot rolling and laminar cooling (Pushkareva, 2009). b) Cold rolling and intercritical annealing (Figure 4). After cold rolling, material is subjected to treatment in continuous annealing often followed by the hot dip galvanizing. The continuous annealing, involves heating and soaking of cold rolled sheets in the intercritical temperature range ( +  two–phase region), followed by two–stage cooling. The first stage with moderate cooling rate is intended to produce the required volume fraction of ferrite in the microstructure. The aim of the second stage is to transform the remaining austenite into martensite.. Figure 4 Cold rolling and intercritical annealing (Pushkareva, 2009). At the same time for a specific intercritical temperature, the amount of austenite fraction increases with increasing carbon content (Figure 5).. 9.

(12) Chapter 2. Dual phase steel. Figure 5 Fe–C diagram of the annealing temperature used during the dual–phase steel production (Tsipouridis, 2006). The austenite develops from ferritic–perlitic microstructure during the intercritical annealing and is composed of several steps according to (Pushkareva, 2009):  instantaneous nucleation of austenite at perlite/carbide particles followed by the rapid growth of austenite until the pearlite/carbide phase is dissolved. At the end of the first step, a high– carbon austenite is generated, which is not in equilibrium with the ferrite,  slower growth of austenite into ferrite at a rate that is controlled by carbon diffusion in austenite at high temperature (e.g. 850°C). Growth of the austenite into ferrite leads to achieving partial equilibrium with the ferrite constitutes,  very slow final equilibration of ferrite and austenite at a rate that is controlled by manganese diffusion in austenite. It should be pointed out that the kinetics of austenite formation in the intercritical annealing is influenced by the initial ferrite grain sizes. It is reported in (Pushkareva, 2009) that the effect of the ferrite grain size is related with the increase of the nucleation sites of austenite at ferrite and perlite/carbide grain boundaries. In general the austenite/ferrite transformation from intercritical region is similar to conventional transformation from the fully austenitic region, however there are two distinctive features. The first, is connected with austenite hardenability. It is directly related to the intercritical processing temperature. The solubility of carbon in the ferrite decreases with increasing intercritical temperature. The martensite carbon content in typical DP steels ranges between 0.4–0.7 wt.%. In that case two types of martensite morphologies can occur: lath or mixed, respectively. Other elements, which influences the hardenability can be the alloying elements (Pushkareva, 2009). Lack of ferrite nucleation process is the second feature. In the intercritical region, ferrite is present, thus the transformation proceeds by its growth without the additional nucleation stage. After the transformation, different morphologies and transformation products are expected to form from the austenite phase. However, due to these complex microstructures, different failure mechanisms can occur during DP steel processing.. 10.

(13) Chapter 2. Dual phase steel. 2.2. Failure mechanisms in DP steels Literature findings suggest that failure in dual phase steels in general have ductile character, however 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. 2.2.1. Cracking of martensite Fracture takes place inside or along the martensite grains. The martensite can be classified as brittle phase, thus as mentioned, failure of DP steels is a combination of brittle and ductile failure mechanisms. Authors of (Szewczyk & Gurland, 1982) pointed out that the formation of voids is closely related to the morphology of the martensitic phase. For the coarse martensite islands failure occurs by cleavage. On the contrary for small distributed martensitic islands, void nucleation occurs along the phase boundaries. Additionally in (Speich & Miller, 1979) authors related fracture mechanisms to martensite volume fraction. For low martensite fraction, voids nucleate due to decohesion of the ferrite/martensite interface, while for high martensite fractions brittle failure of these islands is the major fracture mechanism. Similar observations are reported by other researchers e.g. (Szewczyk & Gurland, 1982; Balliger, 1982; Ahmad, et al., 2000). Very detailed studies on failure mechanisms were conducted in (Maire, et al., 2008) with the use of in–situ tests and X–ray tomography techniques. Both, martensite cracking and interface decohesion were observed. Tensile tests results demonstrated that cracks in martensite already occur at very low local equivalent strain levels around 0.029. These cracks instantly propagate across the volume occupied by the cracked martensite regions. It is reported in (Avramovic–Cingara, et al., 2009) that as deformation proceeds, the voids initiated at martensite continue to grow by further separation of the cracked martensite areas rather than by void coalescence. Finally, at higher strain levels voids open up between adjacent martensite islands and at ferrite/martensite interface where the compatibility of plastic deformation can no longer be maintained. Two types of martensite failure mechanisms can be observed in Figure 6. As mentioned, another fracture mechanism in DP steels is decohesion at ferrite/martensite interface.. Figure 6 Voids initiating in the martensite (Avramovic–Cingara, et al., 2009; Kadkhodapour, et al., 2011).. 11.

(14) Chapter 2. Dual phase steel. 2.2.2. Decohesion at ferrite/martensite interface The SEM observations reported in (Avramovic–Cingara, et al., 2009) present possible sites for the void nucleation as ferrite/martensite interface.. Figure 7 Decohesion between martensite and ferrite phases. It is stated that a dominant percentage of voids nucleate at higher strain levels at the ferrite/martensite interface by decohesion (Figure 7). With increasing strain levels these voids grow along the grain boundaries, i.e. parallel to the direction of the applied tensile load. However, it was also reported in (Avramovic–Cingara, et al., 2009) that small number of voids can be formed at low strain levels on the titanium–nitride particles and calcium– aluminium–silicate inclusions (Figure 8). The propagation of these voids eventually leads to decohesion along the ferrite/martensitic interfaces. More information on this fracture mechanism is presented in the following chapter.. Figure 8 Fracture caused by inclusions. Failure mechanism by interface decohesion is typical for steels with the martensite volume fraction lower than 32%. On the other hand, when martensite volume fraction is higher than 32% decohesion process is not present and main failure mechanism is brittle failure of martensite islands. 2.2.3. Void nucleation at non–metallic inclusions Two types of particles can be distinguished in the DP steels: ceramic inclusions (1μm ≪ 2μm) e.g. aluminium oxide, titanium nitride or manganese sulphide and small carbide particles. The latter, generally have a positive effect on material behaviour leading to increase in strength by blocking motion of dislocations. However, ceramic inclusions are undesirable in the microstructure and are related with the manufacturing process of raw material.. 12.

(15) Chapter 2. Dual phase steel. Void nucleation at the second–phase particles can occur by fracture of particles or by decohesion of the particle–matrix interface. The values of the void nucleation stains as well as interface strength were reported in (Qiu, et al., 1999; LeRoy, et al., 1981; Kwon, 1988; Kosco & Koss, 1993; Poruks, et al., 2006). Based on these works it can be concluded that the void nucleation strains are higher for the carbide particles than for the non–metallic inclusions. The low void nucleation strains in the case of non–metallic inclusions are related with pre–existing cracks and weakly bonded interfaces. Figure 9 shows typical fractures initiated around different inclusions.. Figure 9 Inclusions generated on the a) calcium–aluminium–silicate and b) titanium–nitride particles (Avramovic–Cingara, et al., 2009). 2.2.4. Decohesion between ferrite/ferrite interfaces Because of differences in hardness of the investigated phases, stress concentration perpendicular to loading direction usually occurs, leading to delamination of the ferrite grains (Figure 10). This is often visible in case when ferrite grains have a long contact surface with the martensite (Okayasu, et al., 2008). This is due to the fact that ferrite deforms to a large extend while martensite islands remain undeformed. Due to this deformation mismatch, the two ferrite grains start to delaminate in the neighbourhood of martensitic island. It was also reported in (Zok & Embury, 1990) that ferrite/martensite interface is stronger than ferrite/ferrite as a result of the carbide particles location (Kadkhodapour, et al., 2011). Carbide particles in the ferrite phase during tempering process travel towards grain boundaries. Martensite is free of carbide particles and as a result density of carbide particles in the ferrite/ferrite interface is about two times higher than in the ferrite/martensite interface, what makes these zones a precursor of delamination process.. Figure 10 Delamination between ferrite/ferrite interface (Kadkhodapour, et al., 2011; Avramovic–Cingara, et al., 2009).. 13.

(16) Chapter 2. Dual phase steel. 2.2.5. Voids coalescence Finally, voids nucleated on the basis of above mentioned mechanisms start to coalescence leading to fracture propagation. Voids propagate across the microstructure and elongate in the direction of straining, giving an increase in the void density and total void area fraction. It was also reported (Avramovic–Cingara, et al., 2009) that the microstructure observations reveal two different routes for void coalescence depending on the stress triaxiality. At low triaxiality, only few voids exhibited coalescence, showing the linkage at 45° orientation to the tensile direction. Thus, it seems that under these conditions, interactions between voids are rather weak. On the contrary, at high triaxiality conditions, void growth and coalescence occur quickly mainly in the plane normal to the applied axial load. Example of void coalescence process is presented in Figure 11.. Figure 11 Coalescence of voids during tension (Avramovic–Cingara, et al., 2009). Based on the presented information, it can be summarized that the failure mechanisms in the DP steels are classified mainly by the amount of the martensite volume fraction. Large amount of the hard martensite islands changes ductile mode of fracture to more brittle behaviour. Initiated large number of new voids weakens the microstructure leading to voids coalescence especially in the places where shear deformation occurs what finally results in material failure. Presented information regarding failure mechanisms will be used in the present work to develop reliable numerical model that can predict failure in these complex dual phase microstructures. An extensive literature review of the available numerical fracture models with different levels of complexity that can be used to support experimental research are presented in the next part of this work.. 14.

(17) Chapter 3. Fracture. 3. Fracture 3.1. Basic concept Fracture can be explained as total or partial separation of material as a result of applying specific stress state. Another definition states that fracture is a situation in material when structure is starting to lose the continuity. From the material point of view there are no more existing atomic bonds between specific parts of the material. In metalforming applications, propagation of fracture usually ends material deformation process. Processes of fracture initiation and propagation are related to many factors: history of deformation, type of material, mechanical stress state, thermal stresses, rate of deformation etc. Therefore one of the most important steps for understanding and modelling material behaviour is taking specific fracture parameters connected with investigated material into account. The most commonly used classification in metals science distinguishes two types of fracture types: brittle or ductile (Figure 12).  Ductile fracture requires stable deformation energy increase prior to fracture initiation. Before ductile fracture, plastic strain exceeds the limit of yield strength in the material. Fracture propagation is combined with large plastic strain and grows along the direction of the shear stress and it ends with full material degradation and separation.  Brittle fracture in contrast to ductile fracture is not a very energy consuming process. It is rather a sudden process, which does not need to exceed macroscopic yield strength. Energy needed for fracture propagation comes from elastic energy released during offload of the material. Brittle fracture initiates with very low plastic deformation zone, which is usually neglected in mathematical calculations.. a) b) Figure 12 Schematic representation of stress evolution with a) ductile and b) brittle fracture. Brittle and ductile fractures in metals strongly depend on microstructural features and external parameters (temperature, stress state, loading rate etc.). When microstructural aspects of fracture are considered two major mechanisms can be distinguished:  Transcrystalline fracture, which is classified as a standard route of fracture propagation in single crystals and polycrystalline structures in ductile and brittle conditions (Figure 13a).  Intercrystalline fracture, which occurs only during brittle fracture when strength of the grain boundaries is not sufficient or in high temperatures as a result of creep process (Figure 13b).. 15.

(18) Chapter 3. a). Fracture. b) Figure 13 a) Transcrystalline fracture b) Intercrystalline fracture. As presented, fracture phenomenon is a complex process, thus for its better understanding a new field of science namely fracture mechanics was introduced in the 20th century. Fracture mechanics in based on quantitative relation between fracture toughness, critical fracture size or type of applied loading conditions. Three major approaches are usually used in material science to model fracture: empirical fracture criteria, fracture mechanics and continuum damage mechanics (CDM) : a) Empirical fracture criteria are based on generally simple integral form equations. The major drawback of these models is that they require well defined calibration procedures based on the experimental data. b) Fracture Mechanics:  Linear fracture mechanics describes quantitative fracture propagation under elastic and linear material deformation. Sometimes it can be used to describe fracture during plastic deformation in situation when a crack tip is much smaller than the size of the investigated sample. It can be used only for calculating brittle fracture.  Nonlinear fracture mechanics describes fracture in materials where plastic deformation plays an important role and material behaviour cannot be calculated by means of linear fracture mechanics. c) Continuum damage mechanics describes fracture initiation and propagation in the framework of continuum numerical based models. 3.2. Empirical fracture criteria For practical industrial applications simple damage models are usually required. In the work (Freudenthal, 1950) author proposed first crack criterion based on the critical damage value per unit volume. Presently the crack initiation criteria are classified into several different types (Li, 2001):  stress and strain type crack criteria,  energy type crack criteria,  damage crack criteria. Modelling failure on the basis of these criteria is often used in commercial numerical applications. Mentioned models are incorporated e.g. in to the finite element solution, that provides detailed information on local stress or strain states. These criteria usually have integral form that is calculated during deformation until the critical value is reached. Critical values for fracture initiation are set on the basis of preceded experimental investigation. Equation (1) shows typical example of the crack initiation criterion in the integral form:. 16.

(19) Chapter 3. Fracture 𝜀𝑓 (𝑡). ∫ 𝑓(𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑝𝑎𝑟𝑎𝑚𝑡𝑒𝑟)𝑑𝜀 ≥ 𝐶. (1). 0. where: 𝐶 – constant, which specifies crack initiation criteria in the material. In these criteria crack initiate when integral reaches the value equal or larger than the parameter 𝐶. One of the most commonly used crack initiation criterion is the Latham–Cockcroft one (Cockcroft & Latham, 1968): ∫. 𝜀𝑓 (𝑡). 𝜎 𝑚𝑎𝑥 𝑑𝜀 ≥ 𝐶. 0. where: 𝜎𝑚𝑎𝑥 – maximal principal stress. As well as modified Latham–Cockcroft (Alexandrov & Jeng, 2013): 𝜀𝑓 (𝑡) 𝜎 𝑚𝑎𝑥 ∫ 𝑑𝜀 ≥ 𝐶 𝜎𝑖 0 Other kinds of the similar criteria are listed below (Trębacz, 2011).  equivalent stress criterion: 𝜎𝑖 ≥ 𝐶  mean stress criterion: 𝜎𝑚 ≥ 𝐶  maximal shear stress criterion: 𝜎1 − 𝜎2 𝜎2 − 𝜎3 𝜎3 − 𝜎1 𝜏𝑚𝑎𝑥 = 𝑚𝑎𝑥 { , , }≥𝐶 2 2 2  principal strain criterion: 𝜀1,2,3 ≥ 𝐶  equivalent strain criterion: 2 𝜀𝑖 = √ √𝜀12 + 𝜀22 + 𝜀32 ≥ 𝐶 3. (2). (3). (4) (5) (6) (7). (8). Other types of the criteria used for the ductile fracture investigation can be specified as: ∫. 𝜀𝑓 (𝑡). 𝐷𝑑𝜀 ≥ 𝐶. (9). 0. The 𝐷 coefficient may take a different form depending on the available parameters:  Mohr and Coulomb criterion (Mohr & Henn, 2007): 𝜎𝐼 𝜎1 − 𝜎2 𝐷 = 𝜙 + (1 − 𝜙) ( ) 𝜎𝑖 𝜎𝑖 where: 𝜙 – angle of internal friction, 𝜎1 , 𝜎3 – principal stresses, 𝜎𝑖 – equivalent stress.  Rice and Tracey criterion (Rice & Tracey, 1969): 3𝜎𝑚 𝐷 = 𝛼𝑒𝑥𝑝 ( ) 2𝜎𝑖 where: α – material constant.  Bao and Wierzbicki (Bao & Wierzbicki, 2004): 1. where: 𝜂 =. 𝜎𝑚 𝜎𝑖. 𝐷=𝜂. (10). (11). (12). – stress triaxiality.. 17.

(20) Chapter 3. Fracture.  Lou (Lou & Huh, 2013): 𝑏. 2𝜏𝑚𝑎𝑥 𝑎 ⟨1 + 3𝜂⟩ (13) 𝐷=( ) ( ) 𝜎𝑖 2 where: 𝜏𝑚𝑎𝑥 – maximum shear stress; 𝑎, 𝑏 – material dependent parameters characterizing the overall mechanisms of the ductile damage. Interesting example of the criterion based on the constant parameters, which have to be defined during experimental procedure is the Johnson–Cook (JC) crack criterion: (14) 𝜀𝑖 ≥ 𝐶1 + 𝐶2 𝑒𝑥𝑝(𝐶3 𝜂) where: 𝐶1 , 𝐶2 , 𝐶3 – materials constants. Fractures can also be described as form of the voids that occur on the existing inclusions and second phases particles (Kossakowski, 2012). The growth and coalescence of these voids result in the development of localized plastic deformations. This phenomenon was described with damage criterion based on the Gurson–Tvergaard–Needleman model and is referred to as the GTN model. The model (Gurson, 1977) takes into consideration microdamages (e.g. pores, voids), and assumes that the proportion of voids in the plastic potential function is dependent on the void volume fraction 𝑓. The original Gurson condition was modified by (Tvergaard, 1981) and (Tvergaard & Needleman, 1984) as: 𝜎𝑖 2 3𝜎𝑚 (15) 𝜙 = ( ) + 2𝑞1 𝑓 ∗ cosh (−𝑞2 ) − (1 + 𝑞3 𝑓 ∗ ) = 0 𝜎𝑌 2𝜎𝑌 where: 𝜎𝑖 – von Mises effective stress according to the HMH hypothesis, 𝜎𝑌 – yield stress of the material, 𝜎𝑚 – hydrostatic pressure (mean stress), 𝑓 ∗ – actual void volume fraction, 𝑞𝑖 – Tvergaard coefficients describing the plastic properties of the material. All those criteria have been implemented in many commercial numerical applications (e.g. finite element codes) that are used to model material behaviour under deformation prior and after damage. 3.3. Fracture mechanics 3.3.1. Linear fracture mechanics As mentioned, linear fracture mechanics describes quantitatively fracture propagation in the elastic region of material deformation. Several mathematical theories were suggested to describe this phenomenon. a) Griffith energy crack criterion In 1920 A.A. Griffith (Griffith, 1921) based on the research focused on brittle crack initiation in glass introduced feature called the crack tip as the location responsible for crack initiation and growth. The main idea of this theory is based on the assumption that the growth of a crack requires development of two new surfaces and an increase in the surface energy. This approach is only correct for the perfectly brittle materials. Griffith formulated the crack propagation criterion as: the critical crack propagation occurs when released elastic energy 𝑈𝐸 is higher that surface energy growth 𝑈𝑆 . 𝜕𝑈 𝜕(𝑈𝐸 + 𝑈𝑆 ) (16) = =0 𝜕𝑐 𝜕𝑐 where: 𝜕𝑐 – crack length. Elastic strain energy 𝑈𝐸 can be calculated by:. 18.

(21) Chapter 3. Fracture. 𝜎2 (17) 𝐸 where: 𝑇 – thickness of the sample, 𝜎 – external stress applied to the sample, 𝐸 – Young modulus. As mentioned, crack propagation creates two new surfaces in material. Thus, energy accumulated at both surfaces is described as: (18) 𝑈𝑆 = 4𝑐𝑡𝛾𝑠 where: 𝛾𝑠 – surface specific energy. When equations (17) and (18) are combined with (16) the standard Griffith formula, the crack initiation criterion can be obtained. The Griffith crack criterion for plain stress and plain strain conditions can be described as: 𝑈𝐸 = −𝜋𝑐 2 𝑇. 2𝐸𝛾𝑠 𝜎𝑐 = √ 𝜋𝑐. (19). 2𝐸𝛾𝑠 𝜎𝑐 = √ 𝜋𝑐(1 − 𝜈 2 ). (20). where: 𝜈 – is a Poisson ratio. However, experimental investigation confirmed this criterion only for the quartz glass, which cracks in a perfectly brittle way. In consequence, for the crystalline materials other crack models combined with plastic deformation before crack initiation have to be taken into account. That is why for the metallic and isotropic materials where crack initiation energy is much higher than surface energy, different crack models were recommended. In 1948 G.R. Irwin and E. Orowan (Irwin, 1948; Orowan, 1949) independently worked on the quasi–brittle fracture model, which takes into account local plastic deformation in materials prior nucleation and propagation of the brittle cracks. Plastic deformation is generated because of the dislocations slip, which can be observed in the small area near the crack tip. This is a location where equivalent stress value is close to the yield strength of the material. That is why Orowan included plastic strain energy in the investigation. The main difference to Griffith theory was to include plastic energy and combine it with (19). Because of the fact that plastic energy is four times bigger than the elastic energy, the elastic part can be neglected: 2𝐸𝛾𝑝 𝜎𝑐 ≈ √ 𝜋𝑐. (21). where: 𝛾𝑝 – energy release after braking atoms connections. On the other hand, Irwin introduced the parameter 𝐺, which characterizes intensity of the absorption of energy 𝛾𝑠 . The 𝐺 parameter can be explained as a force needed for crack propagation and gives information about energy cumulated in the crack tip. The crack initiates when intensity of the energy release reaches critical value 𝐺𝑐 : 𝜎𝑐 = √. 𝐸𝐺𝑐 𝜋𝑐. (22). where: 𝐺𝑐 = 2(𝛾𝑠 + 𝛾𝑝 ).. 19.

(22) Chapter 3. Fracture. Both Orowan and Irwin approaches are based on the Griffin theory. In both models when stress reaches some critical value the stage of the uncontrolled crack propagation initiates and then crack propagates in a catastrophic way. b) Irwin force crack criterion Along with the energetic crack criteria a series of the force crack criteria were developed. In these approaches three different crack opening modes were defined as seen in Figure 14. Mode I denotes a symmetric crack opening with respect to the 𝑥, 𝑧–plane. Mode II is characterized by an antisymmetric separation of the crack surfaces due to the relative displacement in 𝑥–direction (normal to crack front). Mode III describes a separation due to relative displacement in 𝑧–direction.. a). b) c) Figure 14 a) Mode I b) Mode II c) Mode III of the crack opening.. However, mentioned Griffith model does not take into account stress field in the area of the crack tip. That is why Irwin modified force interpretation in the crack criterion of the quasi–brittle fracture and defined stress field near crack tip for the mode I (Figure 15). Equation (23) explaining exemplary stress distribution can be calculated in the point situated in some distance from the crack: 𝑐. Θ. Θ. 𝜎11 = 𝜎√(2𝑟) cos 2 (1 − sin 2 sin 𝑐. Θ. Θ. 𝜎22 = 𝜎√(2𝑟) cos 2 (1 + sin 2 sin 𝑐. Θ. Θ. 𝜎12 = 𝜎√(2𝑟) sin 2 cos 2 cos. 3Θ 2. ). 3Θ. 3Θ. 2. ) (23). 2. 𝜎33 = 𝜈(𝜎1 + 𝜎2 ) { 𝜏13 = 𝜏23 = 0 where: 𝑟 – distance between crack tip and single point, Θ – angle between radius 𝑟 and axis situated at the surface where crack propagates.. 20.

(23) Chapter 3. Fracture. Figure 15 Illustration of the stress state near the crack tip. In reality due to the growing thickness of the sample stress state is changing from plain stress to plain strain condition. Thus Irwin modified (23) by adding external parameter called stress intensity factor 𝐾𝐼𝐶 . This parameter defines stress state in the crack tip during crack propagation. It is a function of the material and crack geometry as a result of external stresses applied to the material. This parameter is not sensitive to the coordinates of the point near the crack tip. Stress intensity factor parameter can be calculated as: (24) 𝐾𝛼 = 𝜎√𝛼𝜋𝑐 where: 𝛼 – parameter defining shape of the sample and crack geometry (takes values from 1 to 4). In this case stress for the e.g. 𝑥 direction can be calculate by: 𝐾𝐼 𝛩 𝛩 3𝛩 𝜎11 = 𝑐𝑜𝑠 (1 − 𝑠𝑖𝑛 𝑠𝑖𝑛 ) (25) 2 2 2 √2𝜋𝑟 where: 𝐾𝐼 – stress intensity factor, Θ – angle between radius 𝑟 and axis situated at the surface where crack propagates. Irwin also suggested critical value for the stress intensity factor 𝐾𝑐 . If in some areas of material parameter 𝐾𝑐 reaches the critical value then crack propagates in an unstable manner. Plain strain condition is the most dangerous for the initiation and propagation of brittle fracture. It is also possible to calculate the size of areas 𝑟𝑝2 where material deforms in a plastic and elastic manner for example in plain stress state (𝜎22 ≥ 𝜎11 ≥ 0, 𝜎33 = 0), respectively: 1 𝐾𝐼 2 (26) 𝑟𝑝2 = ( ) 2𝜋 𝜎𝑌 where: 𝐾𝐼 – stress intensity factor, 𝜎𝑌 – stress state received during plastic deformation. 3.3.2. Nonlinear fracture mechanics a) Crack tip opening displacement (CTOD) All criteria described above are used for the linear–elastic conditions. When ductile fracture is taken into account these criteria are not sufficient. To deal with the ductile failure M. Leonov and W. Panasiuk suggested crack tip opening criterion (CTOD) (Leonov & Panasuk, 1959). The main assumption of this method is to consider microcrack in the area where material. 21.

(24) Chapter 3. Fracture. deforms in a plastic manner. Authors defined the microcrack parameter, which describes size of the crack in the investigated part of material and parameter defining the maximum crack dimension that can occur due to external stresses applied to the sample. Figure 16 shows schematic crack geometry defined in the CTOD theory.. Figure 16 Schematic crack geometry defined in the CTOD theory. In the CTOD, size of propagating crack 𝛿𝑐 can be defined as: 𝜋𝜎𝑐2 𝑐 𝛿𝑐 = 𝐸𝜎0 𝐸𝜎0 𝛿0. where: 𝜎𝑐 = √. 𝜋𝑐. (27). , 𝐸 – Young modulus, 2𝑐 – crack length.. Usually for the quasi–static materials 𝜎0 parameter takes the value equal to plastic yield strength 𝑅0,2 . Parameter 𝛿𝑐 as opposed to 𝐾𝐼𝑐 is dependent on the dimensions of the sample (Neimitz, 1998; Gross & Seelig, 2006). b) J integral criterion One of the most commonly used approaches in crack modelling, is the Rice integral energy method, which is devoted to materials with elastic–plastic behaviour (Rice, 1968). Changes in the potential energy as a result of the crack propagation give opportunity to define the 𝐽 parameter. This parameter characterizes material fracture toughness: 𝐽 = ∫ (𝑤𝑛1 − 𝜎𝑖𝑗 𝑛𝑗 𝑢𝑖,1 ) 𝑑𝑠 𝑐. (28). where: 𝑑𝑠 – increase in the length of the arc along the contour 𝐶 (Figure 19), 𝜀. 𝑤 = ∫0 𝑖𝑗(𝜎𝑖𝑗 𝑑𝜀𝑖𝑗 ) (𝑖, 𝑗 = 1,2,3) – density of the strain energy, 𝑢𝑖,1 – displacement vector, 𝑛𝑗 – component of the unit vector 𝑛 normal to external to contour 𝐶), 𝑛1 = 𝑛𝑗 𝛿1𝑗 .. Figure 17 Schematic illustration of the integration path in the Rice method.. 22.

(25) Chapter 3. Fracture. Critical value of the Rice integral parameter 𝐽𝑐 corresponds to the crack initiation in the material. Crack starts to propagate when potential energy is greater or equal to 𝐽𝑐 . For the small plastic strain in the area of crack tip (28) this criterion can be expressed for the plane stress conditions as: 2 𝐾𝐼𝑐 (29) 𝐽𝐼𝑐 = 𝐸 or for the plane strain conditions as: 2 (1 − 𝜈 2 )𝐾𝐼𝑐 (30) 𝐽𝐼𝑐 = 𝐸 As seen in the above described conditions, initiation of the crack is related with the state of the material in the area of the crack tip. However, the underlying microstructure with various defects and flaws is not considered in these approaches. Thus, to deal with these issues that are present in real microstructure the continuum damage mechanics was developed. 3.4. Continuum damage mechanics (CDM) The CDM is based on the work by Kachanov (Kachanov, 1976; Gross & Seelig, 2006) who introduced the geometrical measure of fracture. A simple means of describing the state of damage consists in its geometrical quantification. Three different definitions of geometrical measure of fracture for material not deformed, deformed and weakened by damages were introduced. Based on these assumption three different stress definitions can be used: the apparent stress 𝜎 𝑎 , true stress 𝜎 and effective stress 𝜎̃ as seen in Figure 18. In a cross section of the damage body an area 𝐴𝜔 is considered for the investigation. Scalar measure of damage ω describes current reduction of cutting flank (𝐴 − 𝐴𝜔 ), with respect to the nominal surface 𝐴. Where 𝜔 = 0 corresponds to the undamaged material and 𝜔 = 1 formally describes the totally damaged material with the complete loss of stress carrying capacity. Kachanov model is commonly used during numerical modelling to deal with dual phase fracture. Review of these application can be found in the following chapter.. Figure 18 Geometric measure of damages proposed by Kachanov (Kachanov, 1976) a) undeformed material b) deformed material c) material deformed with damage. 3.5. DP fracture models based on the FEM approach This section reviews different research works dealing with failure in dual phase (DP) steels. Modelling fracture in DP steels is a complex task because of the composite character of the. 23.

(26) Chapter 3. Fracture. investigated microstructure that is composed of two phases with significantly different mechanical properties. Multiphase microstructure in these steel grades does not allow to define universal stress intensity factor (𝐾𝐼𝐶 ) or J–Integral factor. There are many experimental investigations where the problem of searching the appropriate fracture toughness parameter for different dual phases steel grades was undertaken (Kleemola & Pelkkikangas, 1977; Sudhakar & Dwarakadasa, 2000; Bag, et al., 2001; Chao, et al., 2007; Lacroix, et al., 2008; Murakawa, et al., 2009; Giri & Bhattacharjee, 2012). Numerical modelling with applied fracture toughness parameter based on the real experimental procedure gives representative results for this steel grade. Such parameter can be used during numerical investigation but has to be supported by a series of real experimental tests. Data received from these models do not give information about specific behaviour combined with fracture process in DP steels. Another approach to modelling DP fracture is based on forming limit diagrams (FLD) (Stoughton & Yoon, 2011). The criterion can be simply adopted into numerical application but FLD has to be determined for every single DP steel grade under investigation. Examples of modelling fracture with FLD approach are presented in (Luo & Wierzbicki, 2010; Wu-rong, et al., 2011; Wu, et al., 2012). Other works (Ramazani, et al., 2013) based on the maximum shear strain criterion take into account fracture generated during mechanical shearing process. Lack of accuracy of these models and discrepancies observed between model prediction and experimental measurements forced scientists to develop multiscale models with accurate description of microstructure morphology. Thus, recently the multiscale models based on the virtual material representation of microstructure are being intensively developed. There are three usually used virtual material descriptions called RVE, Unit Cell and DMR. The Representative Volume Elements (RVE) is a model of the material to be used to determine the corresponding effective properties for the homogenized macroscopic model. As already mentioned the RVE should be large enough to contain sufficient information about the microstructure in order to be representative, however it should be much smaller than the macroscopic body (Hashin, 1983; Serafin & Cecot, 2013). The Unit Cell (UC) is a part of RVE that enables to obtain results for particular part of the material. Thus the Unit Cell is not representative for the whole numerical model. Application of the Unit Cell idea, enables analysing material behaviour in particular location, e.g. crack initiation along the inclusion, without focusing on the rest of the material. As a result, the Unit Cell provides data only for local analysis, not for the whole material behaviour (unlike the RVE). Usually several Unit Cells can be considered as the RVE. However, both of the presented approaches can have a detailed or simplified geometry of microstructure features. In the simplified model called statistically similar RVE (SSRVE) e.g. only similar volume fraction of particular phase is considered while the shape of this phase is not regarded. Such model can provide representative global response, while morphology is significantly simplified and only statistically similar (Balzani, et al., 2010; Rauch, et al., 2011; Brands, et al., 2011). When detailed representation of morphology of real microstructure is needed, the term of the Digital Material Representation (DMR) is used. The DMR has already been introduced in many research works (Bernacki, et al., 2007; Miller, et al., 2008; Ivanov, et al., 2009; Uthaisangsuk, et al., 2011). The definition according to (Senkov, et al., 2003) states that Digital Material is a material description based on measurable quantities that provide the necessary link between simulation and experiment. 24.

(27) Chapter 3. Fracture. So the main objective of the DMR is to create digital representation of microstructure with its features represented explicitly to match real microstructure morphology. In that sense the DMR can be used as Unit Cell or, if it meets the above mentioned criterion, it can be considered as the RVE as presented in Figure 19. Details on the DMR idea are depicted in the following chapter.. Figure 19 Differences between the RVE, DMR, UC and SSRVE. Modelling of fracture initiation and propagation based on the DMR approach is becoming more popular and development of different models, which take into account fracture with various fracture criteria in DP steels can be found in scientific literature. The first group of models resolves fracture in dual phase steel as a result of the maximum strain that is localized in the small area. Searching for maximum strain localization zones in material is the main subject of (Sodjit & Uthaisangsuk, 2012). Authors compared experimental and numerical results obtained on DP 800 steel grade. Real macroscopic tension experiment showed that crack initiates in places where the sample starts necking and where strain reaches maximum value. Microscopic experiments showed small cracks existing in the close proximity between martensite phase islands. Numerical investigation based on the representative digital microstructure combined with elasto–plastic material model revealed large stress localization, which generates accumulative shear strain. Similar experimental and numerical investigation was made on different DP grades in (Sun, et al., 2009; Asgari, et al., 2012). In these works authors prepared series of the DMR structures where martensite volume fraction contained from 7% to 90%. Numerical experiment based on the elasto–plastic material model without fracture criteria showed places with maximum localized strain values. These places can be treated as future crack initiation locations. Other works (Paul & Kumar, 2012; Sodjit & Uthaisangsuk, 2012) in comparison to previous ones are not based on the microstructures received from real microscope pictures. Digital microstructures were prepared with numerical algorithms. However, results of stress and strain distributions give similar results to the previous models based on the real microstructure images.. 25.

(28) Chapter 3. Fracture. The second group of fracture models consider damage initiation on voids. Those models take into account creating, growing and coalescence of voids during material deformation. For this reason Gurson–Tvergaard–Needleman (GTN) is often used. Soyarslan et al. (Soyarslan, et al., 2012) prepared numerical free bending test for simulation of macrocrack formation in DP1000 steel grade. Calculation was supported by real experimental results, which were used to determine GTN model parameters. Dalloz et al. (Dalloz, et al., 2009) presented similar work on effect of shear cutting in DP steel, where GTN model was used to determine crack propagation directly during cutting process. Authors performed two plastometric tests for tension and crack opening processes to predict size and shape of micro voids observed through fractography. Experimental procedures were used to determine coefficients of the GTN damage model. Similar work was done by Ramazani et al. (Ramazani, et al., 2012). Authors prepared simulation of cross–die test with GTN model. The model was verified through FLD curve recorded during experimental and numerical tests. In works (Uthaisangsuk, et al., 2009; Uthaisangsuk, et al., 2008; Uthaisangsuk, et al., 2011) authors took statistical representative RVE of the material into account. First, authors made simulations at the macro scale (the hole expansion or stretching tests), where deformation state was used as input parameter to create simulations at the micro scale. Maximum force leading to crack initiation was compared to the force received during experimental investigation. Widely used, in DP steels modelling, fracture criterion that is calibrated through real experimental procedure, is called Johnson–Cook (JC) criterion. There are a lot of works where authors prepared experiments and determine material state parameters for DP steel grades based on this criterion. The JC criterion is implemented in almost all of the commercial numerical applications and gives opportunity for qualitative explanation of the observed fracture phenomena. In (Golovashchenko, et al., 2013) authors take JC criterion for modelling cracks in different DP steel grades during electro–hydraulic forming conditions. Different work by (Prawoto, et al., 2012) resolves problem consisting multiscale fracture modelling in the sample during impact test. Deformation state from macro model was set as boundary condition to a micro model. DMR used during simulations was built with three different ferrite volume fractions. Materials constants for the JC criterion were determined during two plastometric tests: tension and impact, respectively. Behrens et al. (Behrens, et al., 2012) considered failure of two cold–rolled hot–dip galvanised dual phase steel grades for cold forming HCT600XD and HCT780XD. In (Vajragupta, et al., 2012) the authors prepared simulation of the microstructure where ductile fracture was calculated by JC model. This work emphasizes the large influence of the mesh sensitivity required for the successful execution of fracture calculations. As mentioned, the fracture modelling techniques in commercial numerical applications are mostly based on the FE method. The above paragraph presented a great number of models for resolving fracture problems in dual phase steel grades as a result of microvoids coalescence, material ductility or shear deformation behaviour. All of those methods require well defined parameters obtained on the basis of experimental investigation. However, taking into account all discontinuous phenomena responsible for crack initiation and propagation at the microstructure level is not possible through standard FE models. In that case the various alternative numerical approaches have been proposed.. 26.

(29) Chapter 3. Fracture. 3.6. Fracture models based on the XFEM, cellular automata, Monte Carlo and molecular dynamic approaches 3.6.1. Extended Finite Element Method (XFEM) One of the commonly used methods that is based on the modified FE approach is called extended Finite Element Method (XFEM). The method can deal with numerical simulation of materials where strong discontinuities can be found. That is the reason why, the method was recently widely used for simulation of material failure (Yu & Liu, 2011; Jovicic, et al., 2010). For modelling fracture with XFEM approach critical crack parameters such as maximal principal stress or strain are required. For the dual phase steels Vajragupta et al. (Vajragupta, et al., 2012) was using XFEM to model failure within the martensite phase. Similar work, but performed on more accurate DMR of dual phase steel grade is presented in (Madej & Perzyński, 2013). The main advantage of this method is related to the simplicity in defining parameters needed for crack initiation and propagation. Author in (Madej & Perzyński, 2013) also used XFEM method for resolving problem of brittle cracks in DP steel grades (Figure 20). This method is also used by the Author of the dissertation, thus more detailed information about the applied idea will be mentioned in the following chapters. 3.6.2. Cellular automata approach Completely different approach to modelling failure is based on discrete cellular automata (CA) approach. Cellular automata was originally invented by Ulam and von Neumann in the 1940’s to provide a formal framework to investigate the behaviour of complex extended systems (von Neumann, 1966). A cellular automata consists of a regular grid of cells, where each cell can be in one of a finite number of possible states, updated synchronously in discrete time steps according to a local, identical interaction rules. The state of a cell is determined by the previous states of a surrounding neighbourhood of the cell. Over the years cellular automata have been applied to study different phenomena occurring in materials including behaviour of dual phase steels during e.g. recrystallization (Bos, et al., 2010; Seyed Salehi & Serajzadeh, 2012; Peranio, et al., 2012), phase transformation (Thiessen, et al., 2007), etc.. Figure 20 Modelling brittle (XFEM) and ductile fracture process in DMR of dual–phase steel (Madej & Perzyński, 2013). 27.

(30) Chapter 3. Fracture. Cellular automata is widely used when disproportion in properties of e.g. steel microstructure can be found. Nowadays, there is limited amount of works where modelling fracture with the CA is used in dual phase steel grades. However, some investigations with other kinds of industrially applied steel grades can be found in scientific literature. Wang et al. (Wang, et al., 2011) took into account modelling shear fracture in tungsten alloy. Movable Cellular Automata (MCA) method was implemented in that work to simulate the interior structure of tungsten alloy where shear deformation failure can be analyzed as a result of applied at meso scale level shear loading. Another application of the MCA approach can be found in work by Pak et al. (Pak, et al., 2007). Authors used CA algorithm to investigate fracture existing in the contact area between two AISI 1040 steel parts. Similar work by (Dmitriev & Osterle, 2010) took contact failure between perlitic and ferritic phases into account. In a different way CA was used in (Franklin, et al., 2008) where the author generated accurate DMR for calculation of rolling contact fatigue (RCF) parameter. The CA calculated micro scale influence of the perlite volume fraction on the mechanical properties of the macro scale model. To address this issue, a CA was used to generate realistic microstructure. Cellular automata in (Khvastunkov & Leggoe, 2004; Khvastunkov & Leggoe, 2004) were also used for finding strain localization zones in the structure that can be a precursor of material failure. A three–dimensional cellular automata were adopted to investigate the effect of the nature of spatial heterogeneity on macro scale porous ductile alloys. Including porosity in the DMR gave opportunity for finding strain localization which leads to failure. Two dimensional cellular automata stochastic model presented in Matic et al. (Matic & Geltmacher, 2001) was implemented for investigation micro cracks and voids propagation during material deformation at macro scale. This model can be adopted to all of the steel microstructures, where fractures are caused by the void formation and coalescence. The CA models can also be the basis of the multiscale models. When advantages provided by the CA are combined with the advantages of the FE approaches the CAFE (Cellular Automata Finite Element) is established. The basic assumption of this method is that each method resolves a single physical problem in different scales. There are two types of multiscale models based on the CA: upscaling and concurrent (Madej, et al., 2008) (Figure 21). Upscaling (hierarchic) model exists in the situation when constitutive model from the higher scale (FE) is created on the basis of results received from simulation executed at the lower scale (CA). In the concurrent (hybrid) one existing problem is resolved in the parallel scales and the results are transferred between models during simulation time. The main advantage of the CAFE approach is that it gives an opportunity to simulate metal forming operations in different length scales. This is important especially during modelling the fracture phenomena, where complex deformation state at the macro scale has an influence on the fracture initiation at micro scale and vice versa. In the CAFE model, finite element part calculates in every single integration point or mesh node macroscopic parameters such as stress, strain or temperature. After that information is sent to the cellular automata part that is associated with each integration point/node or with a set of finite elements in the upscaling and concurrent approaches, respectively. The CA, on the basis of macro scale information, calculates micro parameters, which are then send back to the FE mesh.. 28.

(31) Chapter 3. Fracture. Figure 21 Concurrent and upscaling communication methods used in the CAFE approaches (Madej, et al., 2008). The first use of the CAFE approach in crack modelling was suggested by Beynon and Das in (Beynon, et al., 2002; Beynon, et al., 2002). They used upscaling CAFE approach in modelling fracture during rolling. Using the CA gave a possibility to consider specific microstructure elements such as grain boundaries, phases boundaries, inclusions, grain interiors etc. and investigate influence of these elements on the deformation process. The DMR microstructure was created using the CA grain growth algorithm. Macro parameters such as stress and temperature were used as initial parameters of CA cells. The CA model was used to calculate density of dislocation, crack invitation and propagation direction. Defined FFI (Final Failure Index) coefficient gave information, on amount of CA cells, which were cracked. Based on this information FE element method was modifying values of the yield stress, friction coefficient and thermal exchange coefficient. Shterenlikht (Shterenlikht & Howard, 2006) suggested similar upscaling CAFE model for investigation brittle and ductile crack in steel during the Charpy test. To model brittle and ductile cracks existing in different length scales Shterenlikht used two separate cellular automata spaces. The first resolved brittle cracks in micro scale area from 0.005 to 0.05 millimetre. The second one calculated ductile cracks at the meso scale from 0.1 to 0.5 millimetre. In (Shterenlikht & Howard, 2006) each cell in ductile and brittle spaces is described by several state variables. The most important is the variable that refers to the state of a particular cell. It is assumed that in ductile CA space each cell can be in two different states: alive or dead. For this case, a proper transition rule, which describes real material behaviour during deformation, is defined to control ductile fracture initiation and propagation. The same. 29.

(32) Chapter 3. Fracture. analysis was performed for brittle CA space, although the situation concerning this CA space is more complex. Each cell in the automata is described by four different states: alive, aliveC, deadD, deadB. State aliveC describes crack carbide. In each time step the information is being sent between CA spaces and also between CA and FE model as seen in Figure 22.. Figure 22 Communication protocols between FE and CA models in Shterenlikht’s CAFE approach (Shterenlikht & Howard, 2006). As a result, information about cracks from micro scale, is returned to the FE solver and whenever a specific part of the dead cell in one of the CA spaces exceeds a declared critical value, then a particular FE element is removed from the FE mesh during further calculations in the next time step. The major advantage of the CAFE model in comparison to the FE approach is to take into account discontinuous and stochastic phenomena, which occur during material deformation. The cellular automata model can be connected to the finite element method or to other kind of similar approaches e.g. finite difference method (Makarov & Romanova, 2000). As presented, the CA is an example of the discrete model often used in modelling material behaviour in metalforming applications. Similar approach based on slightly different assumptions but also often used is called Monte Carlo. 3.7. Monte Carlo (MC) approach The MC is the name of the group of approaches based on similar principles of completely random sampling of the solution space Ω for application in mathematical and physical simulations (Metropolis, et al., 1953; Blikstein & Tschiptschin, 1999). However, most of the MC approaches (auxiliary field Monte Carlo, dynamic Monte Carlo, kinetic Monte Carlo, quantum Monte Carlo, quasi Monte Carlo, Makarov Chains etc.) are based on the same major steps:  Definition of a domain of possible input parameters for the model.  Generation of defined input parameters in a random manner over the investigated domain/space.  Realization of required deterministic calculations using generated input parameters.  Aggregation of obtained solutions.. 30.