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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. Analysis of the incremental forming process based on multiscale modelling approach by. MSc. Joanna Szyndler. Supervisor: dr hab. Inż. Łukasz Madej, prof. n. AGH. Cracow, 2017.

(2) Akademia Górniczo-Hutnicza Wydział Inżynierii Metali i Informatyki Przemysłowej. Katedra Informatyki Stosowanej i Modelowania. Rozprawa Doktorska. Analiza procesu kształtowania przyrostowego z wykorzystaniem modelowania wieloskalowego. mgr inż. Joanna Szyndler. Promotor: dr hab. Inż. Łukasz Madej, prof. n. AGH. Kraków, 2017.

(3) To my Parents and Henryk for their never-ending support..

(4) Content. Content ………………………………………………………………………………………………….2 List of Symbols ………………………………………………………………………………….………3 1. Introduction ...………………………………………………………………………………….…..4 2. Integral elements …………………………………………………………….………………….….7 2.1. Forming methods of integral elements ……………………………………………….8 3. Incremental forming methods ………………………………………………………………….…14 4. Multiscale modelling concept ……………………………………………………………….……22 5. Digital material representation concept…………………………………………………………...32 6. The aim of work …………………………………………………………………………….…….38 7. Development of the FE macro model of the IF process…………………………………….……..40 7.1. Finite element software ……………………………………………………………...40 7.2. Material data evaluation ……………………………………………………………..42 7.3. Analysis of maximum anvil indentation per one roll pass …………………………..45 7.4. Macro scale FE model of IF process …………………………………………………47 8. Development of the FE micro model …………………………………………………….…….…59 8.1. DMR model preparation …………………………………………………………….59 8.2. Material data evaluation for microscale model ………….…………………………..66 9. Development of the 2D multiscale model of the IF process….………………………….……..….69 10. Development of the 3D multiscale model of the IF process ……………………………...………79 10.1. FE mesh density analysis – macro model ………………………………………….80 10.2. FE mesh density analysis – micro model ………...……………………………..…84 10.3. Results from 3D multiscale calculations………………………………..………....86 11. Validation of the IF numerical model…………………………..…………………………..……..91 11.1. Comparison between FE simulation and laboratory results …………………...…..99 12. Process window evaluation …………………………………………………………….…….….105 13. Conclusions ………………………………………………………………………………….….109 14. References ………………………………………………………………………………………112. 2.

(5) List of Symbols L – velocity gradient tensor 𝑭̇ – rate of change of F 𝑹̇∗ – rate of change of 𝑹∗ 𝑼̇𝑒𝑙 – rate of change of 𝑼𝑒𝑙 𝑭̇𝑝 – rate of change of 𝑭𝑝. ε – equivalent strain σ – equivalent stress u – displacement T – temperature D – grain size σp – flow stress αp – fraction of the α phase γp – fraction of the γ phase  ij – deviatoric stress tensor. Lp – plastic velocity gradient tensor α – slip system superscript 𝑴𝛼 – Schmid tensor bα – slip direction nα – slip plane 𝛾̇ 𝛼 – rate of dislocation slip E – elastic strain tensor T – second Piola-Kirchhoff stress σ – Cauchy stress c – fourth order elasticity operator τα – shear stress 𝜏𝑐 – critical resolved shear stress (CRSS) 𝛾̇ 0 – reference slip rate a – rate sensitivity 𝜏𝑐0 , 𝛤0 , n – material parameters describing the evolution of the CRSS (𝜏𝑐 ) 𝛤𝑡𝑜𝑡 – accumulated plastic slip V – material volume.  ij – strain rate tensor.  – equivalent strain rate. m – strain rate sensitivity coefficient K – material consistency ρ – material density c – specific heat capacity t – time k – thermal conductivity factor W – internal energy dissipation η – strain efficiency (heat conversion/deformation efficiency) τ – friction shear stress µ – friction coefficient 𝑚 ̅ – Tresca friction coefficient σn – normal stress A, m1, m2, m3, m4, m5, m7, m8, m9 – HS model coefficients Φ(x) – goal function x – vector of the material model coefficients dcalc – calculated output parameters dmeas – measured output parameters Qi – cell state Ei – calculated energy of the single cell Jgb – coefficient for grain boundary determination δSiSj – the Kronecker delta ΔE – energy difference of the selected cell prior and after random change of the state kB – Boltzmann constant F – deformation gradient tensor Fel – elastic part of the deformation gradient tensor F Fp – plastic part of the deformation gradient tensor F R* – an orthogonal tensor that represents crystal Uel – a symmetric tensor of the elastic stretch 3.

(6) 1. INTRODUCTION In recent years natural environment becomes more and more endangered by different sources of pollution. The rising necessity of its protection is one of the driving forces for rigorous European Union regulations for carbon dioxide and noise emissions as well as electricity consumption during subsequent production stages and further exploitation conditions. A worldwide environmental protection policy insists on limitation of factors that are dangerous for natural environment, what can be especially seen in goals of the European Framework Program of Research and Innovation (2014 – 2020) – „Horizon 2020”. One of the main objectives of this program is production of “eco- friendly” (Oxford Dictionary of English) aircrafts, vehicles and ships. It is expected that this will contribute to improvement of environmental protection by noticeable reduction in noise, vibrations and emissions of harmful substances produced by an air and ground transport. There are several opportunities to meet mentioned goals, especially reduction in weight of the commonly used conveyances (e.g. cars, trucks, passenger airplanes, transport aircrafts), development of new materials as well as implementation of innovative manufacturing solutions. These approaches result in decrease of e.g. the amount of consumed fuel and in consequence lead to reduction of the amount of carbon dioxide emission into the atmosphere. Thus, there is a high need to cut weight of vehicles, but at the same time customers demand reduction in prices (production and materials costs) and improvement in the quality of products. Also restrictions of passengers welfare becomes tighter and customers’ expectations raises, so the safety features are constantly enriched, what in consequence increases the overall mass of vehicles. During last several years different solutions have been proposed to address the question of how to lower the vehicle weight. One of them, is to develop new, advanced steels, aluminum alloys, magnesium alloys, carbon fibers, composites or plastics that can be used to construct the final product (Smojver & Ivancevic, 2010). Another, is to replace some constructional parts with lighter customized products e.g. tailored blanks or tailored tubes that can have different properties in different zones (Merklein et al., 2014). Fuel reduction can also be obtained by better shape optimization of chassis or fuselages, by e.g. using numerical simulations. Another idea is to develop new methods of material forming that can provide products specifically designed for particular applications (Ding et al., 2011). Finally, the last concept that can be used to reduce weight and increase safety of constructional elements is to replace several components that were joined together with an integral element, made from one piece of material. The last two approaches are the main motivation for novel incremental forming (IF) process proposed in (Grosman et al., 2012a), which seems to be an efficient solution for obtaining light and durable integral elements for automotive and aerospace industries. To successfully apply this innovative forming technology at the industrial scale, engineers have to gain detailed knowledge on mechanisms that control deformation and microstructure evolution during complex deformation conditions occurring during the 4.

(7) incremental process. Unfortunately, experimental research is expensive and time consuming, especially when material behavior at the microstructure scale is investigated. However, thanks to the continuous and dynamic technological progress, especially in the information technology (IT) field, presently there are many possibilities for an efficient numerical support of experimental research. One option is to use an advanced computer modelling techniques, which give a possibility to obtain a fully virtual process that joins the digital material, simulation of deformation conditions and computer analysis of obtained results. This leads to the cost reduction of laboratory research, facilitates new materials design and also enables an analysis of materials behavior under both manufacturing and exploitation conditions. Thus, the problem of expensive and time consuming laboratory research on incremental forming of integral elements, can be solved by mentioned numerical modelling. Such an approach is less expensive and can not only support but also broaden an experimental analysis. Most of numerical models, currently used in industry, predict material behavior only at the macroscale level with material data that is uniform across the whole sample, without taking into account local material behavior at the microscale. In novel materials it is easily noticeable that the microstructure is inhomogeneous and contains e.g. inclusions, precipitates, varying grain sizes with different crystallographic orientations or different phases (Bueno & Varela, 2006; Shi et.al., 2016). Such diversified character of microstructure is not taken into account in the conventional approaches, where material behavior at the microscale is averaged. This, in consequence may affect the reliability of obtained results. In the case of mentioned innovative incremental forming technology, local inhomogeneous material behavior at the level of subsequent grains cannot be neglected. Thus, Author decided to support experimental investigation on the IF process by advanced numerical simulations. Development of a concurrent multiscale numerical model supported by the digital material representation (DMR) concept combined with the crystal plasticity (CP) theory may give a possibility to understand in detail material flow during incremental forming of integral elements at various scales. As mentioned, the microscale model will be based on the digital material representation concept, which explicitly takes into account material morphology and enables detailed analysis of material behavior at the level of single grains. DMR models will be additionally connected with Crystal Plasticity framework what allows an investigation of crystallographic orientation changes during considered incremental forming. Finally, DMR models coupled with CP concept will be connected with the macroscale model and a numerical multiscale concurrent model of IF process will be established. Because the proposed numerical approach will be based on the finite element method (FEM), an analysis of influence of mesh density at the microscale level on quality of obtained results is required to ensure model predictive capabilities. An analysis of minimum data transfer nodes between micro and macroscale in a 3D multiscale model will also be presented within the work. Results obtained from numerical investigation will be then compared with laboratory research based on: optical and electron microscopy 5.

(8) imaging of microstructure, micro hardness maps and geometrical characteristics of obtained parts, to prove the model robustness. Finally, the thesis will conclude with development of process window, evaluated after detailed numerical analysis of material flow, for practical applications.. 6.

(9) 2. INTEGRAL ELEMENTS Many constructional elements in the aerospace industry e.g. in fuselage, are typically made from several smaller components that are somehow joined (e.g. welded, riveted) (Wiślicki, 1964). Unfortunately, introduction of joints into the material structures weaken the final part, and may cause failure initiation within these areas. The easiest solution to prevent this behavior is to replace complex constructional elements with integral parts, that are made from a single piece of material (figure 1). Integral element is lighter, more durable and less susceptible for cracking in vital locations during exploitation. Additionally, application of integral elements in an aerospace industry enables reduction of exploitation costs of other airplane components (e.g. tires), what extends their lifespan and lowers conservation costs and production outlay even up to 40% (Wiślicki, 1964). Reduction of energy consumption during manufacturing stages, as less elements have to be manufactured, is another important aspect of forming technologies used during manufacturing of integral components.. Fig. 1. Example of a) conventional assembly, b) integral element concept, c) real aircraft frame (Altair, 2010). As presented, advantages of integral parts are vast. Unfortunately, because of complicated shapes and large area sizes, forming process of these components is not a trivial task (Wiślicki, 1964). During the years, different manufacturing technologies based on e.g. machining, rolling, extrusion, pressing/forging, casting, chemical pickling and the recent technology based on powder metallurgy have been developed to obtain integral parts as presented in figure 2 (Wiślicki, 1964).. 7.

(10) Fig. 2. Methods of forming integral elements for the aerospace industry. Regrettably, most of these operations are connected with high costs and technical problems, small efficiency, high material loses or limited applications. Detailed description of mentioned forming methods of integral elements with its application examples is presented in Chapter 2.1. 2.1. FORMING METHODS OF INTEGRAL ELEMENTS Machining is a broad term used to describe removal of material from a workpiece and generally can be divided into cutting (figure 3a), abrasive processes such as grinding (figure 3b), or nontraditional processes e.g. utilizing chemical etchants, electrical or other sources of energy. Machining is the oldest method of manufacturing integral elements dedicated to the aerospace industry. To obtain large, flat and thin elements, that can replace many smaller parts in airplanes construction, series of dedicated machines have been developed. Unfortunately they are quite expensive and usually complex in use. At the same time large quantities of material are wasted what results in small efficiency in obtaining integral elements. What is more, this method can also be characterized by small dimensional accuracy when hard or high strength materials are processed.. a). b) Fig. 3. Schematic illustration of machining process: a) cutting, b) grinding.. It is also economically and technically proven that machining is a good way of finishing treatment for elements that are obtained during other methods, e.g. forged or extruded.. 8.

(11) A lot of work on development of numerical models capable to simulate machining were published in scientific literature e.g. (Zenia et al., 2015; Poniatowska, 2015; Rotella & Umbrello, 2014), where finite element method and non-uniform rational B- spline (NURBS) modelling were primarily used. The easiest and fastest way of forming integral elements is shape rolling (figure 4). Rolling of metal sheet with stiffeners (figure 5) dedicated for the airplane skin panels was the first step in introduction of integral elements into the fuselage construction. Additional ribs provide a sheet that is characterized by a higher stiffness parameter, in comparison with the flat rolled product of the same mass. Ribbed sheets can then be deformed similarly as common flat sheets – by cutting, bending, forming in two dimensions or stretching.. Fig. 4. Schematic illustration of a rolling process.. Fig. 5. Schematic illustration of metal sheet with stiffening ribs. Obtained products are applied in different locations in the airframe construction e.g. skins of less loaded airplane elements, ribs, bottoms or walls of containers, dashboards, covers and doors, etc. Unfortunately, with respect to forming integral elements, rolling method has some application limitations caused by difficulties in obtaining expected height and thickness ratio of stiffening ribs. Thus, the FEM is most commonly used to support designing of such rolling technology that meets customers’ expectations, see e.g. (Cawthorn et al., 2014; Sun et al., 2014; Pesin & Pustovoytov, 2014). Although simple slab theory model, the classic inhomogeneous model of Orowan (1943) and a more recent, mathematical approach introduced by Domanti and McElwain (1995) are also used. Another popular method of obtaining semi-finished products for the aerospace industry is extrusion (figure 6). Looking from the technical aspect point of view, extrusion enables 9.

(12) obtaining elements with complex shapes in a small number of operations, high dimensional accuracy and smooth surfaces, considerable length, difficult to obtain by other methods and reduced weight in comparison with e.g. casting. Also economically, there are several advantages of extrusion, e.g. significantly lower cost of die preparation and material savings due to possibility of manufacturing near net shape products. The method of backward extrusion can be very easily applied to obtain integral elements with additional strengthening ribs, similar to the ones after rolling or machining.. Fig. 6. Schematic illustration of a backward extrusion process. Unfortunately, when room temperature deformation conditions have to be imposed to obtain proper final properties, there is a necessity to use extreme loads, especially when large area parts are considered. That is a limiting factor of wider application of the method. Extrusion at the hot deformation conditions, enables to lower press loads, however, unfavorable decrease in material strength, smaller process productivity (lower speed) and surface oxidation can be observed. Also too large extrusion velocity can cause significant internal stress increase and surface roughness, what requires application of additional finishing operations. All these issues were extensively tackled from both experimental (Matsumoto et al., 2014; Mahmoodkhani et al., 2014a; Alharthi et al., 2014) and numerical points of view e.g. (Matsumoto et al., 2014; Mahmoodkhani et al., 2014b; Tingting et al., 2014; Alharthi et al., 2014). Another way of obtaining integral parts characterized by very good strength properties is pressing/forging (figure 7) (Schongen et al., 2014; Chval & Cechura, 2015; Luri et al. 2013). However, similar to extrusion, the technology also requires presses with significant load carrying capabilities to obtain elements with wide dimensions. In comparison with machining, pressing and forging provides possibility to control mechanical properties of obtained elements and as a result, lowers its final weight. Slightly, better precision can be reached by pressing in comparison to forging, what gives the possibility of employing staff with lower qualifications, and that is important from economical point of view.. 10.

(13) Fig. 7. Schematic illustration of a pressing process. Despite advantages of pressing/forging technologies, several problems can be identified during integral elements production. The first, is related with inaccuracy in filling the grooves for high and densely distributed stiffening ribs. The second, is associated with failure that may occur at the bottom of the rib due to high contact pressures with the die. Another problem is material distortion during cooling after hot forming due to e.g. phase transformations, what in consequences causes the necessity of application of additional finishing operations (Xu et. al., 2016; Cho et. al., 2016; Feng et. al., 2015; Li et. al., 2016). These problems have also been investigated numerically with e.g. FEM or boundary element method (BEM) (Schongen et al., 2014; Chval & Cechura, 2015; Luri et al. 2013). From economical point of view, casting (figure 8) is the best solution of obtaining integral parts with complex shapes. It can also be used to obtain large integral skins of airframes. The method is less time consuming than other methods, what in consequences enables to shorter the production time. What is more, the weight of obtained elements can be lowered notably, if novel light materials are used. The benefit of casted components is often a very good weldability and machinability. Unfortunately, a lot of practical experience is needed to use casting method, because of problems related with obtaining thin-walled parts and with shrinkage phenomenon, which can lower strength of casted parts (Reis et.al., 2012; Palumbo et. al., 2016). Also internal stresses can cause inhomogeneities in material structure (Jakobsen et. al., 2016; Keste et.al., 2016).. Fig. 8. Schematic illustration of a sand mold casting process.. 11.

(14) Physical and anti-corrosion properties of casted parts, are very similar to properties of forged material with the same chemical composition. However, it has to be emphasizes that significantly worst mechanical properties (lower elongation and impact resistance) caused by preservation of original casting structure are obtained in comparison to e.g. forged components (Trosch et al., 2016; Jiang et al., 2012; Kang & Ostrom, 2008). To properly design mentioned process, numerical simulation solutions are quite commonly used e.g. (Jie et al., 2014; Bidhar et al., 2015; Bouzakis et al., 2012; Petrenko et al., 2014, Seo et al., 2007). Another interesting method of manufacturing integral elements is a chemical pickling (figure 9) (Sohlberg, 2005), which is based on removing selected parts of metal by dipping the element in baths with corrosive properties. Zones that should remain unaffected during the pickling process are protected with special covers resistant to the etching factor.. Fig. 9. Schematic illustration of a chemical pickling process. Chemical pickling method has a lot of advantages, e.g.: a). simplicity in forming large and complex parts, conditioned only by apparatus dimensions, b) easiness in treatment curved surfaces, c) high quality of surface smoothness, with no necessity of additional treatment, d) possibility of complete process automation, e) possibility of employing less qualified staff. The main disadvantage of chemical pickling is necessity of providing appropriate safety measures, more complicated than during machining or metal forming operations. The most recent approach, that is still under development, is a group of processes connected with incremental manufacturing from powders (Qiu et al., 2015; Guo et al., 2015), which can also be named as additive fabrication, freeform fabrication, solid freeform fabrication or digital fabrication. Approaches based on fast manufacturing of physical parts, prototypes or patterns from virtual 3D- CAD models, are called Rapid Prototyping (RP), while fast fabrication of forms or dies is named as Rapid Tooling (RT). Finally, fast production of finished product is called Rapid Manufacturing (RM).. 12.

(15) Incremental forming from powders enables obtaining rapid-prototypes at early stages of product manufacturing and almost completely eliminates necessity of additional and expensive treatment at later stages of production. Incremental forming of powders can be used in the direct (RM methods) or indirect (RT methods) way. The direct incremental forming can be realized by selective laser melting of metal powders (e.g. SLM, LaserCusing, Laser Engineered Net Shaping – LENS)(Fischer et.al., 2016), electron beam melting of metal powders (EBM process)(Peng et. al., 2016) or selective laser sintering of metal powders (Direct Metal Laser Sintering – DMLS)(Olakanmi et. al., 2015). The indirect incremental forming is used by RP to form casting models, e.g. for precision casting or with RT method to obtain forms and casting cores. Indirect forming methods involve stereolithography (Weng et. al., 2016) that uses pointed, layered photopolymerization of liquid nanomer with laser radiation (SLA process) or UV light (Polyjet process), selective laser sintering of moulding sands (Direct Croning Process – DCP)(Wen et. al., 2015) and selective three dimensional powder printing (3DP) (Utela et. al., 2008) where powder particles are joined by sprayed layers of liquid binder process. An example of integral element obtained with the 3D printing technology is presented in figure 10b. Incremental forming type processes also include methods where different materials are applied on already formed surfaces, by using e.g. padding (Chen et.al., 2016), metallization (Park et. al., 2015) or coating (Shypylenko et.al., 2016) to obtain required element properties. All methods of incremental forming from powders that were presented above, are schematically shown in figure 10a.. Fig. 10. a) Examples of incremental forming processes from powders (Oczoś & Kawalec, 2012), b) example of 3D printed integral element with stiffening ribs (Alec, 2015). As presented, different forming methods have their advantages as well as limitations in integral parts production. Thus, to solve some of mentioned problems, fast development of modern incremental metal forming of sheet and bulk products is observed. Selected processes of incremental forming methods of solids are presented in the following Chapter. 13.

(16) 3. INCREMENTAL FORMING METHODS Concept of incremental forming methods is based on obtaining large deformations by adding many small deformations. This enables to increase material deformation limits and lower expected press loads. Thus, incremental forming of metals enables obtaining shapes impossible to get from conventional metal forming methods and is dedicated to materials difficult to deform. Processes of an incremental metal forming can be divided into two main groups: the sheet forming (Hussain & Gao, 2007; Isekia & Naganawab, 2002; Hussain et al., 2007; Yoon & Yangl, 2005; Filicel et. al., 2002) and the bulk forming (Muszka et al., 2013; Stanistreet et al., 2006; Groche et al., 2007; Wong et al., 2004; Jin & Murata, 2004). Examples of commonly used incremental forming processes are presented in figure 11.. Fig. 11. Examples of incremental forming processes. One of the most widely researched process of incremental sheet forming is called singlepoint incremental forming (SPIF) (Guzman et al., 2012; Senthil & Gnanavelbabu, 2014; Malwad & Nandedkar, 2014). In the approach a sheet is clamped rigidly around its edges and formed by a single spherical-ended indenter (figure 12a). Other variants of the process exist and are widely referred to as two- point incremental forming (TPIF), in which the sheet is formed against full or partial dies using one or more indenters (figure 12b).. a) 14.

(17) b) Fig. 12. Schematic illustration of a a) single point incremental forming, b) two- point incremental forming processes. The variants of the incremental sheet forming process setup also allow to form the sheet without any supporting tools, with a full or partial die, or with a kinematical counter tool (Xu et al., 2014; Malhotra et al., 2012). Another process of incremental sheet forming is spinning (Watson & Long, 2014; Xia et al. 2014) (figure 13a), also called spin forming. It is a metal forming process dedicated to obtain cylindrical parts by a rotating piece of sheet metal while forces are applied to one side. A sheet metal disc rotates at high speed while rollers press the sheet against a tool, called a mandrel, to form the desired part. Obtained metal parts have a rotational symmetry and hollow shape, such as a cylinder, cone or hemisphere. A variation of conventional spinning process is a shear spinning method (Xia et al. 2014) and is also known as flow turning or spin forging (figure 13b). Shear spinning involves forming the sample over the mandrel, causing metal flow within the sample, what will reduce its thickness. Contrary to the conventional spinning, the initial diameter of the work in shear spinning can be smaller. Proper limits to the thickness reduction exist in order to prevent fracture. Coolants are normally used in shear spinning, since this manufacturing process can generate a lot of heat. One or two rollers may be used, where variant with two provides better balance of forces during the operation. Shear spinning of some materials is often conducted at elevated temperatures. Similar to shear spinning is a tube spinning (Xia et al. 2014) presented in figure 13c. Again, the metal flow enables thickness reduction and length growth of formed cylindrical parts. This process can be performed externally with the tube over a mandrel or internally with the tube enclosed by a die. In some cases the die can be moved during the process in order to obtain features or contours on the inside/outside of the tube.. 15.

(18) Fig. 13. Schematic illustration of a) spinning, b) shear spinning, c) external tube spinning. Despite the fact that incremental forming is usually related to sheet forming, variety of processes with an incremental character can also be found in bulk forming. The first, is a ring rolling process in which a ring of smaller diameter is rolled into a precise ring of larger diameter with a reduced cross section (figure 14). This is done by the use of two rollers, one driven and one idle, acting on either sides of the ring's cross section (Xiaotao & Fan, 2012; Parvizi & Abrinia, 2014; Malinowski et al., 2005).. Fig. 14. Schematic illustration of a ring rolling process. Edging rolls are typically used to ensure that the part will maintain a constant width throughout the radius. The workpiece will essentially retain the same volume, therefore the geometrical reduction in thickness will be counterbalanced by an increase in the ring's diameter. Ring rolling enables obtaining not only flat rings, but also rings of different cross sections. Another advantage is a possibility to produce very precise seamless parts with little waste of material. The most popular parts produced by ring rolling are rings for machinery, aerospace industry, pipes, turbines, ball bearing races etc. Another incremental process that uses rotating rolls is a cross rolling (Obayi et al., 2015) (figure 15), which is a near net-shape rolling process. It can be classified as high temperature incremental forming that gives cylindrical products through multi-step forming by rotating dies. 16.

(19) Fig. 15. Schematic illustration of a cross rolling process. During cross wedge rolling the billet spins as dies make one revolution and the contour on the die forces the billet material into the desired axi-symmetric preform shape (Liu et al., 2014; Wengfei et al., 2014). Variation is a process where dies are flat instead of round and they move past each other while rolling the billet between them. Different type of incremental forming process is a rotary swaging (figure 16), which is dedicated for precision forming of cylindrical workpieces e.g. tubes, bars, in many small processing steps (Zhang et al., 2014; Moumi et al., 2014). The forming dies of the swaging machine are arranged concentrically around the workpiece. The swaging dies perform high frequency radial movements with short strokes and apply compressive force onto the enclosed workpiece. Depending on the application, between two and eight dies can be used. Rotary swaging belongs to the category of net-shape forming processes, where the finished shape of the formed workpiece is obtained without, or with only a minimum amount of processing and cutting. In comparison to the continuous forming process, during rotary swaging the homogenous material properties are obtained, what is achieved by high forming ratios. Rotary swaging has all advantages of cold forming: reduced production time, close tolerances, continuous grain pattern, high surface quality and material savings (Gan et. al., 2014; Abdulstaar et. al., 2013).. Fig. 16. Schematic illustration of a rotary swaging process. Another group of incremental forming processes can be generally named as open die forging processes, where the sample is compressed between two dies, that do not constrains 17.

(20) the material flow during the deformation (Zahalka, 2014). Common open die forging process performed in industrial conditions, uses flat die to round and elongate the ingot and is called cogging (figure 17). With the use of mechanical manipulators, a workpiece is compressed and rotated in a series of steps eventually forming the metal into a cylindrical or square shaped part. The compressions affect the material of the forging, closing up holes and gaps, breaking down and reforming weak grain boundaries as well as creating a wrought grain structure. As the open die forging process progresses the material of the part is altered from the outside and progresses inward. The part should be properly worked to change the structure of the material in the center of the workpiece. Large shafts for motors and turbines are good examples of products forged this way from cast ingots (Song et al., 2014).. Fig. 17. Schematic illustration of a cogging process. Cogging allows to apply smaller machinery with less power and forces and still obtain great length of products (Wolfgarten & Hirt, 2016). Often cogging may be just one of the metal forging process in manufacturing chain required to form a desired part. In case of modelling forging processes it can be seen that the most common is the FEM approach (Zahalka, 2014; Song et al., 2014). One of the incremental forging process, that is often used at the industrial scale is the orbital forming (rotary forging) based on the Marciniak press concept (figure 18). In this technology a sample is placed between fixed lower die and an orbiting upper die that moves towards the sample (Ziółkiewicz & Garczyński, 2008; Feng et al. 2014; Nam et al., 2014). The lower die is properly shaped and gives final shape of the deformed workpiece.. 18.

(21) Fig. 18. Schematic illustration of an orbital forming process. Load reduction and the possibility to obtain large deformations are the main advantages that can be reported during this forming process. Recorded forces necessary for material deformation are much smaller than during conventional forging. High smoothness of the sample surface, material economy, simple design and easily exchangeable tools are also advantages of the forming technology. Unfortunately, some surface deformation may occur in front of the moving upper die, what may cause a micro crack initiation, especially for hardly deformable materials. The most popular simulation method for rotary forging analysis is again the FEM approach (Nowak et al., 2008; Nam et al., 2014). To overcome limitations of orbital forging a new incremental technology was proposed for materials that are considered as hardly deformable (Grosman et al., 2012a). This new process is based on small incremental deformations realized by a series of thin anvils pressed into the material by set of rolls or conical rotating die. The subsequent accumulation of these small deformations finally results in the expected deformation. As a consequence, the pressure is not applied by the upper die directly to the material, but is transferred through series of thin anvils as seen in figure 19.. Fig. 19. Schematic illustration of modified orbital forming process (Grosman et al., 2012a). As a result, a reduction of necessary loads during plastic deformation is obtained. This process joins advantages of orbital forging connected with multi point sheet forming. Typical applications of the process are ring forgings and ribbed wheels. It is also possible to manufacture products with small thickness, deep indentations or with wide surfaces with additional stiffening ribs. Development of this process was closely supported by a series of 19.

(22) numerical simulations realized within the finite element framework e.g. (Grosman et al., 2012a; Grosman et al., 2012b). As presented, the latter technology is a good alternative that can be used to manufacture integral elements mentioned in Chapter 2. However, the method is limited only to cylindrical shapes of products. The solution may be a recently proposed modification (Grosman et al., 2012b) of the technology that can be applied to manufacture different shapes of final products. Excessive loads recorded on the presses during e.g. conventional forging are eliminated in the approach by division the single upper die into a series of small anvils that realize complex deformation in a sequential manner, however in the rectangular setup as seen in figure 20 (Grosman et al., patent). As a result, widely available presses, with lower press loads, can be used to obtain integral elements. Such technology can also be successfully used to manufacture products from materials that are considered as a hardly deformable (Grosman et al., 2012a).. Fig. 20. Idea of an incremental forming approach. The proposed idea for obtaining integral elements is thus, based on combination of incremental forming with additional die in the form of roll with reciprocating movement, that exerts load on subsequent anvils. The roll that moves from one side to the other and backwards, presses subsequent anvils into the deformed component. This approach enables obtaining thin parts with additional strengthening ribs especially useful in the aerospace industry. Schematic illustration of the technology is presented in figure 21.. Fig. 21. Illustration of novel incremental forming process.. 20.

(23) Therefore, development of an innovative incremental forming process seems to be a good solution to obtain integral elements. However, as mentioned in Chapter 1, to successfully apply this innovative forming technology at the industrial scale, engineers have to gain detailed knowledge on mechanisms that control deformation and microstructure evolution during complex deformation conditions occurring during the incremental process. Unfortunately, experimental research is expensive and time consuming, especially when material behavior at the microstructure scale is investigated. Thus, Author within the thesis decided to support experimental investigation on the IF process by advanced numerical simulations. To make a detailed analysis of material flow at different levels, not only at the macroscale but also deeper, at the microscale level, a multiscale numerical models can be used.. 21.

(24) 4. MULTISCALE MODELLING CONCEPT Multiscale modeling is an approach dedicated to numerical simulations, where multiple models at different scales are simultaneously used to describe an investigated system. By definition, multiscale modelling entails application of modelling techniques at minimum two different length or time scales, which are often dissimilar in their theoretical character due to the change in scale (Elliott, 2011). The foundation for multiscale modelling originates from the fact that simplified macroscale models are not accurate enough and do not take into account different microscale features that occur in materials and have influence on local behavior. Also microscale models alone are not efficient enough and may not deliver sufficient information of the macroscopic behavior. Thus, the combination of models that represents material behavior at different scales should improve the quality of obtained results with reasonable compromise between accuracy and efficiency of modeling. Concept of multiscale modelling consists of three basic and closely related parts: multiscale analysis, multiscale models and multiscale algorithms. The first, enables understanding of the relations between models at different scales. Multiscale models give a possibility to formulate single models that are coupled at different scales. The last one, multiscale algorithms, allow to use multiscale ideas to elaborate computational algorithms (E & Lu, 2011). Multiscale approach combines a variety of models at different scales and complexity to study one system. Developed models are linked with each other analytically or numerically (E & Lu, 2011). There are various approaches for multiscale modeling, that generally can be classified into two groups: upscaling (sequential, hierarchical) and concurrent (hybrid) solutions as seen in figure 22.. 22.

(25) Fig. 22. Schematic representation of the concurrent and upscaling multiscale model (Madej et. al., 2008). In the upscaling class of methods, constitutive models at higher scales are constructed from observations and models at lower, more elementary scales. By a sophisticated interaction between experimental observations at different scales and numerical solutions of constitutive models at increasingly larger scales, physically based models and their parameters can be derived at the macroscale, see for example (Da et al., 2002). Methods of the computational homogenization, e.g. (Mieche, 2003), are considered to belong to this group of methods. In these approaches solution at the microscale level usually is insensitive to the mesh density at the macroscale as these micro models are connected to particular integrations points. Generally, two approaches of data transfer direction can be identified: coupled and uncoupled, where the second one can be divided into top-down and bottom-up approach as seen in figure 23 (Pietrzyk et. al., 2015).. 23.

(26) Fig. 23. Classification of concurrent and upscaling approaches. In the upscaling top-down uncoupled multiscale method all or only selected points at macroscale level can be connected with the microscale model (figure 24). An information, regarding calculated macroscale quantities, can be sent to the microscale model in each time step or the entire history of deformation can be sent where macroscale calculations are completed. Such information is used during subsequent calculations at the microscale level (Madej, 2014). An example of effective top-down upscaling bridging method is the use of thermodynamically constrained internal state variables (ISVs) that can be physically based upon microstructure-property relations (Horstemeyer, 2009).. Fig. 24. Concept of the top-down uncoupled upscaling approach. In the upscaling bottom-up uncoupled multiscale method again all macroscale points can be associated with microscale model or only selected ones (figure 25). The second solution is proper only when the interesting area of material deformation is identified in advance. Information with calculated microscale quantities is sent and used during subsequent steps of 24.

(27) macroscale simulation. Because of uncoupled character of this model, the information can be send in each time step or again the entire history of deformation can be send at the end of the microscale analysis (Madej, 2014).. Fig. 25. Concept of the bottom-up uncoupled upscaling approach. In the upscaling coupled multiscale method (figure 26), separate microscale models are connected with each selected material point (e.g. Gaussian point) in the macroscale model. The main difference between uncoupled and coupled approach is that in the second one all macroscale points have to be connected with model at the microscale level. The information between scales is send in every time step and cannot be send at the end of the simulation. Data with calculated macroscale quantities is send and used during subsequent calculations at the microscale level. Then, calculated information from the micro model is send back to the macroscale model and used during next time step. To obtain reliable results, calculated information have to be send between scales in each time step (Madej, 2014).. Fig. 26. Concept of the coupled upscaling approach.. 25.

(28) In the second class of multiscale modelling methods - concurrent multiscale approaches - one strives to solve the problem simultaneously at several scales by an a priori decomposition of the domain (Madej, 2014). Two-scale methods, whereby the decomposition is made into coarse and fine scales, have been considered so far (Fish, 2009). Contrary to the upscaling approaches, mesh density at the macroscale can have an influence on microscale results as several elements in the macroscale are used to transfer data into the microscale model. In concurrent models, particular material point in selected scale is associated with a part of the complete numerical model of lower length scale. Both models describe the same material area, but are based on different phenomena (Zeng & Li, 2010). Again coupled (figure 27) and uncoupled models can be found in concurrent multiscale models, where the second one can be divided into top-down and bottom-up approaches (figure 28), similarly to the upscaling models (Zeng & Li, 2010). In the concurrent multiscale models the entire domain can be described by both macro and micro models, but such solution is extremely time consuming and can be impossible to apply in practical research. Thus, usually selected part of the macroscale model is described with microscale model, where material behavior can be interesting, e.g. material cracking can be observe. Calculated information at the macroscale is send to the microscale model and used during subsequent calculations. Finally, microscale data is send back to the macro model in every time step. Examples of joining methods used in concurrent multiscale methods are MAAD (macroscopic atomistic ab initio dynamics) described and implemented by Abraham, Broughton, Bernstein and Kaxiras (Abraham et. al., 1998; Broughton et. al., 1999; Rudd & Broughton, 1999), HMM (heterogeneous multiscale method) proposed by Li and E (Li & E, 2005), BSM (bridging scale methods) presented by Liu and co-workers (Wagner et. al., 2004; Wagner & Liu, 2003), PMMS (perfectly match multiscale method) given by To and Li (To & Li, 2005; Li et. al., 2006), or a MCFT (multiscale continuum field theory) (Chen & Lee, 2005; Xiong et.al., 2007).. Fig. 27. Concept of the coupled concurrent approach.. 26.

(29) a). b) Fig. 28. Concept of the uncoupled concurrent a) top-down, b) bottom-up approach.. The concurrent multiscale models are presently used in many research works (Wen & Zabaras, 2012; Farrugia & Cheong, 2009; Vereney & Kabiri, 2012; Hollerer, 2014; Vernerey & Kabiri, 2014; Benedetti & Alibadi, 2015; Khoei et al., 2013; Wu et al., 2015), e.g. for parametric study on the fracture behavior of a crack in functionally graded materials where crack-driving forces are evaluated (Chakraborty & Rahman, 2009). Similar example of work based on the concurrent multiscale approach is (Wen & Zabaras, 2012), where a multiscale model reduction scheme based on the bi-orthogonal KLE (Karhunen–Ločve expansion) is presented. Another interesting work (Farrugia & Cheong, 2009) presents a methodology, which combines mechanical testing and modelling that has been developed to account for local microstructural heterogeneities of low ductility steels. A 5-step modeling approach from macroscale damage criteria to mesoscale elastoplastic and viscoplastic constitutive material models combined with microscale FEM of inclusion behavior has been developed to deal with aspect of physical length scale. In (Vereney & Kabiri, 2012) authors introduced a concurrent adaptive multiscale methodology for elasticity problems in which macroscopic deformation strongly interacts with microscopic deformation fields at the scale of the microstructure. A finite element method is introduced such that continuum elements can be replaced by explicit RVEs through properly defined macro-micro kinetic conditions, reminiscent to those used in classical homogenization. Different example of concurrent multiscale usage is presented in (Hollerer, 2014) where the buckling analysis of carbon nanotubes is performed. Continuum mechanics and a molecular statics formulation are used simultaneously and a total potential of the structure is specified by properly weighting the individual contributions. Additional kinematics constraints enforce the compatibility between designated atoms and the continuum body. Three different methods are taken into consideration for the kinematics coupling and the corresponding governing equations. Another work that bases on the concurrent multiscale idea is (Vernerey & Kabiri, 2014). In this work an adaptive concurrent multiscale methodology was introduced to handle situations in which both macroscopic and microscopic deformation fields strongly interact near the tip of a crack in heterogeneous media. Presented method is based on 27.

(30) the balance between numerical and homogenization error. The first one states that elements should be refined in regions of high deformation gradients, the second implies that element size may not be smaller than the threshold determined by the size of the unit cell that represents the material’s microstructure. Thus, a finite element framework was built where the unit cells can be embedded in the continuum region through appropriate macro-micro boundary coupling conditions. Also in (Benedetti & Alibadi, 2015) a multiscale concurrent model was implemented where the three-dimensional micro-representative volume elements (RVE), at the polycrystalline grain-scale level, are analyzed employing a specifically developed threedimensional grain-boundary cohesive-frictional approach. The approach is able to capture intergranular degradation and failure through cohesive laws embodying an irreversible damage parameter. The coupling between two scales is achieved by down-scaling macro-strains as periodic RVE boundary conditions for the micro-RVEs and up-scaling damage information through suitable volume stress averaging. Presented model provides fully 3D homogenization based BEM dynamic link between the two scales for modeling degradation and failure in polycrystalline materials. The formulation involves the engineering component at the macroscale level and the material grain level (micro-scale). The macro-continuum is modeled using a 3D boundary element formulation where the presence of damage is formulated through an initial stress approach to account for the local softening in the neighborhood of points experiencing degradation at the micro-scale. Very interesting work is presented in (Khoei et al., 2013), where multiscale technique was developed for concurrent coupling of atomisticcontinuum domains in modeling nano-mechanical behavior of atomic structures. In the proposed coupling approach, the mass and stiffness matrices of the continuum domain are calculated by a direct-bridging operator that maps the displacement of the atomic lattice laid underneath the continuum grid onto the continuum nodal displacements. A Lagrange-multipier method was applied to couple continuum nodal velocities in the overlapping domain. A multitime-step domain decomposition method was employed to decouple the solution of equilibrium equations for the atomic and continuum domains. Interesting research work that bases on advantages of the concurrent multiscale approach is (Wu et al., 2015). In that work a multiscale model was developed to model a quasi-static crack propagation in elastic material. The EMsFEM (extended multiscale finite element method) serves as a multiscale framework and was extended to take into consideration material failure. Next, the combination of the XFEM (extended finite element method) and LSM (level set method) was integrated into the multiscale framework to simulate the crack propagation at a fine scale. Interesting comparison between two different multiscale solutions is presented in (Silani et al., 2014). In that work a semiconcurrent multiscale method was introduced, where proposed model bridges the meso- and macroscale by a damage parameter. A fully exfoliated clay/epoxy nanocomposite was examined and a simple tension of dog-bone sample in macroscale was simulated. Obtained results were validated with the multiscale hierarchical multiscale model.. 28.

(31) Examples of upscaling multiscale solutions are presented in (Guo & Zhao, 2016a; Guo & Zhao, 2016b; Hajibeygi et al., 2011; Wurm & Ulz, 2016; Jeong et al., 2013; Rahman & Foster, 2014; Wudtke et al., 2015). In (Guo & Zhaoa, 2016a) a 3D hierarchical multiscale concept was used to investigate strain localization under three typical loading conditions: CTC (conventional triaxial compression), CTE (conventional triaxial extension) and PSBC (plane-strain biaxial compression) in granular media. A hierarchical coupling of finite element method and discrete element method was used. FEM was employed to treat a boundary problem of the granular material and the required constitutive relation for FEM was derived directly from the discrete element method (DEM) solution of a granular assembly embedded at each Gauss integration point as the representative volume element. The same Authors in (Guo & Zhao, 2016b) present an extension of the hierarchical multiscale approach applied to the coupled hydro-mechanical behavior for saturated granular soils. Based on a hierarchical coupling of the FEM and DEM, the approach employs the FEM to solve a boundary value problem while using DEM to derive the required nonlinear material responses at each Gauss integration point. Another interesting work, which bases on multiscale hierarchical approach is (Hajibeygi et al., 2011), where an iterative multiscale finite volume method (i-MSFV) is used for the solution of fractured porous media. The fracture modeling approach is extended to become suitable for the MSFV (multiscale finite volume) framework and involves splitting of the fracture pressure into an average value and a deviation. Numerical results for verification and validation for proposed new iterative multiscale solution for fractured porous media are also presented. Another example of research focused on multiscale solutions is (Wurm & Ulz, 2016), where an improvement of information exchange in hierarchical atomistic-to-continuum settings is investigated by applying stochastic approximation methods. On the macroscale level balance equations of continuum mechanic are solved using nonlinear finite element formulation. The microscale level, where canonical ensemble of statistical mechanics is simulated using molecular dynamics, replaces a classic material formulation. In (Jeong et al., 2013) the sequential multiscale analysis is made, where the numerical homogenization method based on the asymptotical expansion framework has been applied to predict elastic properties of micro/nano-sized honeycomb structures. The continuum finite element analysis is carried out by introducing the surface elasticity. The surface elastic property for very simple structure is identified by using the molecular dynamic (MD) simulations, and then the numerical homogenization method in the continuum level is applied to nano-sized periodic components including honeycomb structures. Obtained homogenized stiffness over the representative volume element is used to predict the global behaviors of the honeycomb structures. Another advanced multiscale approach is presented in (Rahman & Foster, 2014), where hierarchical multiscale modeling scheme PFHMM (peridynamics based hierarchical multiscale modeling) is modified in order to link the real atomistic model of polymers with the coarsened PD (peridynamics) models. The generalized scheme is implemented to a complex heterogeneous polymer: ultra high molecular weight polyethylene (UHMWPE). Using extended PFHMM, the 29.

(32) atomistic model of UHMWPE is linked with the coarser PD unit cells. Different phases (e.g. highly oriented unidirectional, amorphous or semicrystalline) of UHMWPE were blended during upscaling of polyethylene (PE) microfibrils. A hierarchical multiscale approach is also used in (Wudtke et al., 2015) to capture the variation of the mechanical properties in the heat affected zone (HAZ) in welded connections. Micro-graphs of different points along the HAZ are used to build numerical RVEs, which took into account different microstructure morphologies of the material in various locations. A classic homogenization method is used as bridging technique. The stress-strain curves of the HAZ as well as the base material are calculated and validated with the experimental results of miniature tensile tests. As can be seen, multiscale modelling become more and more popular to investigate different materials behavior at different scale levels. There are also some recent literature examples, strictly concentrated on multiscale methods in application to the processes with incremental character described in Chapter 3. In literature, multiscale models specially dedicated for incremental forming methods can already be found in (Hol et al., 2015; Nakhoul et al., 2015; Soho et. al., 2014; Pletz et al., 2014; Madej et. al., 2007; Komori, 2014; Franz et al., 2009; Zhu et al., 2012; Chen & Zabaras, 2014). In (Hol et al., 2015) a mixed multiscale lubrication friction model is presented to accurately account for friction in sheet metal forming FE simulations. Two deep-drawing applications are discussed to demonstrate the performance of the friction model. Results show friction coefﬁcients that vary in space and time, and depend on external process settings like the amount and type of lubricant. Another interesting work is (Nakhoul et al., 2015), where the capacity of the multiscale enriched continuum method has been confirmed for thin sheet buckling modeling. Wavy edge and wavy center flatness defects are addressed by a multiscale buckling model (MSBM). An interesting work is also (Soho et. al., 2014), where multiscale numerical model of sheet metal forming process is coupled with polycrystalline plasticity model to estimate the evolution of the mechanical properties, predict the texture evolution of the material as well as other parameters related to its microstructure. A multiscale approach is also used in (Pletz et al., 2014) to predict the performance of three different materials (Manganese steel, Hardox and Marage 300) in view of the development of rolling contact fatigue (RCF) cracks. A model of the whole crossing (crossing model) is used for the calculation of the dynamic forces and movements of wheel and crossing. Multiscale model is also used to evaluate the ductile fracture in the simulation of sheet metal forming process based on the ellipsoidal void model (Komori, 2014). Another example of multiscale adaptation to the sheet metal forming process is presented in (Franz et al., 2009). In that work to analyze material formability, a ductility loss criterion is coupled with a multiscale model. The behavior at the mesoscopic (grain) scale is modeled by a large strain micro-mechanical constitutive law, which is then used in a self-consistent scale transition scheme.. 30.

(33) Finally, some examples of typical bulk incremental forming processes modeled with multiscale approach can be found. In (Madej et. al., 2007), a CAFE (Cellular Automata Finite Element) concept is introduced to simulate the strain localization during cold rolling process. In (Zhu et al., 2012) a physical-based internal state variable microstructure model is adopted to predict the microstructure evolution of TA15 titanium alloy in hot ring rolling. Multiscale approach for incremental forming can be also found for disk forging (Chen & Zabaras, 2014). However, most of research works are based on the multiscale approach, where the material morphology is taken into account in an implicit way. To improve the quality of obtained results, the microstructure should be involved explicitly, because of its influence on material behavior in larger scales. Due to that, more and more research works is focused on application of the multiscale approach for material behavior description connected with the digital material representation (DMR) concept, which is described in the following Chapter.. 31.

(34) 5. DIGITAL MATERIAL REPRESENTATION CONCEPT Numerical models based on a digital material representation become more and more popular in many research works and are used to describe evolution of material morphology under complex loading conditions. The concept of the DMR was proposed recently and is constantly evolving. The definition according to (Senkov, 2004) states that the digital material representation is a material description based on measurable quantities that provides the necessary link between a simulation and an experiment. Such approach makes it possible to directly include inhomogeneities in polycrystalline material, in the form of e.g. different grain sizes, crystallographic grains orientation, inclusions or precipitates, unlike in conventional models, which are based on closed-form equations, where material is treated as homogenous in whole specimen volume (figure 29).. Fig. 29. Comparison between conventional models, real structures and digital material representation concept. The DMR is expected to create a possibility for analyzing behavior of complex materials in difficult, or in some cases impossible, to monitor experimentally conditions at the present state of research equipment. Thus, the DMR approach is mainly used for modeling materials characterized by highly elevated properties, which are the result of sophisticated and complex microstructures with combinations of different microscale features, e.g. for modern steel grades (TRIP – transformation inducted plasticity steel, CP – complex phase steel, DP – dual phase steel, etc.). Generation of microstructure morphology with its specific features and properties is one of the most important algorithmic parts of systems based on the DMR. There are several experimental and numerical methods that can provide accurate representation of microstructure morphology, e.g. based on microscopy imaging (optical or electron microscopy) (Uchic et al., 2006; Uchic et al., 2007; Yazzie et al., 2012) or on less time consuming algorithmic approaches 32.

(35) e.g. Voronoi tessellation, voxel method, cellular automata, sphere growth or Monte Carlo (MC) methods etc. (Danielsson et al., 2007; Madej et al., 2011a; Baxter & Behringer, 1991). The problem of proper generation of the DMRs with various algorithms was extensively studied in (Madej et al., 2009; Madej, 2010). The second step of preparing DMR model is connected with a FE mesh generation, necessary for further numerical calculations. Based on the obtained DMR morphology, the generation of the non-uniform triangular/quadratic or tetrahedral meshes (2D or 3D) can be performed using e.g. the DMRmesh software developed in (Madej et al., 2012) (figure 30). This approach enables obtaining coarse mesh inside microscale features and fine at its boundaries, what accelerates numerical calculations and maintain good quality of numerical results.. Fig. 30. Digital microstructure with non-uniform FE mesh obtained using DMRmesh software (Madej et al., 2012). Assignment of material properties to subsequent features is the next step of generation of digital microstructures. Two approaches are commonly used. The first includes assignment of crystallographic orientations to each microstructure element and then use of crystal plasticity models (Kilian et al., 2011; Devincre & Kubin, 2010; Wajda et al., 2013). These crystallographic orientation can be taken directly from the EBSD photographs or can be randomly generated in predetermined angles range. To properly use crystal plasticity theory in numerical calculations, material parameters for CP equations have to be specified. Identification of these material parameters can be realized by coupling an inverse analysis with laboratory plastometric tests of e.g. single crystals deformation. The second, simplified approach, is based on defining the appropriate flow stress curves, which describe particular features of the investigated microstructure. Stress-strain curves can be obtained from inverse analysis after again e.g. a single crystals compression tests of the investigated material. Then these curves can be described with a single flow stress model (e.g. Hansel-Spittel) additionally differentiated by one selected parameter. This enables to obtain. 33.

(36) slightly different stress-strain curves for each grain in the DMR microstructure, which make an impression of different crystallographic orientation of every grain. Finally, the last step of the DMR model development is an incorporation of such digital model, with DMR morphology, FE mesh and material properties into a commercial FE software. The digital material model can be treated as a representative volume element (RVE) or a unit cell (UC) during numerical calculations (figure 31). The difference between the RVE and the UC is related to a type of information required during investigation, local or global. The RVE is a model of the material to be used to determine the corresponding effective properties for the homogenized macroscopic model. The RVE should be large enough to contain sufficient information about the microstructure in order to be representative for a large volume of the investigated material. However it should be much smaller than the macroscopic body (Gitman et al., 2007; Hashin, 1983). Unlike the RVE, unit cell is not representative for the whole numerical model and enables obtaining results accurate for particular part of the specimen. UC is a small section of the RVE and enables analyzing material behavior in some interesting locations (e.g. crack initiation along the inclusion), without focusing on the rest of the model.. Fig. 31. Schematic relations between digital material representation, representative volume element and unit cell concepts. It is commonly assumed that several UCs can be considered as the RVE. However, both of presented approaches can have a simplified or detailed shape of microstructure features. In 34.

(37) a simplified model e.g. only volume fraction of particular phase is considered, while geometry of this phase is not regarded. Such digital model can provide representative, global material response, while morphology is simplified and gives results only statistically similar to real specimens (Brands et al., 2011; Brands et al., 2010; Rauch et al., 2011). These digital microscale models are often used in multiscale solutions that predict global material behavior as well as can analyze microstructure changes during deformation in some critical/interesting locations, e.g. where the material failure or delaminating can be observed. As mentioned, the DMR approach becomes increasingly popular and is applied in many research works focused on different, new and modern materials, e.g. metals, composites or other structures (Ivanov et al., 2009; Piezel et al., 2012; Flores et al., 2011; Xue et al., 2010; Madej et al., 2011b; Larsson et al., 2011; Różański & Łyżba, 2011; Zeman & Sejnoha, 2007; Liu, 2004; Simonovski & Cizelj, 2007; El Houdaigui et al., 2006, Gurgul et al., 2013a; Gurgul et al., 2013b; Goik et al., 2013; Sieniek et al., 2011). In (Efstathiou et al., 2010) RVE was used to characterize the spatial distribution of residual deformation at the meso- and microscale levels in titanium subjected to cyclic tensile loading. Digital image correlation (DIC) was used to compare the axial residual strain fields obtained at different magnifications ranging from 3.2x to 50x. The strain fields obtained at different magnifications were used to estimate the length scale of a representative volume element based on the standard deviation of the average residual strain. Another work based on DMR approach is (Ha et al., 2016), where a new numerical methodology to build RVE of a wide range of 3D composites was presented. Emphasis was put on the difficulties of creating a mesh of highly complex weaves embedded in a resin. A conforming mesh at the numerous interfaces between yarns was created by a multi-quadtree adaptation technique, which makes it possible thereafter to build an unstructured 3D mesh of the resin with tetrahedral elements. In (Harris and Chiu, 2015) several models were described to determine the RVE size for different common microstructural properties, i.e. volume fraction and particle size. Also an extensive synchrotron X-ray nanotomography imaging of a multiphase composite gas separation membrane was used to provide an experimental comparison to the model predictions. Also in (Savvas et al., 2016) the RVE size was investigated, where novel computational procedure was proposed for the determination of the RVE size for random composites. Proposed approach takes into account local volume fraction variation by processing computer-simulated images of composites with randomly scattered inclusions. Series of microstructure models were derived directly from images using the moving window technique. Each microstructure model contains diversified amount of inclusions, measured by image analysis tools. Proposed procedure is based on extended finite element method coupled with the Monte Carlo simulation. In (Tian et al., 2015) a fiber growth method (FGM) for generating the RVE was presented with the spatially randomly distributed discontinuous fibers and with a relatively high fiber volume fraction. The axes distance method is also presented within the paper. Using FEM and numerical homogenization technique, the mechanical properties of composites reinforced by 35.

(38) spatially randomly distributed discontinuous fibers were simulated within the framework of elasticity. FEM results show that composites behave similar to the homogeneously isotropic materials at the macroscopic scale due to the spatial random distribution of fibers. The DMR approach was again used in (Simoneau et al, 2014), where the main attention was focused on the development of an algorithm capable of generating morphologically representative foam structures. Stereology was used to characterize the pore size and shape distributions. Using the morphology generation algorithm, the smallest RVEs corresponding to the numerically convergent foam morphologies were calculated for different foam porosities. RVE size for composite materials was also discussed in (Zhang et al., 2014). The average thermoelastoplastic properties of particle reinforced metal matrix composites (PRMMC) including the average coefficient of the thermal expansion (CTE) Poisson’s ratio, Young’s modulus and isotropic hardening function were investigated. A computational homogenization method for 3D realistic microstructures (RMs) was employed. Unit cells were compared with RMs with different domain sizes to determine the minimum RVE size. DMR concept was also used to predict microstructure evolution during incremental forming processes e.g. (Muszka, 2013), where multiscale model was connected with DMR and CP-FEM concept to simulate angular accumulative drawing (AAD) process characterized by a significant level of deformation inhomogeneity. It was presented that the inhomogeneity of microstructure refinement plays an important role in the texture formation. The multiscale model was used to predict the strain and microstructure inhomogeneities that exist in the drawn wire. Also the texture was predicted by using a crystal plasticity based modelling. Obtained results showed that the effect of complex strain path history that exists in AAD process can be effectively controlled by help of computer simulation. Comparison of obtained results with experimental observation gave very good agreement. This works has been extended in (Muszka et. al., 2013) were the effect of processing routes and parameters on inhomogeneity of deformation during AAD was studied. Investigated materials were an aluminum, copper, low carbon and micro alloyed steels. Another interesting work based on DMR approach is (Muszka et. al., 2014), where the aim of work was to develop an industry guidance that concerns production of ultrafine-grained (UFG) high strength low alloy (HSLA) steels using straininduced dynamic phase transformations during advanced thermomechanical processing. The effect of processing parameters on the grain refinement was studied as well as multiscale numerical model was developed to predict the mechanical response of studied structures. Another interesting research work based on multiscale approach based on FEM and cellular automata is presented in (Majta et. al., 2016). In that work a combined metal forming process for manufacturing ultrafine-grained and multilayered steel wires with numerical and experimental results is shown and discussed. The combination of AAD, wire drawing (WD) and wire flattening (WF) processes enabled to obtain highly refined structure. Application of multiscale models with DMR concept is also presented in (Madej et. al., 2011b), where the main focus was put on the application of image processing and cellular automata techniques to 36.