Prediction of hardening, localization and fracture of multiphase microstructure in boron alloyed steel

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(1)L ICE N T IAT E T H E S I S. Luleå University of Technology 2014. Prediction of Hardening, Localization and Fracture of Multiphase Microstructure in Boron Alloyed Steel. ISSN 1402-1757 ISBN 978-91-7583-133-6 (print) ISBN 978-91-7583-134-3 (pdf). Stefan Golling. Department of Engineering Sciences and Mathematics Division of Mechanics of Solid Materials. Prediction of Hardening, Localization and Fracture of Multiphase Microstructure in Boron Alloyed Steel. Stefan Golling.

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(3) Prediction of hardening, localization and fracture of multiphase microstructure in boron alloyed steel Stefan Golling. Division of Mechanics of Solid Materials Department of Engineering Sciences and Mathematics Luleå University of Technology SE-971 87 Luleå, Sweden. Licentiate Thesis in Solid Mechanics.

(4) Printed by Luleå University of Technology, Graphic Production 2014 ISSN 1402-1757 ISBN 978-91-7583-133-6 (print) ISBN 978-91-7583-134-3 (pdf) Luleå 2014 www.ltu.se.

(5) Preface This work has been carried out in the Solid Mechanics group at the division of Mechanics of Solid Materials, Department of Engineering Sciences and Mathematics at Luleå University of Technology (LTU), Lule a Sweden. The work has been nancially supported by Vinnova, Volvo Car Corporation in Gothenburg and Gestamp HardTech in Lule a. I would like to give a grateful acknowledgement for their nancial support to this project. Completion of this work, was made possible through help and support from many people. First of all, i would like to thank my supervisor, Mats. Oldenburg and assistant supervisors Karl-Gustaf Sundin and Hans-Ake Häggblad for help and guidance during the course of this work. The assistance of Dr. Greger Bergman as industrial partner is greatly appreciated. I would also like to thank Jan Granström for supporting the experimental work. This work is typeset with LATEXand a modied version of the template provided by Yogeshwarsing Calleecharan. Finally, I would like to thank Rickard Östlund for fruitful discussions.. ´efa® Gol¬in§ Lule a, December 2014. i.

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(7) Abstract In recent years the demand of hot stamped ultra-high strength steel (UHSS) for safety structures in automobiles has increased and continuation of this trend is expected. Vehicle components with tailored material properties can be manufactured by controlling the cooling rate of the blank by heating certain regions of the tool. Controlling the cooling rate of the blank formation of dierent microstructures with varying mechanical properties within a single component are feasible. The structural response in a crash situation can be altered by the design of the component with formation of dierent material grades based on the microstructure in designated areas of the component. A prerequisite for the introduction of high performance materials is the availability of ecient and accurate models for deformation and failure in crash simulations. In this work, tensile test specimens with dierent phases composition are produced. The material studied is the low alloy boron steel 22MnB5. This high strength steel is common in hot stamping applications due to its good hardenability. The specimens are austenitized before starting the heat treatment at dierent temperatures and holding times. In total fourteen dierent microstructures are produced. Reference material grades for pure phases are ferrite, bainite and martensite.. The pro-. duced samples consist of ferrite-bainite, ferrite-martensite and bainitemartensite with dierent volume fractions, additionally a microstructure consisting of three phases, ferrite-bainite and martensite, is available. Using measured mechanical properties of pure phases and the volume fraction of formed phases dierent homogenization methods are compared in their ability to represent the mechanical response of mixed microstructures. The homogenization methods are used to describe the elastic and plastic deformation of the material. A damage model is used for strain localization and a maximum shear stress criterion to predict. iii.

(8) Abstract fracture.. Strain localization is mesh dependent, therefore an analysis. length scale is introduced to account for dierent element sizes.. The. material model for the homogenization of mixed microstructures including damage has been implemented in the commercial nite element code LS-Dyna.. Validation by comparison with experimental results shows. good agreement for most phase compositions.. The main diculty has. been the reliable microstructure characterization.. iv.

(9) Thesis This thesis consists of a summary of the following appended papers.. Paper A: S. Golling and M. Oldenburg, A study on homogenization methods for steels with varying content of ferrite, bainite and martensite. To be submitted for journal publication.. Paper B: S. Golling, R. Östlund and M. Oldenburg, Implementation of homogenization scheme for hardening, localization and fracture of a steel with tailored material properties in Proceedings of the International conference on Hot Sheet Metal Forming of High-Performance Steel CHS 2 on June 9-12, 2013.. v.

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(11) Contents Preface Abstract Thesis 1 Introduction 1.1 1.2 1.3 1.4 1.5. Safety and crashworthiness . . . . . . Hot sheet metal forming . . . . . . . . Material for hot stamping . . . . . . . Analysis of passive safety components Objective and scope . . . . . . . . . .. 2 Method 2.1. 2.2. 2.3. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. Material modeling . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Homogenization schemes . . . . . . . . . . . . . . . 2.1.2 Localization and fracture . . . . . . . . . . . . . . 2.1.3 Modelling of failure . . . . . . . . . . . . . . . . . . 2.1.4 Inuence of carbon on the properties of martensite 2.1.5 Strengthening eect of small amounts bainite in bainite-martensite composites . . . . . . . . . . . . Experimental approach . . . . . . . . . . . . . . . . . . . . 2.2.1 Heat treatment of test specimens . . . . . . . . . . 2.2.2 Tensile testing with strain eld measurement . . . 2.2.3 Microstructure characterization . . . . . . . . . . . Numerical simulation . . . . . . . . . . . . . . . . . . . . .. 3 Results 3.1 3.2. . . . . .. i iii v 1. 1 3 5 7 8. 11. 11 12 19 21 22 23 23 24 27 29 30. 33. Results of four homogenization models . . . . . . . . . . . 33 Inuence of the inclusion geometry . . . . . . . . . . . . . 36 vii.

(12) Table of Content 3.3. Evaluation of fracture in mixed microstructures . . . . . . 37. 4 Summary of appended papers. 45. 5 Discussion and Conclusions. 47. 4.1 4.2. 5.1 5.2. Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Key results . . . . . . . . . . . . . . . . . . . . . . . . . Future work . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Evaluation of fracture for mixed microstructures 5.2.2 Study on detailed models of spot-welds . . . . . . 5.2.3 Evaluation of material model on component level. . . . . .. 47 49 49 49 50. References. 51. Appended Papers. 57. Paper A A study on homogenization methods for steels with varying content of ferrite, bainite and martensite. 61. Paper B Implementation of homogenization scheme for hardening, localization and fracture of a steel with tailored material properties. viii. 99.

(13) Chapter 1 Introduction This thesis begins with a background introduction to help the general reader who may be unacquainted with the research topic. This will hopefully at the same time aid to situate the need for this work in a context of simulation within in the eld of the development of passive safety components in the automotive industry. This chapter is intended for the general reader of this thesis, it provides an overview of the research topic and the context it aims to contribute to.. 1.1 Safety and crashworthiness The aim of every auto manufacturer is to increase the passenger safety without increasing the cars weight. Nowadays consumers are aware of safety relevant components in the automobile purchased and good rankings in, for example the consumer test EURO NCAP, is a selling argument. Legislative regulations on safety are another driving force for 1.

(14) Introduction. manufacturers to improve their products. Passenger safety is one important factor, another factor is fuel consumption. To lower the fuel consumption many dierent techniques and possibilities are used and under development. One factor commonly used to reduce fuel consumption is weight reduction. Reducing the vehicles weight without losses in passenger safety is a challenge for manufacturers. In automotive safety the term "active" is used to describe a component which is intended to prevent the passenger from a crash situation, the term "passive" is used for components preventing the passenger from injuries during a crash. Well known passive safety components are the seat belt and the airbags. Less known in public is the eect of the physical structure of the vehicle as safety component. Crashworthy components prevent or reduce the severity of injuries when a crash is imminent or actually happening. During a crash situation a controlled deformation of the vehicle structure dissipates energy. This part of the structure is called crumple zone, an important component is here the crash box. Modern cars are usually build with a safety cell, this is by reinforcing the passenger compartment with high strength steels at places subjected to high loads in a crash. In Fig. 1.1 structural parts of the car body are colored depending on the material used. Obviously the car body is built mainly of steel with dierent strength, most of the components are manufactured using the technique of hot stamping. Using hot stamping, also called press-hardening, ultra high strength components components are manufactured. In Fig. 1.1 these components are colored red. Hot stamped, ultra high strength parts can contain soft zones using dierent cooling rates during manufacturing. An introduction to the process 2.

(15) 1.2 Hot sheet metal forming. of press-hardening is given in the next section.. Figure 1.1. Body-in-white of a Volvo XC90. The color coding relates to dierent materials and material grades in the safety cell (image courtesy Volvo Cars).. 1.2 Hot sheet metal forming During the last decade, hot sheet metal forming technology, also called hot stamping or press hardening, has become omnipresent in automotive body-in-white design. The driving force for the development of this technology is the demand for further improvement of fuel eciency by lightweight design and passenger safety. The demand for hot sheet metal forming technology is steadily increasing and we are now experiencing an outstanding growth in variety of applications, mainly in the automotive sector. For the design of automotive vehicle structures, hot stamping has become the leading technology for solutions with the aim to reduce weight in combination with main-. 3.

(16) Introduction. tained or increased passenger safety [1]. Hot stamping is a production process for the hot forming of sheet metals. It combines both the shaping and the heat treatment of sheet metal components into one single process step.. The process involves. inserting sheets, which have been austenitized, into a cooled forming tool, in which they are then quenched. The thermal integrated processing produces a martensitic structure that gives the press-hardened parts an extremely high tensile strength [2]. Fig. 1.2 illustrates the production process of a press hardened component.. From a coil blanks are. cut, either in pre-cut shape or unprocessed.. The blanks are austeni-. tized in a furnace prior to quenching. Depending on the blank geometry at the beginning of the process nishing steps are added after forming. Within the technology of hot stamping a method called tailored prop-. Figure 1.2.. Schematic representation of the hot stamping process, from decoiling to nished product (image courtesy voestalpine [3]). erties evolved. Components with tailored properties have varying material properties in desired zones within the part. Mechanical properties within the blank are altered through forming tools which are sequentially heated and cooled.. This type of heat treatment causes regions. 4.

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(24) Introduction. lems with corrosion. The hot stamping process increases the material strength with up to 250-300% compared to the base material. Reason for this increase is the superior strength of martensite compared to ferritic-pearlitic microstructures. In Fig. 1.4 dierent types of steel are compared in elongation and tensile strength.. The steel in focus here is in unhardened condition lo-. cated in the eld CMn, carbon steel alloyed with manganese.. After. processing the steels properties are changed and it locates in the eld MART, which stands for martensitic. All samples used in the experimental work are cut perpendicular to the rolling direction of a coil and austenitized in a furnace. After austenitization dierent holding temperatures and times are utilized to create microstructures consisting of ferrite, bainite and martensite. Using this approach various volume fraction of phases and combinations of them are produced.. Figure 1.4.. Strength elongation relationship for Ultra High Strength Steel in comparison to conventional High Strength Steel [5].. 6.

(25) 1.4 Analysis of passive safety components. 1.4 Analysis of passive safety components. A crash simulation is a virtual recreation of a destructive crash test of a car or a component using a computer simulation in order to examine the level of safety of the car and its passengers. Crash simulations are used by manufacturers and suppliers during the development of new cars. During a real crash or a simulation of it the kinetic energy of the vehicle before the impact is transformed into deformation energy after the impact, mostly by plastic deformation of the body-in-white. Data obtained from a crash simulation indicate the capability of the car body to protect the passengers during a collision against injury. Important results are the deformations of the occupant space and the decelerations the passengers undergo. Decelerations in a crash situation are legally regulated and must fall below threshold values. The history of crash simulations dates back to 1970'ties. First crash analyses were performed on aircraft impacts and with beginning of the 1980'ties car manufacturers became interested in this technique. During the last 30 years many improvements to simulation methods and the increasing capacity of computational resources lead to a wide use of crash simulations during development of new safety structures and cars. To solve numerical models of a crash the nite element analysis (FEA) is used. The nite element method (FEM) is a numerical technique for nding approximate solutions to partial dierential equations. Due to the availability of faster computers the mesh size could be reduced and in its turn minimize the numerical error. This leads to the necessity of improved material models to represent the material in an appropriate manner. 7.

(26) Introduction. 1.5 Objective and scope The objective of this work is to study and establish the relations between phase composition and the localization and fracture failure behavior in a boron alloyed steel. The studied steel is used for UHSS components in the automotive industry. Knowledge and presence of methods for failure predictions in e.g. crashworthiness analyses are necessary prerequisites for optimal use of UHSS materials in car structures. Heat treatment and welding are important industrial procedures in the manufacturing of boron-alloyed high-strength steel components. Due to the temperature history in these processes the mechanical properties of the material is changed. Accurate modeling is essential for simulation of loaded components in automotive applications. The present work is based on earlier research conducted within the subject of solid mechanics at Lule a University of Technology, see Åkerström [6], Eriksson [7], and Bergman [8]. Concerning experimental studies and modeling of fracture failure in boron steel this work is based on Eman [9]. In these studies, experiments with digital speckle photography (DSP) were used to establish criteria for localization and fracture based on measurements on a small length scale. In Häggblad et al. [10] a model has been proposed that compensates for dierent length scales in numerical analyses of localization and fracture. The main scope of the project is: •. Study and evaluate homogenization methods for the prediction of the elasto-plastic response of multi-phase ultra-high strength steel 8.

(27) 1.5 Objective and scope. With use of strain-eld measurements, study and establish relations between the microstructure composition and the mechanical behavior with respect to localization and fracture • Develop a model for prediction of localization and fracture failure of ultra-high strength steel sheet metal components with locally varying microstructure • Application and validation of the composite fracture model on detailed models of spot welds The research question can be formulated as: How is the fracture strain value on a small measurement length scale inuenced by each micro-structural constituent and how is this inuence combined into a failure model based on the stress-, strain- and materialstate as well as the actual analysis length scale? •. 9.

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(29) Chapter 2 Method 2.1 Material modeling. Deformation in which stress and strain are proportional is called elastic deformation. A plot of stress versus strain results in a linear relationship, the slope of this linear segment corresponds to the modulus of elasticity. Elastic deformation is nonpermanent, which means that when the applied load is released, the piece returns to its original shape. For most metallic materials, elastic deformation persists only to small strains. As material is deformed beyond this point, the so-called initial yield strength, the stress is no longer proportional to strain and permanent or plastic deformation occurs. After yielding, the stress necessary to continue plastic deformation increases to a maximum, and then decreases to the eventual fracture. The point after where the load starts to decrease marks the onset of necking i.e. the strain starts to localize into a small region and a neck starts to form. In Fig. 2.1 a typical result for tensile test with annotations for regions named in this work 11.

(30) Method is presented. Throughout the study the relationship between strain and. Engineering stress. [M P a]. 800 Localization. dσ d. 600 Plastic Fracture. 400 200 0 0.00. Elastic. 0.05. 0.10. Engineering strain. Figure 2.1.. 0.15 [−]. 0.20. Stress-strain curve obtained by a tensile test.. stress for pure phases during plastic deformation is described by an exponential function.. (r) (r) n σ (r) = σy0 + h(r) 1 − exp −(r) p The superscript yield strength,. (r) (r). p. is a placeholder for the phase.. is the plastic strain and. h(r) , n(r). (2.1). (r). σy0. is the initial. are parameters to. describe the hardening curve of respective phase. A comparison of curves tted to experimental data is given in gure 2.11.. 2.1.1. Homogenization schemes. Homogenization methods are widely used if a material consists of two or more constituents, phases or materials which exhibit dierent mechanical properties.. The aim of the homogenization for alloys is the esti-. mation of the macroscopic response depending on properties of phases or constituents.. 12.

(31) 2.1 Material modeling. 2.1.1.1. Phenomenological homogenization schemes. Two simple models are the iso-strain and the iso-work assumption. For the iso-strain model it is assumed that all present phases are submitted to the same strain. The iso-work assumption uses the mechanical work increment which is equal in every phase. This method was rst proposed by Bouaziz and Buessler [11], who used the model for a study on ferrite and pearlite and an alloy consisting of iron and silver. The described phenomenological models use a linear mixture rule for stress and strain to obtain the composite response. The iso-strain model assumes equal strain in each present phase and a linear mixture rule for the stresses. The iso-strain model represents an upper bound, a lower bound would be the iso-stress assumption where equal stress in each phase is assumed. The iso-strain homogenization is formulated as c = 0 = 1. σc (c ) = c0 σ0 (0 ) + c1 σ1 (1 ). (2.2). where c and σc are the strain and stress in the composite, 0 , 1 and σ0 , σ1 are the strain and stress in phase one and two, respectively. The variables c0 and c1 are the volume fractions of the phases. The iso-strain model assumes an equal strain distribution in a material. This is true for materials consisting of phases with similar mechanical properties. For materials where mechanical properties of the phases vary a strain partition is more realistic. A more realistic approach keeps the linear volume fraction mixture law for stresses and applies the same mixture law for strains, this approach was rst pro13.

(32) Method posed by Tomota et al. [12]. c = c0 0 + c1 1. σc (c ) = c0 σ0 (0 ) + c1 σ1 (1 ). (2.3). An equi-incremental mechanical work assumption, see Eq. 2.4, is used to decompose the strain into components applied to the dierent phases. The rule of mixture 2.3 and a suitable description for the stress depending on the strain is used to predict the composite response. (2.4). σ0 d0 = σ1 d1. A graphical representation of the two phenomenological models is depicted in Fig.2.2. 2,000 σ2 (2 ) Stress [M P a]. 1,500. 1,000 σ1 (1 ). σ2 d2 500 σ1 d1 0. Figure 2.2.. 0. 0.05. 0.1 Strain [−]. 0.15. 0.2. Schematic representation of the iso-strain and iso-work assumption.. 2.1.1.2 Micromechanical based homogenization schemes Except for the two phenomenological models introduced, all further methods are based on the work of Eshelby [13]. The Eshelby solution solves the problem of a single ellipsoidal inclusion in an innite matrix.. 14.

(33) 2.1 Material modeling. An inclusion strain concentration tensor is used to compute the strain in the inclusion and the matrix. The interaction between phases must be taken into account if the volume fraction of inclusion exceeds a few percent. An assumption in this model is the perfect bonding between inclusion and matrix phase. Mori and Tanaka [14] used the Eshelby solution for an ellipsoidal inhomogeneity but it includes certain eects of the inhomogeneity by taking the average strain in the matrix when all inhomogeneities are present. This is in contrast to the Eshelby solution where the average strain in the matrix is taken when none of the inhomogeneities are present. Weng [15] proposed an analytic model for the estimation of the composite response. To estimate the elastic constants the secant moduli are computed, therefore its name secant method. The method in its original version assumes spherical inclusions in a homogeneous matrix. The user needs to decide which phase is the matrix phase, permuting the properties of matrix and inclusion yields fairly dierent results. The secant method is divided into three deformation stages. During the rst stage both phases deform elastic. This stage ends if the yield criterion for one of the phases is reached. The second stage stands for elastic deformation of one phase and plastic deformation for the other, at this point it is essential to dene which of the phases is the matrix and which the inclusion. The third deformation stage is reached if both phases exceed the respective yield criteria. Depending on the volume fraction of each phase, the shape of the inclusion and the stage of deformation a strain concentration factor b0 for the matrix and b1 for the inclusion are. 15.

(34) Method calculated. The shape of the inclusion is assumed to be spherical.. b0 =. β0s (μ1 − μs0 ) + μs0 (c1 + c0 β0s ) (μ1 − μs0 ) + μs0 (2.5). bs1 =. (c1 +. μ1 s c0 β0 ) (μ1. − μs0 ) + μs0. Where β is the shape parameter describing the shape of the inclusion and μ is the shear modulus of matrix, subscript 1, or inclusion, subscript 2, respectively. The superscript s denotes that the variable is calculated using the secant modulus. The stress in the composite depending on the plastic strain and the stage of deformation can be evaluated in an analytic way by solving a set of equations. The model is in detail described by Weng [15], a method to incorporate a third phase is described by Rudiono and Tomota [16]. The double-inclusion method is as well based on Eshelby's inclusion theory and the Mori-Tanaka method and was rst proposed by NematNasser and Hori [17]. Similar to the secant method a strain concentration tensor, B, is calculated.. −1 B = I + E : D−1 0 : D1 − I. (2.6). Here I is the fourth order identity tensor, E is the Eshelby tensor. The isotropic stiness of matrix and inclusion are denoted D0 and D1 , respectively. The Eshelby tensor depends on the geometry of the inclusion and its stiness. For spheroid inclusions and isotropic stiness it only depends on the aspect ratio of the semi axes of the inclusion and the. 16.

(35) 2.1 Material modeling. Poisson's ratio. In the present study the elastic constants used in Eshelby's tensor are computed using the tangent moduli. If in Eq. 2.6 the stiness of matrix and inclusion are reversed the strain concentration tensor corresponds to an inverse Mori-Tanaka method. Using this two tensors an upper and lower bound for the macroscopic stiness is found. Lielens [18] proposed a material model with an interpolation between upper Bu and lower strain concentration tensor Bl . This approach can be seen as strengthening eect of the real inclusion on the matrix. The resulting strain concentration tensor B is given by −1 B = (1 − ξc1 ) B−1 l + ξ c1 Bu. (2.7). with the quadratic interpolation function 1 ξ(c1 ) = c1 (1 + c1 ) 2. (2.8). A detailed discussion on computation and implementation aspects of the double inclusion model is given by Doghri and Ouaar [19]. For a composite with more than two phases the double inclusion method is extended in a straight forward manner as the strain concentration tensor is additive. The upper and lower strain concentration tensors are computed for each phase. Lielens interpolation function relates each inclusion phase to the matrix phase, this implies that for every new phase taken into account an additional interpolation is used. The strain concentration tensors are then used to compute the strain in respective phase until the residual criteria of the phase is fullled.. 17.

(36) Method 2.1.1.3. Eshelby solution for inclusions and inhomogeneities. Many mean eld methods used in the micromechanical modeling of materials are based on the work of Eshelby [13]. Eshelby studied the stress and strain distributions in homogeneous media and assumed that inhomogeneities in the composite are far apart from each other and no interactions between them occur. With these assumptions every inhomogeneity can be treated as if it exists in a homogeneous matrix. An inhomogeneity is perturbing the stress and strain eld in a body, to solve for the stresses and strains the so-called equivalent inclusion method is applied introducing an eigenstrain into the calculation. The eigenstrain ∗ is related to the perturbed strain eld by the Eshelby tensor E . (2.9). = E∗. Eshelby's inclusion tensor is computed using the aspect ratio of the axes of an ellipsoid. A property of the Eshelby tensor is its independence from the size of the inclusion, the only variables used to calculate the tensor are the elastic properties of the phases. For some inclusion geometries, analytical solutions for the Eshelby tensor exist e.g. spherical and penny shape. For a general ellipsoidal shape volume integrals need to be solved numerically. In paper B three dierent geometry cases are investigated and graphical representations of the inclusions used are presented in Fig. 2.3. A detailed discussion on Eshelby's solution for inclusions and inhomogeneities, its use for dierent types of materials and alternative values for inclusion shapes is found in e.g. Mura [20] and Qu and Cherkaoui [21]. 18.

(37) 2.1 Material modeling. Z. Z X Y. (a). Z X. X. Y. Spherical shape.. (b). Y. Penny shape.. (c). Ellipsoidal shape.. Figure 2.3. Three dierent inclusion shapes used in paper B. 2.1.2. Localization and fracture. The homogenization scheme is extended with a phenomenological localization and fracture model. Only major parts of the damage model are given here. A detailed description and a discussion are given by Östlund et al. [22]. The localization and fracture model is an extension of a commonly used radial return mapping algorithm for isotropic von Mises plasticity as described in e.g. Ottosen and Ristinmaa [23]. Fracture occurs when the localization function reaches its critical value or the maximum shear criterion is fullled. The phenomenon of localized deformation is typical for a wide range of solids. A strain localization, or shear band, usually develops during severe plastic deformation of ductile materials. During loading of a body the deformation is homogeneous until a point where it starts to conne to a narrow region. In this region the intense straining occurs in the material. Usually, strain localization precedes fracture in tensile loading of ductile materials. Results of nite element analysis of localization problems are mesh dependent. Loss of ellipticity of the governing equations causes numerical solutions to be inherently mesh dependent, as the width of the localized band is set by the mesh spacing, Needlemen [24]. 19.

(38) Method The localization model used aims to predict the load response after onset of necking using elements larger than the localized zone. A modied von Mises yield equation was rst suggested by Häggblad et al. [10], f =σ ¯ − σy (1 − L) ,. σ ¯=. . 3J2. (2.10). where L is termed localization function. The localization function is introduced into the yield function to reduce the load bearing capacity of the material. L is dependent on the element size of the mesh, termed analysis length l, and evolutes with the equivalent plastic strain ¯p ,. L = A eB(¯p −¯0 ) − 1 ,. ¯p ≥ ¯0. (2.11). the parameters A and B are functions of the analysis length l, A=. A0 , l. (2.12). B = B0 l. and ¯0 is an equivalent threshold strain. The constants A0 and B0 are constants calibrated to testing results. The analysis length is calculated using the initial shell element area, Ainit , and the initial shell thickness, tinit √ l=. Ainit tinit. (2.13). Analysis length is a dimensionless characteristic element size and describes the level of spatial discretization, therefore the initial geometric properties are used. Introducing an analysis length scale into the constitutive equation allows treating the mesh dependency concerning load 20.

(39) 2.1 Material modeling response and failure prediction.. 2.1.3. Modelling of failure. Callister [25] denes fracture as the separation of a body into two or more pieces in response to an imposed stress.. Most fracture damage. theories are characterized by hydrostatic stress dependence. drostatic stress. p. is normalized to the eective stress. the dimensionless stress triaxiality parameter. η.. σef f. The hy-. leading to. The stress triaxiality. parameter characterizes all loading conditions in a plane state of stress.. η=. p. (2.14). σef f. For von Mises plasticity the eective stress is dened using the second deviatoric invariant of the stress tensor, The fracture strain. ¯f. σef f =. √. 3J2 .. of the pure phase is determined by its maxi-. f. mum shear stress τmax . To compensate for mesh size eects the analysis length scale. l. is introduced into the calculation of the fracture strain ¯f ,. ¯f = ¯f0 − ¯0 e−Cl + ¯0 The variable i.e.. ¯f =. ¯f0. ¯f0 for. (2.15). is fracture strain at an analysis length. l → 0.. l. equal to zero,. The maximum shear stress of respective phase is. found by use of experimental data. Using the maximum shear stress a maximum stress is calculated with use of the Lode angle. A weakest link assumption is applied for the modeling of fracture. It is postulated that the composite fails if one phase fullls the fracture. 21.

(40) Method criteria or reaches its critical value Lf .. tf 0. ¯˙p dt = 1, f. tf 0. L˙ p dt = 1 Lf. (2.16). where ¯˙p is the plastic strain increment accumulated over simulation time. The variable Lf is calculated comparable to equation 2.11 but with the modication that the fracture strain ¯f is used instead of the plastic strain ¯p .. 2.1.4. Inuence of carbon on the properties of martensite. During the formation of ferrite and bainite carbon migrates into the remaining austenite. Pure ferrite, even at elevated temperatures, can not hold considerable amount of carbon. The carbon migrates therefore to the remaining austenite and enriches it to higher carbon content than the bulk material. Phases formed after ferrite thus contain higher amounts of carbon.. The inuence of carbon on the hardness and strength of. martensite is well documented in literature, see for example Krauss [26] and Hutchinson et al. [27]. Carbon partitioning during the formation of ferrite with the eect of increasing the amount of carbon in remaining austenite is calculated by a relationship derived from a mass balance consideration as described by Bergman and Berglund [28]. For the homogenization methods the input for the martensite hardening curve is therefore dependent on the carbon content.. 22.

(41) 2.2 Experimental approach. 2.1.5. Strengthening eect of small amounts bainite in bainite-martensite composites. For microstructures consisting only of bainite and martensite literature reports a strengthening of the composite above pure martensite for bainite volume fractions up to 25%, see Tomita and Okabayashi [29]. The eect behind the strengthening consists of two factors, the rst is the increase of carbon in remaining austenite during bainite formation, and the second is an enhancement of the bainite via a plastic constraint by the surrounding martensite. For small volume fractions of bainite the yield strength of it is virtually equivalent to the strength of martensite with its dependency on carbon content.. 2.2 Experimental approach To compare the predictive capabilities of the homogenization methods described in section 2.1.1, samples with dierent microstructures and volume fractions of phases are produced. To obtain force-elongation data tensile tests are performed. The tensile test specimens are cut perpendicular to rolling direction of the blank. The blank thickness is t = 1.2mm and cross-section width w = 12.5mm for straight and w = 15.0mm for notched specimens. All samples are made from the boron alloyed steel 22MnB5. The elongation during testing is measured with an extensometer with a gauge length of L0 = 50mm. Additional to conventional elongation measurements digital speckle photography is used to determine displacement on a small length scale. The tests are performed at a constant speed of v = 0.1 mm s using a hydraulic test machine. 23.

(42) Method 2.2.1. Heat treatment of test specimens. In section 1.2 an introduction to the industrial process of hot stamping is presented. Hot stamping utilizes the possibility of altering the mechanical properties of steel due to heat treatment. Using time-transformation diagrams it is possible to estimate the amount of formed phases. In general two types of diagrams are useful for planning experiments, the timetemperature-transformation (TTT) and the continuous cooling transformation (CCT) diagram, see Fig. 2.4a and 2.4b respectively. Isothermal heat treatments are not used in industrial applications concerning hot stamping as they are not the most practical. Most industrial processes involve continuous cooling of a specimen to room temperature. For continuous cooling, the time required for a reaction to begin and end is delayed.. Thus the isothermal curves are shifted to longer times and. lower temperatures, see 2.4. The most important heat treatment used in hot stamping is quenching. The denition of quenching is the rapid cooling of a workpiece to obtain certain material properties, in the case of hot stamping an austenitized blank is rapidly cooled to form martensite. In a CCT diagram, see Fig. 2.4b, this is visible if the cooling rate is chosen in a way that neither the ferrite nor the bainite eld is entered, this quenching rate is termed critical cooling rate. Cooling rates below the critical form other phases than martensite. For rates passing through several elds formation of dierent phases is possible. A tool with plane surfaces and the possibility of being heated is used for quenching to martensite and isothermal bainite transformations. To achieve simultaneously double sided contact of the specimen with the tool, spring supported holders are used.. 24. In Fig.. 2.5 a schematic.

(43) 2.2 Experimental approach Austenite. 600. Bainite. 400 200 0 10−1. (a). 800. Ferrite. Temperatur [◦ C]. Temperatur [◦ C]. 800. 100. 101. Time [s]. 102. Austenite Ferrite Pearlite. 600. Bainite 400. Martensite. 200 0 10−1. 103. Time temperature transformation (TTT) diagram. Reproduced from He et al. [30]. (b). 100. 101. 102. Time [s]. 103. 104. Continuous cooling transformation (CCT) diagram. Reproduced from Tang et al. [31]. Figure 2.4. Phase transformation diagrams for 22MnB5.. drawing of tool and specimen is shown. During heat treatment in the tool, a pressure of 20MPa is applied on the sample. The temperature in the tool is measured at six points two millimeters below the tool surface. In addition the specimen temperature is measured in three points along the gauge length. In total fourteen phase compositions were produced. Three phase compositions are reference data for pure phases and eleven heat treatment schemes are used to produce samples consisting of ferrite-martensite, ferrite-bainite, bainite-martensite and ferrite-bainite-martensite. All samples are austenitized at 900◦ C for four minutes prior to the cooling procedures, a graphical illustration of the process is shown in Fig. 2.6. The pure phases are in the continuation labeled F730, B1015 and M1660. The material grade F730 is after austenitization air cooled, and the characterization shows an irregular ferritic microstructure. As reference material for bainite the material B1015 is used. The grade is produced by cooling in the tool to 430◦ C and held there for one hundred 25.

(44) ◦. ◦. ◦. ◦. ◦.

(45) 2.2 Experimental approach. martensitic (BM) microstructures the samples are after austenitization cooled in the tool, dierent holding times are used to form varying volume fractions of bainite, the remaining austenite is transformed into martensite by quenching in water. All previously described samples consist of 1,000. Temperature [◦ C]. 800 tf 600 tb. 400 200 0 Time. Figure 2.7.. Schematically representation of the heat treatment used to produce dierent volume fractions of phases in dual- and multi-phase microstructures.. two desired phases and small amounts of austenite which did not transform. A sample containing three phases is produced using two holding temperatures, one for the ferrite and one for the bainite transformation. To form ferrite the same procedure as for samples FB is used but the holding time is changed. Bainite is formed isothermally in the heated tool similar to samples BM, the remaining austenite is transformed into martensite by quenching in water. 2.2.2. Tensile testing with strain eld measurement. The full-eld measurement provides direct information about the local planar deformation eld at the region of interest for a number of time instants during deformation. If the specimen surface exhibit a random 27.

(46) Method pattern the in-plane displacement of any small unique region can be determined by a cross-correlation procedure of the digital images taken before and after deformation. The digital image correlation (DIC) is performed stepwise using the previous image as reference state. The in-plane strain and shear components are calculated from the displacement eld. A detailed description of the measurement procedure and the determination of the strain eld is given by Kajberg and Lindkvist [32]. Tensile tests are performed in a servo-hydraulic testing machine,. Figure 2.8.. Strain eld on a speckled specimen and experimental setup.. pictures for the image correlation are taken with a standard CCD camera. In advance to testing the coating on the specimens is removed by sandblasting. The reason for removing the coating is its brittle behavior. During deformation of the specimen the AlSi coating would ake of the specimen and disturb the measurement or cause the loss of the pattern which makes it impossible to compute the strain eld properly. The specimen surface after sandblasting showed enough contrast and pattern which made it unnecessary to spray paint the specimens prior to testing. During testing three pictures per second were taken.. 28.

(47) 2.2 Experimental approach. 2.2.3. Microstructure characterization. Besides the mechanical properties the volume fraction of phases in the mixed microstructure is needed for the models. The main characterization methods were SEM/EBSD and SEM imaging of slightly etched surfaces with subsequent image analysis. The measurements are made at about one quarter of the blank thickness to avoid surface specic phenomena and center segregations.. (a). Ferrite-martensite sample FM-1. Brighter areas are martensite, darker areas correspond to ferrite.. (b). Ferrit-bainite sample FB1. Ferrite is gray colored and brighter areas are bainite. The white parts in the bainite grains is cementite.. (c). Bainite-martensite sample BM-3. Brighter areas are martensite, darker areas with dot pattern correspond to bainite.. (d). Ferrit-bainite-martensite sample FBM-1. The dark gray areas are ferrite, darker areas with dot pattern are bainitic and light gray areas are martensite.. Figure 2.9. Examples of microstructures from scanning electron microscope, magnication 2500x.. 29.

(48) Method The estimation of phase volume fractions is made using SEM images and manual image analysis. All samples contain small amounts of residual austenite beside the main phases. The content of ferrite in samples FM is measured using EBSD data. Using the band slope information it is possible to separate the ferrite from the martensite. The sample characterization was performed by Swerea Kimab in Kista, Sweden. In Fig. 2.9 example SEM pictures are shown.. 2.3 Numerical simulation The homogenization and damage model is implemented in the commercially available nite element code LS-Dyna [33] via a user dened subroutine. LS-Dyna solves the fundamental conservation equations in continuum mechanics using an explicit time algorithm. In the time integration loop the user routines are called after having calculated strain and strain rates. The user routine calculates the stress eld using these input arguments [34]. The intention of the material model is the use in sheet metal applications and therefore it is implemented for shell elements. Thickness reduction is taken into account using plane stress iterations. The material model is validated using two dierent types of test geometries, see Fig. 2.10. The geometries represent two dierent stress triaxialities. Furthermore the samples were discretized with three different element sizes. The element denition is of type Belytschko-Tsay (LS-Dyna, type 2). The boundary conditions are xed in tensile direction on one side of the specimen, on the opposite side the displacement 30.

(49) 2.3 Numerical simulation is prescribed in accordance to experimental conditions.. Figure 2.10. Geometry and mesh used for the comparison of experi-. mental to nite element results. The width of the straight specimen is 12.5mm and of the notched specimen 15mm. The elements in the critical cross section are square in shape.. For the modeling of the hardening some assumptions are made. The secant method and the double-inclusion model need a decision which phase is the matrix phase, therefore the phase formed rst is assumed as matrix phase. During the formation of ferrite and bainite the remaining austenite is enriched with carbon and therefore the mechanical properties of martensite change. To model the increased strength of martensite the initial yield strength is scaled.. Stress. [M P a]. 1500. 1000. 500. 0 0.00. 0.05. 0.10 [−]. 0.15. 0.20. Strain Ferrite. Bainite. Martensite. Ferrite, exp.. Bainite, exp.. Martensite, exp.. Figure 2.11. Yield curve of ferrite, bainite and martensite obtained. from pure phase measurement, comparison between experimental result and tted curve. 31.

(50) Method Input data for the material model are the volume fractions of present phases, the yield curve for pure phases and parameters used in the damage model. In Fig. 2.11 a comparison between experimental result of pure phases and the curve tted for use in the model is shown.. 32.

(51) Chapter 3 Results In total fourteen samples with dierent volume fraction of phases and microstructures are produced and compared to results from homogenization schemes described in Sec. 2.1.1. Three samples are assumed as pure phase consisting of a single microstructure, ten samples are assumed as dual phase microstructure while one sample consists of three phases with signicant volume fractions of formed phase. For samples consisting of ferrite and martensite a study on the inuence of the inclusion shape on the prediction of the localization and fracture is performed.. 3.1 Results of four homogenization models A general observation of experimental results is the behavior of the initial yield stress throughout all samples. Increasing amount of tougher phase increases the initial yield strength but not in a linear way as the phenomenological models predict. Concerning the prediction of the onset of plastic deformation the micromechanical models show better results. The. 33.

(52) Results computation of the composite response has a strong dependency on the dierence of mechanical properties of pure phases. Ferrite and martensite show the largest dierence of the pure phases, representing the upper and lower bound for all composites. The proper representation of these compositions has therefore the highest sensitivity to changes in the phase volume fraction. Looking at ferrite-bainite and bainite-martensite mixtures the dierences in mechanical properties of the phases in these composites is less pronounced. In Fig. 3.1, two example plots for ferritebainite and bainite-martensite composites are shown, plots of other compositions are found in paper A. The prediction of ferrite-bainite samples showed good agreement with experimental results for the yield curve. However, the onset of necking is too late for all samples. Comparable to ferrite and bainite, bainite and martensite do not exhibit large contrast in mechanical properties. Though, a phenomena similar to a braze joint alters the mechanical response signicant. Bainite-martensite composites with a volume fraction of less than 25% bainite in martensite show a stronger response than the model can predict. In this case the use of the carbon dependent martensite curve alone yields the best result. Additional to the two phase composites one mixture containing three phases is produced and compared to the models. In consistency with modeling assumptions taken for two phase samples the results is presented in Fig. 3.2. All models are in range of the experimental results. The secant method is the only model underestimating measurements. This is expected as this model usually showed the lowest resultant yield curve.. 34.

(53) 3.1 Results of four homogenization models. 2000. 1200. 1500. 800. Stress [M P a]. Stress [M P a]. 1000. 600 400. 1000. 200 0 0.00. Isostrain F730. (a). 0.02. 0.04. 0.06 Strain [−]. Isowork B1015. 0.08. 500 0 0.00. 0.10. 0.02. 0.04. 0.06. 0.08. 0.10. Strain [−]. Secant Experiment. DI. Isostrain B1015. Samples consisting of 50% ferrite and 50% bainite.. (b). Isowork M1660. Secant Experiment. DI. Sample consisting of 22% bainite and 78% martensite.. Figure 3.1. Comparison of yield stress of two phase samples.. 2000. Stress [M P a]. 1500 1000 500 0 0.00. 0.02. 0.04. 0.06. 0.08. 0.10. Strain [−] Isostrain F730. Isowork B1015. Secant M1660. DI Experiment. Figure 3.2. Comparison of yield stress of samples consisting of ferrite, bainite and martensite.. 35.

(54) Results. 3.2 Inuence of the inclusion geometry In paper B a study on the inuence of the inclusion geometry on the hardening, localization and fracture of ferrite-martensite samples was conducted. To show the inuence of changes in the geometry as clear as possible the dependency of martensite on its carbon content is not used in this study. For more detailed results see paper B. ·104. 3. ·104. 3. x20. x20 x35. [N ]. x50. Force. Force. [N ]. x35. 2. 1. 0. 0. 2. 4 Elongation. (a). 6 [mm]. 8. 2. 1. 0. 10. x50. 0. 2. 4 Elongation. 6 [mm]. 2 Elements. 3 Elements. 2 Elements. 3 Elements. 4 Elements. Experiment. 4 Elements. Experiment. Spherical inclusion shape.. (b). 8. 10. Ellipsoidal inclusion shape.. Figure 3.3. Result from FEM analysis for three dierent ferritemartensite compositions, mesh sizes and inclusion shapes. The Eshelby tensor oers several possibilities for the alteration of the inclusion geometry.. In this study symmetric inclusion geometries. in the x-y-plan are used. This means the inclusion is only changed in its z-direction, which corresponds to the sheet thickness. The inuence of varying the inclusion radius in z-direction is less pronounced in the hardening part of the yield curve, see Fig. 3.3. After the onset of necking the response is quite dierent for spherical and ellipsoidal inclusions. The strain state around the inclusion is dierent and leads to a dierent. 36.

(55) 3.3 Evaluation of fracture in mixed microstructures strain concentration tensor which causes a dierent stress state. This leads numerical to dierent response in localization and elongations as the fracture strain is reached.. 3.3 Evaluation of fracture in mixed microstructures To test the damage function with mesh size compensation and the fracture model for a steel containing ferrite, bainite and martensite four samples with dierent volume fractions were produced and compared to nite element computations. The material model and its implementation in the commercial available nite element code LS-Dyna is described in previous sections. From the digital speckle photography measurements and an evaluation routine using a standard elasto-plastic material model in a radial return algorithm, the fracture strain and the stress triaxiality during testing is obtained. Fracture observed in the experiment is always failure of the composite, i.e.. on macroscopic scale.. In the -. nite element implementation a weakest link criteria is applied. In this case it is assumed that the composite fails if the fracture strain in one of the phases is reached. An example for an evaluation of a sample is given in Fig. 3.4. Here the plastic strain is plotted against the stress triaxiality calculated with help of an elasto-plastic material model. The point plotted corresponds to the position on the sample where the rst visible crack is observed. The fracture model for pure phases was calibrated in an earlier study by Östlund et al. [22]. The calibration was performed using six dierent specimen geometries representing dier-. 37.

(56) Results Eective plastic strain p. 0.6. 0.4. 0.2. 0 −0.40 −0.20 0.00. 0.20. 0.40. 0.60. 0.80. Stress triaxiality η. Figure 3.4. Plot of the eective plastic strain versus the stress triaxiality of sample FB-2 consisting of 50% ferrite and 50% bainite. The plot represents the complete experiment from initial loading until fracture.. ent stress triaxialities. Using digital speckle photography the fracture strain could be evaluated and pure phase data calibrated to it by least squares t. In Fig. 3.5 the fracture strain versus the stress triaxiality for the three pure phases are plotted based on experimental results from on-going work and an assumption of a maximum shear stress failure criterion. Using the mean eld homogenization from section 2.1.1.2 and the. Fracture strain f [−]. 1 0.8 0.6 0.4 0.2 −0.40 −0.20 0.00. 0.20. 0.40. 0.60. 0.80. Triaxiality η [−] Ferrite. Bainite. Figure 3.5. Fracture strain. Martensite. f versus stress triaxiality phases ferrite, bainite and martensite.. 38. η. for the pure.

(57) 3.3 Evaluation of fracture in mixed microstructures. damage model described in 2.1.2 the mechanical response of a composite material can be estimated. In Fig. 3.6a and 3.6b fracture strain versus triaxiality is plotted for composites corresponding to sample FB-2 and FBM-1, respectively. Due to the strain decomposition in the homogenization procedure the stress state in matrix and inclusion changes compared to the pure phases. This causes that stress triaxiality in the matrix phase to exceed the values obtained in the pure phase. In the same way the triaxiality in the inclusion phase shifts further to the left, i.e. to lower stress triaxialities. For a three phase composite the resulting fracture strain is less intuitive. Depending on the volume fractions of present phases, pure phase curves are shifted to lower or higher stress triaxialities. For the only sample consisting of three phases, FBM-1, the fracture strain versus the stress triaxiality is shown in Fig. 3.6b.. 1. 0.6 Fracture strain f [−]. Fracture strain f [−]. 0.8. 0.4. 0.2. −0.40−0.20 0.00 0.20 0.40 0.60 0.80 1.00. 0.8 0.6 0.4 0.2. Triaxiality η [−] Matrix (ferrite) Composite. (a). −0.40−0.20 0.00 0.20 0.40 0.60 0.80 1.00. Triaxiality η [−]. Inclusion (bainite) 2nd. Sample FB-2 consisting of ferrite and bainite.. (b). Matrix(ferrite) inclusion (martensite). 1st inclusion(bainite). Composite. Sample FBM-1 consisting of ferrite, bainite and martensite.. Figure 3.6. Result of the calculation of the fracture strain stress triaxiality η.. f. versus. The material model is implemented in the commercial available nite element code LS-Dyna via a user dened subroutine. To test the. 39.

(58) Results. mesh size compensation three dierent mesh sizes are generated for two dierent specimen geometries, see Fig. 2.10. Eight dierent phase compositions were produced for both specimen geometries, see table A.1 in paper A. Subsequent the result of the nite element analysis for three dierent mesh sizes is compared to the experimental results. Input data for the material model are the hardening curve, the damage parameters and the phase volume fractions of the present phases. In the following Fig. 3.7a to 3.9, show the result of the nite element simulation for three selected phase compositions which serve as an example. ·104. ·104 1.5. Force [N ]. Force [N ]. 1.5. 1. 0.5. 1. 0.5. 0 0.00. 1.00. 2.00. 3.00. 4.00 5.00 [mm]. 0 0.00. 6.00. Displacement 2 elements, A50 3 elements, A50 4 elements, A50 Experiment, A50. (a). Standard sample.. 0.50. 1.00. 1.50. 2.00. 2.50. Displacement [mm] 2 elements, R30 3 elements, R30 4 elements, R30 Experiment, R30. (b). Notched geometry.. Figure 3.7. Comparison of experimental and numerical result for sample FB-2 consisting of ferrite and bainite.. In Fig. 3.7a and 3.7b the result for a ferrite-bainite composite with equal volume fractions is given. For this sample the prediction of the yield curve up to necking was already shown to be in good agreement in the study of dierent homogenization schemes. From this study it was already expected that the onset of necking would be overestimated and 40.

(59) 3.3 Evaluation of fracture in mixed microstructures therefore the localized part of the yield curve would not be well predicted. The general observation is that the qualitative shape of the localization is in good agreement, for both, straight and notched tensile specimen. 3. ·104. ·104. [N ]. 2. 1. 0 0.00. 2. Force. Force. [N ]. 3. 1. 1.00. 2.00. 3.00. Displacement. 4.00 5.00 [mm]. 0 0.00. 6.00. 0.50. 1.00. 1.50 2.00 [mm]. 2.50. Displacement. 2 elements, A50. 3 elements, A50. 2 elements, R30. 3 elements, R30. 4 elements, A50. Experiment, A50. 4 elements, R30. Experiment, R30. (a). Standard sample.. (b). Notched geometry.. Figure 3.8. Comparison of experimental and numerical result for sample BM-1 consisting of ferrite and bainite. In Fig. 3.8a and 3.8b a comparison of experimental and numerical results for samples consisting of bainite and martensite is presented. For this type of microstructure composition the homogenization schemes did not approach yield stresses as found in experimental results. For modeling purposes literature suggests using the yield curve of martensite with the strengthening eect of elevated carbon content for bainite volume fractions up to 25%. The dierence between the homogenization and the martensite yield curve is depicted in the previous section, see Fig. 3.1b. This approach leads to better agreement between experiment and numerical result. For the notched specimen the numerical result and the experiment are good agreement while the result for the standard specimen deviates. A possible explanation for this dierence is the production. 41.

(60) Results process. The bainite transformation is, compared to the ferrite transformation, fast as visible in the time-transformation diagrams Fig. 2.4. Due to manual handling of the samples small dierences in time can occur and change the phase composition. However, the dierence between experimental and calculated result is within the error margin of the phase characterization. From the study on homogenization methods it is known that the onset of necking is better predicted for this microstructure. The localization and fracture are close to experimental observations. ·104. Force [N ]. 2 1.5 1 0.5 0 0.00. 1.00. 2.00. 3.00. 4.00. Displacement [mm] 2 elements 4 elements. 3 elements Experiment. Figure 3.9. Comparison of nite element and experimental result for sample FBM-1 with a standard, lower curve, and notched tensile specimen geometry, upper curve. The sample consisting of three phases is for both test specimen geometries depicted in Fig. 3.9. Numerical results clearly underestimate the experimental observation but in terms of elongation before fracture good agreement is obtained. The material model uses for all composites the same pure phase input data and spherical inclusion shape, the only parameter to be changed is the volume fraction of present phases. Microstructure characterization of the samples showed a signicant error. 42.

(61) 3.3 Evaluation of fracture in mixed microstructures. margin and therefore the only parameter possible to adjust underlies an uncertainty.. 43.

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(63) Chapter 4 Summary of appended papers 4.1 Paper A In paper A, four homogenization methods are evaluated concerning their ability of predicting the yield curve of mixed microstructures. The homogenization methods are taken from literature without modications. One microstructure shows a dependency on its carbon content, this is taken into account for the input data of the models.. Strengthening. eects of a combination of two phases are as well incorporated. To compare the capabilities of the models fourteen temperature histories are used to produce samples for experimental evaluation. Three samples are assumed as pure phases, while the remaining are mixed microstructures. Taking the broad eld of mechanical properties of the pure phases into account, correlation between predicted and experimental results are in agreement to an extent that varies between the dierent phase contents and models used.. Author contribution:. The present author performed all experi-. 45.

(64) Summary of appended papers. mental work and wrote the paper. Homogenization methods were jointly implemented in Matlab. 4.2 Paper B. In paper B the homogenization scheme yielding best results in paper A, is combined with a damage model. The damage model predicts localization and fracture. Both, localization and fracture are mesh size dependent. Introducing an analysis length scale into the damage model compensates to some extend for this eect. In this publication the inuence of the shape of the inclusion on the hardening, localization and fracture is studied. For validation of numerical results, three ferrite-martensite microstructures with varying amount of phase content are used. The inuence of the inclusion geometry on the response is signicant and gives a possibility to alter the response of the model. Author contribution: The authors jointly planned the paper. The present author performed all experimental work, implemented the constitutive model into LS-Dyna and performed the numerical simulations.. 46.

(65) Chapter 5 Discussion and Conclusions 5.1 Key results Fourteen samples with dierent microstructure compositions and volume fractions of formed phases are compared to four homogenization methods. The pure phases taken into consideration are ferrite, bainite and martensite, remaining austenite is neglected and its volume fraction is added to the softest present phase, pearlite was not found in any sample. The majority of samples consist of two distinct phases, one sample contains three phases in considerable amount. For all compositions the same modeling assumptions are used and therefore not the best t is achieved for every single case. In general the double-inclusion model yielded the most promising results. In order to develop a localization and fracture model the material response after the onset of necking is of interest. Ferrite-martensite mixtures showed a brittle behavior. The ferrite-bainite samples showed necking in testing and a clear point for the onset of necking is found. 47.

(66) Discussion and Conclusions. All models miss this point and indicate the onset of necking at a later point. The bainite-martensite samples showed less pronounced necking with exception of the sample with highest bainite content. The onset of necking is well predicted compared to the previous composites. For the composite consisting of three phases the onset of necking is in good agreement for all models. The double-inclusion method is used as homogenization scheme to calculate the hardening response and distinct strain in each phase of the composite. It is combined with a phenomenological localization and fracture model to predict the composite response under tensile loading. The localization and fracture model is possible to combine with the double-inclusion model. Variation of the half axis ratio of inclusion shape used in Eshelby's tensor is a way to achieve reasonable agreement between experimentally obtained elongations before fracture. A possible explanation for dierences between experiment and simulation is the parameters used. Initially all parameters were determined on measurements of sheets with a thickness of 1.2mm but all tensile specimens were cut from sheets with thickness 1.8mm. Numerical results for the composition labeled FM-3 show a scattered result for the fracture strain. A possible explanation is the analysis length scale factor. The localization band is most likely within the element size for 2 elements but for smaller elements, the band is larger than one element in width. Applying the implemented material model on other microstructure compositions, taking some modeling assumptions into account, it is possible to get reasonable results for most composites. The main diculty found is the reliable estimation of present phase volume fractions. Mi48.

(67) 5.2 Future work crostructures consisting of two distinct phases are easier to characterize and the error margin is comparable small. The three phase sample showed a larger error margin and therefore the model result is inuenced as the phase volume fraction is the only parameter possible to change without altering modeling assumptions.. 5.2 Future work 5.2.1. Evaluation of fracture for mixed microstructures. A set of samples consisting of ferrite and bainite with varying amounts of respective phase are produced. For all compositions ve dierent specimen geometries, representing distinct stress triaxialities are available. This study aims to improve the understanding and modeling of fracture in this type of mixed microstructures. Due to the variety of specimen geometries it is intended to improve the calibration of fracture models.. 5.2.2. Study on detailed models of spot-welds. In this work fracture strains of ultra-high strength steel in spot welds and their proximity will be investigated and compared to nite elements simulations. This study is motivated by the use of spot welding as major method in joining components of a body-in-white. The experimental procedure includes blanks with two dierent base microstructures, one ferritic and the other fully hardened martensitic. These types of microstructures are commonly found in press hardened components with tailored material properties. On both types of samples a dummy blank is spot welded. The welding process introduces heat into the blank and. 49.

(68) Discussion and Conclusions changes the microstructure and due to this the mechanical properties of it. During tensile testing of the spot welded samples the behavior of the material will be recorded by digital speckle photography (DSP) and force measurement. The experimental method of DSP measurement allows resolving displacement and strain elds in the specimen and the vicinity of the spot-weld up until fracture.. 5.2.3. Evaluation of material model on component level. To further evaluate the material model it will be tested on component level. In a rst stage a test specimen with a more component like geometry will be developed. The requirement of the specimen is the in-house production and testing in a standard tensile test machine. A precondition of the choice of component geometry is that changes in the pre-cut of the specimen change the stress triaxiality in a designated area of the geometry. Experimental results will be compared to the material model implemented in LS-Dyna.. 50.

(69) References. References [1]. Hot sheet metal forming of high-performance steel, CHS2: 4th International Conference M. Oldenburg, K. Steinho, and B. Prakash. Editorial.. . Ed. by M. Oldenburg, K. Steinho, and B. Parkash.. Verlag Wissenschaftliche Scripten, July 2013.. [2]. Fraunhofer Institute for Machine Tools and Forming Technology IWU.. [3]. Press Hardening of Sheet Metal and Closed Proles. Voestalpine Steel Division.. benchmark [4]. . 2012.. phs-ultraform The press-hardening steel. . www.voestalpine.com/steel. 2013.. R. Erhardt and J. Böke. Industrial application of hot forming. Hot sheet metal forming of high-performance steel, CHS2: 1st International Conference press simulation.. . Ed. by M. Oldenburg,. K. Steinho, and B. Parkash. Verlag Wissenschaftliche Scripten, Oct. 2008.. [5]. International Iron & Steel Institute. Advanced High Strength Steel (AHSS) Application Guidlines.. [6]. Application Guidlines. . Mar. 2005.. P. Åkerström. Modelling and Simulation and Simulation of Hot Stamping. PhD thesis. Luleå University of Technology, September 2006.. [7]. M. Eriksson. Modelling of Forming and Quenching of Ultra High Strength Steel Components for Vehicle Structures. PhD thesis. Luleå University of Technology, May 2002.. 51.

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(75) Appended papers.

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(77) Paper A.

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(79) A study on homogenization methods for steels with varying content of ferrite, bainite and martensite S. Golling1 , R. Östlund1 and M. Oldenburg1. Division of Mechanics of Solid Materials, Lule a University of Technology, SE-97187 Lule a, Sweden 1. Abstract The demand of ultra high strength steel (UHSS) components increased in the last decade due to their high strength to weight ratio. The driving force in this development is the automotive industry and regulations concerning passenger safety and fuel consumption. The use of ultra high strength steel enables design of lighter car bodies with equal or better passenger safety compared to earlier 61.

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