Modelling and Characterisation of

application of new material grades is the characterization of its mechan-ical properties. Post-localization and fracture predictive technologies great...

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L I C E N T I AT E T H E S I S

Department of Engineering Sciences and Mathematics Division of Mechanics of Solid Materials

ISSN: 1402-1757 ISBN 978-91-7439-320-0 Luleå University of Technology 2011

Modelling and Characterisation of Fracture Properties of Advanced High Strength Steels

Rickard Östlund

Modelling and characterisation of fracture properties of advanced high strength steels ¨ Rickard Ostlund

Division of Mechanics of Solid Materials Department of Engineering Sciences and Mathematics Lule˚ a University of Technology SE-971 87 Lule˚ a, Sweden

Licentiate Thesis in Solid Mechanics

¨ c Rickard Ostlund  ISSN 1402–1757 ISBN 978–91–7439–320–0 Published: October 2011 Printed by Universitetstryckeriet Lule˚ a Tekniska Universitet www.ltu.se

Abstract Growing demands for passenger safety, vehicle performance and fuel economy is a continuous driving force for the increase in use of advanced high strength steels (AHSS) in the automotive industry. These steels area characterised by improved formability and crash worthiness compared to conventional steel grades. An important prerequisite of the application of new material grades is the characterization of its mechanical properties. Post-localization and fracture predictive technologies greatly facilitate the design of components which make optimal use of these steel grades. In this thesis, press hardened boron alloyed steel subjected to differential thermo-mechanical processing is characterized. Fracture properties in relation to the different microstructures obtained is studied. Furthermore a dual phase (DP) cold forming steel is chosen for evaluation of ductility limit in shear loading. throughout this work a strategy for modelling post-localization response and predicting ductility limit using shell elements larger then the typical width of the localized neck is used. The studied material is assumed to be in a state of plane stress. Mesh dependency is alleviated by the introduction of a element size dependent parameter into the constitutive description. This parameter acts as a hardening parameter, controlling the evolution of the yield surface depending on loading, strain history and shell element size. Model calibration relies on a full field measurement technique, Digital Speckle Photography (DSP), to record the plane deformation field of tensile specimens. Quantitative measurements of the severely localized deformation preceding crack initiation are feasible. With the proposed strategy, mesh sensitivity in terms of post localization load response and fracture elongation predictions is reduced significantly compared to results obtained without the element size dependent parameter. It was found that high strain hardening favours strain localization of shear band v

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Abstract

type, and accelerates the formation of a localized neck. The hardening characteristics is determinant to which deformation mode dissipates the minimum energy. For the DP steel, the Tresca yield surface more accurately describes the yielding point compared to the von Mises plane stress elipse. Furthermore, the exponential ductility function dependent on the stress triaxiality parameter agrees well with experimental fracture data in the ductile loading regime for both DP and boron steel. In shear loading, the maximum shear (MS) stress criterion successfully describes the ductility limit. Due to the significantly different ductility of the various microstructures obtainable by the thermo-mechanical processing of boron alloyed steel, a modelling strategy is needed. It was found that in ductile loading, local equivalent fracture strain can be related to the hardness of that material point. An exponential decrease in ductility with increased hardness describes experimental data collected for five different microstructures.

Preface The work presented in this thesis has been carried out in the Solid Mechanics group at the division Mechanics of Solid Materials, Department of Engineering Sciences and Mathematics at Lulea˙ University of Technology (LTU), Lulea˙ Sweden. The work has been conducted within and funded by the Centre for High Performance Steel (CHS). CHS is gratefully acknowledged for supporting the present research. Without the help, guidance and support of many people directly or indirectly involved, completion this study would be unassailable. Firstly i would like to thank my supervisor, Prof. Mats Oldenburg and assis˙ tant supervisor Prof. Hans-Ake H¨aggblad for help and guidance during the course of this work. The assistance of Dr. Daniel Berglund as industrial partner is greatly appreciated, and Dr. Greger Bergman, Dr. ˙ Paul Akerstr¨ om and their colleagues at Gestamp R&D. I would also like to thank Jan Granstr¨ om for supporting the experimental work, and Yogeshwarsing Calleecharan for providing the template for this thesis. Many thanks to my friends and colleagues at the division for stimulating discussions and making the office a pleasant place to come to. Finally, many thanks to family and friends for support and encouragement.

¨ Rickard Ostlund Lulea˙ , September 2011

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Contents Abstract

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Preface

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Publications list

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1 Introduction 1.1 Passive safety for automotive vehicles 1.2 Advanced high strength steels . . . . . 1.2.1 Press-hardening . . . . . . . . . 1.3 Mathematical modelling in design . . . 1.4 Objective and scope . . . . . . . . . .

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3 Method 3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Deformation and fracture 2.1 Physical mechanisms of deformation 2.2 Physical mechanisms of fracture . . . 2.3 Modelling of localized deformation . 2.4 Modelling of fracture initiation . . .

4 Results 5 Summary 5.1 Paper 5.2 Paper 5.3 Paper

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appended papers 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Conclusions and outlook

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Table of contents

Acknowledgement

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References

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Appended papers

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Paper A

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A Evaluation of localization and failure of boron alloyed steels 25 Paper B B hardness based failure model Paper C

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C Localization and fracture properties of a dual phase steel 61

Publications list Paper A ¨ ˙ Sundin K-G and Berglund Ostlund R, Oldenburg M, H¨ aggblad H-A, D. Evaluation of localization and failure in boron alloyed steels with different microstructure compositions. To be submitted for journal publication.

Paper B ¨ ˙ and Berglund D. Failure model Ostlund R, Oldenburg M, H¨ aggblad H-A evaluation for varying microstructure based on material hardness. In proceedings ofHot Sheet Metal Forming of High-performance Steel. June 13-17, 2011, Kassel, Germany

Paper C ¨ ˙ Analysis of deformation, Ostlund R, Oldenburg M and H¨ aggblad H-A. localization and fracture properties of a dual phase steel. To be submitted for journal publication.

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Chapter 1 Introduction This thesis begins with an introduction to acquaint the general reader with the research topic, and addresses the more general context to which the present work aims to contribute to.

1.1

Passive safety for automotive vehicles

Improvements of automobile safety have steadily reduced injury and death rates in all first world countries. Nevertheless, auto collisions are the leading cause of injury-related deaths, an estimated total of 1.2 million in 2004, or 25% of the total from all causes [1]. This public health challenge requires concerted efforts from the automotive industry, research community, governmental legislators and associated organizations. The term passive safety refer to components of the vehicle that help to protect occupants during a crash. Crashworthiness systems prevent or reduce the severity of injuries in a crash situation, by dissipating the energy of collision and reducing the impact force on the vehicle occupants. To meet emissions legislations and reduce fuel consumption, mass minimization of automotive body structures is vital. The combination of maximum crashworthiness and minimum structural weight constitute a challenging engineering task. This increase in strength to weight ratio has been partially addressed by the application of new advanced materials, which in turn require efficient engineering tools in component design. Deformation and fracture predictive technologies are prerequisites for optimal use of the properties of the material.

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1.2

Introduction

Advanced high strength steels

Among automotive body structure materials, advanced high strength steels (AHSS) groups a number of different steel grades. They are characterized by improved properties in relation to conventional mild steel, but relatively low alloying content compared to other high strength steels. Instead, mechanical properties are improved by microstructural design through thermomechanical processes. This work centres around a hot forming steel type which is boron alloyed, and a Dual Phase (DP) cold forming steel. As delivered, the microstructure of dual phase steel is composed of a soft ferrite matrix with embedded hard martensite particles. The boron steel exhibit homogeneous distribution of pearlite and ferrite, however this is often changed in the manufacturing process, which is described in the following section.

1.2.1

Press-hardening

Simultaneous forming and heat treatment of thin sheet structures, termed press-hardening can be summarized as a four step process. Blanks of boron alloyed steels are cut, then austenitized at a temperature of ∼ 900◦ C, followed by forming and cooling and eventual post-treatment. Due to high formability in the austenitic state, complex geometries are feasible. By controlled tool temperature, local microstructure can be designed such that a wide range of mechanical properties are obtainable within the same component. As such, press-hardening offers an technological advantage in relation to conventional cold working. Component design and material design can be performed concurrently. Much effort is currently invested into the development of components with spatially varying microstructure [2, 3], often termed “tailored properties”. Zones of ferritic-bainitic microstructure with continuous transition to ultra high strength martensite combines high energy absorption and high intrusion protection within the same component.

1.3

Mathematical modelling in design

Mathematical modelling of the structural behaviour of safety components subjected to crash loading is commonly used, with non-linear Finite Element Analysis (FEA) as the numerical method to solve the formulated problem. The finite element method is a numerical tech-

1.4 Objective and scope

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nique for finding approximate solutions to partial differential equations. The present work is concerned with the constitutive behaviour of the studied materials, not with developments on governing equilibrium formulations nor the finite element method as such. The reader is referred to textbooks such as [4–6] for more information on the subject.

1.4

Objective and scope

The objective of this thesis is to evaluate and improve numerical engineering tools for predicting fracture initiation of sheet metal subjected to inplane loading. A general research question can be formulated as “How does thermomechanical process history influence product functionality in terms of fracture properties ?”. This work is a part of past and ongoing efforts concerning modelling of the press-hardening process [7–9] and in-service product functionality [10–12] of crashworthiness systems. Specifically, the developments in the current research project focuses around components with tailored components. Using results from process simulations, such as phase fraction content, the aim is to be able to predict fracture of tailored components.

Chapter 2 Deformation and fracture In this chapter the physical mechanisms of metal deformation and fracture are briefly discussed. A few fracture theories and models of the vast variety available in literature are reviewed, and models used in the present work are presented.

2.1

Physical mechanisms of deformation

For metals, two types of deformations are identified. Elastic, reversible deformations which occurs at the atomic level and inelastic permanent deformations occurring at the crystal level. In addition to elastic deformation, they correspond to relative displacement of atoms which remains when the load is removed. The mobility of dislocations, defects in the crystal lattice, is the essential cause of permanent deformations. Irreversible movement of dislocations initiates when the stress acting within the elasto-plastic solid reaches a critical level, termed yield stress, and the original geometry is not recovered upon unloading. Large plastic deformation will in turn increase the density of dislocations, which increases the number of obstacles in the crystal lattice contributing to further hardening of the material. Detailed descriptions of dislocations and strengthening mechanisms can be found in textbooks such as [13]. To further acquaint the reader with the topic of this thesis, some elaboration of localized deformation is necessary. Localized deformation is an example of a material instability, corresponding to an abrupt loss of homogeneity of deformation occurring in a solid sample subject to a loading path compatible with continued uniform deformation. Initially homogeneous deformation confines to a narrow region of intense strain4

2.2 Physical mechanisms of fracture

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ing. This phenomenon usually precedes fracture in tensile loading of ductile metals, particularly for plane sheets. Terms like diffuse and localized necking are often used to describe the various stages of tensile deformation of sheet metals.

2.2

Physical mechanisms of fracture

Elastic and plastic deformation both maintain the cohesion of matter. Fracture destroys this cohesion by creating surface or volume discontinuities within the material. Therefore, fracture occurs at several length scales, from the scale of crystals to micro-cracks and to the scale of structures and components. The two main basic mechanisms of local fracture are brittle fracture by cleavage and ductile fracture resulting from large localized plastic deformations. This work focuses on the latter. Ductile fracture is due to nucleation, growth and coalescence of voids in the material. Void nucleation arises in the vicinity of crystal defects, e.g. inclusions which causes stress concentrations followed by decohesion.

2.3

Modelling of localized deformation

An important issue in nonlinear mechanics is the stability of the material model. Material or constitutive models are the mathematical description of material behaviour, which gives the stress response as a function of the deformation history of the body. See [14] for in-depth treatment of constitutive modelling. Material model instabilities are usually associated with localized growth of deformation, corresponding to the phenomena earlier described occurring in nature. Theoretical and numerical studies of the conditions of localized necking for a variety of material models can be found in [15–17]. One conclusion from these papers is that for the rate-independent elasto-plastic solid, the condition for strain localization coincides with the condition of singularity (propagation at null speed) of an acceleration wave. This conditions represents the so-called ’loss of ellipticity’ of the differential equations governing the rate equilibrium. Material instability inherent in constitutive descriptions will cause numerical issues when discretized, manifesting themselves as being strongly dependent on the mesh. When the material reaches threshold of localization, deformation localizes to a set of measure zero. This is most often solved with some regularization technique, which can be gradi-

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Deformation and fracture

ent type, nonlocal or by introducing rate dependent material description, see [18–20]. With regularization, difficulties still remain due to the very fine mesh needed to accurately resolve sharp gradients. There are techniques to improve coarse-mesh accuracy, such as embedding of discontinuities and enriched fields in the element [21]. The objective of this thesis is not to model the phenomenon of localized deformation as such, but rather the effects on load response and subsequent fracture using coarse meshes, .i.e. elements significantly larger than the size of the zone of localized necking. The methodology adopted here suggests that constitutive parameters should be dependent on a discretization parameter, the element size. Similar ideas has been used earlier in cohesive crack modelling of concrete with finite elements [22], and more recently in the context of crashworhiness simulations [12, 23]. Element size dependency is introduced into the constitutive model by a weakening factor L, termed localization function, coupled to the yield function Φ = f (σ) − σy (1 − L),

(2.1)

where f (σ) is some isotropic function of the stress tensor and σy is the yield strength. The localization function evolutes exponentially with equivalent plastic strain εp , L = A exp[B(εp − ε0 ) − 1],

εp ≥ ε0 .

(2.2)

A and B are functions of the characteristic element size l, A=

A0 , l

B = B0 l,

(2.3)

and ε0 is equivalent localization threshold strain, A0 , B0 are calibration constants. This approach is similar to that of damage induced weakening, see e.g. [24], however L is not viewed as to be solely composed of material damage. Due to inability of the coarse mesh to resolve localization, the stress state is altered by L so that post-localization load and deformation response is invariant with element size. Further details are found in appended papers A and C.

2.4

Modelling of fracture initiation

The present thesis is concerned with problems involving ductile fracture of crack-free bodies. As earlier described, material separation is a result

2.4 Modelling of fracture initiation

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of complex processes which initiates at the microscale. Micromechanical models such as [25, 26] have been developed on physical basis to model this process, however in this work the phenomenological route of failure prediction is undertaken. On a macroscale, the variables controlling fracture are the current values and history of the stress and strain tensors. This class of local ductile fracture criteria can be written as an integral over equivalent plastic strain εp , weighted by a function of the stress reaching a threshold value C,  g(σ) dεp = C. (2.4) εp

The essential difference between various fracture models proposed [24, 27–30] is the weighting function g(σ). Within the framework of the thermodynamics of irreversible processes, continuum damage mechanics [31] have been developed. Damage is an internal variable quantifying the area fraction of microcracks and cavities in a damaged body. However, the term damage is quite widely used for any type of accumulated failure indicator. Common for almost all fracture damage theories is the hydrostatic stress dependence. Hydrostatic tension accelerates void growth process, while pressure retards formation and growth. Theoretical [32, 33], and experimental studies [29] shows that fracture strain increases when hydrostatic pressure increases. In most model formulations, the hydrostatic pressure is normalized with equivalent stress rendering to the dimensionless stress triaxiality parameter. In a plane state of stress, σ3 = 0, the stress triaxiality parameter uniquely characterizes all loading conditions. In this thesis a fracture model with exponential dependence on the stress triaxiality parameter in conjunction with the maximum shear stress criterion is evaluated. Furthermore, equivalent fracture strain is formulated as a function of element size to alleviate mesh dependency on fracture predictions. All three appended papers addresses various aspects of the fracture modelling conducted within this work.

Chapter 3 Method The Oxford English Dictionary says that scientific method is: “a method of procedure that has characterized natural science since the 17th century, consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses. ” The work presented in this thesis has been carried out much in spirit of phenomenology, where experimental observations of studied phenomena are essential. Models are formulated and evaluated based on these observations, together with a view on the practical engineering application.

3.1

Experiments

Experiments are of paramount importance in mechanics, acting as a vehicle for observations, measurements and model calibration. In this thesis, much of the methodology relies on full-field measurement of displacements. Conventional methods of material characterization commonly involve the standardized tensile test, where strain is calculated from extensiometer measurements by dividing extension with initial gauge length. This method requires a uniform deformation field within the gauge length. The results presented in this thesis rely on the optical measurement method Digital Speckle Photography (DSP). Given that the specimen surface exhibit a random pattern, the inplane displacement of any small subregion can be determined by a cross-correlation procedure of the images taken before and after deformation. Randomness of the of the surface pattern ensures that the evaluated pointwise displacement field is 8

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3.1 Experiments

unique. The procedure is repeated throughout the deformation history, giving the evolution of displacement field through a number of time instants. This method does not require a uniform deformation field for the evaluation of constitutive parameters. A typical full-field measurement result is illustrated in Figure 3.1, which shows contours of equivalent plastic strain for two different test specimen geometries. Figure 3.1a shows the localized strain field preceding fracture of a notched tensile specimen and Figure 3.1b shows the strain distribution of a shear test at incipient fracture. More information on the experimental technique can be found in [11, 34].

(a) Notched tensile specimen

(b) Shear specimen

Figure 3.1: Illustration of the DSP method by contours of equivalent plastic strain at incipient crack initiation for two specimen geometries. The material used is DP600 steel sheets with nominal thickness of 1.5 mm.

Chapter 4 Results This chapter briefly summarizes the key results of this work, which are elaborately described in the appended papers. • Concerning post-localization and fracture modelling, the localization function approach used in this work shows promising results. As shown in Figures A.8, C.6, Force displacement response and fracture elongation predictions are practically invariant with mesh size. Furthermore, coarse-mesh discretization relative to the size of localized zone is feasible. • Several material grades are developed by differential thermal treatment of the boron steel. These show different localization and fracture behaviour, which the constitutive and fracture model is able to predict. • An extensive experimental program of five press-hardened material grades shows that local fracture ductility can be related to material hardness, see Figures B.2,B.3. • Results illustrated by figure C.6 shows that the proposed modelling strategy is readily expanded to loading conditions in the shearing regime.

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Chapter 5 Summary of appended papers This chapter summarizes the investigations and findings of the appended papers. Common for all papers is that the DSP experimental method described in section 3.1 has been used for the study and calibration of constitutive and fracture parameters. Furthermore, a modelling strategy for predicting post-localization response and fracture using shell elements larger then the typical width of the localized neck is used, with slight variations, in all papers. Issues concerning localization is described in section 2.3.

5.1

Paper A

In paper A, three different material grades, labelled HT400, HT550 and HT800 are developed and characterized. By a controlled tool temperature process in a plane hardening tool, the hot blanks of boron alloyed steel are subjected to different thermal cycles. Metallographic study shows that HT400 is mainly ferritic, HT800 is bainitic and HT550 is a mixture of those phases. It is observed that a higher strain hardening accelerates the formation of a localized neck. Model parameters are determined for the three grades, and computed force-displacement response illustrates that fracture prediction and localization load response is not mesh dependant.

5.2

Paper B

Paper B focuses on the fracture properties in relation to thermal treatment. In addition to the material grades studied in paper A, HT700 11

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Summary of appended papers

and a fully martensitic grade is included in this paper. A strategy to account for varying phase composition in failure modelling based on material hardness is evaluated. Here the fracture model used in paper A is further developed. Fracture strain is taken to be dependent on stress state represented by the stress triaxiality parameter and phase fraction content characterized by material hardness. It is found that an exponential decrease in fracture strain with hardness well describes experimental data.

5.3

Paper C

In paper C a dual phase steel is chosen for evaluation of plasticity and fracture in shear loading. The von Mises plane stress ellipse used in paper A is replaced by the Tresca yield surface. The fracture model from the two above papers is further developed to include predictive capabilities in the shear loading regime, by introducing the maximum shear stress criterion. Furthermore, the restrictions on the localization function is imposed, so that yield stress reduction only takes place in the ductile loading regime.

Chapter 6 Conclusions and outlook This thesis concerns deformation and fracture modelling of Advanced High-Strength Steel sheets for automotive applications. Experimental observations of post-localization crack initiation phenomena are facilitated by a full field measurement technique. Due to the introduction of the localization function and element size dependent fracture model, post-localization and fracture predictions are not mesh dependent. This applies when coarse mesh discretization is used, i.e. elements significantly larger then the physical size of the localized zone. Several material grades are investigated, primarily centred around the boron alloyed steel. Trough a controlled tool temperature manufacturing process, five different material grades based on the boron steel are evaluated. Localization characteristics and strain hardening are studied, and fracture properties are related to material hardness. With reference to the research question formulated in section 1.4, the three contributions in this thesis will hopefully shed light on some aspects on modelling of steels with varying properties. There is however still much work to do, where the following ideas for future investigations are put forward: • Development of a rule of mixture for capturing the composite localization behaviour depending on the phase content of the material. This could then be used together with a mixture rule defining the composite flow behaviour. • Further development of the fracture model. It could be investigated whether the hardness based model can predict fracture in a larger span of loading conditions, e.g. shear loading.

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Conclusions and outlook

• Micromechanical studies of the fracture properties of different phase mixtures using multi-scale methods could be useful in the further development of a modelling strategy for tailored components. Apart from the items directly related to this work stated above, a more comprehensive material characterization considering anisotropy, yield surface shape, rate dependence etc. could be useful.

Acknowledgement The authors wish to acknowledge the European Union Structural Funds and the Centre for High Performance Steel at Lulea˙ University of Technology for providing financial support.

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References [1] Peden M, editor. World report on road traffic injury prevention. World health Organization, 2004. [2] Oldenburg M, Steinhoff K, and Prakash B, editors. 2nd International Conference on Hot Sheet Metal Forming of HighPerformance Steel. Verlag Wissenschaftliche Scripten, June 15-17 2009. Lulea˙ , Sweden. [3] Oldenburg M, Steinhoff K, and Prakash B, editors. 3nd International Conference on Hot Sheet Metal Forming of HighPerformance Steel. Verlag Wissenschaftliche Scripten, June 13-17 2011. Kassel, Germany. [4] Belytschko T, Liu W K, and Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley, 2000. [5] Fung Y.C. and Tong P. Classical and computational solid mechanics, volume 1. World Scientific Publishing, 2001. [6] Bonet J and Wood R D. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 2008. [7] Bergman G. Modelling and simulation of simultaneous forming and quenching. PhD thesis, Division of Solid Mechanics, Department of Applied Physics and Mechanical Engineering, Lulea˙ University of Technology, 1999. [8] Eriksson M. Modelling of forming and quenching of ultra high strength steel components for vehicle structures. PhD thesis, Division of Solid Mechanics, Department of Applied Physics and Mechanical Engineering, Lulea˙ University of Technology, 2002.

16

References

17

˙ [9] Akerstr¨ om P. Modelling and simulation of Hot Stamping. PhD thesis, Division of Solid Mechanics, Department of Applied Physics and Mechanical Engineering, Lulea˙ University of Technology, 2006. [10] Kajberg J and Lindkvist G. Characterization of materials subjected to large strains by inverse modelling based on in-plane displacement fields. Int .J. Solids and structures, 2004. [11] Eman J. Study and characterization of localization and failure behaviour of ultra high strength steel. Licenciate thesis, 2007. Lulea˙ University of Technology. ˙ Berglund D, Sundin K-G, and Oldenburg M. For[12] H¨ aggblad H-A, mulation of a finite element model for localization and crack initiation in components of ultra high strength steels. In proceedings of Hot sheet metal forming of high performance steel, June 15-17 2009. Lulea, ˙ Sweden. [13] Callister W D. Materials science and engineering: an introduction. John Wiley & sons, 2007. [14] Ottosen N S and Ristinmaa M. The Mechanics of Constitutive Modeling. Elsevier, 2005. [15] Hill R. Acceleration waves in solids. Journal of the Mechanics and Physics of Solids, 10(1):1 – 16, 1962. [16] Rice J R. The localization of plastic deformation. Theoretical and Applied Mechanics (Proceedings of the 14th International Congress on Theoretical and Applied Mechanics, Delft, 1976, ed. W.T. Koiter), 1:207–220, 1976. North-Holland Publishing Co. [17] Tvergaard V, Needleman A, and Lo K.K. Flow localization in the plane strain tensile test. Journal of the Mechanics and Physics of Solids, 29(2):115 – 142, 1981. [18] De Borst R and M¨ uhlhaus H-B. Gradient-dependent plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering, 35(3):521–539, 1992. [19] Baˇazant P Z and Jir´ asek M. Nonlocal integral formulations of plasticity and damage: Survey of progress. J. Eng. Mech., 128(11):1119–1149, 2002.

18

References

[20] Needleman A. Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng., 67:69– 85, March 1988. [21] M. Ortiz, Y. Leroy, and A. Needleman. A finite element method for localized failure analysis. Comput. Methods Appl. Mech. Eng., 61:189–214, March 1987. [22] A. Hillerborg, M. Mod´eer, and P.-E. Petersson. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 6(6):773 – 781, 1976. [23] Kessler L, Gese H, Metzmacher G, and Werner H. An approach to model sheet failure after onset of localized necking in industrial high strength steel stamping and crash simulations. SAE Int. J. Cars -Mech. Syst., (1):361–370, April 2009. [24] Xue L. Damage accumulation and fracture initiation in uncracked ductile solids subject to triaxial loading. International Journal of Solids and Structures, 44(16):5163 – 5181, 2007. [25] Gurson A L. Continuum theory of ductile rupture by void nucleation and growth: Part i—yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology, 99(1):2–15, 1977. [26] Needleman A and Tvergaard V. An analysis of ductile rupture in notched bars. Journal of the Mechanics and Physics of Solids, 32(6):461 – 490, 1984. [27] Wilkins M L, Streit R D, and Reaugh J E. Cumulative-straindamage model of ductile fracture: Simulation and prediction of engineering fracture tests. Technical report, Lawrence Livermore National Laboratory, October 1980. Technical Report UCRL-53058. [28] Cockcroft M G and Latham D J. Ductility and the workability of metals. Journal of the Institute of Metals, 1968. [29] Johnson G R and Cook W H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1):31 – 48, 1985.

References

19

[30] Wierzbicki T, Bao Y, Lee Y-W, and Bai Y. Calibration and evaluation of seven fracture models. International Journal of Mechanical Sciences, 47(4-5):719 – 743, 2005. A Special Issue in Honour of Professor Stephen R. Reid’s 60th Birthday. [31] Lemaitre J. A continuous damage mechanics model for ductile fracture. Journal of Engineering Materials and Technology, 107(1):83– 89, 1985. [32] McClintock F A. A criterion of ductile fracture by the growth of holes. Journal of applied Mechanics, (35):363–371, 1968. [33] Rice J R and Tracey D M. On the ductile enlargement of voids in triaxial stress fields. Journal of the Mechanics and Physics of Solids, 17(3):201 – 217, 1969. [34] Kajberg J. Displacement field measurement using digital speckle photography for characterisation of materials subjected to large deformations and high strain rates. PhD thesis, Division of Solid Mechanics, Department of Applied Physics and Mechanical Engineering, Lulea˙ University of Technology, 2003.

Appended papers

Paper A

Paper A Evaluation of localization and failure in boron alloyed steels with different microstructure compositions 1 ¨ ˙ H¨ R Ostlund , M Oldenburg1 , H-A aggblad1 , K-G Sundin1 and D Berglund2 1 Mechanics

of Solid Materials, Lulea˙ University of Technology, SE 971 87 Lulea, ˙ Sweden 2 Gestamp R&D, SE 971 25 Lulea, ˙ Sweden

Abstract Within the press hardening technology, where hot sheet blanks are simultaneously formed fixed and quenched, new methods with differential thermal treatment are emanating. With controlled tool temperature variation components with tailored properties can be produced. Automotive components combining high energy absorption and intrusion protection in a crash situation are feasible. In the present work the mechanical properties of three different material qualities, emanating from the same base sheet metal subjected to different thermal histories, are investigated. A strategy for modelling post-necking response and crack initiation using shell elements larger then the typical band width of the localized neck is used. The model relies on a sequence of full field measurements throughout a tensile test, i.e. Digital Speckle Photogra25

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Paper A

phy(DSP). The full field experimental method allows for evaluation of mechanical and failure properties at different analysis lengths, providing parameters for a model which accounts for shell element size. Additionally the model contains a strain based failure criteria as a function of stress triaxiality. Good agreement between a simulated tensile test and experimental results are found. A detailed metallographic study of the three grades is performed and presented.

A.1

Introduction

The increasing demand to reduce vehicle weight in the automotive industry has led to a increase in use of ultra high strength press hardened safety structures. The press hardening process uses boron steel blanks which are first austenitized at a temperature of ∼ 900◦ C and then formed and quenched between cold tools. The coming generation of these components can have tailored mechanical properties, providing additional means for designing the structural response in a crash situation. Soft zones in certain areas of the component can give an advantageous crash deformation mode, whilst retaining homogeneous thickness. Varying properties can be obtained by controlling the cooling rate during the forming process. By controlled tool temperature variation, zones of bainitic-ferritic microstructure with continuous transition to ultra high strength martensite gives the desired mechanical properties. By Using finite element simulations of the forming process together with models for phase transformation kinetics, the mechanical properties of the final product can be predicted [1–3]. An important precondition for the application of new materials is the characterization of its failure properties. The aim of the current work is to establish a failure criterion and to characterize mechanical properties including post-necking behaviour of three material grades. The material grades are obtained from the same base steel but subjected to different thermal histories. They are labelled HT400, HT550 and HT800 according to their yield strength. During initial tensile plastic deformation of a sheet metal, plastic strain is generally distributed uniformly within the volume. If further loaded the plastic strain often localizes, first forming a diffuse neck and finally a localized neck preceding rupture. Once a neck has formed, subsequent straining is confined within the neck. This causes an instability, when the increase in stress due to decrease in load-carrying area is greater then the increase in load-carrying ability of the metal

A.1 Introduction

27

due to strain hardening. The rate of decrease in cross-sectional area may also be accelerated due to material deterioration caused by ductile plastic damage. Localized necking is the dominant mechanism leading to fracture of safety components in a crash. In this work the localization process is observed experimentally using full field Digital Speckle Photography (DSP) measurements in accordance with the method used in [4]. The DSP method produces a two-dimensional displacement field determined at a very small measuring length compared to ordinary tensile testing with extensiometer. In this work a measurement length of 0.5 mm is used, however higher resolutions are possible [4]. Calculation of the strain field is described in [4], where the theoretical background can be found in textbooks such as [5]. Hencky’s logarithmic strain tensor is used. With this method the evolving strain field during a tensile experiment can be measured over time, and the strain levels within the localized band surface up to rupture can be determined. Digital image correlation is described in detail in for instance [6]. Finite element analyses using various yield criteria and constitutive relations have been widely applied to analyse plastic deformation of ductile materials and predict strain localization behaviour. Strategies to alleviate mesh-dependence includes rate dependant material description [7], and non-local (gradient) constitutive relations [8] in which a material characteristic length scale is introduced. The aim of the current work is to be able to predict the mechanical response of the three investigated material qualities under large-scale component simulation circumstances cost-effectively, using the finite element method. In general this precludes an element size small enough to spatially resolve localized necking, since the size of the localized zone is comparable to sheet thickness. Approaches concerning the modelling of post localization material behaviour under these circumstances, with spatial discretization large compared to the studied phenomena, has been presented in literature. See e.g. [9] where a post instability strain is calculated based on element dimensions, to minimize mesh dependency on failure prediction. The objective of this work is similar in the sense that the intent is not to model the phenomenon of localized necking, but rather its effects on load response and subsequent rupture. This leads to a methodology adopted here which suggests that constitutive and failure parameters should be adapted to element size, by introducing an ’analysis’ length

28

Paper A

scale into the yield equation. Analysis length is not viewed as a material parameter, but rather as a coupling between local plastic strain and the plastic strain calculated with at certain mesh size. From the experimental DSP evaluation at a small analysis length, parameters for a model which minimizes mesh dependency on both post localization response and failure prediction can be determined.

A.2

Material

The base steel material investigated is USIBOR 1500 P produced by ArcelorMittal. Prior to heat treatment, USIBOR 1500 P exhibit homogeneous distribution of pearlite and an equiaxed grain ferritic matrix. By a controlled tool temperature process in a plane hardening tool, the hot blanks are subjected to different thermal cycles. The sheet is first heated to austenitization temperature and then cooled at a controlled rate and finally post-cooled, see Figure A.1. Three different multi-phase microstructures were developed and characterized by a combination of light-optical metallography(LOM) and electron backscattered diffraction (EBSD). The steel grades are labelled HT400, HT550 and HT800 according to their yield strength. The microstructural phase fraction content of the materials is shown in Table A.1. The sheets measure 150 x 270 Ta

Temperature

Decreased yield strength

Tr Prehandling

Tool cooling

tph

ttc

Postcooling

tpc = ∞ Time

Figure A.1: Schematic illustration of temperature history with the definitions of process sequences.

29

A.2 Material

x 1.2 mm but only a 60 mm wide band is heat treated according to the method described, and the residual area of the sheets is quenched between cold tools. Vickers indention tests are performed within the heat treated area to estimate the material homogeneity. The largest variation found is for HT 800, shown in Figure A where the heat-treated region is outlined with dashed lines and the position where the specimen is cut with thick lines. Variations within the body of the test specimen is ∼ 10%. Three different plasticity and fracture test specimen geometries are cut from the sheets, one with notch radius 30 mm, second with notch radius 10 mm and a third straight sample. Table A.1: Phase fraction content. Grade

Phase content (%) Austenite

HT800 HT550 HT400

0.75 ± 0.3 1.50 ± 0.5 1.75 ± 0.5

Ferrite Polygonal

Irregular

Bainite Upper

Lower

Martensite

− − ≤ 0.50

− 25.0 ± 15.0 95.0 ± 5.0

− 75.0 ± 25 ≤ 3.0

97.0 ± 2.0 ≤ 5.0 −

1.00 ± 0.5 2.25 ± 0.5 3.50 ± 1.0

Hardness Vickers (HV)

600 550 500 450 400 350 300 250 50 0 -50 y-coordinate (mm)

-100

-50

0

50

100

x-coordinate (mm)

Figure A.2: Hardness Vickers results for one HT800 measurement and the outline of the test specimen (thick line). The heat treated area is denoted by dashed lines.

30

A.3

Paper A

Modelling

Mesh dependency issues in conventionally conducted finite element analysis of localization phenomena incorporates several aspects. 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 [7]. The ability of the finite element mesh to resolve localized necking clearly influences numerical results. Both of these effects have a significant impact on computed stiffness and deformation characteristics. As earlier mentioned the aim of the current work is to include predictive capability of post-localization load response using elements larger then the localized zone. A modification of von Mises yield equation is suggested,  σ = 3J2 , (A.1) f = σ − σy (1 − L), where L is termed localization function introduced to reduce load bearing capability of the material. L is dependent on analysis length, denoted l, and evolutes with equivalent plastic strain εp , L = A(exp[B(εp − ε0 )] − 1),

εp ≥ ε0 .

(A.2)

A and B are functions of l, A=

A0 , l

B = B0 l,

(A.3)

and ε0 is equivalent localization threshold strain. The localization function approach is adopted from [10], but with the modification that the L function causes weakening, by reduction of the current yield strength, instead of reduction of Youngs modulus. L can be viewed as to be composed of both geometric effects unresolvable by the mesh, and microvoids induced by plastic damage, although to a lesser extent. A strain based failure criterion from [10] is used in which failure strain, εf , is also dependant on analysis length and the state of stress, εf = (εf0 − ε0 )e−Cl + ε0 .

(A.4)

The analysis length dependence is introduced to comply equivalent strain to failure with element size. The variable εf0 of Equation (A.4) is failure

31

A.4 Experimental procedure

strain at zero l, i.e. εf = εf0 for l → 0. Failure is postulated to occur when the localization function reaches its critical value Lf ,  tf ˙ L dt = 1, (A.5) f L 0 where the rate form of L yields L˙ = B(L + A)ε˙ p ,

Lf = A(exp[B(εf − ε0 )] − 1).

(A.6)

To account for the state of stress on failure any suitable equation where equivalent failure strain is a function of the stress invariants can be used within this framework. The ductile failure function of [11] is adopted here and inserted in Equation (A.4) εf0 = d0 e−dη + d1 edη ,

η=

σm . σ

(A.7)

where η is the stress triaxiality parameter, the ratio of mean hydrostatic stress to equivalent stress. Analysis length, l in Equations A.2 to A.7, is taken to be a dimensionless characteristic element size, e.g. square root of shell element area, Ainit , divided by thickness, tinit . Observe that it is the initial area and thickness that is used i.e. l describes the level of spatial discretization, not a measure of deformation. √ Ainit (A.8) l= tinit The introduction of l into the constitutive equations allows for an effective treatment of mesh sensitivity in terms of load response and failure prediction.

A.4

Experimental procedure

The full-field measurements provides direct information about the local planar deformation field at the region of interest for a number of time instants during deformation. If the specimen surface exhibit a random 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 is performed stepwise using the previous image as reference state. The inplane strain and shear components are calculated from the displacement field. A detailed description of the digital image correlation procedure and the method of determining the strain field can be found in [6].

32

Paper A

A servo-hydraulic testing machine and a charge-coupled device (CCD) camera is used to perform tensile plasticity and failure experiments. The speckle pattern is applied with black and white spray paint. Approximately 40 images are taken from initial unloaded specimen up to final fracture. All tests are run under quasi-static displacement control at room temperature. Three repetitions of each test are performed. Three material qualities and two specimen geometries render in an experimental program of 18 tests. The experimentally determined strain field together with the specimen force recordings constitute the basis for evaluation of parameters for the material model. To determine the flow curve and experimental values of the localization function, denoted Lexp , a strategy that closely follows [4] is adopted. With an assumption of plane state of stress and the above discussed material relationship, associative J2 plasticity and isotropic piecewise linear hardening, the stress acting within a crosssection of the tensile specimen is calculated using a hypo-elastoplastic rate formulation: ep ε˙kl σ˙ij = Dijkl

(A.9)

ep is the elasto-plastic tangent stiffness. If the yield stress where Dijkl (L)

term of Equation (A.1) is rewritten as σy (1 − L) = σy , the derivation ep follows standard procedure for J2 plasticity and can be found of Dijkl in textbooks such as [12]. Equation (A.9) is integrated using the radial return algorithm [13] and a iterative Newton method to find the through-thickness strain increment which satisfies the plane stress constraint. When the stress is known, the specimen force can be evaluated by integration over the current cross-sectional area. While the strain field is homogeneous i.e. the Consid´ere stability criterion [14], H − σ ≤ 0,

(A.10)

is met the plastic modulus H is determined in each step by minimization of the difference between recorded and calculated force with respect H and Lexp = 0. If Equation (A.10) is not met, H is kept constant at H = σ c , the effective stress value at the point of instability, and Lexp is the function value determined. The resulting flow and localization curves are piecewise linear with a number of points equal to the number of experimentally determined strain fields. The purpose of the introduction of Lexp is to eliminate eventual softening in the flow-curve which is

A.5 Calibration

33

instead represented by the localization curve. An advantage is that depending on the analysis length at which the strain field is observed, a corresponding Lexp curve can be determined whilst the flow curve remains the same. This is taken advantage of by averaging the strain field at a number of different grid sizes and repeating the described procedure to produce several experimental localization curves each corresponding to a certain analysis length. A schematic evaluation at 5 mm analysis length is shown in Figure A.3a with an outline of the specimen in thick line and crosses illustrating the points where the strain field is determined. The strain field is averaged within each grid square and the stresses are calculated at mid-point designated by circles. Figure A.3b shows the resulting normalized major strain produced by the averaging procedure for three different analysis lengths.

A.5

Calibration

The choice of failure loci, Equation (A.7), dictates the number of test geometries required for calibration. From an extensive experimental program performed in [4] and [10], it was observed that the results from different specimen thicknesses are consistent if normalized by thickness. Since the equation for analysis length, Equation (A.8), contains a normalizing thickness parameter, only one specimen thickness is used for calibration in this work. The flow curve is constructed in a piecewise linear manner as described, using one of the specimens preferably with the highest failure strain to define a flow curve up to the highest possible equivalent plastic strain. Failure is assessed as the image directly preceding a to the eye visible crack opening. When a flow curve is obtained, experimental localization curves are computed for all test geometries at a number of different experimental measuring distances. Once experimental localization curves are obtained, calibration of the localization and failure model is performed in a stepwise procedure. In total the model contains 7 parameters, A, B, C, ε0 , d, d0 and d1 . 1. Localization threshold strain, ε0 , is observed as the value of equivalent strain when H − σ = 0, i.e at point of tensile instability. 2. Using Equation (A.4) together with experimentally obtained failure strains for a number of different analysis lengths and triaxiality values, εf0 and C can be obtained by best fit.

34

Paper A

Normalized major strain (-)

y-coordinate (mm)

10 5 0 -5 -10 -15

0 -5 5 x-coordinate (mm)

-10

10

1 0.8 0.6 0.4 0.2 0 20

l=0.38 mm l=0.78 mm l=4.1 mm

0

15 y-coordinate (mm)

(a)

-20 -40 -20

0

20

40

x-coordinate (mm)

(b)

Figure A.3: Schematic illustration of the strain field averaging procedure at 5 mm analysis length. a: Strains within each of the grid squares are averaged to the center, where the stresses are calculated. b: Normalized major strain averaged at 0.5,1 and 5 mm analysis length. 3. When εf0 is known, which is failure strain extrapolated to zero l, for at least 3 triaxiality values, d, d0 and d1 in Equation (A.7) can be determined. 4. B is found from the normalized localization function LLf , in which B is the only unknown. A best fit by least squares to the normalized experimental localization curves is performed. 5. Finally the function A/l is the only remaining unknown and is determined with Equation (A.2) together with experimental localization curves at a number of different analysis lengths. Since the triaxiality parameter may vary during the loading process, additional assumptions are needed to calibrate the failure model. To overcome this issue, the failure loci is calibrated to a weighted average of the experimentally determined stress triaxiality at the location of subsequent crack initiation,  η =

0

εf

ΔLexp η dεp . max(Lexp )

(A.11)

The weight used is the normalized increment of the experimental localization value, introduced to give a larger significance of η at the end of the loading process closer to fracture. The hypothesis is that the

35

A.6 Results

stress state during localized necking and ductile fracture process has a larger influence on the ductility limit than the stress state during initial plastic loading. To summarize the calibration procedure it is clear that the experimental localization curves, determined from the full-field measurements, are the key quantities. The calibration result is shown in table A.2. Since only two specimen geometries where used for model calibration the second term of Equation (A.7) is unused. Table A.2: Model parameters. Grade

A0

B0

C

ε0

d

d0

d1

HT800 HT550 HT400

2.2e-01 1.1e-01 1.4e-01

3.3e+00 3.1e+00 2.7e+00

2.7e-01 2.5e-01 2.5e-01

5.0e-02 8.0e-02 1.0e-01

1.1e+01 3.0e+00 2.9e+00

1.8e+02 2.3e+00 2.3e+00

0.0e+00 0.0e+00 0.0e+00

A.6

Results

Primary experimental results are the time-histories of the tensile force and elongation, together with the plane displacement field of the speckled subimages. The presented method produces stress and strain fields at the region of interest at known time instants. The main results of the calibration is the model parameters, shown in Table A.2. The performance of the model is validated through comparison of forcedisplacement response in the following section. It is possible to follow the development of certain quantities at the critical point where a crack eventually initiates. The change in principal strain directions during deformation gives insight to the localization processes of the material grades, shown in Figure A.4. This deviation is synchronous with the development of a preferred shear direction. A higher degree of strain hardening seem to favour localization in the form of a shear band. From the localization threshold value, ε0 in Table A.2, it is observed that a larger strain hardening accelerates the formation of a localized neck. The hardening characteristics play an important role in which deformation mode dissipates the minimum energy. An illustration of the localization and failure model is shown in Figure A.5. The thick solid line is the flow curve. Solid, dashed and dashdot curves are the localization curves L for different analysis lengths, and stars, squares, circles etc. are the corresponding experimental lo-

36

Paper A

12 HT400 HT550 HT800

Angle (Degrees)

10 8 6 4 2 0 0

0.1

0.2 0.3 0.4 0.5 Major principal strain (-)

0.6

0.7

Figure A.4: Principal angle versus major strain. calization values. The failure locus, Equation (A.7), is shown in Figure A.6. The dots represents the calibration points, which are the weighted average of the stress triaxiality, Equation (A.11), together with equivalent strain to failure. The thick line is the failure function, and the dashed and dash-dotted lines are experimental results without averaging. It is observed especially for the two softer material grades that the local stress triaxiality approaches that of transverse plane strain tension irrespective of notch radius.

A.7

Error assessment and model validity

Apart from the displacement field calculation by digital image correlation, a major experimental uncertainty is the equivalent failure strain assessment. As mentioned the strain field is evaluated incrementally through the set of digital images from the first one to the one directly preceding a visible crack opening. Increments of equivalent plastic strain at the end of the tensile test are quite large due to severe strain localization, which results in an underestimate of the failure strain. The presented failure loci can therefore be viewed as a lower bound. Accuracy of the DSP method is discussed in [15], [16], and determination of failure locus using digital image correlation is further commented in [17]. The quality of the experimentally determined stress state relies on the validity of the material model used (as described in section A) and the

37

A.7 Error assessment and model validity

l l l l l

400 300

= 0.38 = 0.78 = 2.5 = 4.1 = 6.2

Localization value (-)

True stress (MPa)

500

0.30 0.25 0.20

200

0.15 0.10

100

0.05 0

0

0.1

0.2 0.3 0.4 0.5 True equivalent plastic strain (-)

0.6

Figure A.5: Flow curve (thick line) and localization curves for a number of analysis lengths, HT400. Stars, squares circles etc. are experimental localization curves which the localization model (Equation (A.2)) is fitted to.

Equivalent plastic strain (-)

0.6 0.5 0.4 0.3 0.2 30 mm radius

10 mm radius

0.1 0 0.4

0.5 Stress triaxiality (-)

0.6

Figure A.6: Failure envelope for HT400 (thick line), Dots indicate calibration points. The dashed and dash-dot lines are the time-histories of stress triaxiality and equivalent plastic strain obtained from experiments of the two notched samples.

38

Paper A

plane stress assumption. Associative Von Mises plasticity is well renown in metal plasticity and used here for its simplicity however a more advanced theory, e.g. three invariant plasticity model [18], is feasible within the suggested experimental procedure. Once a localized neck has formed the state of stress is no longer planar. However as the model is intended for plane stress shell elements, the experimentally determined and with FEM calculated stress triaxiality can be viewed as the same quantity, but not necessarily the true stress triaxiality. Post localization load response and fracture initiation analysis of shell structures using the finite element method is the intended application of the presented model. Mesh size independent predictions of the aforementioned phenomena is the objective however certain limits are imposed on l. Combining Equations A.2 and A.3 it is observed that lim (A0 /l)(eBl(εp −ε0 ) − 1) = ∞

(A.12)

l→0

which makes no sense. However with the intended use of shell elements which are large compared to the width of a eventual zone of localized plastic strain, Equation (A.2) fits well to experimental values obtained within 0.38 ≤ l ≤ 6.2 mm (Figure A.5). The failure model , Equation (A.7) with the parameters in table A.2 represents a monotonic function of failure strain with respect to the stress triaxiality parameter. In a larger span of loading conditions, experimental evidence has shown that material ductility is not a monotonic function of η which is discussed in [19]. Therefore the calibration points of Figure A.6 should be viewed as the validity limits on stress triaxiality in this work. As a validation example a finite element simulation of a tensile fracture experiment is performed using different mesh sizes. The purpose is to review the predictive capability in terms of overall load response, failure elongation and failure load compared to the corresponding experiment. The conformity of results using different element sizes are investigated. A straight initially uniaxial specimen is chosen, which has a lower stress triaxiality than both of the specimens used for calibration. These specimens are cut perpendicular to the one shown in Figure A , i.e. along the y-direction in the center of the heat treated area. The hardness gradient caused by the heat treatment process is smaller in this direction, providing a specimen that is micro-structurally more homogeneous. The presented localization and failure model together with associative J2 plasticity and piecewise linear isotropic hardening material

39

A.8 Conclusion

routine is implemented into the commercial finite element solver LSDYNA [20] through a user-defined material subroutine. The specimen is discretizised with fully integrated quadrilateral shell elements (Type 16) using three different element sizes, see Figure A.7. Displacement boundary conditions are prescribed to zero at one end of the specimen and linear displacement on the other corresponding to experimental conditions. The resulting force displacement curves are shown in Figure A.8.

2 elements through specimen width

3 elements through specimen width

4 elements through specimen width Figure A.7: Tensile test specimen discretized by 3 different mesh sizes. With of the test specimen body is 12.5 mm.

A.8

Conclusion

Three different multiphase steel grades with significantly different behaviour in terms of mechanical and failure properties are characterized. Experimental observations of post-localization crack initiation phenomena are facilitated by a full field measurement technique. Using these experimental results a phenomenological localization and failure model is parametrized. The predictive capacity in terms of stiffness and crack initiation is not mesh-dependent within a characteristic shell element size of 2.6 to 5.2 mm. The introduction of the localization function

40

Paper A

15

Force (kN)

HT800

HT550

10

HT400 5

0

0

2

4 6 Displacement (mm) 3 elems 4 elems 2 elems

8 exp

Figure A.8: predicted and measured force-displacement response of a uniaxial tensile test specimen, for three material grades. Three mesh sizes are used designated by 4, 3 and 2 elements through the width of the specimen. in the constitutive relations controls the response of the element where strain localizes in a manner consistent with experiments. The model is not applicable for numerical analyses of the localization phenomena itself using a discretization smaller than the width of the localized band. Only ductile failure is considered in this work. It is however possible to use a more comprehensive failure function by expanding the dependence of failure strain on stress state within this strategy. Material parameters are evaluated based on the measured local deformation field, and the predicted force-displacement curves in Figure A.8 are calculated with an assumption of material homogeneity. This could be a reason for the over-prediction of the global failure elongation, caused by the gradient in material properties showed in Figure A. It is also possible that this gradient, with its minimum in the center of the specimen, acts as a trigger for localized deformation. Furthermore, since the material parameters are determined from specimens cut along rolling direction, and the experimental force-displacement curves shown in Figure A.8 originates from specimens cut across, plastic anisotropy could also contribute to the over-prediction in failure elongation. Comparing the force-displacement response obtained with different mesh sizes, the results show that the approach of adapting material pa-

A.8 Conclusion

41

rameters to element size with the presented model is a feasible method for modelling post-localization and failure in e.g. large-scale crashworthiness simulations. The press hardening process offers a technological advantage, where varying material properties within a component are obtainable by differential thermal treatment during forming. To accurately predict product performance a failure model concept based on varying microstructure characteristics is needed. A continued development in failure modelling of heterogeneous material states is intended by the authors.

Acknowledgement The authors wish to acknowledge Swerea Kimab for performing the metallurgical investigation. Furthermore the authors wish to acknowledge the European Union structural funds and and the Centre for High Performance Steel at Lulea˙ University of Technology for providing financial support.

References ˙ [1] Akerstr¨ om P, Wikman B, and Oldenburg M. Material parameter estimation for boron steel from simultaneous cooling and compression experiments. Modelling Simulation Mater. Sci. Eng., 13(8):1291– 1308, 2005. ˙ [2] Akerstr¨ om P and Oldenburg M. Austenite decomposition during press hardening of a boron steel–computer simulation and test. Journal of Materials Processing Technology, 174(1-3):399 – 406, 2006. ˙ [3] Akerstr¨ om P, Bergman G, and Oldenburg M. Numerical implementation of a constitutive model for simulation of hot stamping. Modelling Simulation Mater. Sci. Eng., 15(2):105, 2007. [4] Eman J. Study and characterization of localization and failure behaviour of ultra high strength steel. Licenciate thesis, 2007. Lulea˙ University of Technology. [5] Bonet J and Wood R D. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 2008. [6] Kajberg J and Lindkvist G. Characterization of materials subjected to large strains by inverse modelling based on in-plane displacement fields. Int .J. Solids and structures, 2004. [7] Needleman A. Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng., 67:69– 85, March 1988. [8] De Borst R and M¨ uhlhaus H-B. Gradient-dependent plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering, 35(3):521–539, 1992. 42

References

43

[9] Kessler L, Gese H, Metzmacher G, and Werner H. An approach to model sheet failure after onset of localized necking in industrial high strength steel stamping and crash simulations. SAE Int. J. Cars -Mech. Syst., (1):361–370, April 2009. ˙ Berglund D, Sundin K-G, and Oldenburg M. For[10] H¨ aggblad H-A, mulation of a finite element model for localization and crack initiation in components of ultra high strength steels. In proceedings of Hot sheet metal forming of high performance steel, June 15-17 2009. Lulea, ˙ Sweden. [11] Hooputra H, Gese H, Dell H, and Werner H. A comprehensive failure model for crashworthiness simulation of aluminium extrusions. International Journal of Crashworthiness, 9(5):449–464, 2004. [12] Ottosen N S and Ristinmaa M. The Mechanics of Constitutive Modeling. Elsevier, 2005. [13] Simo J C and Taylor R L. Consistent tangent operators for rateindependent elastoplasticity. Comput. Methods Appl. Mech. Eng., 48(1):101 – 118, 1985. [14] Consid´ere A G. Annales des Ponts et Chausses, 6(9):574, 1885. [15] Sj¨ odahl M and Benckert L R. Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy. Appl. Opt., 32(13):2278–2284, May 1993. [16] Sj¨ odahl M. Electronic speckle photography: increased accuracy by nonintegral pixel shifting. Appl. Opt., 33(28):6667–6673, Oct 1994. [17] Mohr D and Ebnoether F. Plasticity and fracture of martensitic boron steel under plane stress conditions. International Journal of Solids and Structures, 46(20):3535 – 3547, 2009. [18] Bai Y and Wierzbicki T. A new model of metal plasticity and fracture with pressure and lode dependence. International Journal of Plasticity, 24(6):1071 – 1096, 2008. [19] Wierzbicki T, Bao Y, Lee Y-W, and Bai Y. Calibration and evaluation of seven fracture models. International Journal of Mechanical Sciences, 47(4-5):719 – 743, 2005. A Special Issue in Honour of Professor Stephen R. Reid’s 60th Birthday.

44

Paper A

[20] Hallquist J O. LS-DYNA Theory Manual. Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, California 94551, 2006.

Paper B

Paper B

Failure model evaluation for varying microstructure based on material hardness 1 ¨ ˙ H¨ R Ostlund , M Oldenburg1 , H-A aggblad1 and D Berglund2 1 Mechanics

of Solid Materials, Lulea˙ University of Technology, SE 971 87 Lulea, ˙ Sweden 2 Gestamp R&D, SE 971 25 Lulea, ˙ Sweden

Abstract The growing demands to reduce vehicle weight in the automotive industry has led to an increase in the use of press hardened safety structures. Using new thermo-mechanical processes with differential heating and cooling, components with tailored mechanical properties can be produced. Specific zones of a component can be subjected to a slower cooling rate, producing a bainitic-ferritic multiphase microstructure with continuous transition to fully quenched ultra-high strength martensite. This combination of properties within a component gives additional means for designing the structural response in a crash situation. An important prerequisite for the application of new material grades is the characterization of its failure properties. In this work post-necking and rupture of five different grades is studied using full-field measurement techniques. A strategy to account for varying phase composition in failure modelling based on material hardness is evaluated. This is in47

48

Paper B

corporated in the modelling method where mesh size dependency of post-localization and failure prediction is accounted for. Constitutive and failure parameters are adapted to element size by introducing an analysis length scale parameter into the constitutive and failure model equations. Failure strain is taken to be dependant on stress state represented by the stress triaxiality parameter and phase fraction content characterized by material hardness. With the mechanical properties predicted by a thermo-mechanical phase-transformation simulation and the failure mixture law based on material hardness, the aim is to be able to predict the structural response of components with tailored properties.

B.1

Introduction

Simultaneous forming and heat treatment of thin sheet structures offers many interesting technological advantages compared to ordinary cold working. A high strength of the final part together with high level of geometrical complexity makes it suitable for light weight and highperforming car crash protection systems. Due to low ductility of the ultra-high strength martensitic micro-structure further increase in crash performance can be obtained by producing components with tailored properties. Generally two different approaches to realize components with tailored properties within the press hardening technology can be identified. The first one applies tailored welded blanks consisting of heat-treatable and non heat-treatable steel grades, and in the other approach the homogeneous blank is subjected to differential heat treatment. By adjusting local micro-structural phase compositions a wide range of mechanical properties are obtainable within the same component. High energy absorption and high intrusion protection in a crash situation can be combined. Trough recent advances of process modelling within the press hardening technology, predictions of the workpiece final constituent properties are feasible. Micro-structure evolution with its related change in deformation properties is determined by finite element (FE) simulations of the thermo-mechanical problem together with models for phase transformation kinetics. Details of the process modelling strategy can be found in [1], and a study of process parameters to obtain soft zones in a typical press hardened component is presented in [2]. To achieve a crashworthiness modelling concept for tailored components, a failure criterion which accounts for the significantly different ductility of various phase mixtures obtainable in the manufacturing process is

B.1 Introduction

49

needed. The aim of the current work is to evaluate a failure model [3], in context of tailored components with spatially varying micro-structure composition. It is investigated whether local failure strain can be related to material hardness. During initial tensile plastic deformation of a sheet metal, plastic strain is generally distributed uniformly within the volume. If further loaded the plastic strain often localizes, first forming a diffuse neck and finally a localized neck preceding rupture. Once a neck has formed, subsequent straining is confined within the neck. This causes an instability and the ductility limit of the material is rapidly reached. Plastic strain rate increase may also be accelerated due to material deterioration caused by ductile plastic damage. Localized necking is a dominant mechanism leading to fracture of safety components in a crash situation. Instability is also associated with the analysis of localization problems, where under quasi-static loading conditions the equations of incremental equilibrium for rate-independent material descriptions lose ellipticity [4]. As a consequence finite element (FE) numerical solutions of localization problems are mesh dependent [5]. The discretization decides the width of the localized band and therefore affects calculated stiffness and strain level within the localized zone. Another issue is the ability of the mesh to accurately resolve localized necking. These matters must be addressed in order to successfully use a strain based failure criterion for problems where localized necking is a probable mode of deformation. A methodology suggested in [3] is adopted here where constitutive and failure parameters are adapted to element size by introducing an ’analysis’ length scale into these equations. Analysis length is not viewed as a material parameter but rather as a coupling between local plastic strain and the plastic strain calculated with a certain mesh size. Material indentation resistance in relation to mechanical properties have been numerously investigated in literature, see e.g. [6], [7]. In the present work the correlation between Vickers hardness HV and equivalent plastic failure strain is phenomenologically investigated. Tensile failure experiments of five material grades subjected to different thermal histories are performed. A full-field measurement technique using Digital Speckle Photography (DSP) is used to record the displacement field at a small length scale compared to conventional tensile testing with extensiometer.

50

B.2

Paper B

Failure model

The failure criterion suggested in [3] is chosen for the present evaluation. Failure strain, εf is formulated as a monotonically decreasing function of analysis length, l εf = (εf0 − ε0 )e−Cl + ε0

(B.1)

and ε0 represents localization threshold strain. The quantity εf0 is failure strain at zero l, i.e. εf = εf0 for l → 0. C is a material dependent parameter that describes the exponential decay in failure strain with respect to analysis length. Analysis length is defined as the square root of element area, Ainit , divided by sheet thickness, tinit , √ Ainit . (B.2) l= tinit Observe that it is the initial area and thickness that is used i.e. l describes the level of spatial discretization, not a measure of deformation. It is assumed that shell element representation of the workpiece is used. The localization threshold strain is here identified as the equivalent plastic strain value at instability in uni-axial tension according to the Consid´ere criterion [8] h(ε0 ) − σ = 0,

h=

∂σ . ∂εp

(B.3)

Where the von Mises equivalent stress, σ at instability in Equation (B.3) can be approximated as the ultimate tensile strength σ t of the material. A linear relationship between hardness and strength is normally assumed as a rule of thumb in estimating the strength of steels from hardness measurements. With this assumption threshold strain can be solved from the relation σ t = kHV ,

h(ε0 ) − kHV = 0

(B.4)

where h is the flow curve tangent of the composite phase-mixture. The quantity εf0 in Equation (B.1) is taken to be a function of the stress triaxiality parameter, η, and the local microstructure of the material point characterized by hardness Vickers, HV .

51

B.2 Failure model

Among assumptions in literature, [9] suggested that local equivalent failure strain is inversely proportional to the hardness of the matrix phase of high-speed steels. [10] investigated the relation between critical damage value and hardness and concluded that critical damage decreases linearly with hardness for AISI 52100 bearing steel. Based on experimental observations for the boron alloyed steel it is proposed that local failure strain is exponentially decreasing with hardness 2

εf0 = (aH V + b)e−(cH V +d)η ,

HV =

HV , HV∗

η=

σm , σ

(B.5)

where a, b, c, and d are model parameters, σm is mean stress and σ is equivalent stress. Equation (B.5) states that failure strain decreases monotonically with η which is assumed to be sufficiently accurate within 0.4 ≤ η ≤ 0.58, the validity limits on stress triaxiality in this work. In a larger span of loading conditions, experimental evidence has shown that material ductility is not a monotonic function of η which is discussed in [11]. Hardness is normalized with the maximum measured hardness, HV∗ , of the fully quenched martensitic microstructure.

B.2.1

Hardness estimation model

The equations for hardness calculation presented in [1] in context of press hardening of boron alloyed steels are recapitulated here for conceptual completeness: HVf p = Kf p1 + Kf p2 log10 V HVb HVm

= 259.4 − 254.7C + 4834.1C

(B.6) 2

= 181.1 + 2031.9C − 1940.1C

(B.7) 2

(B.8)

where the superscripts fp, b and m stands for ferrite/pearlite, bainite and martensite. The different K for the ferrite/pearlite products depends on the alloying elements and can be found in [12]. V is the average cooling rate between 800 and 5000 C and C is the carbon content. To determine the the hardness of the final microstructure a linear mixture law is used HV =

4 

HVi xi ,

0≤x≤1

(B.9)

i=1

where xi is the i’th phase (or phase mixture) relative mass content of the steel grade.

52

B.3

Paper B

Experimental methods

The full-field measurements provides direct information about the local planar deformation field at the region of interest for a number of time instants during deformation. If the specimen surface exhibit a random 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 is performed stepwise using the previous image as reference state. With an incompressibility condition the strain field is calculated. A detailed description of the digital image correlation procedure and the method of determining the strain field can be found in [13]. Failure strain is assessed as the highest equivalent strain of the image directly preceding a to the eye visible crack opening. Due to highly localized deformation directly preceding fracture, failure strain is dependent on which measuring length the strain field is observed. The strain field is averaged to several different measuring lengths and corresponding failure strain values are extracted, to find the dependence of failure strain to measuring length.

B.3.1

Material

The base steel material investigated is USIBOR 1500 P produced by ArcelorMittal. Prior to heat treatment, USIBOR 1500 P exhibit homogeneous distribution of pearlite and an equiaxed grain ferritic matrix. By a controlled tool temperature process in a plane hardening tool, the hot blanks are subjected to different thermal cycles. The sheet is first heated to austenitization temperature and then cooled at a controlled rate and finally post-cooled in air, see Figure B.1. Four different multi-phase microstructures were developed and labelled HT400, HT550, HT700 and HT800 according to their yield strength. The three qualities HT400, HT550 and HT800 together with a fourth fully martensitic material are used to calibrate the presented failure function, Equations B.1 to B.5. The predictive capability is then reviewed by comparison to the intermediate quality HT700. Two different fracture test specimen geometries are cut, one with a notch radius of 30 mm and the second is a plane strain specimen with notch radius 10 mm.

B.3.2

Calibration

The DSP full field measurements constitutes the basis for model calibration, where the equivalent strain within the localized neck from

53

B.4 Results

Ta

Temperature

Decreased yield strength

Tr Prehandling

Tool cooling

tph

ttc

Postcooling

tpc = ∞ Time

Figure B.1: Schematic illustration of temperature history with the definitions of process sequences. unloaded state to macro crack initiation is directly measured. The first step of the calibration procedure is to fit the parameters C and εf0 of Equation (B.1) to experimental values of equivalent failure strain at a number of different l , experimental measuring length. The variable l is taken as a experimental characteristic length, and corresponds to element size when the calibrated model is then used in a finite element simulation. When εf0 and the hardness values are known for all investigated material qualities and specimen loading conditions, the failure surface Equation (B.5) can be fitted by least squares. Concerning specimen loading conditions the initial triaxiality values are used, that is η = 0.4 for the 30 mm notch radius specimen and η = 0.58 for the 10 mm notch radius specimen. Average values of three repetitions of each experiment are used for final calibration.

B.4

Results

The aim for the current evaluation is to investigate a possible correlation between local failure strain and material hardness for steel grades of the same chemical composition but subjected to different thermal treatment. The primary result is the calibrated failure surface , Equation (B.5), shown in Figure B.2 with parameters in table B.1. The concern is now whether a failure curve can be determined from a

54

Paper B

thermo-mechanical phase transformation simulation of a heat treatment process. The accuracy of the hardness calculation of the final microstructure is reported in [1] and is not investigated further in this work. Instead the measured hardness of the intermediate quality HT700 is used to compare measured failure strain to computed failure strain versus stress triaxiality using Equation (B.5). The resulting failure curve and measured equivalent failure strain is shown in Figure B.3. The agreement is acceptable in the low triaxiality range however as the triaxiality increases towards plastic plain strain the failure function overestimates the ductility of HT700. An interesting phenomenon is that according to these measurements the local equivalent failure strain of the HT700 grade is almost identical for η = 0.4 compared to the softest grade, HT400. This fact is however difficult to establish due to the large standard deviation of the experimental results. Table B.1: Failure model parameters for zero analysis length, l = 0, Equation (B.5). Parameter a b c d Value 3.7 0.52 6.8 −1.2

B.5

Conclusion

Local fracture ductility in relation to material indentation resistance is phenomenologically investigated, facilitated by full-field measurements performed on fracture specimens of five different material grades. A failure function is defined in space of analysis length, stress triaxiality and material hardness. From the calibrated surface, a failure curve of equivalent failure strain and stress triaxiality for zero analysis length, l = 0, and HV = 270 is compared to corresponding experimental results for HT700. The suggested strategy is considered as a first approach towards a failure model concept for steels with varying microstructure. A more thorough study of the different microstructure morphologies failure properties, subjected to a larger span of different loading conditions could be fruitful in the work towards establishing a failure criterion for tailored safety components.

55

Equivalent failure strain (-)

B.5 Conclusion

0.6 0.4 0.2 0 184 276 0.55

368

0.5 460

Hardness Vickers (HV)

0.45 0.4

Stress triaxiality (-)

Figure B.2: Calibrated failure surface, Equation (B.5), in space of stress triaxiality and Hardness Vickers. Black dots are calibration points.

Pred. failure curve Experiment

Failure strain (-)

0.6 0.5 0.4 0.3 0.2 0.1 0 0.35

0.4

0.45 0.5 Stress triaxiality (-)

0.55

Figure B.3: Predicted failure curve and experimental failure strain results of HT700, HV = 270 HV. Error bars are one standard deviation symmetrically around the mean-value for three repetitions of the same experiment.

References ˙ [1] Akerstr¨ om P. Modelling and simulation of Hot Stamping. PhD thesis, Division of Solid Mechanics, Department of Applied Physics and Mechanical Engineering, Lulea˙ University of Technology, 2006. [2] Oldenburg M. and Lindkvist G. Study of micro-structure evolution in the press hardening process with respect to tool and contact thermal properties. In Metal Forming, 2010. ˙ Berglund D, Sundin K-G, and Oldenburg M. For[3] H¨ aggblad H-A, mulation of a finite element model for localization and crack initiation in components of ultra high strength steels. In proceedings of Hot sheet metal forming of high performance steel, June 15-17 2009. Lulea, ˙ Sweden. [4] Needleman A. Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng., 67:69– 85, March 1988. [5] Tvergaard V, Needleman A, and Lo K.K. Flow localization in the plane strain tensile test. Journal of the Mechanics and Physics of Solids, 29(2):115 – 142, 1981. ´ A. V. Rusakov, and A. I. El ´ ` kin. [6] D. M. Belenkii, Relationship between hardness and mechanical properties. Strength of Materials, 8:1177–1181, 1976. 10.1007/BF01533564. [7] M. Umemoto et al. Relationship between hardness and tensile properties in various single structured steels. Materials science and technology, 17:505–511, 2001. [8] Consid´ere A G. Annales des Ponts et Chausses, 6(9):574, 1885.

56

References

57

[9] Leskovˇsek V., Ule B., and Liˇsˇci´c B. Relations between fracture toughness, hardness and microstructure of vacuum heat-treated high speed steel. Journal of materials processing technology, 127(3):298–308, 2002. [10] Umbrello D., Hua J., and Shivpuri R. Hardness-based flow stress and fracture models for numerical simulation of hard machining aisi 52100 bearing steel. Materials science and engineering A, 374(12):90–100, 2004. [11] Wierzbicki T, Bao Y, Lee Y-W, and Bai Y. Calibration and evaluation of seven fracture models. International Journal of Mechanical Sciences, 47(4-5):719 – 743, 2005. A Special Issue in Honour of Professor Stephen R. Reid’s 60th Birthday. [12] Maynier Ph., Jungman B., and Dollet J. Creusot-loire system for the prediction of the mechanical properties of low alloy steel products. In Hardenability Concepts with Application to Steel, pages 545–578, 1978. [13] Kajberg J and Lindkvist G. Characterization of materials subjected to large strains by inverse modelling based on in-plane displacement fields. Int .J. Solids and structures, 2004.

Paper C

Paper C Analysis of deformation, localization and fracture properties of a dual phase steel ¨ ˙ H¨ R Ostlund, M Oldenburg and H-A aggblad Mechanics of Solid Materials, Lulea˙ University of Technology, SE 971 87 Lulea, ˙ Sweden

abstract The use of advanced high-strength steels are continuously increasing in the automotive industry, as are the demands for efficient numerical tools in component design. Post-localization deformation and fracture predictive technologies greatly facilitate the design of components which make optimal use of these steel grades. In this paper, a dual phase sheet metal is chosen for the study and characterization of deformation and fracture. A strategy for modelling post-localization response and predicting ductility limit using shell elements larger then the typical width of the localized neck is used. Mesh dependency is alleviated by the introduction of a element size dependent parameter in the constitutive description. Model calibration relies on full field measurements of the plane deformation field of four different test geometries. It was found that the Tresca yield surface more accurately describes the yielding point compared to the von Mises plane stress ellipse. Furthermore, the maximum shear stress criterion in conjunction with an exponential ductility function dependent on the stress triaxiality parameter agrees well with experimental frac61

62

Paper C

ture data. Mesh sensitivity in terms of post localization load response and fracture elongation predictions is reduced significantly compared to results obtained without the element size dependent parameter.

C.1

Introduction

The increasing demands for higher passenger safety and reduction of vehicle weight in the automotive industry has led to an increase in use of Advanced High strength steels (AHSS) [1]. One steel type which falls in to that category is Dual Phase (DP) steel. The microstructure of dual phase steels is composed of soft ferrite matrix and 10-40 % of hard martensite or martensite-austenite (M-A) particles. Due to strong distinctions in mechanical properties between these constituents, superior mechanical properties compared to conventional steels are obtained. The aim of the current work is to characterize the mechanical response and fracture properties of DP 600 subjected to plane stress loading, ranging from pure shear to plane strain tension. In the ductile loading regime, despite high strain hardening, localized deformation precedes final fracture. Localization is initiated when the increase in stress due to decrease in load-carrying area is greater then the increase in loadcarrying ability of the metal due to strain hardening. This reduction of load-carrying area might also be caused by material deterioration, i.e. initiation of cracks caused by ductile plastic damage. The localization process is observed experimentally using full field Digital Speckle Photography (DSP) [2,3], in accordance with the method used in [4]. From the DSP measurements, a two-dimensional displacement field is determined at a very small measuring length compared to extensiometer measurements. The strain levels within the localized neck directly preceding rupture can me measured. The difficulties associated with analysis of localization problems using standard constitutive equations, such as loss of ellipticity in quasistatic conditions [5] causes mesh dependant finite element numerical solutions [6]. This is often addressed with non-local type constitutive relations [7]. The objective of this work is not to model the phenomenon of localized necking as such, but rather its effects on load response and subsequent rupture using element sizes larger then the typical size of the localized zone. This leads to a methodology adopted here which suggests that constitutive and failure parameters should be adapted to element size [8], by introducing an ’analysis’ length scale into material

63

C.2 Modelling

descriptions. Analysis length is not viewed as a material parameter, but rather as a coupling between local plastic strain and the plastic strain calculated with at certain mesh size. From the experimental DSP evaluation at a small analysis length, parameters for a model which minimizes mesh dependency on both post localization response and ductile failure prediction can be extracted.

C.2

Modelling

The constitutive relations used, strain localization with associated mesh dependency as well as the used fracture criteria are reviewed in this section

C.2.1

Plasticity

The Tresca elasto-plastic model, featuring a linear elastic law, nonassociative flow rule, isotropic hardening and a second hardening parameter controlling post-localization response is used here. The yield surface is written  √  √  1 σy 3 3  J3  J2 − √ (1 − L) = 0, θ = arccos (C.1) f= cos θ 3 2  J 3/2  3 2 where J2 , J3 are the second and third deviatoric stress invariants, θ is the lode angle and σy is the uniaxial flow resistance. L is termed localization function and is described in the following section. A non-associative flow rule is chosen, thus avoiding the complexity of determining a unique flow direction in the corner regions, ˙ ij , ε˙pij = λN

Sij Nij = √ . J2

(C.2)

This expression corresponds to the gradient of the von Mises yield potential, where ε˙pij is plastic strain rate, λ˙ the incremental plastic multiplier and Sij the stress deviator. Experimental results favour the Tresca surface compared to von Mises, which will be shown later. This strategy is used as an alternative to associated flow rules with generalized yield surfaces such as in [9], or flow direction constructed by two vectors in the corner regions, see e.g. [10, 11]. Isotropic hardening is introduced by postulating that σy is a function

64

Paper C

of the accumulated equivalent plastic strain, p

σy = σy (ε ),

C.2.2

p

 t

ε = 0

3 p p ε˙ ε˙ 2 ij ij

1/2 dt.

(C.3)

Localization

To alleviate mesh dependency on post-localization response, a second hardening parameter termed localization function, L, is introduced in the yield potential, Equation (C.1). When the loading is in the ductile regime, L compensates the yield strength with respect to element size. L is defined L = A(exp[B(εp − ε0 )] − 1),

εp ≥ ε0 .

(C.4)

A and B are functions of element size, denoted l, A=

A0 , l

B = B0 l,

(C.5)

where localization threshold strain, ε0 , and A0 , B0 are material parameters. The localization function approach is adopted from [8], but with the modification that the L function causes weakening, by reduction of the current yield strength, instead of reduction of Youngs modulus. L can be viewed as to be composed of both geometric effects unresolvable by the mesh, and micro-voids induced by plastic damage, although to a lesser extent. Since localized deformation caused by material instability occurs only in the ductile loading regime, further restrictions on L are needed. This is done by requiring that the rate of L fullfills the following condition L˙ = B(L + A)ε˙p , L˙ = 0,

η > 1/3,

(C.6)

η < 1/3

(C.7)

where η is the stress triaxiality parameter, I1 , η= √ 3 3J2

−2/3  η  2/3 (for plane stress)

(C.8)

and I1 is the first stress invariant. During integration of the constitutive relations, an increment of L is non-zero only if the accumulated plastic strain is larger then ε0 , and the loading is in the ductile regime as indicated by η. The element size l is taken to be a dimensionless

C.2 Modelling

65

characteristic element size, square root of shell element area, Ainit , divided by thickness, tinit . Observe that it is the initial area and thickness that is used i.e. l describes the level of spatial discretization, not a measure of deformation. √ Ainit (C.9) l= tinit The introduction of l into the constitutive equations allows for an effective treatment of mesh sensitivity in post-localization loading.

C.2.3

Fracture

Fracture prediction is a vital tool in the design of lighter and more efficient steel components. Considering ductile fracture due to void growth and coalescence, the CrachFEM criterion [12] is used, ε0f = d0 exp[−dη] + d1 exp[dη],

(C.10)

in conjunction with the method proposed in [8] to normalize equivalent fracture strain with element size εf = (ε0f − ε0 ) exp[−Cd l] + ε0 .

(C.11)

From this equation equivalent fracture strain, εf , is determined for a specific element size, with ε0f calculated from Equation (C.10). ε0f is equivalent fracture strain for l = 0, and d0 , d1 , d and Cd are calibration constants. Ductile failure is postulated to occur when the localization function reaches its critical value Lf ,  tf ˙ L dt = 1, Lf = A(exp[B(εf − ε0 )] − 1). (C.12) f L 0 Following the argument of [12], a distinction between ductile and shear fracture is made. For low triaxialities, the maximum shear (MS) stress criterion is used, i.e. fracture occurs when the maximum shear stress reaches a critical value. This failure criterion among others were successfully evaluated in [13]. Since Tresca flow theory is used, this criterion translates directly to a constant equivalent strain criterion, √ σy (εp ) J2 f = τmax = √  εp = εfshear . (C.13) cos θ 3 As previously discussed, shear fracture is generally not preceded by strain localization caused by material instability. However, the ability

66

Paper C

of the mesh to resolve sharp shear modes will affect the fracture predictive capabilities of a finite element calculation. To be able to use large elements in relation to such deformation modes, the equivalent shear fracture strain is expressed as a function of l εfshear = ε0f,shear exp[−Cs l],

η < 1/3

(C.14)

εfshear = ∞,

η > 1/3

(C.15)

where ε0f,shear is shear fracture strain for l = 0 and Cs a calibration constant. For the general case of non-linear strain path, the shear failure criterion is cast in an integral form  tf ˙ εp dt = 1. (C.16) f 0 εshear Adding the contributions from ductile and shear fracture the final expression render to   tf  ˙ ε˙ p L + f dt = 1 (C.17) Lf εshear 0 where the bracketed terms constitute two failure modes. It is assumed that there is no interaction between fracture mechanisms. The ductile ˙ f , will only have non-zero contributions when the loading is term, L/L in the ductile range and the shear term only when η is less then 1/3. This type of incrementally linear relationship is often termed damage indicator, where a value of unity will determine the fracture limit. In the ductile regime the point of damage initiation coincides with incipient localized necking, whereas in shearing damage growth is initiated with plastic yielding. In total the fracture criterion and localization model involves nine free parameters to be calibrated from experiments.

C.2.4

Algorithmic aspects

The algorithm used for integrating the constitutive relations described in the previous section is the standard elastic predictor – plastic corrector scheme widely employed in computational plasticity, see e.g. [14]. A brief description of the implementation is presented here, where (·)(1) denotes current state, (·)(t) elastic trial state and (·)(2) the updated state consistent with the elasto-plastic rate evolution equations. In this work, the constitutive relations are implemented in two different settings:

67

C.2 Modelling

• Firstly, to calibrate the material parameters using the experimen(2) tally determined deformation fields. Given εij , the major un(2)

(2)

knowns are the stresses σij , the hardening parameters σy , L(2) and internal variable εp(2) . This procedure is described further in section C. (2)

• Secondly, a finite element implementation is performed. Now, σy and L(2) are given functions of equivalent plastic strain and the (2) only unknown are the stresses σij and internal variable εp(2) . Explicit time integration of the equilibrium equations are assumed, hence no elasto-plastic tangent operator is needed. The constitutive model and fracture criterion is implemented into the commercial finite element software LS-DYNA [15] through a user-defined material subroutine. Under the assumption that the total strain rate can be additively decomposed into elastic and plastic parts, the integration scheme is written p(2)

εij

p(1)

= εij

(2)

+ ΔλNij

(2)

(2)

p(2)

σij = Dijkl (εkl − εkl ) (2)

(C.18) (C.19)

f (2) = f (σij , σy(2) , L(2) ) = 0.

(C.20)

L(2) = L(εp(1) + Δεp )

(C.21)

σy(2)

(C.22)

= σy (ε

p(1)

p

+ Δε )

where Dijkl is the elastic stiffness and Nij the plastic flow direction. Enforcing the plastic consistency condition at the end of the timestep , Equation (C.20), gives the following expression  f

(2)

=

(t)

J2 − 2GΔλ cos(θ (t) )

1 − √ σy (εp(1) + Δεp )(1 − L(εp(1) + Δεp )) = 0, 3 (C.23)

from which the incremental plastic multiplier Δλ is calculated. The choice of flow rule gives a purely elastic lode angle response, i.e. θ (2) = θ (t) . Once Δλ is determined from Equation (C.23) using a Newton

68

Paper C

iterative method, the state and internal variables are updated. (t)

σ Δλ (t) = √ (t) (1 − 2G)Sij + kk δij 3 J2 2 εp(2) = εp(1) + √ Δλ. 3

(2) σij

(C.24) (C.25)

The secant method is used for calculating the necessary normal strain increment enforcing the plane stress constraint, see [16]. In the finite element implementation, the rate of localization, Equations (C.6,C.7) are integrated in time as L(2) = L(1) + B(L(1) + A)Δεp ,

η  1/3

(C.26)

and the flow curve is represented by a piecewise linear function. In the calibration procedure, both L and σy are piecewise linear functions.

C.3

Experimental procedure

The full-field measurements provides direct information about the local planar deformation field at the region of interest for a number of time instants during deformation. If the specimen surface exhibit a random 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 is performed stepwise using the previous image as reference state. The inplane strain and shear components are calculated from the displacement field. A detailed description of the digital image correlation procedure and the method of determining the strain field can be found in [17].

C.3.1

Experimental program

A servo-hydraulic testing machine and a charge-coupled device (CCD) camera is used to perform plasticity and failure experiments. The speckle pattern is applied with black and white spray paint. Approximately 40 images are taken from initial unloaded specimen up to final fracture. All tests are run under quasi-static displacement control at room temperature. The recorded data are the digital images, the force, the cross-head displacement and elongation A50 measured with extensiometer. Three repetitions of each test are performed. From an extensive experimental program performed in [4] and [8], it was observed that fracture strain

69

C.3 Experimental procedure

from different sheet thicknesses are consistent if normalized by thickness. Since the equation for element size, Equation C.9, contains a normalizing thickness parameter, only one sheet thickness of 1.5 mm is used for calibration in this work. Table (C.1) summarizes the experimental program, Table C.1: Experimental program summary Test number

Specimen description

ηav

ε0f

1 2 3 4

Flat tensile Notched tensile, r=30 mm Notched tensile, r=10 mm Shear

0.36 0.39 0.51 0.07

0.79 0.58 0.50 0.69

where ηav is the accumulated average stress triaxiality and ε0f is equivalent fracture strain for l = 0. All four specimens are used for calibration.

C.3.2

Parameter evaluation

The experimentally determined strain field together with the specimen force recordings constitute the basis for evaluation of parameters for the material model. To determine the flow curve and experimental values of the localization function, L, a strategy that closely follows [4] is adopted. With the algorithm discussed in Section C, the stress acting within a cross-section of the specimen is calculated. Then the cross-sectional force is evaluated, and the difference between the calculated force and the experimentally recorded force constitutes an object function to be minimized in incremental optimization of σy and L. While the strain field is homogeneous i.e. the Consid´ere stability criterion [18], H − σ ≤ 0,

(C.27)

is met the plastic modulus H is determined in each step by minimization of the difference between recorded and calculated force with respect H and Lexp = 0. If Equation C.27 is not met, H is kept constant at H = σ c , the effective stress value at the point of instability, and Lexp is the function value determined. The resulting flow and localization curves are piecewise linear with a number of points equal to the number of experimentally determined strain fields. For more details, see [4]. The purpose of the introduction of L in the constitutive equations is to eliminate eventual post-localization softening in σy . Instead, the

70

Paper C

softening branch is represented by the element size dependent function L. Since the experimental method is not finite element based, element size, l, defined in Equation (C.9) need some explanation. The deformation field is averaged within a certain area, and that area determines l through Equation (C.9). A schematic evaluation at 5 mm2 is shown in Figure C.1a with an outline of the specimen in thick line and crosses illustrating the points where the strain field is determined. The strain field is averaged within each grid square and the stresses are calculated at midpoint designated by circles. Figure C.1b shows the resulting normalized major strain produced by the averaging procedure for three different l.

Normalized major strain (-)

y-coordinate (mm)

10 5 0 -5 -10 -15

-10

-5 0 5 x-coordinate (mm)

(a)

10

1 0.8 0.6 0.4 0.2 0 20

l=0.38 mm l=0.78 mm l=4.1 mm

0

15 y-coordinate (mm)

-20 -40

-20

0

20

40

x-coordinate (mm)

(b)

Figure C.1: Schematic illustration of the strain field averaging procedure at 5 mm. a: Strains within each of the grid squares are averaged to the center, where the stresses are calculated. b: Normalized major strain averaged at 0.38, 0.78 and 4.1 mm2 . The described procedure is repeated to produce several localization curves each corresponding to a certain element size. The strain localization and its size effect is directly observable from experiments, providing means for calibrating the localization function, Equation (C.4) to replicate experimental observations. The efforts of determining localization functions only applies to tests #1,#2 and #3 which are in the ductile loading range.

C.4 Results

C.3.3

71

Calibration

The flow curve is constructed in a piecewise linear manner as described, using test specimen #1. Failure is assessed as the image directly preceding a to the eye visible crack opening. When a flow curve is obtained, experimental localization curves are computed for test geometries #1,#2 and #3 at a number of different experimental averaging areas. Once experimental localization curves are obtained, calibration of the localization and failure model is performed in a stepwise procedure. In total the model contains nine parameters, A, B, Cd , Cs , ε0 , d, d0 , d1 , and ε0f,shear . 1. Localization threshold strain, ε0 , is observed as the value of equivalent strain when H − σ = 0, i.e at point of tensile instability. 2. Using Equation (C.11) together with experimentally obtained failure strains for a number of different analysis lengths and average triaxiality values, εf0 and C can be obtained by best fit. 3. When εf0 is known, which is failure strain extrapolated to zero l, d, d0 and d1 in Equation (C.10) can be determined. 4. B is found from the normalized localization function LLf , in which B is the only unknown. A best fit by least squares to the normalized experimental localization curves is performed. 5. Finally the function A/l is the only remaining unknown and is determined with Equation (C.4) together with experimental localization curves at a number of different analysis lengths. 6. The parameters ε0f,shear and Cs are obtained from test #4, using measured failure strains for a number of different l, and fitting Equation (C.14) to these values. a procedure similar with item 2 above. To summarize the calibration procedure it is clear that the experimental localization curves, determined from the full-field measurements, are the key quantities.

C.4

Results

Primary results of this work are the calibrated material and failure model, where parameter values are shown in Table C.2. Calibration

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revealed that the exponential factors Cd and Cs of equations (C.11,C.14) can be equated, thereby reducing one parameter. In this section, features of the presented model will be illustrated. Yield stress versus equivalent plastic strain, and the localization curves for a number of different element sizes, determined from test #2, are shown in Figure C.2. The thick solid line is the flow curve. Magenta, blue green and red curves are the localization curves L for different analysis lengths, and stars, dots, circles etc. are the corresponding experimental localization values. This figure implicates how the effective flow strength, the product σy (1 − L) is controlled by the dimensionless element size parameter l. Table C.2: Model parameters. A0 1.4e-02

B0 5.7e+00

Cd , Cs 2.7e-01

ε0 1.2e-01

Parameter: Value:

d 4.8e+00

d0 3.3e+00

d1 1.5e-02

ε0f,shear 6.9e-01

1200

True stress (Mpa)

1000

l l l l

= = = =

1.0345 2.069 3.4483 5.1724

800

Localization value (-)

Parameter: Value:

600

0.15

400

0.10

200

0.05

0

0

0.1

0.2 0.3 0.4 0.5 True equivalent plastic strain (-)

Figure C.2: Flow curve (black line) and localization curves (coloured) for a number of characteristic element sizes (l). Stars, dots circles etc. are experimental localization values determined by the procedure described in section C. Full coloured lines are the calibrated localization function, Equation (A.5) for four different characteristic element sizes. Results shown are obtained from test #2.

73

C.4 Results

The fracture criteria, equations (C.10,C.11,C.13), represented by a fracture locus in the space of stress triaxiality and equivalent plastic strain is shown in Figure C.3. The fracture limit for three different characteristic element sizes are shown. 0.8

equivalent plastic strain

0.7 0.6 0.5 0.4

l = 1.0 l = 2.0

0.3

l = 4.0

0.2 0.1 0

0

0.1

0.2

0.3 0.4 0.5 Stress triaxiality

0.6

Figure C.3: Fracture locus in the space of stress triaxiality and equivalent plastic strain, equations (C.10,C.11,C.14,C.15). Solid lines indicate fracture strain limit for three different element sizes. Dots , circles and triangles are the corresponding experimental values. All points have been used for calibration. Finite element simulations of tests #1 and #4 are performed to illustrate model predictions of post-localization load response and fracture elongation. Test specimen #1 is discretizised with fully integrated quadrilateral shell elements (LS-DYNA type 16) using three different element sizes. For test specimen #4, results using two element sizes are presented. The finite element discretizations are shown in Figure C.4. Displacement boundary conditions are prescribed to zero at one end and linear displacement at the other, corresponding to experimental conditions. The results are presented in terms of force and elongation for test #1, and force and crosshead displacement for test #4. Two cases are demonstrated, with plasticity model and localization and fracture prediction shown in Figure C.6, and with only plasticity accounted for shown in Figure C.5. Experimental values are included for comparison.In the finite element implementation, if a material point reaches

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its ductility limit i.e. Equation C.17 equals unity, the strength is reduced significantly.

(a) 2 elements through(b) 3 elements through(c) 4 elements through width, sim #21 width, sim #31 width, sim #41

(c) 10 elements through width, sim #14 (d) 5 elements through width, sim #54 Figure C.4: Tensile test specimen discretized by 3 different mesh sizes, and shear specimen with 2 different mesh sizes. With of the test specimen body is 12.5 mm, and 5.15 mm for the shear specimen.

75

C.4 Results

14 12

Force (kN)

10 8 6 4

exp test #1 sim #41 sim #31 sim #21 exp test #4 sim #54 sim #14

2 0

2 8 0 4 6 10 Elongation A50 / Crosshead displacement (mm)

Figure C.5: Calculated and measured force-displacement response of tensile test specimen #1 and shear specimen #4. without localization and fracture model. Results are obtained using the different element sizes depicted in figure C.4

14 12

Force (kN)

10 8 6 4

exp test #1 sim #41 sim #31 sim #21 exp test #4 sim #54 sim #14

2 0

2 4 8 10 0 6 Elongation A50 / Crosshead displacement (mm)

Figure C.6: Calculated and measured force-displacement response of tensile test specimen #1 and shear specimen #4. Results are obtained using the different element sizes depicted in figure C.4, with the localization and fracture model proposed.

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C.5

Paper C

Discussion

The advanced high-strength steel type DP600 is analysed and characterized in terms of mechanical properties and ductility limit. Experimental observations of localized deformation and crack initiation are facilitated by a full field measurement technique. A Tresca yield criterion is chosen, due to the lower yield stress and flow resistance observed for the shear test as compared to the uniaxial tensile test. Figure C.6 confirms the validity of this choice by showing good agreement between calculated and measured force response for both tests. Furthermore plastic flow is assumed to be non-associated, obeying a flow rule associated with the von Mises yield potential. Yielding occurs on a plane with the maximum shear stress but the plastic flow direction vector is normal to the von Mises cylinder, radial in the deviatoric plane. This simplifies construction of the algorithm, and experimentally observed phenomena are captured. A element size dependent localization function denoted L is introduced into the yield equation, alleviating post localization mesh dependency in the ductile loading regime. A prerequisite of the presented localization function approach is that the element size is larger then the physical size of the band of localized deformation, typically comparable to the sheet thickness. Within the element size range 3.2  l ∗ tinit  6.5 mm, fracture ductility predictions vary by less then 5 %, see figure C.6. This is appealing for large scale crashworthiness analysis where typical element sizes used often falls within that range. For the shear fracture finite element simulation, smaller elements were necessary to represent the geometry of the shear specimen. Concerning fracture, the material is studied and characterized in plane stress loading, ranging from shearing to transverse plastic plane strain. A combination of constant equivalent plastic strain based on the hypothesis of maximum shear stress and a exponential ductility function is used to construct the failure locus. Mesh dependency of the damage indicator D is alleviated by the introduction of l in failure equations. The method of separation of ductile and shear fracture used in this work gives a ductility discontinuity around the chosen value, η = 1/3, as seen in Figure 3. No experimental data were collected in the combined shear-tensile regime.

C.5 Discussion

77

Acknowledgement The authors wish to acknowledge the European Union Structural Funds and the Centre for High Performance Steel at Lulea˙ University of Technology for providing financial support.

References [1] Kuziak R., Kawalla R., and Waengler S. Advanced high strength steels for automotive industry. Archives of civil and mechanical engineering, 8(2):103–117, 2008. [2] Sj¨ odahl M and Benckert L R. Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy. Appl. Opt., 32(13):2278–2284, May 1993. [3] Sj¨ odahl M. Electronic speckle photography: increased accuracy by nonintegral pixel shifting. Appl. Opt., 33(28):6667–6673, Oct 1994. [4] Eman J. Study and characterization of localization and failure behaviour of ultra high strength steel. Licenciate thesis, 2007. Lulea˙ University of Technology. [5] Needleman A. Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng., 67:69– 85, March 1988. [6] Tvergaard V, Needleman A, and Lo K.K. Flow localization in the plane strain tensile test. Journal of the Mechanics and Physics of Solids, 29(2):115 – 142, 1981. [7] De Borst R and M¨ uhlhaus H-B. Gradient-dependent plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering, 35(3):521–539, 1992. ˙ Berglund D, Sundin K-G, and Oldenburg M. For[8] H¨ aggblad H-A, mulation of a finite element model for localization and crack initiation in components of ultra high strength steels. In proceedings of Hot sheet metal forming of high performance steel, June 15-17 2009. Lulea, ˙ Sweden. 78

References

79

[9] Hosford W. F. A generalized isotropic yield criterion. Journal of Applied Mechanics, 39(2):607–609, 1972. [10] Crisfield M. A. Plasticity computations using the mohr-coulomb yield criterion. Engineering Computations, 4:300–308, 1987. [11] Peric D. and de Souza Neto E. A. A new computational model for tresca plasticity at finite strains with an optimal parametrization in the principal space. Computer Methods in Applied Mechanics and Engineering, 171(3-4):463 – 489, 1999. [12] Hooputra H, Gese H, Dell H, and Werner H. A comprehensive failure model for crashworthiness simulation of aluminium extrusions. International Journal of Crashworthiness, 9(5):449–464, 2004. [13] Wierzbicki T, Bao Y, Lee Y-W, and Bai Y. Calibration and evaluation of seven fracture models. International Journal of Mechanical Sciences, 47(4-5):719 – 743, 2005. A Special Issue in Honour of Professor Stephen R. Reid’s 60th Birthday. [14] Ortiz M and Popov E P. Accuracy and stability of integration algorithms for elastoplastic constitutive relations. International Journal for Numerical Methods in Engineering, 21(9):1561–1576, 1985. [15] Hallquist J O. LS-DYNA Theory Manual. Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, California 94551, 2006. [16] Schoenfeld S. and Benson D. Quickly convergent integration methods for plane stress plasticity. Communications in Numerical Methods in Engineering, 9(4):293–305, 1993. [17] Kajberg J and Lindkvist G. Characterization of materials subjected to large strains by inverse modelling based on in-plane displacement fields. Int .J. Solids and structures, 2004. [18] Consid´ere A G. Annales des Ponts et Chausses, 6(9):574, 1885.