11th Forming Technology Forum Zurich 2018 Experimental … › content › dam › ethz ›...

July 2 & 3, 2018Zurich, Switzerland

swddrgSwiss Group of IDDRG

Experimental and numericalmethods in the FEM based

crack prediction

11th Forming Technology Forum Zurich 2018

Proceedings

EditorProf. Dr. Pavel Hora Institute of Virtual Manufacturing, ETH Zurich, Switzerlandwww.ivp.ethz.ch

© Institute of Virtual Manufacturing | ETH Zurich | 2018 ISBN 978-3-906916-26-2

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IPREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III

CONTENTS

CRITICAL ASPECTS OF THE EXPERIMENTAL AND THEORETICAL CRACK PREDICTION IN SHEET AND BULK METAL FORMING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1P. Hora*, B. Berisha, S. Hirsiger, T. Komischke, R. Schober

MEASUREMENT AND ANALYSIS OF FRACTURE STRESSES AND STRAINS OF SHEET METALS AND TUBES USING A MULTIAXIAL TUBE EXPANSION TESTING METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15T. Kuwabara*

ON THE DEVELOPMENT OF EXPERIMENTAL TECHNIQUES FOR CHARACTERIZING THE EFFECT OF STRESS STATE ON DUCTILE FRACTURE IN SHEET METAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17D. Mohr*, C. Roth

ON THE DEVELOPMENT OF A NEW GENERALIZED ORTHOTROPIC DAMAGE AND FRACTURE MODEL . . . . 19D. Koch*, F. Andrade, P. DuBois, M. Feucht, A. Haufe

PREDICTION OF FAILURE IN SHEET METAL FORMING SIMULATION - AN INTEGRATED APPROACH FOR SHELL AND SOLID DISCRETIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25H. Gese*, H. Dell, M. Reissner, F. Brenner

ANISOTROPIC FRACTURE CRITERION AND ITS CALIBRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27J. Yoon*, S. Zhang, T. Stoughton

A NECKING AND FRACTURE ALGORITHM FOR PREDICTION OF FORMABILITY OF ANISOTROPIC METALS SUBJECTED TO COMPLEX DEFORMATION HISTORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29T. Stoughton*

ON THE MODELING OF MULTI-STAGE SHEET METAL DEFORMATION PROCESSES INCLUDING HEAT TREATMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31F. Neukamm*, M. Feucht, F. Andrade

INVERSE CALIBRATION OF A SHEAR CUTTING SIMULATION AND INVESTIGATION OF VARIOUS SHEAR CUTTING PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35W. Volk, M. Krinninger, J. Stahl*

LIMITATIONS OF FORMING LIMIT DIAGRAMS: CONSIDERATION OF BENDING STRAIN, SURFACE AND EDGE CRACKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37S. Ertürk*, M. Sester, M. Selig

PLASTICITY AND DUCTILE FRACTURE BEHAVIOR OF AN ADVANCED HIGH-STRENGTH STEEL SHEET UNDER QUASI-STATIC AND DYNAMIC STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43W. Liu, J. Lian*, S. Münstermann

ON ANISOTROPY AND NON-PROPORTIONALLOADING IN DAMAGE AND FRACTURE MODELS . . . . . . . . . . 45A.H. van den Boogaard*

SMART STAMPING: IMPROVED QUALITY IN STAMPING BY MODEL DRIVEN CONTROL . . . . . . . . . . . . . . . . . 47M. Sigvant*, J. Pilthammar, S. Tatipala, E. Andreasson

FORMING FRACTURE LIMITS OF AHSS SHEETS AS RELATED TO DIFFERENT CHARACTERIZATION TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51L. Wagner*, E. Berger, P. Larour, H. Pauli

A NEW STRAIN RATE- AND TEMPERATURE-DEPENDENT DUCTILE FRACTURE CRITERION . . . . . . . . . . . . 57Q. Hu, X. Li, M.W. Fu, J. Chen*

TOWARDS A CONSTRUCTION OF GENERAL FRAMEWORK FOR FORMING LIMITS PREDICTION USING BIFURCATION THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59T. Oya*, J. Yanagimoto, K. Ito, G. Uemura, N. Mori

A NEW EXPERIMENTAL METHOD FOR THE EVALUATION OF FRACTURE CRITERIA IN BULK FORMING OPERATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65T. Komischke*, P. Hora, G. Domani

INVESTIGATION OF MICROSTRUCTURAL FEATURES ON DAMAGE ANISOTROPY . . . . . . . . . . . . . . . . . . . . . . 71E.E. Asik*, E.S. Perdahcioglu, A.H. van den Boogaard

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ON THE INFLUENCE OF MICROSTRUCTURE ON FAILURE INITIATION IN THE FRAMEWORK OF CRYSTAL PLASTICITY AND A FFT-BASED SPECTRAL SOLVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73B. Berisha*, S. Hirsiger, P. Hora

NUMERICAL INVESTIGATION ON DAMAGE EVOLUTION AND VOID CLOSURE IN HOT FLAT ROLLING . . . . . 79C. Liebsch*, G. Hirt

FINITE ELEMENT ANALYSIS OF CALIBER ROLLING PROCESSES TO INVESTIGATE POSSIBILITIES TO INFLUENCE THE DAMAGE EVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85S. Wang*, J. Pöplau, M. Grüber, G. Hirt

MODELING CRACK INITIATION IN AL-SI COATING DURING HEATING/QUENCHING PHASE OF HOT STAMPING PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87S.B. Zaman*, J. Hazrati, M.B. de Rooij, D.T.A. Matthews, J. Venema, A.H. van den Boogaard

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PREFACEExperimental and numerical methods in the FEM based crack predictionStrain localization, which precedes fracture, is used as limiting formability factor for many forming processes. In some cases, however, process constraints or material properties may prevent localization and directly deform up to the fracture limits. As far as sheet metal forming is considered such situations are bending cracks, edge cracks and the so-called shear fracture phenomenon. In the current state of the art, a universally accepted fracture criterion or, to make an analogy to the FLC a Crack Limit Curve (CLC) is still lacking. A number of experimental setups have been proposed to measure fracture strains but none of these received the acceptance enjoyed by Nakajima tests. Furthermore the correct treat-ment of the problem in FEM including mesh dependence and crack propagation is also still unsolved.

An even more complex situation is observed in the fracture modeling of bulk metal form-ing processes. The recent literature discusses a large number of coupled and uncoupled damage evolution laws which aim capturing the gradual degradation of the material before fracture. These however mostly stay at a theoretical level, lacking direct measurements and are often characterized inversely to deliver the observed macroscopic behavior. A more widely used approach is the identification fracture strains in function of stress triaxiality as proposed by Johnson and Cook. This in combination to damage evolution approaches is better able of modelling the experimental outcomes. In addition to the stress triaxiality an increasing number of approaches nowadays also include the effect of the Lode parameter. Although the theoretical foundations of this choice are well understood, it is often challeng-ing to create reliable experiments, especially in the low triaxiality regimes.

The goal of the conferences is to discuss the current state of the art in the «crack predic-tion and modeling field» and to explore new ideas for better numerical approaches. An im-portant contribution should be the proposal of better standardized experimental methods which are applicable to industrial circumstances.

Prof. Dr. Pavel Hora | Institute of Virtual Manufacturing, ETH Zurich

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Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland

* Corresponding author: Tannenstrasse 3, +41446327198, [email protected]

CRITICAL ASPECTS OF THE EXPERIMENTAL AND THEORETICAL CRACK PREDICTION IN SHEET AND BULK

METAL FORMING

P. Hora1*, B. Berisha1,2, S. Hirsiger1 T. Komischke1, R. Schober1

1 ETH Zürich, Institute of virtual Manufacturing, 8092 Zurich, Switzerland 2 inspire AG, inspire-ivp, 8092 Zurich, Switzerland

ABSTRACT: In contrast to the necking prediction in sheet metal forming process, which bases on well- defined, standardized experiments and numerical methods, the prediction of cracks and ruptures, is more com-plex and still not generally mastered. The presented contribution discusses the most significant points as (1) types of cracks, (2) validity of experimental tests, (3) impact of loading history on damage accumulation, (4) idealization degree of mathematical failure criteria and last but not least (5) the problems of correct FEM model building. KEYWORDS: Fracture Forming Limit Diagram (F-FLD), Damage accumulation, Finite Element Methods, Regularization

1 INTRODUCTION 1.1 SHEET METAL FORMING LIMITS Forming of sheet metal is in the most cases done un-der plane stress condition with a free surface. In this case the forming limits for ductile materials are given by the initiation of diffuse necking (DN) which changes then to localized necking (LN). In the stress range of 0.0 < 𝜎𝜎2𝜎𝜎1 < 1.0 the limits are mapped by the Forming Limit Curve (FLC) which can be evaluated experimentally with the Nakajima or Marciniak tests. They can be also calculated nu-merically based on different criteria like Marciniak-Kuczynski (MK), Hutchinson-Neal (HN) or the Modified Maximum Force (eMMFC) criterion. The necking limits (Case I) are given by structural instabilities and are in this sense not to be under-stood as material specific forming limits - their de-scription as forming limits may be misconducting! If the structural instability can be suppressed by sur-rounding boundary conditions (BC) the strains reach much higher values. Such deformation cases are given for example for bending (hemming) states and especially in the case of a “single point” incre-mental forming (Case II). The crack initiation and crack propagation is then the effective forming limit of the material. Both limits – the first one of necking (FLC-N) and the later one (FLC-F: Forming Limit Curve - Fracture) are schematically plotted in Fig. 1.

Fig. 1 Necking and fracture limits in a forming

limit diagram

Some examples of such “sheet” fractures are illus-trated in Fig. 2.

a) Bending

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* Corresponding author: postal address, phone, fax, email address

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b) Hole expansion test. Edge cracks

c) Die curvature bend-ing crack

d) Compression-tension shear crack

Fig. 2 Different types of cracks in sheet metal forming applications.

A special, and in sheet metal forming rarely observ-able, forming limit occurs under tension-compres-sion conditions. Due to the compression-tension case the highest shear stress becomes 𝜏𝜏12 which leads to shear fracture lines in the sheet plane, see Fig 2d. In Fig. 1 this fracture case was schematically mapped as Case III.

1.1.1 Ductile fracture types in sheet metal forming

For ductile sheets the final fracture behaviour is an out-of-plane shear state as visible in Fig. 3a and an in-plane shear as visible in Fig. 3 b. These types of fracture significantly differ from the fracture strain fields covered with the specimens proposed by Wierzbicki et al. (2005), s. Fig. 19.

a) Branch I: out of plane shear 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜏𝜏13 = 1/2(𝜎𝜎𝐼𝐼 − 𝜎𝜎𝐼𝐼𝐼𝐼𝐼𝐼)

b) Branch II: in-plane shear 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜏𝜏12 = 1/2(𝜎𝜎𝐼𝐼 − 𝜎𝜎𝐼𝐼𝐼𝐼)

Fig. 3 a) Out of plane shear; b) In-plane shear

cracks developing from the edge boundary under compression-tension conditions (Case III).

Fig. 4 demonstrates that an out-of-plane shear crack occurs in the case of edge cracks as well.

Fig. 4 Development of out-of-plane fractures un-

der hole expansion conditions. Internal IVP report “FUSION”.

A different behaviour can be observed by very soft materials. Tests with materials like AA5005 showed that diffuse thinning will not be stopped by a crack but will rather continue up to a remaining thickness of nearly zero, see Fig. 5 or Cu tensile test in Fig. 7.

Fig. 5 Material AA5005. Extreme diffuse necking

behaviour without an out-of-plane fracture.

1.2 BULK METAL FORMING FRACTURES In bulk metal forming the development of cracks ap-pears to be more complex. The cracks can develop after an initial necking (mostly on the surface) or di-rectly without a recognizable necking inside of the parts. Fig. 6 shows some examples of appearance of such fractures.

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b) Hole expansion test. Edge cracks

c) Die curvature bend-ing crack

d) Compression-tension shear crack

Fig. 2 Different types of cracks in sheet metal forming applications.

A special, and in sheet metal forming rarely observ-able, forming limit occurs under tension-compres-sion conditions. Due to the compression-tension case the highest shear stress becomes 𝜏𝜏12 which leads to shear fracture lines in the sheet plane, see Fig 2d. In Fig. 1 this fracture case was schematically mapped as Case III.

1.1.1 Ductile fracture types in sheet metal forming

For ductile sheets the final fracture behaviour is an out-of-plane shear state as visible in Fig. 3a and an in-plane shear as visible in Fig. 3 b. These types of fracture significantly differ from the fracture strain fields covered with the specimens proposed by Wierzbicki et al. (2005), s. Fig. 19.

a) Branch I: out of plane shear 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜏𝜏13 = 1/2(𝜎𝜎𝐼𝐼 − 𝜎𝜎𝐼𝐼𝐼𝐼𝐼𝐼)

b) Branch II: in-plane shear 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜏𝜏12 = 1/2(𝜎𝜎𝐼𝐼 − 𝜎𝜎𝐼𝐼𝐼𝐼)

Fig. 3 a) Out of plane shear; b) In-plane shear

cracks developing from the edge boundary under compression-tension conditions (Case III).

Fig. 4 demonstrates that an out-of-plane shear crack occurs in the case of edge cracks as well.

Fig. 4 Development of out-of-plane fractures un-

der hole expansion conditions. Internal IVP report “FUSION”.

A different behaviour can be observed by very soft materials. Tests with materials like AA5005 showed that diffuse thinning will not be stopped by a crack but will rather continue up to a remaining thickness of nearly zero, see Fig. 5 or Cu tensile test in Fig. 7.

Fig. 5 Material AA5005. Extreme diffuse necking

behaviour without an out-of-plane fracture.

1.2 BULK METAL FORMING FRACTURES In bulk metal forming the development of cracks ap-pears to be more complex. The cracks can develop after an initial necking (mostly on the surface) or di-rectly without a recognizable necking inside of the parts. Fig. 6 shows some examples of appearance of such fractures.


Fig. 6 Different types of cracks in bulk metal

forming. a) fine blanking; b) rolling c)cut-ting; d) extrusion profiles - thermal induced cracks.

1.2.1 Influence of hydrostatic pressure In bulk metal forming the stress conditions change from a 2D plane stress case to a complex 3D stress state, where the hydrostatic pressure develops to the most significant parameter. In this case it’s common to change the forming limit description space to an equivalent strain 𝜀𝜀𝑓𝑓– normalized hydrostatic pres-sure space 𝜂𝜂 = 𝜎𝜎𝐻𝐻/𝜎𝜎𝑒𝑒𝑒𝑒 . The influence of hydrostatic pressure on the fracture behaviour is illustrated in Fig. 7 for a Cu alloy tested under (a) tensile conditions, (b) torsion and (c) com-pression conditions.

a) Tensile test:

𝜀𝜀𝑓𝑓 ≈ 0.5

b) Torsion test:

𝜀𝜀𝑓𝑓 ≈ 5.0

c) Compression test: 𝜀𝜀𝑓𝑓 𝑛𝑛𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

Fig. 7 Influence of hydrostatic pressure on frac-ture behaviour.

1.2.2 Influence of microstructure Under 3D stress conditions it is assumed that a com-bination of normal and shear stresses - relatively to the weakest material orientation – defines the criti-cal state. In this sense the microstructure may play a dominant role for the fracture development.

Fig. 8 By microstructural properties induced crack

orientation.

The normal stress tN leads to a void nucleation and a void growth. The shear stress tt develops to a shear decohesion failure. Fig. 9 illustrates the fracture sur-faces in the different loading cases, Danas et al. (2012).

tensile mixed shear

Fig. 9 Fracture surface development under differ-ent loading conditions, Danas et al. (2012).

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Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland 2 EXPERIMENTAL METHODS 2.1 EXPERIMENTAL METHODS FOR

FRACTURE DETECTION ON SHEETS As mentioned before the weakness of fracture mod-elling is the lack of standardized experiments and evaluation methodology. In industrial research the “Daimler bending test” (Fig. 10) and different hole expansion tests (Fig. 11) for detecting the edge formability [ISO/DIS 16630] have been established.

Fig. 10 Daimler bending test (internal report)

Fig. 11 Hole expansion test, cone and flat

Those tests are definitely insufficient to cover the whole stress range −1 < 𝛼𝛼 = 𝜎𝜎2𝜎𝜎1 < 1 and to con-struct all three fracture lines as schematically given in Fig.1. Due to this fact other methods have to be mentioned as well. Takuda et al. (2009) proposed an approach for meas-uring fracture strains based on Marciniak-type biax-ial specimens for various strain ratios. In the princi-pal strain space the measured fracture strains showed a linear behaviour. Lee (2005) published in his PhD thesis fracture line measurements based on former investigations of Embury and LeRoy (1977) and LeRoy et al.(1981). Later on, the FFLD was analytically constructed by Atkins (1985, 1996a), Dyrli HK (1999), and Lee and Wierzbicki (2003). The fracture strain data have been evaluated based mostly on fracture thinning values 𝜀𝜀33

𝑓𝑓

Fig. 12 Normalized FFLDs for eleven different ma-

terials in classical punch indentation prob-lems, Lee (2005).

Fig. 13 Normalized fracture loci for five different

materials in upsetting tests, Lee (2005).

Note that all curves are made to pass through the same point (𝛽𝛽 = 0 , 𝜀𝜀1𝐼𝐼 = 1.0) on the 𝜀𝜀1𝐼𝐼 axis and the data for AA 2014 and 1045 steel are taken from At-kins (1985), see Lee (2005). The combination of both fracture regions delivers the combined fracture locus of Fig. 14.

Fig. 14 Combination of the fracture lines to a frac-

ture locus, Lee (2005).

From Fig. 14 it is clear that the fracture limits in the branch II (combined tension-compression stress) are not covered. Tests to detect the critical values in this range are proposed by Wierzbicki et al. (2005), see Fig. 19, or can be detected by the Tension-Torsion-Test (TTT) described later in chapter 2.1.7.

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Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland 2 EXPERIMENTAL METHODS 2.1 EXPERIMENTAL METHODS FOR

FRACTURE DETECTION ON SHEETS As mentioned before the weakness of fracture mod-elling is the lack of standardized experiments and evaluation methodology. In industrial research the “Daimler bending test” (Fig. 10) and different hole expansion tests (Fig. 11) for detecting the edge formability [ISO/DIS 16630] have been established.

Fig. 10 Daimler bending test (internal report)

Fig. 11 Hole expansion test, cone and flat

Those tests are definitely insufficient to cover the whole stress range −1 < 𝛼𝛼 = 𝜎𝜎2𝜎𝜎1 < 1 and to con-struct all three fracture lines as schematically given in Fig.1. Due to this fact other methods have to be mentioned as well. Takuda et al. (2009) proposed an approach for meas-uring fracture strains based on Marciniak-type biax-ial specimens for various strain ratios. In the princi-pal strain space the measured fracture strains showed a linear behaviour. Lee (2005) published in his PhD thesis fracture line measurements based on former investigations of Embury and LeRoy (1977) and LeRoy et al.(1981). Later on, the FFLD was analytically constructed by Atkins (1985, 1996a), Dyrli HK (1999), and Lee and Wierzbicki (2003). The fracture strain data have been evaluated based mostly on fracture thinning values 𝜀𝜀33

𝑓𝑓

Fig. 12 Normalized FFLDs for eleven different ma-

terials in classical punch indentation prob-lems, Lee (2005).

Fig. 13 Normalized fracture loci for five different

materials in upsetting tests, Lee (2005).

Note that all curves are made to pass through the same point (𝛽𝛽 = 0 , 𝜀𝜀1𝐼𝐼 = 1.0) on the 𝜀𝜀1𝐼𝐼 axis and the data for AA 2014 and 1045 steel are taken from At-kins (1985), see Lee (2005). The combination of both fracture regions delivers the combined fracture locus of Fig. 14.

Fig. 14 Combination of the fracture lines to a frac-

ture locus, Lee (2005).

From Fig. 14 it is clear that the fracture limits in the branch II (combined tension-compression stress) are not covered. Tests to detect the critical values in this range are proposed by Wierzbicki et al. (2005), see Fig. 19, or can be detected by the Tension-Torsion-Test (TTT) described later in chapter 2.1.7.

Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland 2.1.1 Experimental method proposed by Gorji Gorji’s method was developed during his PhD at ETH (2012-2016) and documented by several pub-lications, Gorji (2013, 2014, 2015a, 2015b, 2015c, 2016). At this time the Wierzbicki-Bao (2005) experi-mental method was already well known. The motivation of the IVP was to develop a method which is applicable on standard thin sheet materials typically with a thickness of 0.8 to 1.2 mm which minimizes the experimental effort. The basic idea was to use tests, which are part of the standard test-ing methods – these are the Nakajima tests. Their advantage is that they directly cover the stress range 0.0 < 𝜎𝜎2𝜎𝜎1 < 1.0, which is the most critical for sheet fractures.

Fig. 15 Evaluation of fracture limits in branch I by

thinning method, Gorji (2014).

For the detection of the fracture in branch I (Fig. 14) Gorji (2014) proposed the thinning method based on the micro-thinning measurements on Nakajima tests. The details of the evaluation process are given in his PhD thesis Gorji (2015b). For the detection of the behaviour in the combined tension-compression region (branch II, Fig. 14) a special cup deep drawing (DD) test using a square blank and die radii of 𝑟𝑟 = 3.0 𝑚𝑚𝑚𝑚 was added.

Fig. 16 Detection of fracture free strains in branch

II, (Gorji 2015b)

For this tool geometry the rupture develops on the left side of FLD near to the β= -0.5 line and not as the deep drawing typical button crack at β= 0.0. This

point corresponds to the position of Point (1) in Fig. 16 Gorji applied this method to evaluate the fracture limits for a multi-layer Al-sheet “FUSION” com-posed by both materials AA6016 (core) and AA5005 (clad). The detected fracture limits are given in Fig. 17.

Fig. 17 Fracture lines evaluated by the thinning

method. Materials AA6016 and AA5005.

A comparison with different fracture criteria showed that the best fit could be achieved with a liner fracture line approximation.

2.1.2 Detection of shear fracture limits by a square cup drawing test.

A further “control fracture strain point” (Point 2, Fig. 16) can be detected with the same test, if shear induced “in-plane shear bands” occur. Fig. 18 shows the initiation of such shear bands. It is well known, that the “edge quality” strongly influences this be-haviour.

Fig. 18 Building of in-plane shear band as forming

limits in branch II

5










Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland The use of the proposed sheet tests have different advantages compared to other tests using special specimens as given in Fig. 19: - The deformation states correspond exactly to de-

formation states on real DD parts - The material has not to be changed – for example

by local thickness reduction - The boundary conditions and as consequence the

deformation fields are similar as they appear on real parts

- Evaluation is more accurate than with DIC meth-ods

2.1.3 Experimental methods proposed by Bao and Wierzbicki

An experimentally completely different approach was proposed by Bao (2003). By the use of 11 dif-ferent tests he was able to cover the stress range − 13 < 𝜂𝜂 < 1.0, Fig. 19.

Fig. 19 Detection of the fracture limits. Material Al 2024-T35, Lee (2005).

As already introduced in Fig. 14, Lee (2005) sepa-rates the behaviour in three regions

Branch I: void growth induced crack Branch II: mixed mode Branch III: localized shear band cracks

Branch I covers the tensile stress dominant region approximately mapped with the Johnson-Cook equation.

𝜀𝜀�̅�𝑓 = [𝐷𝐷1 + 𝐷𝐷2 ⋅ exp(𝐷𝐷3𝜂𝜂)] [1 + 𝐷𝐷4 ln (𝜀𝜀̇̅𝑝𝑝𝑝𝑝𝜀𝜀0̇

)]

⋅ [1 + 𝐷𝐷5 (𝑇𝑇 − 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟

𝑇𝑇𝑟𝑟𝑚𝑚𝑝𝑝𝑚𝑚 − 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟)]

(1)

It has to be mentioned that even in branch I, under the localized band conditions, a combined normal and shear crack occurs.

2.1.4 Consideration of intermediate stress given by Lode parameter

Wierzbicki et al. (2005) proposed to introduce the intermediate stress as an additional influence param-eter. The influence of the intermediate stress can be ex-pressed in different ways. Wierzbicki et al. (2005) proposed a formulation using the -angle in the Haigh-Westegaard space with

𝜂𝜂 = 𝜎𝜎𝐻𝐻/𝜎𝜎𝑚𝑚𝑒𝑒 (2)

𝜉𝜉 = 272 ∗𝐽𝐽3

�̅�𝜎3 and 𝜉𝜉 = cos (3𝜃𝜃) (3)

with J3 as the third invariant of the stress deviator. Very often the normalized lode angle parameter �̅�𝜃 =1 − 6𝜃𝜃/𝜋𝜋 is used.

Fig. 20 Triaxiality diagram, Bai (2008).

For the mapping of the surface they gave the equa-tion

6










Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland The use of the proposed sheet tests have different advantages compared to other tests using special specimens as given in Fig. 19: - The deformation states correspond exactly to de-

formation states on real DD parts - The material has not to be changed – for example

by local thickness reduction - The boundary conditions and as consequence the

deformation fields are similar as they appear on real parts

- Evaluation is more accurate than with DIC meth-ods

2.1.3 Experimental methods proposed by Bao and Wierzbicki

An experimentally completely different approach was proposed by Bao (2003). By the use of 11 dif-ferent tests he was able to cover the stress range − 13 < 𝜂𝜂 < 1.0, Fig. 19.

Fig. 19 Detection of the fracture limits. Material Al 2024-T35, Lee (2005).

As already introduced in Fig. 14, Lee (2005) sepa-rates the behaviour in three regions

Branch I: void growth induced crack Branch II: mixed mode Branch III: localized shear band cracks

Branch I covers the tensile stress dominant region approximately mapped with the Johnson-Cook equation.

𝜀𝜀�̅�𝑓 = [𝐷𝐷1 + 𝐷𝐷2 ⋅ exp(𝐷𝐷3𝜂𝜂)] [1 + 𝐷𝐷4 ln (𝜀𝜀̇̅𝑝𝑝𝑝𝑝𝜀𝜀0̇

)]

⋅ [1 + 𝐷𝐷5 (𝑇𝑇 − 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟

𝑇𝑇𝑟𝑟𝑚𝑚𝑝𝑝𝑚𝑚 − 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟)]

(1)

It has to be mentioned that even in branch I, under the localized band conditions, a combined normal and shear crack occurs.

2.1.4 Consideration of intermediate stress given by Lode parameter

Wierzbicki et al. (2005) proposed to introduce the intermediate stress as an additional influence param-eter. The influence of the intermediate stress can be ex-pressed in different ways. Wierzbicki et al. (2005) proposed a formulation using the -angle in the Haigh-Westegaard space with

𝜂𝜂 = 𝜎𝜎𝐻𝐻/𝜎𝜎𝑚𝑚𝑒𝑒 (2)

𝜉𝜉 = 272 ∗𝐽𝐽3

�̅�𝜎3 and 𝜉𝜉 = cos (3𝜃𝜃) (3)

with J3 as the third invariant of the stress deviator. Very often the normalized lode angle parameter �̅�𝜃 =1 − 6𝜃𝜃/𝜋𝜋 is used.

Fig. 20 Triaxiality diagram, Bai (2008).

For the mapping of the surface they gave the equa-tion


𝜀𝜀�̅�𝑓(𝜂𝜂, �̅�𝜃) = [12 (𝐷𝐷1𝑒𝑒

−𝐷𝐷2𝜂𝜂 + 𝐷𝐷5𝑒𝑒−𝐷𝐷6𝜂𝜂) −𝐷𝐷3𝑒𝑒−𝐷𝐷4𝜂𝜂] �̅�𝜃2 +

12

(𝐷𝐷1𝑒𝑒−𝐷𝐷2𝜂𝜂 + 𝐷𝐷5𝑒𝑒−𝐷𝐷6𝜂𝜂)�̅�𝜃 + 𝐷𝐷3𝑒𝑒−𝐷𝐷4𝜂𝜂

(4)

Alternative fracture surfaces as “mapping” func-tions of the experimental data have been developed by different other authors. In Fig. 21 the fracture loci of the following criteria; Johnson-Cook, Mohr-Cou-lomb and Hosford-Coulomb are plotted. Additional investigations have been recently done by Mohr et al. (2015). Remarkable is how strongly they deviate in the extrapolated regions of the triaxility diagram which highlights the uncertainty of the models. It becomes also evident how insufficient the number of experimental measured points still is.

Fig. 21 Different models for the description of the

fracture locus surface shape. A) Johnson-Cook, B) Mohr-Coulomb, c) Hosford-Cou-lomb, Komischke et al. (2018).

2.1.5 Applicability of Wierzbicki-Bao testing method for sheet metal forming applica-tions

As disadvantages or at least a point of discussion the following drawbacks have to be mentioned:

- Observed fractures of the 11 tests differ from the in-plane shear crack respectively the out-of-plane cracks known in sheet metal forming, compare Fig. 3 and Fig. 19

- Specimens cannot be fabricated out of a sheet material

- Transformation of compression-tension case to a shear load case is only allowed for isotropic materials

- The transformation from the strain-space 𝜀𝜀𝐼𝐼(𝜀𝜀𝐼𝐼𝐼𝐼) to a combined equivalent strain – stress space 𝜀𝜀𝑒𝑒𝑒𝑒𝑓𝑓 (𝜂𝜂, 𝜃𝜃) needs a constitutive model for evaluation of the stresses

- The load history influences the strain be-haviour

- The plots in function of average values are a strong simplification of the exact loading history

2.1.6 Differences between simple shear and tension-compression test

In deep drawing processes especially the flange re-gions undergoes a compression tension defor-mation.

Fig. 22 Compression-tension stress states on DD

parts

In numerous investigations the compression-tensile stress state will be tested under “simple shear” con-ditions by use of special specimens. These bases on the stress transformation rule.

Fig. 23 Transformation of the compression-tension

load case into a simple shear case by rota-tion of coordinates.

For the shear stress measurements geometries like specimens 6 an 7 in Fig. 19 are commonly used. Dif-ferent types of such geometries have been proposed by Wierzbicki et al. (2005), Bao (2003) and recently modified by Mohr et al. (2015) and Roth et al. (2016). It can be proved by crystal plasticity models that both deformation states (simple shear and compres-sion-tension), develop to different textures and can-not be interchanged for anisotropic microstructure behaviour of metals. The examples below demon-strates this behaviour for an AA6016 material a) Original (measured) texture

b) Texture after a simple shear deformation

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Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland c) Texture after a compression-tension deformation

Fig. 24 Different texture developments a) initial

texture; b) texture after 𝜀𝜀𝑒𝑒𝑒𝑒 = 0.5 of simple shear; c) texture after 𝜀𝜀𝑒𝑒𝑒𝑒 = 0.5 of com-pression-tension (pure shear) deformation

With the change of the texture also the hardening behaviour will change. Fig. 25 shows the theoreti-cally evaluated yield curves for the 3 different tex-tures. In this case the measured texture (Fig. 24a) has been used for calibration of the hardening curve.

Fig. 25 Influence of the texture on the hardening

behaviour

The reasons why it is important to distinguish be-tween simple shear and “pure shear” are summa-rized in Fig. 26

Fig. 26 Deformation and material specific differ-ences between a) a single point considera-tion, b) simple shear behaviour and c) compression-tension-behaviour (“pure shear”)

2.1.7 Applicability of Tensile-Torsion-Test (TTT)

In many cases the most sensitive change of the frac-ture strain occurs in the range 0 < 𝜂𝜂 < 13. This stress region corresponds to branch II and can be tested by a combined tensile-torsion test, Fig. 27 .

Fig. 27 Tensile-Torsion-Test

In the range 𝜂𝜂 > 13 additional data points can be achieved by tensile tests on notched specimens. The advantages of this type of tests are:

- Identical specimen geometry for all load cases

- Test can be done for RT up to 1100C - Nonlinear loading tests - Cyclic loading can be tested

As disadvantage the geometrical restriction has to be mentioned. 3 NUMERICAL FEM MODELS 3.1 FRACTURE CRITERIA For the prediction of necking in sheet metal forming, in addition to phenomenological models, physically based models like MK or MMFC are known as well. Much more complex is the definition of the fracture criteria. As discussed in chapter 1, different types of fracture can occur.

8










Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland c) Texture after a compression-tension deformation

Fig. 24 Different texture developments a) initial

texture; b) texture after 𝜀𝜀𝑒𝑒𝑒𝑒 = 0.5 of simple shear; c) texture after 𝜀𝜀𝑒𝑒𝑒𝑒 = 0.5 of com-pression-tension (pure shear) deformation

With the change of the texture also the hardening behaviour will change. Fig. 25 shows the theoreti-cally evaluated yield curves for the 3 different tex-tures. In this case the measured texture (Fig. 24a) has been used for calibration of the hardening curve.

Fig. 25 Influence of the texture on the hardening

behaviour

The reasons why it is important to distinguish be-tween simple shear and “pure shear” are summa-rized in Fig. 26

Fig. 26 Deformation and material specific differ-ences between a) a single point considera-tion, b) simple shear behaviour and c) compression-tension-behaviour (“pure shear”)

2.1.7 Applicability of Tensile-Torsion-Test (TTT)

In many cases the most sensitive change of the frac-ture strain occurs in the range 0 < 𝜂𝜂 < 13. This stress region corresponds to branch II and can be tested by a combined tensile-torsion test, Fig. 27 .

Fig. 27 Tensile-Torsion-Test

In the range 𝜂𝜂 > 13 additional data points can be achieved by tensile tests on notched specimens. The advantages of this type of tests are:

- Identical specimen geometry for all load cases

- Test can be done for RT up to 1100C - Nonlinear loading tests - Cyclic loading can be tested

As disadvantage the geometrical restriction has to be mentioned. 3 NUMERICAL FEM MODELS 3.1 FRACTURE CRITERIA For the prediction of necking in sheet metal forming, in addition to phenomenological models, physically based models like MK or MMFC are known as well. Much more complex is the definition of the fracture criteria. As discussed in chapter 1, different types of fracture can occur.

Forming Technology Forum 2018 July 2 & 3, 2018, Zurich, Switzerland 3.1.1 Basic modelling techniques A basic differentiation of the models shows Fig. 28

Fig. 28 Different types of fracture models

More or less the following methods can be distin-guished: a) Phenomenological integral models

Strain limit models o FFLD o Johnson-Cook o Triaxiality (Wierzbicki, Bao, Bay,

Lee) Stress limit models

o Crach modell (MATFEM) o Mohr-Coulomb model (linear and

non-linear) o 3D stress locus model (Hora)

Combined strain-stress models

b) Void growth models (GTN)

c) RVE based models

3.2 MACRO FEM MODELLING TECHNIQUE

For the accurate FEM prediction material specific as well as numerical aspects have to be considered. Fig. 29 summarizes the most important aspects.

Fig. 29 Macroscopic FEM based fracture model-ling

The specific aspects of the material damage model-ling, the damage accumulation as well as the mesh size influence and the crack opening have to be dis-cussed.

3.2.1 History and accumulation of damage Neither the JC fracture lines nor the triaxiality dia-grams compensate the influence of non-linear strain paths – Both of them are plotted in function of the average values av and av. On the other hand, with virtual tests, it can be simply demonstrated that the non-linearity can be very im-portant. In real applications e.g. like rolling, a non-linear history occurs anyway.

Fig. 30 Triaxiality development under different con-ditions, Effelsberg et al. (2012).

Based on the damage definition

𝐷𝐷 = (𝜀𝜀𝑝𝑝𝜀𝜀𝑓𝑓)𝑛𝑛

(5)

an incremental formulation can be defined by

�̇�𝐷𝑓𝑓 =𝑛𝑛𝜀𝜀𝑓𝑓𝐷𝐷(1−

1𝑛𝑛) 𝜀𝜀�̇�𝑝 (6)

Using the assumption of Lemaitre, Neukamm et al. (2009) introduced additionally a modification of the true stress to an effective stress with the relation

𝜎𝜎∗ = 𝜎𝜎 [1 − (𝐷𝐷−𝐷𝐷𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐1−𝐷𝐷𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)𝑚𝑚] (7)

Fig. 31 Influence of the damage on the effective

stress. Neukamm et al. (2009)

9











Fig. 32 Influence of the fading parameter. Neu-

kamm et al. (2009)

In GISSMO model equations n represents the non-linear damage accumulation exponent and m a mesh-size dependent fading parameter. As fracture locus the simple Johnson-Cook description 𝜀𝜀𝑓𝑓(𝜂𝜂) or the more complex triaxiality fracture surface 𝜀𝜀𝑓𝑓(𝜂𝜂, 𝜃𝜃) can be used. Nowadays the GISSMO model is one of the most widely applied fracture pre-diction method. A significant advantage is the pos-sibility to regularize the mesh size influence.

3.2.2 Impact of FEM mesh size It is well known, that FEM simulation of localizai-tion strain fields suffers from strong mesh size sen-sitivity. In combination with some special constitutive mod-els, like the GISSMO model, an adaptation of the material parameters to the mesh size is possible.

Fig. 33 Influence of the mesh size on the post criti-

cal strain development

In the GISSMO formulation the fading parameter m can be defined in dependency of the mesh size. Fig. 34 demonstrates the effectiveness of the regulariza-tion, compared to the original mesh size dependency in Fig. 33.

Fig. 34 Tensile test simulated with different

meshes und the use of the GISSMO regu-larization method

3.2.3 Effective stress models In many FEM-applications the regularization is still neglected and only the effect of the stress damage weakening is considered. The consideration of dam-age influence on the effective stress, as given in eq. 7, can be applied in a general way. The validation of the models is done based on the control of the force-displacement curves of simple experiments. Fig. 35 demonstrates the drop of the equivalent stress due the effective stress reduction induced by increasing damage.

Fig. 35 Hardening curve with and without consider-

ing damage along percentage of damage in equi-biaxial loading. Gorji et al. (2013)

3.2.4 Impact of crack openings on the stress history

A further influence on the stress development arises when cracks are explicitly modelled. An opening of the crack decreases the stiffness of the structure. The following example of a cutting process demon-strate the impact of crack modelling with respect to stress field changes.

10











Fig. 32 Influence of the fading parameter. Neu-

kamm et al. (2009)

In GISSMO model equations n represents the non-linear damage accumulation exponent and m a mesh-size dependent fading parameter. As fracture locus the simple Johnson-Cook description 𝜀𝜀𝑓𝑓(𝜂𝜂) or the more complex triaxiality fracture surface 𝜀𝜀𝑓𝑓(𝜂𝜂, 𝜃𝜃) can be used. Nowadays the GISSMO model is one of the most widely applied fracture pre-diction method. A significant advantage is the pos-sibility to regularize the mesh size influence.

3.2.2 Impact of FEM mesh size It is well known, that FEM simulation of localizai-tion strain fields suffers from strong mesh size sen-sitivity. In combination with some special constitutive mod-els, like the GISSMO model, an adaptation of the material parameters to the mesh size is possible.

Fig. 33 Influence of the mesh size on the post criti-

cal strain development

In the GISSMO formulation the fading parameter m can be defined in dependency of the mesh size. Fig. 34 demonstrates the effectiveness of the regulariza-tion, compared to the original mesh size dependency in Fig. 33.

Fig. 34 Tensile test simulated with different

meshes und the use of the GISSMO regu-larization method

3.2.3 Effective stress models In many FEM-applications the regularization is still neglected and only the effect of the stress damage weakening is considered. The consideration of dam-age influence on the effective stress, as given in eq. 7, can be applied in a general way. The validation of the models is done based on the control of the force-displacement curves of simple experiments. Fig. 35 demonstrates the drop of the equivalent stress due the effective stress reduction induced by increasing damage.

Fig. 35 Hardening curve with and without consider-

ing damage along percentage of damage in equi-biaxial loading. Gorji et al. (2013)

3.2.4 Impact of crack openings on the stress history

A further influence on the stress development arises when cracks are explicitly modelled. An opening of the crack decreases the stiffness of the structure. The following example of a cutting process demon-strate the impact of crack modelling with respect to stress field changes.


a) With crack opening

b) Without crack opening

Fig. 36 Fine blanking simulation. Influence of the

crack opening on the stress fields

In Fig. 36a the stress state is represented by the Mohr circles shortly after a crack opening. In the second case, Fig. 36b, crack development was not considered and therefore, the stress state achieves much higher values. This example demonstrates the problem, that with-out the simulation of the crack openings, the stress distributions in later stages are wrong and should not be used anymore!

3.3 MICRO FEM MODELING TECHNIQUE Real metals have an inhomogeneous microstructure. The macroscopic FEM models, as described before, are neglecting those effects by assuming a homoge-nous material structure. An interesting approach is the so-called Micro-FEM model, which uses element sizes of 0.01 mm and stochastically distributed material behavior, see Fig. 37. The “doted points” in Fig. 37 visualize single 2D elements of the meshed structure.

The mesh structure in this case is so fine, that delet-ing single elements does not disturb the stress distri-bution. This allows the modelling of a fracture prop-agation in a more appropriate way. Fig. 37 demonstrates the method on a plane stress loading case of a thin sheet (2D plane strain ele-ments). It predicts the development of an out-of-plane crack under 45 to the x-y sheet plane in a cor-rect way.

Fig. 37 Simulation of out-of-plane shear cracks un-

der plane strain conditions.

An analogous modelling approach was applied in Fig. 38 for the simulation of the localized necking in tensile test. The material was AA6016.

Fig. 38 Simulation of LN combined tensile-shear

crack

The pictures show the localization process starting at the stage of diffuse necking and then rapidly lo-calizing. As characteristic for aluminium alloys with no strain rate sensitivity the diffuse necking is not very distinct.

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