Embed Size (px)
Citation preview
Linkoping Studies in Science and Technology.Thesis No. 1475
Improved Material Models forHigh Strength Steel
Rikard Larsson
LIU-TEK-LIC-2011:14
Division of Solid MechanicsDepartment of Management and Engineering
Linkoping University,SE–581 83, Linkoping, Sweden
Linkoping, April 2011
Cover:Coil of rolled steel sheet with test specimens
Printed by:LiU-Tryck, Linkoping, Sweden, 2011ISBN: 978-91-7393-205-9ISSN: 0280-7971
Distributed by:Linkoping UniversityDepartment of Management and EngineeringSE–581 83, Linkoping, Sweden
c© 2011 Rikard LarssonThis document was prepared with LATEX, April 1, 2011
No part of this publication may be reproduced, stored in a retrieval system, or betransmitted, in any form or by any means, electronic, mechanical, photocopying,recording, or otherwise, without prior permission of the author.
Preface
The work presented in this thesis has been carried out at the Division of SolidMechanics at Linkoping University, and was initially funded by the VINNOVAMERA ”FE simulation of sheet metal forming” project. During the last year, workhas been funded by the SFS ProViking ”Super Light Steel Structures” project.
I am grateful to my supervisor Prof. Larsgunnar Nilsson for his support, feed-back and guidance. I would like to thank my colleagues, especially my fellow PhDstudents, for support, friendship and discussions during these years.
Dr Joachim Larsson and Dr Jonas Gozzi at SSAB and Dr. Ramin Moshfegh atOutokumpu Stainless are greatly acknowledged for support and material supply.
All assistance on experimental issues from Bo Skoog, Ulf Bengtsson and SorenHoff at Linkoping University has been appreciated. Their helpfulness and flexibilityhave facilitated the experimental part of this work to a great degree.
Finally, I would like to thank my friends and my family, and especially my deargirlfriend Katrin, for all their support during these years.
Linkoping, April 2011Rikard Larsson
iii
Abstract
The mechanical behaviour of the three advanced high strength steel grades, Do-col 600DP, Docol 1200M and HyTens 1000, has been experimentally investigatedunder various types of deformation, and material models have been developed,which account for the experimentally observed behaviour.
Two extensive experimental programmes have been conducted in this work. Inthe first, the dual phase Docol 600DP steel and martensitic Docol 1200M steel weresubjected to deformations both under linear and non-linear strain paths. Regulartest specimens were made both from virgin materials, i.e. as received, and frommaterials pre-strained in various directions. The plastic strain hardening, as wellas plastic anisotropy and its evolution during deformation of the two materials,were evaluated and modelled with a phenomenological model.
In the second experimental program, the austenitic stainless HyTens 1000 steelwas subjected to deformations under various proportional strain paths and strainrates. It was shown experimentally that the material is sensitive both to dynamicand static strain ageing. A phenomenological model accounting for these effectswas developed, calibrated, implemented in a Finite Element software and, finally,validated.
Both direct methods and inverse analyses were used in order to calibrate theparameters in the material models. The agreement between the numerical andexperimental results are in general very good.
This thesis is divided into two main parts. The background, theoretical frame-work and mechanical experiments are presented in the first part. In the secondpart, two papers are appended.
v
List of Papers
This thesis consists of the following two papers:
I. Larsson, R., Bjorklund, O., Nilsson, L., Simonsson, K., (2011) A study of highstrength steels undergoing non-linear strain paths - experiments and mod-elling. Journal of Materials Processing Technology 211, 122–132.
II. Larsson, R., Nilsson, L., (2011). On the modelling of strain ageing in ametastable austenitic stainless steel, Submitted.
Own contribution
I have been the main contributor for the modelling and writing of both the papers.The biaxial bulge tests have been performed by IUC, Olofstrom, and the measure-ments of the martensite transformation were performed by Outokumpu Stainless,Avesta. All other experimental work have been carried out at Linkoping Universityby Oscar Bjorklund and myself.
vii
Contents
Preface iii
Abstract v
List of Papers vii
Contents ix
Part I Theory and background
1 Introduction 3
2 Material models 52.1 Modelling framework . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Plastic strain hardening . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Plastic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Anisotropic hardening . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Phase transformations . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Dynamic strain ageing . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Static strain ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Material models used in this work . . . . . . . . . . . . . . . . . . . 142.9 Calibration procedures . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Mechanical testing 193.1 Proportional tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Non-linear strain paths . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Jump tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Ageing tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Finite Element modelling 274.1 Non-linear strain paths . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Static strain ageing . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Review of appended papers 33
6 Conclusion and discussion 35
ix
Bibliography 37
Appendix A – Stress update algorithm 43
Part II Appended papers
Paper I – A study of high strength steels undergoing non-linear strainpaths - experiments and modelling . . . . . . . . . . . . . . . . . . 51
Paper II – On the modelling of strain ageing in a metastable austeniticstainless steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
x
Part I
Theory and background
Introduction1
Simulation Based Design, SBD, including Finite Element, FE, analyses, is one ofthe most important methodologies in product development, and it has alreadysignificantly reduced the need for prototypes and physical tests.
Many products, including automotive components, are made of sheet metalcomponents produced by forming operations. During such forming operations, thematerial is subjected to complex deformation modes and in general also strain pathchanges. Successful forming operations require correct tool geometry. Many prob-lems associated with plastic forming operations, e.g. localisation and springback,can to a large extent be predicted and avoided early in the tool design process byusing the SBD methodology, where the tool designer can evaluate the impact ofdesign changes on the outcome. However, there are still some discrepancies be-tween the numerical predictions and the physically produced parts, which to someextent depends on inaccurate material models. Therefore, the development of moreaccurate material models is of a great importance in order to further facilitate theSBD process and decrease the number of prototypes and try-out tools.
There are several important mechanical issues to consider in the developmentof the forming process of high strength steel. Two major issues are springback andmaterial failure. High strength steels generally have lower ductility compared toconventional steel. This fact increases the risk for strain localisation, and a subse-quent rapid material fracture. The occurrence of strain localisation is associatedwith plastic hardening, and even though the yield stress is higher in HSS, the sub-sequent plastic hardening is comparatively lower and, thus, localisations may occurat lower strains. Springback is often considered as an elastic phenomenon, whichdepends on the strength-stiffness ratio and which generally increases with materialstrength. This implies that higher effort must be made to control springback whenforming HSS.
The main objective of this work is to develop material models with the poten-tial to accurately describe observed physical phenomena during deformation. Thematerial models should have an industrial applicability for detailed analyses, bothof the forming process of a product and of its intended use in function.
High strength steels are divided into conventional high strength steel, HSS,and advanced high strength steels, AHSS. Advanced high strength steels are oftenmultiphase steels, where two or more phases are mixed, e.g. dual phase steel. Dualphase steel consists of a ferritic matrix combined with hard martensitic regions.Austenitic stainless steel mainly consists of austenite that transforms to martensiteduring deformation, which significantly increases the plastic hardening. Three
3
CHAPTER 1. INTRODUCTION
steel grades have been considered in this work. Two of them are the cold-rolledmicro alloyed high strength Docol 600DP and Docol 1200M. The former is a dualphase steel with 75% ferrite and 25% martensite, whereas Docol 1200M is a fullymartensitic steel, see Olsson et al. (2006). The third material is a cold-rolledaustenitic stainless steel, HyTens 1000, which is within the EN 1.4310 standard.
Two extensive experimental programmes have been conducted to find the me-chanical properties of these materials. In the first, Docol 600DP and Docol 1200Mwere subjected to deformations both under linear and non-linear strain paths. Theexperimental programme aimed at investigating the influence of pre-straining onthe plastic anisotropy and hardening. An existing material model, comprising ahigh exponent yield surface with a non-linear mixed isotropic-kinematic hardening,was calibrated to the obtained experimental data.
In the second experimental programme, the plastic anisotropy and the anisotropicplastic hardening of HyTens 1000 were investigated. Furthermore, the sensitivitiesto static and dynamic strain ageing were evaluated from two additional test series.A phenomenological material model, including an isotropic-distortional hardeningmodel in combination with models for strain ageing and martensitic transforma-tion, was developed, calibrated and implemented in a Finite Element, FE, software.
Both direct methods and inverse analyses were used in order to calibrate theparameters of the material models. The agreement between the numerical andexperimental results is in general very good.
4
Material models2
Plastic strain hardening and anisotropy are typical phenomena considered in ma-terial models for sheet metals. However, descriptions of several other phenomena,e.g. deformation induced anisotropy, phase transformations and dependency onstrain rate and temperature, may also be incorporated in the material model. Thischapter describes the addressed phenomena and their respective incorporation intoa material model. First, the continuum framework is presented, in which manyelasto-plastic material models are defined. The specific material models that havebeen used and developed in this thesis are briefly described along with the basiccalibration technique. Details of the numerical implementation of the constitutiveequations are given in Appendix A.
2.1 Modelling framework
Sheet metal forming operations result in large deformations and rotations of thematerial, therefore the material models must be defined in this context. By usinga co-rotational material frame, the anisotropy can easily be accounted for, see e.g.Hallquist (2009) and Belytschko et al. (2000).
An additive decomposition of the rate of deformation tensor is assumed, i.e.
D = De
+ Dp
(1)
where ( · ) denotes a corotated quantity ( · ). D denotes the corotated rate ofdeformation tensor, and the superscripts e and p denote the elastic and plasticparts, respectively. In the case of small elastic deformations, a hypo-elastic stressupdate is often assumed, i.e.
σ = C : (D − Dp) (2)
where σ and C are the corotated Cauchy stress tensor and the fourth order elas-tic stiffness tensor, respectively. A major part of a material model is the yieldfunction, f , and the yield criterion
f = σ − σy≤ 0 elastic= σv plastic flow
(3)
where σ, σy and σv denote the effective stress, the current yield stress and a viscousstress, respectively. The yield function determines the elastic region in the stress
5
CHAPTER 2. MATERIAL MODELS
space, where the elastic limit is given by f = 0, the so called yield surface. If thematerial obeys an associative flow rule, the direction of plastic flow is proportionalto the gradient of the yield function, i.e.
Dp
= λ∂f
∂σ(4)
where λ denotes a plastic multiplier.
2.2 Plastic strain hardening
The yield stress, σy, is typically considered as a function of equivalent plastic strain,εp,
εp =
t∫
0
λdt (5)
where t denotes time. Plastic hardening may also depend on other quantities, e.g.equivalent plastic strain rate, martensitic fraction and temperature.
Experimental data can often be used directly, but in some cases, e.g. in the caseof a serrated stress-strain relation, an analytical function can be used to filter thedata. Numerous analytical functions have been proposed over the years. The pow-erlaw hardening function was introduced by Hollomon (1945), but it often offers anunsatisfactory fit to the experimental data. Voce (1948) proposed an exponentialhardening, which was extended by Hockett and Sherby (1975) by adding an expo-nent to the plastic strain. Both the Voce and Hockett-Sherby hardening functionsmay be extended to several components in order to fit well to experimental data.An extended Voce hardening function with m components yields, see Lemaitre andChaboche (2000),
σy(εp) = σ0 +
m∑
i=1
Qi(1− exp(−Ciεp)) (6)
where σ0, Qi and Ci are material constants.Normally, plastic hardening is evaluated from uniaxial tensile tests, i.e. σ(εL),
where σ is the true stress, and εL = ln(L/L0) is the longitudinal logarithmic strain.L and L0 denote actual and initial lengths, respectively, of the extensometer. How-ever, this is only true for strains before diffuse necking. After diffuse necking thestress-strain relation must be extrapolated or realised from other mechanical testsat extended strain levels. Analytical functions can be used for this extrapolation.However, hardening functions calibrated by least square fits can be significantlyincorrect in the extrapolated area. Both the Voce and the Hockett-Sherby func-tions often afford a good fit to experimental data for strains before diffuse necking,but the obtained extrapolated function is often an underestimation compared toexperimental findings, cf. Lademo et al. (2009). One approach, which has been
6
2.3. PLASTIC ANISOTROPY
utilised throughout this work, is joining other analytical functions to an extendedVoce function, e.g. a powerlaw function, i.e.
σy(εp) = σ0 +
m∑
i=1
Qi(1− e−Ciεp
) εp ≤ εt1
A1 +B1(εp)n1 εt1 ≤ εp ≤ εt2A2 +B2(εp)n2 εt2 ≤ εp
(7)
where σ0, A1, B1, n1, A2, B2 and n2 are material constants The transition strains,εt1 and εt2, may be chosen arbitrarily in order to get a good fit to experimentaldata. The joining functions are typically constrained by continuity requirementsand by one additional parameter, e.g. the stress level at εp = 100%, denotedσ100 = A2 +B2.
2.3 Plastic anisotropy
Rolling manufacturing processes of sheet metals may lead to plastic anisotropy,where the axes of orthotropy coincide with the rolling direction, RD, the transversaldirection, TD, and the normal direction, ND.
A distinction is made between anisotropic yield stresses and anisotropic plasticflow. The latter is traditionally described with the Lankford parameter, R, or therelated plastic strain ratio, k, defined as
R =dεpTdεpN
=dεpT
−dεpL − dεpT; k =
dεpTdεpL
=−RR + 1
(8)
where the transversal logarithmic strain, εT , and normal logarithmic strain, εN ,have been introduced. High values of R indicates a higher resistance to thinning,and thus better formability, cf. Marziniak et al. (2002). Anisotropy in yield stresses,on the other hand, can be described with the so called yield stress ratios,
rφ =σy,φσy,ref
(9)
where σy,φ is the uniaxial yield stress in the direction φ with respect to the RD,and σy,ref is the reference yield stress. Similar quantities can be defined for thebalanced biaxial stress state,
rb =σy,bσy,ref
; Rb = kb =dεpTDdεpRD
(10)
where the subscript b denotes the balanced biaxial stress state, i.e. σTD = σRD.In case of an associated flow rule according to Eq. (4), both anisotropic yield
stresses and anisotropic plastic flow are described with an anisotropic effectivestress function, σ. A yield surface for plane stress is shown in Fig. 1(a). Thecorresponding yield locus for τRD−TD = 0 is shown in Fig. 1(b), where some typicalstress states and normal directions for evaluation of plastic anisotropy are shown.
7
CHAPTER 2. MATERIAL MODELS
1
−1
−0.5
0
0.5
1
Tensile test, σ1 = 0, σ2 = 0Balanced biaxial test, σ1 = σ2
Shear, σ1 = −σ2
Plane strainσTD
τRD−TD
σRD
(a)
9
10 Yield locus
σRD
σTD
1k00
1
k90
1
kb
‖ RD
‖ TD
ut
rs
rs
b
brs
rs
(r00σref , 0)
(0, r90σref)(rbσref , rbσref)
Material Characterization
(b)
Figure 1: (a) Yield surface for plane stress state. (b) Corresponding yield locus forτRD−TD = 0. Uniaxial, biaxial, shear and plane strain stress states are indicatedin the figures.
Several anisotropic effective stress functions, σ, have been proposed over theyears. A good review is given by Banabic (2000). Barlat and co-workers havedeveloped anisotropic yield criteria based on a linear transformation of the stress
8
2.4. ANISOTROPIC HARDENING
state into an isotropic domain. In particular the three parameters high exponenteffective stress function by Barlat and Lian (1989), the so called YLD89, has beenwidely used in industrial applications. Further development along this line haslead to the eight parameters effective stress function by Barlat et al. (2003a),generally referred to as YLD2000. Aretz (2004, 2005) and Banabic et al. (2005)independently derived two eight parameters anisotropic effective stress expressionsbased on the work by Barlat and Lian (1989), denoted YLD2003 and BBC2000,respectively. Barlat et al. (2007) showed that they can both be made identical tothe YLD2000. These three eight parameters effective stress functions are designedfor a plane stress state, i.e. σ = σ(σ11, σ22, σ12), where the 11 and 22 directionscorrespond to the RD and the TD directions, respectively. σ33 = σ13 = σ23 = 0in the case of a plane stress state, where σ33 denotes the normal direction. Mostother effective stress functions for plane stress found in the literature correspondto special cases of these three eight parameters functions. One exception is thefunction proposed by Cazacu et al. (2006), which is able to represent a tension-compression asymmetry in yield stresses. For general stress states, one notes the18 parameters effective stress function proposed by Barlat et al. (2005).
2.4 Anisotropic hardening
Plastic deformation may introduce further anisotropy, since the mechanical prop-erties are affected also in other directions than the current loading direction, seee.g. Barlat et al. (2003b). One well known deformation induced anisotropic effectis the Bauschinger effect, which is the phenomenon of a lower yield stress in thecase of reversed loading.
Two basic modelling techniques for deformation induced plastic anisotropy arekinematic and distortional hardening. Kinematic hardening means a translation ofthe yield surface during deformation, and is achieved by introducing a backstresstensor, α. The related overstress tensor is defined as Σ = σ−α, and replaces theCauchy stress tensor in the effective stress function, i.e.
σ = σ(σ −α) = σ(Σ) (11)
If the yield surface simultaneously expands, the hardening is referred to as mixedisotropic-kinematic. Several evolution rules for the backstress tensor have beenproposed, see the review by Chaboche (2008). Among such, one notes the wellknown rule by Frederick and Armstrong (2007), which realises a hardening atmonotonic loading similar to the Voce model. It is able to describe the Bauschingereffect. However, it is unable to describe permanent softening, a phenomenon wherethe yield stress at reverse loading remains significantly lower than in monotonicloading. Geng and Wagoner (2002) addressed this issue by using an additionalbounding yield surface. Moreover, Yoshida and Uemori (2002) considered the workhardening stagnation effect at reverse loading, by using an additional hardeningsurface defined in the stress space.
9
CHAPTER 2. MATERIAL MODELS
A mixed isotropic-kinematic hardening has a great impact on the stress-strainrelation also in the case of a non-linear strain path. An example of such loading isshown in Fig. 2. The figure shows the stress-strain relations and the correspondinggrowth and translation of the yield locus during pre-straining in the RD, unloadingand subsequent reloading in the TD.
0 0.05 0.1 0.150
200
400
600
800
Lon
gitu
dina
lst
ress
σ[M
Pa]
Equivalent plastic strain εp[−]
Initial yield stressYield stress afterpre-straining unloadingYield stress at reloadingYield stress aftersubsequent straining
αασRD
σTD
initial yield locus
yield locus afterpre-straining
yield locusafterre-loading
pre-straining inthe RD
re-loadingin the TD
Figure 2: The predicted stress-strain relation and the corresponding growth andtranslation of the yield locus during pre-straining in the RD, unloading and sub-sequent reloading in the TD
The importance of a mixed isotropic-kinematic hardening in the case of non-linear strain paths has previously been pointed out in literature. Kim and Yin(1997) performed a three-stage deformation test where sheets were first pre-strainedin the RD. A second pre-straining was performed at several angles to the RD, fromwhich tensile test specimens were cut out and tested in various directions. Hahmand Kim (2008) extended this work and found that the Lankford parameters didnot change in accordance with the change in yield stresses. This difference waspartly explained by kinematic hardening. Tarigopula et al. (2008) investigated adual phase steel under non-linear strain paths. They found that the deformationinduced anisotropy required a mixed isotropic-kinematic hardening model.
Unlike what is the case with isotropic and kinematic hardening, the shape ofthe yield surface is allowed to change during deformation in distortional hard-ening. In the most general case, the shape may change arbitrarily. A simplerdistortional hardening can be obtained by allowing the parameters of the effectivestress function to change during deformation. The distortion is then limited to thearbitrariness of the actual effective stress function. Furthermore, if the parametersdepend on a scalar variable only, e.g. the equivalent plastic strain, εp, and neitheron the stress state nor on the strain history, the distortion of the yield surface willnot depend on the loading direction,
σ = σ (Σ, Ai(εp), a(εp)) (12)
10
2.5. PHASE TRANSFORMATIONS
Such a hardening approach is referred to as an isotropic-distortional hardening,Aretz (2008). However, despite its simplicity, such an approach allows for varyingplastic strain ratios and yield stress ratios, i.e. kφ = kφ(εp), kb = kb(ε
p), rφ = rφ(εp)and rb = rb(ε
p). Better accuracy can be achieved in predictions of localisationsin various directions with isotropic-distortional hardening, cf. Aretz (2008). Anexample of a distortion of the yield locus is shown in Fig. 3.
−1 0 1
−1
0
1
σT
D/σ
y,r
ef
σRD/σy,ref
εp = 0εp = 0.05εp = 0.1εp = 0.2εp = 0.3εp = 0.6
Figure 3: Example of evolution of the YLD2003 yield locus in the case of anisotropic-distortional hardening for HyTens 1000.
2.5 Phase transformations
The high strength and deformation hardening in austenitic stainless steels arepartly due to phase transformations during plastic deformation. Martensitic trans-formation can be spontaneous, stress-assisted, or strain induced, see Seethara-man (1984). The γ-austenite, FCC, transforms to ε-martensite, HCP, and to α′-martensite, BCT, during deformation. Transformation to α′-martensite causesa volume expansion, since the austenite is more close packed than the BCT-martensite. The magnitude of the volume expansion depends on the carbon con-tent, see Krauss (2008). Angel (1954) showed that the transformation dependson temperature. Furthermore, the transformation also depends on strain rate, seeHecker et al. (1982), and hydrostatic pressure, see Lebedev and Kosarchuk (2000).
Ramırez et al. (1992) suggested a model for martensitic transformation basedon an energy assumption, where the martensitic fraction was determined by thetemperature and plastic strain. They also presented a non-linear mixture rule basedon the strains of the two phases and their individual hardening. In later work, Tsuta
11
CHAPTER 2. MATERIAL MODELS
and Cortes (1993) reformulated this model into an incremental formulation to beused in multiaxial plasticity applications. Further development lead to the Hanselet al. (1998) model, where the rate of transformation depends on the martensiticfraction, VM , itself, and on the temperature, T ,
∂VM∂εp
= V pM
B
2A
(1− VMVM
)B+1B
eQ/T [1− tanh(C +D · T )] (13)
where A,B,C,D,Q, and p are material constants. The transformation rate doesnot explicitly depend on strain rate, but high strain rates will affect the temperaturedue to adiabatic heating, and thus also affect the transformation rate.
2.6 Dynamic strain ageing
Dynamic strain ageing, DSA, denotes the interaction between mobile solute atomsand temporarily arrested dislocations, cf. van den Beukel and Kocks (1982). So-lute atoms diffuse to temporarily arrested dislocations and strengthen them, cf.Fressengeas et al. (2005). In the special case of a constant plastic strain rate, theaverage ageing time, ta, of the arrested dislocation is assumed to be constant andinversely proportional to the plastic strain rate. Thus, a lower strain rate, i.e. alonger ageing time, implies a stronger resistance to dislocation glide due to diffu-sion of solute atoms to temporarily arrested dislocations, which contributes to anegative strain rate sensitivity. This leads to a competition between the instanta-neous strain rate sensitivity, SRS, which in general is positive, and the DSA, cf.Mesarovic (1995). Depending on the strain rate, temperature and the sensitivityto DSA, this competition leads to a total negative or positive steady state SRS,which herein is denoted ∆σss.
An instantaneous increase in strain rate is in general accompanied by an in-stantaneous stress increase, ∆σi, followed by a transient period, with a significantlylower hardening rate compared to the reference hardening. The transient periodcan be explained by the successive release of arrested dislocations. Since the higherplastic strain rate implies a short ageing time, this results in a decreasing stresscontribution from the DSA during this transient period, which may result in anegative steady state SRS, i.e. ∆σss < 0, see Fig. 4.
A negative SRS, leads to temporal strain localisations, since homogeneous de-formation is unstable, cf. McCormick (1988). In the case of uniaxial tension, anegative SRS leads to a force singularity, i.e. dF = 0, and that the strain is lo-calised into a band, see e.g. Rodriguez (1984). However, unlike diffuse necking,the strain band will propagate along the tensile specimen during deformation, seeFig. 5. The repetitive birth and propagation of such bands result in a serratedstress-strain relation, or so called ”jerky flow”, and a staircase like relation betweenthe total elongation of the specimen and the strain measured by the extensometer.This effect is usually referred to as the Portevin-Le Chatelier, PLC, effect.
Dynamic strain ageing and the PLC effect have been the subject of many re-search projects, and have been identified experimentally for a variety of materials,
12
2.6. DYNAMIC STRAIN AGEING
Tru
est
ress
σ
Equivalent plastic strain εp
˙εp(1)
Δσi
Positive instanteneous SRS
˙εp(2)
Sudden strainrate increase:˙εp(1)
< ˙εp(2)
Transientperiod
|Δσss|Negative steady state SRS
Figure 4: Stress response in the case of a jump in the equivalent plastic strain rate.
1
1 Plane strain dimensions and measure method
2 Tensile test
cross headvelocity
v
strain band propagation
b b
L
extensometer location
Material Characterization
Figure 5: Sketch of the propagation of a strain band in a tensile test specimensubjected to a crosshead velocity, v. The location of the mechanical extensometer,with initial length L0 and current length L is indicated in the figure.
e.g. aluminium, see Clausen et al. (2004), TWIP steel, see Zavattieri et al. (2009)and austenitic stainless steel, see Meng et al. (2009).
Several incorporations of the DSA effect in FE simulations have previouslybeen made. McCormick (1988) developed a model, henceforth denoted the MCmodel, based on the concentration of solute atoms at pinned dislocations, whichis a function of the ageing time, ta, of such dislocations. The evolution of theageing time depends on the plastic strain rate, where the steady state value, ta,ss,depends on strain rate and acts as a target value for ta. An instantaneous strainrate change is followed by a transient period where ta evolutes towards its newsteady state value. The model was further developed by Mesarovic (1995), andhas gained some popularity. Zhang et al. (2001) used the MC model to investigatethe morphology of the PLC bands by Finite Element, FE, analyses. Hopperstadet al. (2007) used the MC model in order to investigate the influence of the PLCeffect on plastic instability and strain localisation. It was shown that the PLC
13
CHAPTER 2. MATERIAL MODELS
effect reduces strain to necking under both uniaxial and biaxial stress states. Thiswork was extended by Benallal et al. (2008), where Digital Image Correlation wasused in order to detect the PLC bands, follow their propagation and validate thematerial model.
The lack of mesh convergence in FE analyses of propagating instabilities havepreviously been shown by e.g. Benallal et al. (2006) and Maziere et al. (2010).Maziere et al. used the MC model and found that the band width, and thus alsothe maximum plastic strain rate within the band, is mesh dependent, whereas theband propagation speed and the plastic strain carried by the band are not. Theysuggested a non-local approach as regularisation in order to overcome the meshdependency.
2.7 Static strain ageing
Static strain ageing, SSA, is a phenomenon which refers to an increased yieldstress observed at re-loading of a specimen which has been unloaded during sometime after pre-straining, see Fig. 6. Static strain ageing is related to pinning ofdislocations by solute atoms and pinning of new dislocation sources at the grainboundaries, cf. Leslie and Keh (1962), but it is distinguished from DSA by theindependency on plastic strain rate. Thus, the ageing time is prescribed in thecase of SSA, whereas it depends on the plastic strain rate in the case of DSA. Theincrement in increased stress at yielding depends both on the level of pre-strain aswell as on the ageing time, cf. Kubin et al. (1992). Ballarin et al. (2009) used anadditional term, ∆σ, in the plastic hardening function in order to model the SSA,
∆σ = Ra +Qa[1− exp(−ba(εp − εp0))] (14)
where Ra, Qa and ba are material constants and εp0 is the equivalent plastic strainat unloading. In this work, a dependency on the ageing time, τ , was added, andthe contribution from the SSA, denoted σSSA, can thus be written,
σSSA = [σsat − (σsat − στ ) exp(−Cε(εp − εp0)](1− exp(−Cττ)) (15)
where the constants στ and σsat govern the amplitude of the static strain ageing atincipient recurring plastic flow and the saturation level of the static strain ageing,i.e. the permanent increase in yield stress, respectively. Furthermore, τ denotesthe ageing time from unloading to reloading. The two material constants Cτ andCε govern the ageing rate and how fast the effect vanishes with recurring plasticstrain, respectively.
2.8 Material models used in this work
Two specific material models have been used in the papers appended to this thesis.A modified version of the effective stress function proposed by Aretz (2004, 2005),
14
2.8. MATERIAL MODELS USED IN THIS WORK
Tru
est
ress
σ
Longitudinal strain εL
unloading, ageingand reloading
pre-straining
permanenteffect
yield stressincrease
Figure 6: Principal effect of static strain ageing on the stress-strain relation
YLD2003, has been used in both of them, i.e.
2σa(Σ) = |Σ′1|a + |Σ′2|a + |Σ′′1 − Σ′′2|a where
Σ′1Σ′2
=A8(εp)Σ∗11 + A1(εp)Σ∗22
2±√(
A2(εp)Σ∗11 − A3(εp)Σ∗22
2
)2
+ A4(εp)2Σ12Σ21
Σ′′1Σ′′2
=
Σ∗11 + Σ∗22
2±√(
A5(εp)Σ∗11 − A6(εp)Σ∗22
2
)2
+ A7(εp)2Σ12Σ21
(16)
where Σ∗11 = Σ11−Σ33 and Σ∗22 = Σ22−Σ33. With this modification, a shell elementwith normal stress and strain components can be used, which enables constraints onthe element thickness, e.g. continuous element thickness across element boundaries,see LS-DYNA Keyword User’s Manual (2007). The effect of a such constraint onan FE analysis is discussed in Chapter 4.
In the first paper, Docol 600DP and Docol 1200M were subjected both to linearand non-linear strain paths. The effective stress function in Eq. (16) was combinedwith a non-linear mixed isotropic-kinematic hardening model, in order to accountboth for the initial and deformation induced plastic anisotropy. The evolution ofthe backstress tensor followed a two components AF rule,
α =2∑
i
αi =2∑
i
CXi
(QXi
Σ
σ−αi
)˙εp (17)
Furthermore, the isotropic part of the plastic hardening, σy(εp), was described by
an extended Voce function followed by a powerlaw type of hardening, i.e.
σy(εp) =
σ0 +2∑
i
QRi(1− e−CRiεp
) εp ≤ εt
A+B(εp)C εp > εt(18)
15
CHAPTER 2. MATERIAL MODELS
where the transition strain, εt, was chosen to be close to the plastic strain at diffusenecking.
In Paper 2, the austenitic stainless steel HyTens 1000 was studied under varioustypes of deformation. Since, in this case, only linear strain paths were considered,no kinematic hardening was included, i.e. α = 0 and thus Σ = σ. Instead, severalother significant improvements were made to the material model. The anisotropichardening in the different directions was described by allowing the yield stress ratiosto depend on the equivalent plastic strain, i.e. an isotropic-distortional hardening,see Eq. (12).
Both SSA and DSA were observed in the experimental results, and were ac-counted for in the yield function,
f = σ − σ0 −R− σSSA − σta =
≤ 0 for ˙εp = 0= σv( ˙εp) for ˙εp > 0
(19)
where σ0 is the initial yield stress and R is the plastic hardening, according to
R(εp) =
2∑
i=1
Qi(1− e−Ciεp
) εp ≤ εt1
A1 +B1(εp)n1 εt1 ≤ εp ≤ εt2A2 +B2(εp)n2 εt2 ≤ εp
(20)
The possible contribution from static strain ageing, σSSA, was described accord-ing to Eq. (15), and the DSA was modelled according to McCormick (1988) andMesarovic (1995),
σta = SH
[1− exp
−(tatd
)α](21)
where S, H, td and α are material constants. The evolution of the average ageingtime, ta, follows
dta =
0 for εp = 0
dt− taΩ
dεp for εp > 0(22)
where Ω is a material property, which in this work was assumed to be a constant.The instantaneous SRS was governed by the viscous stress, cf. Hopperstad et al.(2007),
σv( ˙εp) = S ln
(1 +
˙εp
˙εp0
)(23)
where ˙εp0 is a material constant.Furthermore, the yield surface exponent was allowed to depend on the marten-
sitic fraction, according to the mixture rule
a(VM) = 8 · (1− VM) + 6 · VM (24)
16
2.9. CALIBRATION PROCEDURES
where a = 8 and a = 6 have been used for austenite, FCC, and martensite, BCT,respectively. The transformation rate from austenite to martensite was describedwith the Hansel model, see Eq. (13)
At a reference plastic strain rate, ˙εpref , the total yield stress can be written
σy(εp) =σ0 +R(εp) + σta(ta,ss( ˙εpref )) + σv( ˙εpref ) + σSSA (25)
Since low strains rates were applied, the temperature, T , was assumed to be con-stant and equal to room temperature, and the austenitic plastic hardening wasevaluated from
Rγ = R−∆RMVM −∆RTT (26)
where ∆RM and ∆RT are material constants and T = 293 K.
2.9 Calibration procedures
The calibration procedures have in general been formulated as optimisation prob-lems, where the mean square difference between the numerical predictions, F , andthe experimental results, F , was minimised. A typical structure of such a minisa-tion problem is
find gi
minimise e =m∑
j=1
(Fj − Fj)2
subject to gi ∈ Gi for all i
(27)
where Gi denotes the feasible ranges for the sought variables gi, and m denotes thenumber of quantities regarded in the problem.
Some parameters, e.g. the extended Voce model parameters, effective stress pa-rameters and the dynamic strain ageing parameters, were evaluated directly fromexperimental data, whereas other parameters, e.g. σ100, were identified from inversemodelling. All direct identifications were conducted with the built-in MATLAB(2007) function fmincon, whereas a meta model approach was used in the iden-tification procedures in which case FE analyses are required, see Stander et al.(2009).
17
Mechanical testing3
The testing procedures are briefly described in this chapter, and some importantresults are presented. In general three tests of each kind have been conducted.However, for the readers convenience, the result from just one specimen is presentedin some figures, which is visually and subjectively chosen to be representative atthis test condition.
In the first test series, the two steels, Docol 600DP and Docol 1200M, weresubjected both to linear and to non-linear strain paths. Tensile and shear testspecimens were made both from virgin, i.e. as received, and from pre-strainedmaterial in various directions.
In the second test series, the HyTens 1000 steel was subjected to deformationunder various strain paths and strain rates. In addition to uniaxial tensile tests,plane strain tests, shear tests and a balanced biaxial test were also conducted.Furthermore, the sensitivity both to dynamic and static strain ageing, DSA andSSA, under uniaxial tension was investigated.
3.1 Proportional tests
The in-plane anisotropy was evaluated by conducting tensile tests in several di-rections, i.e. φ = 0, 45 and 90 with respect to the RD, and a balanced biaxialtest. Tensile tests provide information on the stress-strain relation and the relationbetween the longitudinal and transversal plastic strain components, i.e. kφ. Thegeometry of the tensile test specimen, which has been used in this work, is shownin Fig. 7(a), and the stress-strain relations from testings on virgin materials areshown in Fig. 8(a).
The relations between the longitudinal strain, εL, and the total elongation ofthe tensile specimen, approximated with the crosshead displacement, δ, from thetensile tests on HyTens 1000 were found to have a staircase type of behaviour,see Fig.9, due to recurring strain band propagation across the gauge length of theextensometer, see Fig. 5.
The plastic anisotropy can be further evaluated by shear tests, plane straintests an a so called bulge test, where a balanced biaxial stress state is obtained,i.e. σTD = σRD, see Fig. 1. The biaxial stress-plastic normal strain relations fromthe bulge tests are shown in Fig. 8(b).
Significantly higher plastic strain levels can be obtained in a shear test comparedto tensile, plane strain and biaxial tests. The geometry of the shear test specimen
19
CHAPTER 3. MECHANICAL TESTING
3
extenso-meterlocation
Material Characterization
(a)
(b) (c)
Figure 7: Geometry of the (a) tensile test, (b) plane strain test and (c) shear testspecimens. Dimensions in mm.
is shown in Fig. 7(c), and the experimental results from testings on virgin materialsare shown in Fig. 8(c).
Plane strain tests are of great importance since they trigger failure modes sim-ilar to those occurring in sheet metal forming. The design of the plane strain testspecimen is of uttermost importance in order to realise a desired strain path. Thegeometry of the test specimen used in this work is shown in Fig. 7(b), and haspreviously been used by e.g. Lademo et al. (2009), and gives a close to plane strainpath. As in the shear test, the plane strain test requires inverse modelling in orderto evaluate the material properties. The experimental nominal stress-displacementrelations from plane strain tests on virgin materials are shown in Fig. 8(d).
3.2 Non-linear strain paths
Tensile, plane strain, shear and bulge tests, result in rather linear strain paths.Non-linear strain paths can be obtained in several ways. One of the most efficientis the general biaxial test, which however, requires a biaxial test machine. This testcan also be used for evaluations of anisotropy, since a great variety of deformationscan be applied, see Kuwabara et al. (1998). Alternatively, by straining a large sheetunder uniaxial tension, and then cutting out regular small test specimens from it,a number of non-linear strain paths can be achieved. In this work, large specimenswere pre-strained under uniaxial tension both in the RD and in the TD. Thegeometry of the pre-strained sheets used in this work is shown in Fig. 10(a). Tensileand shear test specimens were made from the pre-strained sheets with orientationsas shown in Fig. 10(b). One drawback with pre-straining under uniaxial tension isthat diffuse necking limits the maximum level of the pre-strain, and only a slight
20
3.3. JUMP TESTS
0 0.05 0.1 0.15 0.2 0.250
500
1000
1500Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
φ = 0φ = 45
φ = 90Docol 600DP
Docol 1200M
HyTens 1000
(a)
0 0.1 0.2 0.3 0.4 0.50
500
1000
1500
Bia
xial
stre
ssσ
b[M
Pa]
Normal plastic strain |εpND| [-]
Docol 1200M
HyTens 1000
Docol 600DP
(b)
0 1 2 30
200
400
600
800
1000
1200
Nom
inal
stre
ssF
/A0
[MPa]
Displacement δ [mm]
φ = 0φ = 45
φ = 90Docol 600DP
HyTens 1000Docol1200M
(c)
0 2 4 60
500
1000
1500
Nom
inal
stre
ssF
/A0
[MPa]
Displacement δ [mm]
φ = 0φ = 45
φ = 90
Docol 1200MHyTens 1000
Docol 600DP
(d)
Figure 8: Experimental results from (a) uniaxial tensile tests, (b) balanced biaxialtests, (c) shear tests and (d) plane strain tests.
effect of pre-straining can be observed on the subsequent shear tests, especiallyin the case of a high strength steel with limited ductility, e.g. Docol 1200M. Theeffects of pre-straining on the uniaxial stress-strain relations and on the nominalstress-displacement relations from shear tests on Docol 600DP are shown in Figs. 11and 12, respectively.
3.3 Jump tests
A constant initial crosshead velocity was applied in regular tensile tests. This veloc-ity was then suddenly increased to various significantly higher velocities in a seriesof tensile tests. The jump in crosshead velocity results in a jump in strain rate.Henceforth, these tests are denoted ”jump”tests. The jump tests were conducted in
21
CHAPTER 3. MECHANICAL TESTING
0 5 10 15 20 250
0.1
0.2
Lon
gitu
dina
lstr
ain
ε L[-]
Crosshead displacement δ [mm]
Numerical, Le = 0.25 mmExperimental
Figure 9: The numerical and experimental longitudinal strains as functions ofcross-head displacement for uniaxial loading in the TD.
(a)
(b)
Figure 10: (a) Geometry of the pre-strain specimen. Dimensions are given in mm.(b) Sketch including some cut out specimens.
the TD, and the effect of the strain rate jump on the stress-strain relation, i.e. thestrain rate sensitivity, was assumed to be isotropic. The experimental results arepresented in Fig. 13 together with a reference stress-strain relation. An instanta-neous stress response to the change of strain rate was observed, with a subsequenttransient period, corresponding to the evolution of the ageing time, ta, towardsits new steady state value. It is noticed that the reference stress-strain relation isserrated in Fig. 13(b), since the strain rate jump was applied close to the strainlevel at which the strain bands start to occur and propagate at the nominal strainrate. Thus, only the instantaneous SRS but no steady state SRS was evaluatedfrom the results presented in Fig. 13(b).
22
3.3. JUMP TESTS
0 0.05 0.1 0.150
200
400
600
800Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
Virgin materialPrestrained 4.5% in the TDPrestrained 8% in the TDExperimentNumerical
(a)
0 0.05 0.1 0.150
200
400
600
800
Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
Virgin materialPrestrained 5% in the RDPrestrained 10% in the RDExperimentNumerical
(b)
Figure 11: Experimental and numerical results from tensile tests on virgin andpre-strained Docol 600DP material (a) in the RD and (b) in the TD.
0 0.5 1 1.50
100
200
300
400
500
600
Nom
inal
stre
ssF
/A0
[MPa]
Displacement δ [mm]
Virgin materialPrestrained 4.5% in the TDPrestrained 8% in the TDExperimentNumerical
(a)
0 0.5 1 1.50
100
200
300
400
500
600
Nom
inal
stre
ssF
/A0
[MPa]
Displacement δ [mm]
Virgin materialPrestrained 5% in the RDPrestrained 10% in the RDExperimentNumerical
(b)
Figure 12: Experimental and numerical results from shear tests on virgin andpre-strained Docol 600DP material (a) in the RD and (b) in the TD.
23
CHAPTER 3. MECHANICAL TESTING
0.04 0.045 0.05 0.055
1030
1040
1050
1060
1070
1080
Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
referencev : 0.5 → 2.5 mm/minv : 0.5 → 10 mm/minv : 2.5 → 40 mm/minv : 10 → 40 mm/min
0.084 0.086 0.088 0.09 0.0921140
1150
1160
1170
1180
Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
referencev : 0.5 → 3 mm/minv : 0.5 → 7 mm/minv : 0.5 → 15 mm/min
(a) (b)
Figure 13: The stress-strain relations from the jump tests. (a) ε∗L ≈ 0.04 and (b)ε∗L ≈ 0.084.
24
3.4. AGEING TESTS
3.4 Ageing tests
A series of tensile test specimens were loaded, unloaded and reloaded with a varyinglag in time, i.e. the ageing time, τ , between the loadings, in order to evaluatethe influence of time on the SSA. Contrary to the jump tests, these tests wereconducted in the RD. Test results are shown in Fig. 14. Three specimens weretested for each ageing time, however, only one representative curve for each ageingtime is presented in order to more clearly illustrate the dependency of ageing time.Obviously, the ageing time has a considerable effect on the yield stress at the secondloading.
0 0.02 0.04 0.06 0.08 0.1 0.120
200
400
600
800
1000
1200
Longitudinal strain εL [-]
Tru
est
ress
σ[M
Pa]
0 days3 days11 days56 days
0.09 0.1 0.11
1050
1100
1150
1200
Figure 14: Influence of ageing time on static strain ageing.
25
Finite Element modelling4
This chapter presents the Finite Element, FE, models, and some selected numericalresults. In all FE analyses, a fully integrated element with included normal stressand normal strain components have been used. With this approach, constraintson the normal strain can be applied, e.g. continuous normal strain across elementboundaries, c.f LS-DYNA Keyword User’s Manual (2007), which corresponds toregularisation of the normal strain.
The tensile tests were analysed using the FE mesh shown in Fig. 15. Thelongitudinal strain, εL, was evaluated by measuring the distance between two nodeslocated along the centre of the specimen, with an initial distance of L0 = 12.5 mm,corresponding to the physical extensometer used in the experiments, see Fig. 15.The numerical and experimental stress-strain relations from tensile tests on HyTens1000 are shown in Fig. 16, and the longitudinal strain-crosshead displacement, i.e.εL − δ, relation in the TD is shown in Fig. 9. Finite Element models of the tensilespecimen with three element lengths, i.e. Le = 1, 0.5, 0.25 mm, were used in orderto evaluate mesh dependency on the PLC effect. Only a weak mesh dependencywas identified.
6
b b
location of extensometer nodes
Material Characterization
Figure 15: Finite Element mesh of the tensile test specimen. Element lengthLe = 1 mm.
The FE model used for the simulation of the plane strain test specimen is shownin Fig. 17. The smallest element size was Le = 0.2 mm in the centre parts. Load-ing was applied by a prescribed velocity at the edge nodes, corresponding to theclamps, whereas the deformation was measured between two nodes, correspondingto the extensometer in the physical experiments, see Fig. 17. The experimentaland numerical nominal stress-displacement relations from the plane strain tests inthe RD on Docol 600DP are shown in Fig. 18. The analysis was conducted bothwith a plane stress assumption and with a continuous thickness across the elementedges. Experimental and numerical results from the plane strain tests on Hytens1000 are presented in Fig. 19.
27
CHAPTER 4. FINITE ELEMENT MODELLING
0 0.05 0.1 0.15 0.2 0.250
200
400
600
800
1000
1200
1400
1600
Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
φ = 0 exp.φ = 45 exp.φ = 90 exp.φ = 0 sim.φ = 45 sim.φ = 90 sim.
Figure 16: Experimental and numerical stress-strain relations from tensile tests.
The FE model used in the analyses of the shear tests, cf. Fig. 7(c), is shown inFig. 20. The element length was Le = 0.06 mm in the centre of the shear zone, seeFig. 20(b). Loading was applied as a prescribed velocity at a node in the centreof each hole. Thus, rotation around the bolts was unconstrained, corresponding tothe experimental procedure. The elements were fully integrated in the FE modelof the shear test specimen in order to avoid either so called hourglassing or toohigh artificial hourglass energy. Experimental and numerical results from the sheartests in the TD and in the RD on Docol 600DP are shown in Fig. 12.
4.1 Non-linear strain paths
Due to homogeneous deformation in the pre-straining, just one single element rep-resents the complete pre-strain specimen. This element was pre-strained in the nu-merical analyses, corresponding to the experimental pre-straining. The equivalentplastic strain and the backstress components were mapped from the pre-strainedelement to the FE models of the tensile and shear tests, see Fig. 21. The axes oforthotropy in the element were rotated around the ND in order to account for thedifferent directions in subsequent testing, so that the longitudinal direction in thetest always was parallel to the x-axis in the simulation. Stress-strain relations fromtensile tests on Docol 600DP, both from pre-strained and from virgin material, areshown in Fig. 11. Nominal stress-displacement relations from the shear tests onpre-strained Docol 600DP are shown in Fig. 12.
28
4.2. STATIC STRAIN AGEING
5
b b
location of extensometer nodes
Material Characterization
Figure 17: Finite element model of the plane strain tensile test specimen.
4.2 Static strain ageing
Similarly to the experimental procedure, the tensile test was pre-strained and un-loaded. The SSA model was activated at reloading, with ageing times correspond-ing to the experiments. One such FE analysis was performed for each ageing time,and the results are presented in Fig. 22 together with the corresponding experi-mental results.
29
CHAPTER 4. FINITE ELEMENT MODELLING
0 0.5 1 1.5 2 2.5 30
100
200
300
400
500
600
700
800N
omin
alst
ress
F/A
0[M
Pa]
Displacement δ [mm]
exp.with normal stressplane stress
Figure 18: Experimental normal stress-displacement relations from plane straintests on Docol 600DP in the RD, together with numerical predictions using twotypes of shell elements.
0 1 2 3 4 50
500
1000
1500
Nom
inal
stre
ssF
/A0
[MPa]
Displacement δ [mm]
φ = 0 exp. (mean)φ = 45 exp. (mean)φ = 90 exp. (mean)φ = 0 sim.φ = 45 sim.φ = 90 sim.
Figure 19: Experimental and numerical results from the plane strain tests. Theexperimental curves are based on the mean values of three experimental results ineach direction.
30
4.2. STATIC STRAIN AGEING
(a) (b)
Figure 20: (a) Finite Element model of the shear test. (b) Details of the FE meshin the shear zone.
Materia
l Charac
teriza
tion
RD
ψ
θ
φ
(a)
⇒α, εp
F F F F
RD
TD
ψ
RD
TD
φ
x
y
x
y
F
pre-straining
F
subsequent testing
(b)
Figure 21: (a) Angle definitions. (b) One pre-strained element. The backstress, α,and the equivalent plastic strain, εp, were mapped onto the subsequent model.
31
CHAPTER 4. FINITE ELEMENT MODELLING
0.08 0.085 0.09 0.095 0.1 0.105 0.11
1050
1100
1150
1200
Tru
est
ress
σ[M
Pa]
Longitudinal strain εL [-]
τ =3 days, exp.τ =11 days, exp.τ =56 days, exp.τ =3 days, sim.τ =11 days, sim.τ =56 days, sim.
Figure 22: Experimental and numerical results from the strain ageing tests.
32
Review of appended papers5
Paper I
A study of high strength steels undergoing non-linear strain paths -experiments and modelling
This paper concerns two advanced high strength steels, Docol 600DP and Docol1200M. Plastic anisotropy and its evolution during deformation was experimentallyinvestigated. A material model, which accounts for the observed behaviours, wassubsequently developed.
In addition to tests on virgin materials, tensile and shear tests were performedon pre-strained materials. The resulting deformation induced plastic anisotropywas evaluated and modelled with a mixed isotropic-kinematic hardening functioncombined with a high exponent yield surface. The anisotropy found from the sheartests was very well predicted by the anisotropy evaluated from tensile tests and abulge test. It was concluded that an isotropic-kinematic hardening is necessary foraccurate hardening predictions at non-linear strain paths.
Paper II
On the modelling of strain ageing in a metastable austenitic stainlesssteel
The mechanical behaviour of an austenitic stainless steel, HyTens 1000, withinthe EN 1.4310 standard, was investigated. Three tensile tests, three plane straintests and a bulge test were used in order to evaluate the plastic anisotropy andanisotropic plastic hardening. A significantly different plastic hardening in therolling direction compared to the transversal direction was found. A model with acombination of a high exponent yield surface and an isotropic-distortional harden-ing assumption was successfully used in order to represent the anisotropic plastichardening.
Additionally, two series of experiments were conducted in order to evaluatesensitivity both to dynamic and to static strain ageing. Dynamic strain ageing wasevaluated by conducting so called jump tests, from which both the instantaneousand the steady state strain rate sensitivities were evaluated. The dynamic strainageing was accounted for in the material model by introducing additional terms
33
CHAPTER 5. REVIEW OF APPENDED PAPERS
in the yield function. The prediction of mechanical behaviour in the tensile testsagreed well with the experimental results.
Static strain ageing was evaluated by pre-straining a series of tensile tests, whichwere aged at room temperature. The yield stress after pre-straining and ageing wasfound to depend on the ageing time, whereas the overstress effect partly vanishesduring recurring deformation. A term was added to the plastic strain hardeningfunction in order to account for this phenomenon. This function made it possibleto predict the main characteristics of the static strain ageing phenomenon.
34
Conclusion and discussion6
It was found that the eight parameters effective stress function was able to accu-rately describe the initial anisotropy in all three materials. A very close fit to theplastic hardening in different material directions of the Docol steels was achievedby a simultaneous least square fit both of the extended Voce parameters and of theyield stress ratios. Furthermore, the anisotropy in simple shear, i.e. the differencebetween shear in the φ = 0, 45 and φ = 90 directions, was very well predicted forthe Docol steels.
A satisfactory representation of the plastic hardening of HyTens 1000 in alldirections, i.e. φ = 0, 45 and 90, required an anisotropic hardening, where theyield surface was allowed to distort during deformation. The effective stress pa-rameters were evaluated from uniaxial tensile and bulge test results, and the modelwas able to predict the anisotropy in the plane strain tests. However, it should benoticed that data beyond a displacement δ > 2.3 mm were used for calibration ofthe hardening relation for equivalent plastic strains above εt2, and cannot serve asa validation of the model. However, agreement between the numerical and experi-mental results is very good. The prediction of the force-displacement of the sheartests on HyTens 1000 was reasonable good, despite a slight overprediction of theforces. It is believed that the martensitic transformation is less favored in the caseof simple shear than in the tensile and plane strain tests due to a lower triaxial-ity. No hydrostatic effect on the martensitic development is taken into account inthe Hansel model, which has been calibrated to uniaxial tensile tests. Thus, themartensitic transformation may be overpredicted in the shear tests, which affectsthe shear stress level and so also the shear force.
It was noticed that a force instability, similar to diffuse necking in the uniaxialtensile tests, also occurred in the plane strain tests before strain localisation. Theuse of an element with the normal stress component included, together with con-tinuous thickness across element boundaries, was shown to afford a more physicalnominal stress-displacement relation in the plane strain test than a plane stressassumption without any regularisation. The regular plane stress element predicteda load drop too early, caused by strain localisation in one single row of elementsin the centre of the specimen. On the contrary, the nominal stress prediction ob-tained by using the element with a normal stress component agrees well with theexperimental result.
The kinematic part of the mixed isotropic-kinematic model was fitted to theexperimental results from the tensile tests on pre-strained material. The nominalstress-displacement relation from shear tests of pre-strained materials was predicted
35
CHAPTER 6. CONCLUSION AND DISCUSSION
with good agreement to experimental findings on Docol 600DP. It should be notedthat, these tests were not included in the model calibration, and can thus serve asa model validation. The corresponding results from Docol 1200M were in generalalso good. However, only a small influence of pre-straining on the shear tests wasfound, and the effect of the kinematic part of the hardening is most obvious on thetensile tests on pre-strained material.
The serrated plastic flow and the strain-displacement relations agree fairly wellwith the experimental findings. Moreover, the start of the serration was well pre-dicted in the RD, despite no such test being included in the calibration procedureof the strain rate sensitivity parameters.
The model for static strain ageing, SSA, was able to predict the main observedphenomena, i.e. increased yield stress with ageing time, and that this effect de-creases during deformation after reloading.
The overall agreement between the numerical predictions and experimental re-sults are very good. A good agreement indicates that high accuracy can be expectedin subsequent validations, and thus also in real applications.
36
Bibliography
Angel, T., 1954. Formation of Martensite in Austenitic Stainless Steels. Journalof the Iron and Steel Institute, pp. 165–174.
Aretz, H., 2004. Applications of a new plane stress yield function to orthotropicsteel and aluminium sheet metals. Modelling and Simulation In MaterialsScience and Engineering 12, pp. 491–509.
Aretz, H., 2005. A non-quadratic plane stress yield function for orthotropic sheetmetals. Journal of Materials Processing Technology 168, pp. 1–9.
Aretz, H., 2008. A simple isotropic-distortional hardening model and its applica-tion in elastic-plastic analysis of localized necking in orthotropic sheet metals.International Journal of Plasticity 24, pp. 1457–1480.
Ballarin, V., Soler, M., Perlade, A., Lemoine, X., Forest, S., 2009. Mechanismsand Modeling of Bake-Hardening Steels: Part I. Uniaxial Tension. Metallur-gical and Materials Transactions A 40 (6), pp. 1367–1374.
Banabic, D., 2000. Formability of Metallic Materials. Springer Verlag, Berlin.
Banabic, D., Aretz, H., Comsa, D. S., Paraianu, L., 2005. An improved ana-lytical description of orthotropy in metallic sheets. International Journal ofPlasticity 21 (3), pp. 493–512.
Barlat, F., Aretz, H., Yoon, J., Karabin, M., Brem, J., Dick, R., 2005. Lineartransformation-based anisotropic yield functions. International Journal ofPlasticity 21, pp. 1009–1039.
Barlat, F., Brem, J. C., Yoon, J.-W., Chung, K., Dick, R. E., Lege, D. J., Pour-boghrat, F., Choi, S.-H., Chu, E., 2003a. Plane stress yield function foraluminium alloy sheets - Part I: Theory. International Journal of Plasticity19, pp. 1297–1319.
Barlat, F., Duarte, J. M. F., Gracio, J. J., Lopes, A. B., Rauch, E. F., 2003b.Plastic flow for non-monotonic loading conditions of an aluminum alloy sheetsample. International Journal of Plasticity 19 (8), pp. 1215–1244.
Barlat, F., Lian, J., 1989. Plastic behaviour and stretchability of sheet metals.Part I: A yield function for orthotropic sheets under plane stress conditions.International Journal of Plasticity 5, pp. 51–66.
37
CHAPTER 6. CONCLUSION AND DISCUSSION
Barlat, F., Yoon, J. W., Cazacu, O., 2007. On linear transformations of stresstensors for the description of plastic anisotropy. International Journal ofPlasticity 23 (5), pp. 876–896.
Belytschko, T., Liu, W. K., Moran, B., 2000. Nonlinear Finite Elements forContinua and Structures. Wiley, Chichester.
Benallal, A., Berstad, T., Borvik, T., Clausen, A., Hopperstad, O., 2006. Dynamicstrain aging and related instabilities: experimental, theoretical and numericalaspects. European Journal of Mechanics - A/Solids 25 (3), pp. 397–424.
Benallal, A., Berstad, T., Børvik, T., Hopperstad, O., Koutiri, I., de Codes,R. N., 2008. An experimental and numerical investigation of the behaviourof AA5083 aluminium alloy in presence of the Portevin Le Chatelier. Inter-national Journal of Plasticity 24, pp. 1916–1945.
Cazacu, O., Plunkett, B., Barlat, F., 2006. Orthotropic yield criterion for hexag-onal closed packed metals. International Journal of Plasticity 22, pp. 1171–1194.
Chaboche, J.-L., 2008. A review of some plasticity and viscoplasticity constitutivetheories. International Journal of Plasticity 24 (10), pp. 1642–1693.
Clausen, A., Børvik, T., Hopperstad, O., Benallal, A., 2004. Flow and fracturecharacteristics of aluminium alloy AA5083-H116 as function of strain rate,temperature and triaxiality. Materials Science and Engineering A 364 (1-2),pp. 260–272.
Frederick, C., Armstrong, P., 2007. A mathematical representation of the multi-axial Bauschinger effect. Materials At High Temperatures 24 (1), pp. 1–26.
Fressengeas, C., Beaudoin, A., Lebyodkin, M., Kubin, L., Estrin, Y., 2005. Dy-namic strain aging: A coupled dislocation–solute dynamic model. MaterialsScience and Engineering A 400-401, pp. 226–230.
Geng, L., Wagoner, R. H., 2002. Role of plastic anisotropy and its evolution onspringback. International Journal of Mechanical Sciences 44 (1), pp. 123–148.
Hahm, J., Kim, K., 2008. Anisotropic work hardening of steel sheets under planestress. International Journal of Plasticity 24 (7), pp. 1097–1127.
Hallquist, J., 2009. LS-DYNA Theory Manual, Livermore Software TechnologyCorporation, Livermore.
Hansel, A. H. C., Hora, P., Reissner, J., 1998. Model for the kinetics of strain-induced martensitic phase transformation at non-isothermal conditions forthe simulation of sheet metal forming processes with metastable austeniticsteels. In: Huetink, Baaijens (Eds.), Simulation of Materials Processing:Theory, Methods, and Applications. Rotterdam, pp. 373–378.
38
Hecker, S., Stout, M., Staudhammer, K., Smith, J., 1982. Effects of Strain Stateand Strain Rate on Deformation-Induced Transformation in 304 StainlessSteel Part I. Magnetic Measurements and Mechanical Behavior. Metallurgi-cal Transactions A 13A, pp. 619–626.
Hockett, J. E., Sherby, O. D., 1975. Large strain deformation of polycrystallinemetals at low homologous temperatures. Journal of the Mechanics andPhysics of Solids 23, pp. 87–98.
Hollomon, J. H., 1945. Tensile deformation. Transactions of the American Insti-tute of Mining, Metallurgical and Petroleum Engineers 162, pp. 268–290.
Hopperstad, O., Børvik, T., Berstad, T., Lademo, O.-G., Benallal, A., 2007. Anumerical study on the influence of the Portevin-Le Chatelier effect on neck-ing in an aluminium alloy. Modelling and Simulation in Materials Scienceand Engineering 15 (7), pp. 747–772.
Kim, K. H., Yin, J. J., 1997. Evolution of anisotropy under plane stress. Journalof the Mechanics and Physics of Solids 45 (5), pp. 841–851.
Krauss, G., 2008. Steels: Processing, Structure, and Performance. ASM Interna-tional, Ohio.
Kubin, L., Estrin, Y., Perrier, C., 1992. On static strain ageing. Acta Metallurgicaet Materialia 40 (5), pp. 1037–1044.
Kuwabara, T., Ikeda, S., Kuroda, K., 1998. Measurement and analysis of differ-ential work hardening in cold-rolled steel sheet under biaxial tension. Journalof Materials Processing Technology 80–81, pp. 517–523.
Lademo, O.-G., Engler, O., Keller, S., Berstad, T., Pedersen, K., Hopperstad,O., 2009. Identification and validation of constitutive model and fracturecriterion for AlMgSi alloy with application to sheet forming. Materials andDesign 30, pp. 3005–3019.
Lebedev, A. A., Kosarchuk, V. V., 2000. Influence of phase transformations onthe mechanical properties of austenitic stainless steels. International Journalof Plasticity 16 (7-8), pp. 749–767.
Lemaitre, J., Chaboche J.-L., 1990. Mechanics of Solid Materials. CambridgeUniversity Press, Cambridge.
Leslie, W. C., Keh, A. S., 1962. An electron transmission study of the strainageing of iron. Journal of The Iron and Steel Institute 200, pp. 722–728.
LS-DYNA Keyword User’s Manual, Version 971, 2007. Livermore Software Tech-nology Corporation, LSTC, Livermore.
39
CHAPTER 6. CONCLUSION AND DISCUSSION
Marziniak, Z., Duncan, J., Hu, S., 2002. Mechanics of Sheet Metal Forming.Butterworth-Heinemann, Oxford.
MATLAB, 2007. Version 7.5. The Mathworks Inc., Natick.
Maziere, M., Besson, J., Forest, S., Tanguy, B., Chalons, H., 2010. Numericalaspects in the finite element simulation of the Portevin-Le Chatelier effect.Computer Methods in Applied Mechanics and Engineering 199 (9-12), 734–754.
McCormick, P., 1988. Theory of flow localisation due to dynamic strain ageing.Acta metall. 36 (12), pp. 3061–3067.
Meng, L., Sun, J., Xing, H., Pang, G., 2009. Serrated flow behavior in AL6XNaustenitic stainless steel. Journal of Nuclear Materials 394, 34–38.
Mesarovic, S. D., 1995. Dynamic strain aging and plastic instabilities. Journal ofthe Mechanics and Physics of Solids 43 (5), pp. 671–700.
Olsson, K., Gladh, M., Hedin, J.-E., Larsson, J., 2006. Microalloyed high-strengthsteels. Advanced Materials & Processes 164 (8), pp. 44–46.
Ramırez, J. A. C., Tsuta, T., Mitani, Y., Osakada, K., 1992. Flow Stress andPhase Transformation Analyses the Austenitic Stainless Steel under ColdWorking (Part 1, Phase Transformation Characteristics and ConstitutiveFormulation by Energetic Criterion). JSME International Journal, Series1 35 (2), pp. 201–209.
Rodriguez, P., 1984. Serrated plastic flow. Bulletin of Material Science 6 (4),pp. 653–663.
Seetharaman, V., 1984. Deformation and martensitic transformation. Bulletin ofMaterial Science 6 (4), pp. 703–716.
Simo, J., Hughes, T., 1998. Computational Inelasticity. Vol. 7. Springer Verlag,New York.
Stander, N., Roux, W., Goel, T., Eggleston, T., Craig, K., 2009. LS-OPT usersmanual, Version 4.0. Livermore Software Technology Corporation.
Tarigopula, V., Hopperstad, O.-S., Langseth, M., Clausen, A.-H., 2008. Elastic-plastic behavior of dual-phase, high-strength steel under strain-path changes.European Journal of Mechanics A/Solids 27, pp. 764–782.
Tsuta, T., Cortes, J., 1993. Flow stress and phase transformation analyses inaustenitic stainless steel under cold working. JSME International JournalSeries A 36 (1), pp. 63–72.
van den Beukel, Kocks, U. F., 1982. The strain dependence of static and dynamicstrain-aging. Acta metallurgica 30, pp. 1027–1034.
40
Voce, E., 1948. The relationship between stress and strain for homogeneous de-formation. Journal of the Institute of Metals 74, pp. 537–562.
Yoshida, F., Uemori, T., 2002. A model of large-strain cyclic plasticity describingthe Bauschinger effect and workhardening stagnation. International Journalof Plasticity 18, pp. 661–686.
Zavattieri, P., Savic, V., Jr., L. H., Fekete, J., Tong, W., Xuan, Y., 2009. Spatio-temporal characteristics of the Portevin-Le Chatelier effect in austenitic steelwith twinning induced plasticity. International Journal of Plasticity 25 (12),pp. 2298–2330.
Zhang, S., McCormick, P. G., Estrin, Y., 2001. The morphology of Portevin-LeChatelier bands: finite element simulation for Al-Mg-Si. Acta Materialia49 (6), pp. 1087–1094.
41
Stress update algorithmA
Rotations of tensors from the global to the corotated element coordinate system,and back, are performed before and after the stress update, respectively, see Hal-lquist (2009). Henceforth, the corotational superscript ( ) will be excluded andall subsequent constitutive relations will be related to this configuration. A strainincrement, dε, a time step, ∆t, the current stress, σn, and the history variables,(·)n, including the equivalent plastic strain, εpn, at the beginning of the time stepn, are entered into the stress update algorithm.
First, a trial stress tensor is calculated by assuming an elastic strain increment
σtrialn+1 = σn + C : dε
ttriala,n+1 = ta,n +
0 for εpn = 0∆t for εpn > 0
(28)
Similarly, the trial ageing time is updated according to ttriala,n+1 = ta,n + ∆t. Otherhistory variables are not updated since they remain unchanged if no plastic defor-mation occurs. The yield criterion is then checked according to
f = σ(σtrialn+1 )− σta(ttriala,n+1)− σy(εpn, VM,n)
≤ 0 elastic> 0 plastic deformation
(29)
If the elastic condition is fulfilled, the stress update is finished and thus σn+1 = σtrialn+1 ,ta,n+1 = ttriala,n+1 and (·)n+1 = (·)n where (·) denotes the other history variables. Oth-erwise, plastic flow occurs and the plastic strain increment has to be found.
In this case, a new flow function f is defined in order to include the viscousstress, σv,
f = f − σv = 0 (30)
which is solved by using an iterative method. By assuming
r(k+1)n+1 ≈ r(k)
n+1, r(k)n =
∂σ
∂σ
∣∣∣∣σ(k)
n ,εp(k)n
(31)
where an iteration index, k, has been introduced and where a quantity (·)(1)n+1 =
(·)n, the evaluation of second order derivatives of the equivalent stress function,σ, with respect to the stress components, σ, can be avoided. Furthermore, noequation system has to be solved in each iteration, only one single scalar equation.
43
APPENDIX A. STRESS UPDATE ALGORITHM
The method is generally referred to as the cutting-plane algorithm, c.f. Simo andHughes (1998). Equation (30) is linearised according to
f(k+1)n+1 = f
(k)n+1 +
∂f
∂εp
∣∣∣∣εp(k)n+1
dεp(k)n+1 = 0 (32)
where the current plastic strain, εp(k)n+1, is evaluated according to
εp(k)n+1 = εpn +
k∑
i=1
dεp(i)n+1 = εpn + ∆ε
p(k)n+1 (33)
The derivative of the flow function, f , with respect to the equivalent plastic strain,εp, is evaluated from
∂f
∂εp
∣∣∣∣εp(k)n+1
=− ∂σ
∂σ
∣∣∣∣σ(k)
n+1
: C :∂σ
∂σ
∣∣∣∣σ(k)
n+1
+8∑
i=1
∂σ
∂Ai
∣∣∣∣σ(k)
n+1
∂Ai∂εp
∣∣∣∣εp(k)n+1
+∂σ
∂a
∣∣∣∣σ(k)
n+1
∂a
∂εp
∣∣∣∣εp(k)n+1
− ∂σv∂ ˙ε
p
∣∣∣∣˙εp(k)n+1
∂ ˙εp
∂εp
∣∣∣∣εp(k)n+1
− ∂σy∂εp
∣∣∣∣εp(k)n+1
(34)
where ˙εp = ∆εp(k)n+1/∆t and the derivative of the dynamic strain ageing stress, σta ,
is excluded due to stability reasons.
The plastic strain increment, dεp(k)n+1, is evaluated from Eq. (32), and the equiv-
alent plastic strain and strain rate are updated according to
∆εp(k+1)n+1 = ∆ε
p(k)n+1 + dε
p(k)n+1
εp(k+1)n+1 = εpn + ∆ε
p(k+1)n+1
˙εp(k+1)n+1 = ∆ε
p(k+1)n+1 /∆t
(35)
The average waiting time is obtained from
t(k)a,n+1 = ttriala − t
(k)a,n+1
Ω∆ε
p(k)n+1 ⇒ t
(k)a,n+1 =
ttriala
1 +∆ε
p(k)n+1
Ω
(36)
The stress tensor, the martensitic fraction, the components of the yield stress and
44
the effective stress parameters are updated according to
σ(k+1)n+1 = σ
(k)n+1 −C :
∂σ
∂σ
∣∣∣∣σ(k)
n+1
dεp(k)n+1
V(k+1)M,n+1 = V
(k)M,n+1 + VM,ε|V (k)
M,n+1dε
p(k)n+1
σ(k+1)y,n+1 = σy
(εp(k+1)n+1 , V
(k+1)M,n+1
)
σ(k+1)ta = σta
(t(k+1)a,n+1
)
σ(k+1)v = σv
(∆ε
p(k+1)n+1 /∆t
)
a(k+1)n+1 = a
(V
(k+1)M,n+1
)
A(k+1)i,n+1 = Ai
(εp(k+1)n+1
)i = 1, . . . , 8
(37)
The consistency condition, Eq. (30), is then re-checked with the updated variables.
If the yield function is within the tolerance, i.e. |f (k+1)n+1 | ≤ ftol, the iteration
procedure has finished and (·)n+1 = (·)k+1n+1. The tolerance ftol = 10−4 · σ was
used. On the other hand, if |f (k+1)n+1 | > ftol, the quantities in Eq. (35) are updated
according to (·)kn+1 = (·)k+1n+1, and a new plastic strain increment is evaluated.
Normally, the iteration procedure converges within less than 7 iterations, but oftenin less than 4.
45
Division of Solid Mechanics,Department of Managementand Engineering
2011–04–29
xx
http://urn.kb.se/resolve?urn:nbn:se:liu:diva-
66993
Improved Material Models for High Strength Steel
Rikard Larsson
The mechanical behaviour of the three advanced high strength steel grades, Docol 600DP,Docol 1200M and HyTens 1000, has been experimentally investigated under various types ofdeformation, and material models have been developed, which account for the experimentallyobserved behaviour.Two extensive experimental programmes have been conducted in this work. In the first,the dual phase Docol 600DP steel and martensitic Docol 1200M steel were subjected todeformations both under linear and non-linear strain paths. Regular test specimens weremade both from virgin materials, i.e. as received, and from materials pre-strained in variousdirections. The plastic strain hardening, as well as plastic anisotropy and its evolution duringdeformation of the two materials, were evaluated and modelled with a phenomenologicalmodel.In the second experimental program, the austenitic stainless HyTens 1000 steel was subjectedto deformations under various proportional strain paths and strain rates. It was shownexperimentally that the material is sensitive both to dynamic and static strain ageing. Aphenomenological model accounting for these effects was developed, calibrated, implementedin a Finite Element software and, finally, validated.Both direct methods and inverse analyses were used in order to calibrate the parameters inthe material models. The agreement between the numerical and experimental results are ingeneral very good.
This thesis is divided into two main parts. The background, theoretical framework and
mechanical experiments are presented in the first part. In the second part, two papers are
appended.
plastic anisotropy, non-linear strain paths, mixed isotropic-kinematic hardening,isotropic-distortional hardening, dynamic strain ageing, static strain ageing, marten-sitic transformation
ISSN0280–7971
ISRN
LIU–TEK–LIC–2011:14
ISBN
978–91–7393–205–9
Nyckelord
Keyword
Sammanfattning
Abstract
Forfattare
Author
Titel
Title
URL for elektronisk version
Serietitel och serienummer
Title of series, numbering
Sprak
Language
Svenska/Swedish
Engelska/English
Rapporttyp
Report category
Licentiatavhandling
Examensarbete
C-uppsats
D-uppsats
Ovrig rapport
Avdelning, InstitutionDivision, Department
DatumDate
