Multi-level Modelling of Plastic Anisotropy of Aluminium Alloys Using Crystal Plasticity Models and

216 Pages·2014·10.65 MB·English

by  Kai Zhang

Kai Zhang
Multi-level Modelling of Plastic
Anisotropy of Aluminium Alloys
Using Crystal Plasticity Models
and Advanced Yield Functions
Thesis for the degree of Philosophiae Doctor
Trondheim, June 2014
Norwegian University of Science and Technology
Faculty of Natural Science and Technology
Department of Materials Science and Engineering
NTNU
Norwegian University of Science and Technology
Thesis for the degree of Philosophiae Doctor
Faculty of Natural Science and Technology
Department of Materials Science and Engineering
© Kai Zhang
ISBN 978-82-326-0250-6 (printed ver.)
ISBN 978-82-326-0251-3 (electronic ver.)
ISSN 1503-8181
Doctoral theses at NTNU, 2014:164
IMT-Report 2014:207
Printed by NTNU-trykk
Preface
This thesis is submitted in fulfilmentof the requirements for the doctor of philosophy at
the Norwegian University of Science and Technology (NTNU). The work has been
carried out at the Department of Materials Science and Engineering (DMSE), NTNU
from March 2011 to March 2014. Professor Bjørn Holmedal at DMSE, NTNU was the
main supervisor. Professor Odd Sture Hopperstad at the Department of Structural
Engineering, NTNU, Dr. Stéphane Dumoulin at SINTEF Materials and Chemistry, and
Professor Knut Marthinsen at DMSE, NTNU, were the co-supervisors. The thesis
consists of two parts. The first part (Part I) includes the introduction, a short literature
review and a summary of the work. The second part (Part II) contains articles which
were published or prepared by the candidate during the PhD study. Four articles are
listed in the main body of Part II, which present the main work, results and academic
contributions of this PhD project. Two articles which were presented at international
conferences are attached in the appendixof Part II.
The articlescontained in this thesis:
1. “Multi-level Modelling of Mechanical Anisotropy of Commercial Pure Aluminium
Plate: Crystal Plasticity Models, Advanced Yield Functions and Parameter
Identification”, International Journal of Plasticity, 2014. DOI:
10.1016/j.ijplas.2014.02.003
2. “Use of Plane-strain Tension and Shear Tests to Evaluate Yield Surfaces for AA1050
Aluminium Sheet’’, acceptedafter peer-review forthe 14th International Conference on
Aluminium Alloys (ICAA14). The conference proceeding will appear in the journal:
Materials Science Forum.
3. “A Robust and Efficient Substepping Scheme for the Explicit Numerical Integration
of a Rate-dependent Crystal Plasticity Model”, International Journal for Numerical
Methods in Engineering, 2014.DOI: 10.1002/nme.4671
4. “Modelling the Plastic Anisotropy of Aluminium Alloy 3103 Sheets by Polycrystal
Plasticity Models”, submitted to: Modelling and Simulation in Materials Science and
Engineering.
The articles contained in the Appendix:
I. “An Explicit Integration Scheme for Hypo-elastic Viscoplastic Crystal Plasticity”,
1stAsian Conference on Aluminium Alloys (ACAA-2013), accepted for publication in
the journal:Transactions of Nonferrous Metals Society of China, 2014.
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II. “Crystal Plasticity Calculations of Mechanical Anisotropy of Aluminium Compared
to Experiments and to Yield Criterion Fittings”, 13th International Conference on
Aluminium Alloys (ICAA13), Pittsburgh, USA, 3-7 June, 2012,
doi:10.1002/9781118495292.ch137.
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Abstract
This thesis aims to accurately describe the plastic anisotropy of aluminium alloys
through a hierarchical multi-level method. Robust and efficient integration schemes
have been proposed for the explicit numerical integration of rate-dependent crystal
plasticity models.
On the mesoscale, the plastic anisotropy is modelled by crystal plasticity models
considering a representative volume element (RVE). The RVE consists of a number of
single grains and inherits the microstructuralinformation of the polycrystalline material,
e.g. crystallographic texture, grain size and shape and grain boundary misorientation.
Five crystal plasticity models have been used in this work, namely the full-constraint
(FC) Taylor model, the Alamel model, the Alamel model with so-called type III
relaxation (Alamel Type III), the visco-plastic self-consistent (VPSC) model and the
crystal plasticity finite element method (CPFEM). The accuracy and applicability of
these crystal plasticity models when predicting the plasticity anisotropy have been
investigated for three different aluminium alloys. On the continuum scale, the yield
surface of the material is represented by advanced yield functions. Two yield functions
have been employed and investigated for this purpose, namely the Yld2004-18p yield
function and the Facet yield function. The yield function is a key component of an
anisotropic model in a finite element method (FEM) code, in addition to the flow rule
and work hardening law, for simulating plastic deformations.Advanced yield functions,
like Yld2004-18p, are conventionally identified by experiments, e.g. uniaxial tensile
tests, biaxial tension/compression tests and shear tests. However, the number of
available experimental tests is limited for sheet metals and most of the stress space is
not covered by the experiments. The multi-level modelling was made through
identifying the parameters of the advanced yield functions partially or fully by stress
points at yielding provided by crystal plasticity calculations. The accuracy and
applicability of this multi-level modelling scheme were evaluated for describing the
plastic anisotropy of three aluminium alloy sheetsin this thesis.
In Article 1, the plastic anisotropy of a fully annealed AA1050 aluminium sheet is
studied by the use of five crystal plasticity models and two advanced yield functions.
The in-plane uniaxial tension properties of the sheet were predicted by the FC-Taylor
model, the Alamel–type models, the VPSCmodel andCPFEM. Results were compared
with data fromtensile tests at every 15° from the rolling direction (RD) to the transverse
direction (TD) of the plate. Furthermore, all the models, except CPFEM, were used to
provide stress points in the five-dimensional deviatoric stress space at yielding for 201
plastic strain-rate directions. The Facet yield surface was calibrated using these 201
stress points and compared to the in-plane yield loci and the planar anisotropy which
were calculated by the crystal plasticity models. The anisotropic yield function
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Yld2004-18p was calibrated by three methods: using uniaxial tension data, using the
201 virtual yield points in stress space, and using a combination of experimental data
and virtual yield points (i.e., a hybrid method). Optimal yield surface exponents were
found for each of the crystal plasticity models, based on calibration to calculated stress
points at yielding for a random texture, and used in thelatter two calibration methods. It
wasfound that the hybrid calibration method couldcapture the experimental results and
at the same time ensure a good fit to theanisotropy in the full stress space predicted by
the crystal plasticity models.
Plane-strain tension and shear tests were carried out for the same AA1050 sheet and
described in Article 2. The tests were simulated numerically with a commercial FEM
code using an anisotropic plasticity model including the Yld2004-18p yield function,
the associated flow rule and isotropic hardening. FEM simulations of the tests were
made with parameters of Yld2004-18p identified in Article 1 by three methods, i.e.
using uniaxial tension data combined with FC-Taylor model predictions of the
equibiaxial yield stress and r-value, using 201 virtual yield points in stress space
provided by the Alamel Type III model, and using a combination of experimental data
and virtual yield points. Predicted force-displacement curves were compared to the
experimental data, and the accuracy of the parameter identification methods for
Yld2004-18p was evaluated based on these comparisons. The results showed that the
hybrid method captured the initial yielding most accurately for both the plane-strain
tension and shear tests.
Similar studies as described in Article 1havebeen carried outon AA3103 sheets in the
cold-rolled condition (H18 temper) and in the fully annealed condition (O temper) in
Article 4. The plastic anisotropy of AA3103-H18 and AA3103-O sheets was studied
experimentally and numerically. The microstructure and texture of the two materials
were characterized and the anisotropic plastic behaviour was measured by in-plane
uniaxial tension tests along every 15° from RD to TD of the sheets. The same five
polycrystal plasticity models as used in Article 1 were employed to predict the plastic
anisotropy in the plane of the sheet. Experimentally observed grain shapes have been
taken into consideration. In addition, a multi-level modelling method was employed
where the advanced yield function Yld2004-18p was calibrated to stress points at
yielding provided by CPFEM simulations along 89 strain-paths, and the plastic
anisotropy was then produced by the yield function.Based on comparisons between the
experimental and the predicted results, the multi-level fitting method was found to be
the most accurate way of describing the plastic anisotropy. The Alamel Type III and
Alamel models were also recommended as accurate and time-efficient models for
predicting the plastic anisotropy of the AA3103 sheets in H18 and O tempers.
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Article 3 describes the development of efficient and robust numerical integration
schemes for rate-dependent crystal plasticity models. A forward Euler integration
algorithm was first formulated. An integration algorithm based on the modified Euler
method with an adaptivesubstepping scheme wasthen proposed, where the substepping
was mainly controlled by the local error of the stress predictions within the time step.
Both integration algorithms were implemented in a stand-alone code with the Taylor
aggregate assumption and in an explicit finite element code. The robustness, accuracy
and efficiency of the substepping scheme were extensively evaluated for large time
steps, extremely low strain-rate sensitivity, high deformation rates and strain-path
changes using the stand-alone code. The results showed that the substepping scheme is
robust and in some cases one order of magnitude faster than the forward Euler
algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite
element simulations for quasi-static problems wasalso discussed. Finally, simulation of
the Taylor bar impact test was carried out to show the applicability and robustness of
the proposed integration algorithm for the modelling of dynamic problems with contact.
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Acknowledgements
First of all, I would like to express my deepest gratitude to my principal supervisor Prof.
Bjørn Holmedal. Bjørn showed great patience when I was innocent of theoriesof crystal
plasticity at the beginning of my PhD study. Together with my co-supervisors, Bjørn
helped me find an interestingtopic and made long-terms plansformyPhD project. The
well-defined topic and plans have made my research progresssmoothly most of the time.
During my study, Bjørn has provided very professional and academic guidance. I also
enjoyed the time discussing with Bjørn and watching him illustrating his ideas on the
whiteboard in his office.Prof. Odd Sture Hopperstad,Dr. StéphaneDumoulinand Prof.
Knut Marthinsenare my co-supervisors who have played important roles in my project
and deserve my gratitude.Odd Sture is a prestigious scholar in the field of mechanics of
materials, computational plasticity and fracture. He has broad knowledge covering
topics of thePhD project. In particular, the substepping scheme was suggested by him.
He works in an efficient, structural and careful manner which prompted the revision and
publication of my articles.Stéphanepossesses enormous experience on the development
and application of CPFEM. He was very generous to provide the code and any help I
asked. It was a great pleasure to work with him. Knut has played as a valuable
discussion partner and provided many practical help, for example the workstation
computer on which I preformed most of the CPFEM calculations. Thank you, all my
supervisors!
I would like to thank Dr. Jerzy Gawad, Prof. Albert Van Bael and Prof. Paul Van
Houtte at MTM, KU Leuven, Belgium. I spent two weeks as a visiting scholar at their
group. It was a short but fruitful visit, which improved my understanding about the
Alamel model and the Facet method. They generously provided the Facet software
package which had producedsome interesting results.
Prof. Yanjun Li and Dr. Torodd Berstad are acknowledged for many inspiring
discussions.I would like to thank the staff for their technical assistance: Trond Auestad,
Yingda Yu, Torild Krogstad, and especially Pål Christian Skaret, with whom I did
plenty of mechanical tests.
I want to thank my colleagues in the department who created an environment full of
friendliness, knowledge and vigour. I enjoyed the time sitting in the same office with
Dr. Ke Huang. Discussions with Dr. Qinglong Zhao and Dr. Tomáš Mánik contributed
to my research. I knew more about Norwegian society and culture from Dr. Sindre
Bunkholt. I had a lot of fun together with Dr. Ning Wang, Dr. Nagaraj Vinayagam
Govindaraj and Dr. Emmanuel Hersent in our group. There are also many other
colleagues and friends to be acknowledged: Min Zha, Sarina Bao, Zebin Xu, Shenbao
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Jin, Xiaoguang Ma, Song Zhang, Yu Chen et al., who brought fun to my life in
Trondheim.
Special thanks to my family. Although living in a tiny village of the enormous China,
my parents always encouraged me to dream big and bigger. My wife Jingjing Gao
showed great patience and always encouraged meduring my PhD study.She makes me
feel warm in the deep of my heart every day. Finally, the new-comer in my life, my
beloved daughter Zhiting Zhang, needs a special acknowledgement. Thank you for all
the adventures and fun we had together.
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