Homogenization, linearization and dimension reduction in elasticity with variational methods [Elektronische Ressource] / Stefan Minsu Neukamm

Stefan Neukamm

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2010
• Nombre de lectures: 32
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Technische Universit¨at Mu¨nchen
Zentrum Mathematik
Homogenization, linearization and dimension
reduction in elasticity with variational methods
Stefan Minsu Neukamm
Vollsta¨ndiger Abdruck der von der Fakult¨at fu¨r Mathematik der Technischen
Universit¨atMu¨nchenzurErlangungdesakademischenGradeseines
Doktors der Naturwissenschaften (Dr.rer.nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof.Dr. Folkmar Bornemann
Pru¨fer der Dissertation: 1. Univ.-Prof.Dr. Martin Brokate
2. Univ.-Prof. Gero Friesecke, Ph.D.
3. Univ.-Prof.Dr. Stefan Mu¨ller,
Rheinische Friedrich-Wilhelms-Universit¨at Bonn
Die Dissertation wurde am 19.05.2010 bei der Technischen Universit¨at Mu¨nchen
eingereichtunddurchdieFakulta¨tfu¨rMathematikam20.09.2010angenommen.Acknowledgments
I would like to thank my supervisor Prof.Martin Brokate for helping and encouraging
me — not only during the time I spent on this thesis, but also as a mentor in the inte-
grateddoctoralprogramTopMaththatIjoinedinautumn2005. Duringthisperiodof
time, he accompanied and supported my academic adolescence.
Special thanks go to Prof.Stefan Mu¨ller for his encouraging support and the stimulat-
ing discussions during several visits at the Hausdorff Center for Mathematics at the
University of Bonn. The collaboration with him on the topic of the commutability of
linearization and homogenization hasbeenapreciousandshapingsourceofknowledge
and inspiration.
Furthermore, I would like to thank Prof.Gero Friesecke for his great support and con-
siderate advice. I remember many interesting and valuable discussions, often starting
with a chance meeting, but lasting several hours and spanning various topics. These
encounters have always been refreshing, inspiring and of great importance for this
thesis.
IgratefullyacknowledgethefinancialsupportfromtheDeutsche Graduiertenf¨orderung
through a national doctoral scholarship. The participation in the integrated doctoral
program TopMath was a privilege for me. In this context, I would like to thank
Dr. Christian Kredler, Dr. Ralf Franken and Andrea Echtler for the organizational
effort.
Iwouldliketothankallmyfriendsandcolleaguesfortheirdirectandindirectsupport.
Inparticular,IwouldliketothankPhilippStelzigandThomasRocheforproofreading
partsofthisthesis,andthemembersoftheresearchunitM6forenduringmepracticing
violin in the seminar room.
I would like to thank my sister for tips and suggestions on how to write in English;
and for preparing my parents for the intricacies of writing a doctoral thesis. I would
like to express my sincere gratitude to my girlfriend for her patience, understanding
and encouragement.
My way to mathematics was not a straightforward one. After studying the violin for
twoyearsattheHochschule fu¨r Musik und Theater,Idecidedtoswitchtomathematics
and to continue the violin at the same time. I am very grateful to my parents for
alwaysallowingmetofollowmycuriosity,forunconditionalsupportandthepermanent
encouragement during my path of education.Contents
1. Introduction 3
I. Mathematical preliminaries 11
2. Two-scale convergence 13
2.1. Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . 16
2.2. Two-scale properties of piecewise constant approximations . . . . . . . . 21
2.3. Two-scale convergence and linearization . . . . . . . . . . . . . . . . . . 28
3. Integral functionals 35
3.1. Basic properties and lower semicontinuity . . . . . . . . . . . . . . . . . 35
3.2. Periodic integral functionals and two-scale lower semicontinuity . . . . . 37
3.3. Convex homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4. Gamma-convergence and the direct method 45
4.1. The direct method of the calculus of variations . . . . . . . . . . . . . . 45
4.2. Gamma-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
II. Variational multiscale methods for integral functionals 51
5. Linearization and homogenization commute in finite elasticity 53
5.1. Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2. Simultaneous linearization and homogenization of elastic energies . . . . 58
5.3. Proof of the main results. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.1. Equi-coercivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2. Proof of Theorem 5.1.2 . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.3. Proof of Theorem 5.1.3 . . . . . . . . . . . . . . . . . . . . . . . 72
5.4. Proof of Theorem 5.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6. Two-scale convergence methods for slender domains 77
6.1. Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2. Two-scale convergence suited for in-plane oscillations . . . . . . . . . . . 79
6.3. Two-scale limits of scaled gradients. . . . . . . . . . . . . . . . . . . . . 81
6.3.1. Recovery sequences for auxiliary gradients . . . . . . . . . . . . . 87
6.3.2. A Korn inequality for the space of auxiliary gradients . . . . . . 89
6.4. Homogenization and dimension reduction of a convex energy . . . . . . 93
1Contents
III. Dimension reduction and homogenization in the bending regime 97
7. Derivation of a homogenized theory for planar rods 99
7.1. Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2. The scaled formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3. A qualitative picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3.1. Ansatzes ignoring oscillations . . . . . . . . . . . . . . . . . . . . 113
7.3.2. Ansatzes featuring oscillations . . . . . . . . . . . . . . . . . . . 116
7.4. Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.4.1. Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.4.2. Two-scale characterization of the limiting strain . . . . . . . . . 133
7.4.3. Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.4. Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.4.5. Cell formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.5. Strong two-scale convergence of the nonlinear strain for low energy se-
quences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.6. Interpretation of the limiting models . . . . . . . . . . . . . . . . . . . . 156
7.7. Advanced applications: Layered and prestressed materials . . . . . . . . 158
7.7.1. Sharpness of the two-scale characterization of the limiting strain 158
7.7.2. Application to layered, prestressed materials . . . . . . . . . . . 161
8. Derivation of a homogenized Cosserat theory for inextensible rods 167
8.1. Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . 167
8.2. Compactness and two-scale characterization of the nonlinear limiting
strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.2.1. Proof of the Theorem 8.1.1: Compactness . . . . . . . . . . . . . 174
8.2.2. Approximation of the scaled gradient. . . . . . . . . . . . . . . . 175
8.2.3. Two-scale characterization of the limiting strain . . . . . . . . . 178
8.3. Γ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3.1. Proof of Theorem 8.1.1: Lower bound . . . . . . . . . . . . . . . 183
8.3.2. Proof of Theorem 8.1.1: Recovery sequence . . . . . . . . . . . . 183
9. Partial results for homogenized plate theory 189
9.1. Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . 189
9.2. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
A. Appendix 197
A.1. Poincar´e and Korn inequalities . . . . . . . . . . . . . . . . . . . . . . . 197
A.2. Attouch’s diagonalization lemma . . . . . . . . . . . . . . . . . . . . . . 198
A.3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
21. Introduction
The main objective of this thesis is the derivation of effective theories for thin elastic
bodies featuring periodic microstructures, starting from nonlinear three-dimensional
elasticity. Our approach is based on the variational point of view and the derivation
is expressed in the language of Γ-convergence. A peculiarity of thin elastic objects is
their capability to undergo large deformations at low energy. In this thesis we are par-
1ticularly interested in regimes leading to limiting theories featuring this phenomenon .
Mathematically, this corresponds to a scaling of the energy that leads to a lineariza-
2tion effect in the limiting process.
Ourmainresultistherigorous, ansatzfreederivation of a homogenized Cosserat
3theory for inextensible rods as a Γ-limit of nonlinear three-dimensional elasticity.
The starting point of our derivation is an energy functional that describes an elastic
body with a periodic material microstructure with small period, say ε. We suppose
3that the elastic body is slender and occupies a thin cylindrical domain in R with
small diameter h. A special feature of this setting is the presence of the two small
length scales ε and h. We prove that the associated energy sequence converges to a