Relevant bibliographies by topics / Schwarz Lemma and generalization

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Schwarz Lemma and generalization'

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Published: 10 March 2023

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Journal articles on the topic "Schwarz Lemma and generalization"

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Joseph, James E., and Myung H. Kwack. "A Generalization of the Schwarz Lemma to Normal Selfaps of Complex Spaces." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 1 (February 2000): 10–18. http://dx.doi.org/10.1017/s1446788700001543.

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Svetlik, Marek. "A note on the Schwarz lemma for harmonic functions." Filomat 34, no. 11 (2020): 3711–20. http://dx.doi.org/10.2298/fil2011711s.

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In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given.
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Roth, Oliver. "The Nehari-Schwarz lemma and infinitesimal boundary rigidity of bounded holomorphic functions." Studia Universitatis Babes-Bolyai Matematica 67, no. 2 (June 8, 2022): 285–94. http://dx.doi.org/10.24193/subbmath.2022.2.05.

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"We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self{maps of the unit disk. In particular, we discuss the case of in nitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary."
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Bisi, Cinzia, and Caterina Stoppato. "Landau’s theorem for slice regular functions on the quaternionic unit ball." International Journal of Mathematics 28, no. 03 (March 2017): 1750017. http://dx.doi.org/10.1142/s0129167x17500173.

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During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].
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Zhu, Jian-Feng. "Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings." Filomat 32, no. 15 (2018): 5385–402. http://dx.doi.org/10.2298/fil1815385z.

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In this paper, we first improve the boundary Schwarz lemma for holomorphic self-mappings of the unit ball Bn, and then we establish the boundary Schwarz lemma for harmonic self-mappings of the unit disk D and pluriharmonic self-mappings of Bn. The results are sharp and coincides with the classical boundary Schwarz lemma when n = 1.
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Yang, Yan, and Tao Qian. "Schwarz lemma in Euclidean spaces." Complex Variables and Elliptic Equations 51, no. 7 (July 2006): 653–59. http://dx.doi.org/10.1080/17476930600688623.

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Edigarian, Armen, and Włodzimierz Zwonek. "Schwarz lemma for the tetrablock." Bulletin of the London Mathematical Society 41, no. 3 (March 22, 2009): 506–14. http://dx.doi.org/10.1112/blms/bdp022.

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Ratto, Andrea, Marco Rigoli, and Laurent Veron. "extensions of the Schwarz Lemma." Duke Mathematical Journal 74, no. 1 (April 1994): 223–36. http://dx.doi.org/10.1215/s0012-7094-94-07411-5.

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Xu, Zhenghua. "Schwarz lemma for pluriharmonic functions." Indagationes Mathematicae 27, no. 4 (September 2016): 923–29. http://dx.doi.org/10.1016/j.indag.2016.06.002.

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Huang, Ziyan, Di Zhao, and Hongyi Li. "A boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions." Filomat 34, no. 9 (2020): 3151–60. http://dx.doi.org/10.2298/fil2009151h.

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In this paper, we present a boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions, which extends the classical Schwarz lemma for bounded harmonic functions to higher dimensions.
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Dissertations / Theses on the topic "Schwarz Lemma and generalization"

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Terenzi, Gloria. "Lemma di Schwarz e la sua interpretazione geometrica." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13543/.

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Il tema centrale di questa tesi, suddivisa in tre capitoli, è il Lemma di Schwarz e la sua applicazione nella geometria iperbolica. Il lemma di Schwarz, che prende il nome da Hermann Amandus Shchwarz, descrive una proprietà delle funzioni olomorfe. Nel primo capitolo enuncio il Lemma di Schwarz e la sua versione infinitesimale. Descrivo le mappe conformi del dominio per poi applicare il lemma di Pick che è una forma particolare del lemma di Schwarz.Nel secondo capitolo introduco brevemente la geometria euclidea con i cinque postulati di Euclide, per poi passare a descrivere la geometria iperbolica. Introduco la definizione di forma fondamentale (o forma metrica) di una superficie. Nel terzo capitolo affronto la geometria iperbolica nel disco. Quindi data una forma metrica ho definito distanza iperbolica e lunghezza iperbolica per poi arrivare a dimostrare tramite una reinterpretazione del lemma di Schwarz l'invarianza delle mappe olomorfe.
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Bacca, Salvatore. "Il lemma di Schwarz e la distanza di Kobayashi." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13823/.

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Questa tesi è un approccio elementare alla teoria geometrica delle funzioni, campo che ebbe inizio con i lavori di Poincarè sulla geometria del disco. Affronteremo dapprima il Lemma di Schwarz ed alcune sue generalizzazioni che metteranno in correlazione il risultato analitico di tale asserto con il suo aspetto geometrico-differenziale. Introdurremo poi una distanza invariante su varietà complesse, la distanza di Kobayashi, e tramite questa dimostreremo i teoremi di Picard riguardanti il range dell'immagine di funzioni olomorfe sul piano complesso o su un dominio avente una singolarità isolata.
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Barros, Jéssica Laís Calado de. "O teorema da aplicação de Riemann: uma prova livre de integração." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-13122017-161946/.

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Neste trabalho, seguindo a abordagem de Weierstrass, temos o objetivo de responder a seguinte questão: conhecida a equivalência entre holomorfia e analiticidade no caso complexo, quais propriedades das funções analíticas podem ser obtidas sem assumir tal equivalência? Analisando esta situação, resultados interessantes serão obtidos sem o uso de qualquer teorema de integração complexa e, para alcançar tal objetivo, nossas principais ferramentas serão a teoria de somas não ordenadas de famílias em C e propriedades do índice de caminhos fechados. Entre os resultados apresentados estão os conhecidos Teorema Fundamental da Álgebra, Lema de Schwarz, Teorema de Montel, Teorema da Série Dupla de Weierstrass, Princípio do Argumento, Teorema de Rouché, Teorema da Fatoração de Weierstrass, Pequeno Teorema de Picard e o Teorema da Aplicação de Riemann.
In this work, following the Weierstrass\'s approach, we aim to answer the following question: knowing the equivalence between holomorphy and analyticity in the complex case, which properties of analytic functions can be obtained without assuming such equivalence? Through analyzing this situation, interesting results will be obtained without employing of any complex integration theorem and in order to achieve this goal, our main tools will be the theory of unordered sums in C and properties of winding numbers of closed paths. Among the proven results are the well known Fundamental Theorem of Algebra, Schwarz\'s Lemma, Montel\'s Theorem, Weierstrass\'s Double Series Theorem, Argument Principle, Rouché\'s Theorem, Weierstrass\'s Factorization Theorem, Picard\'s Little Theorem and the Riemann\'s Mapping Theorem.
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N'Doye, Ibrahima. "Généralisation du lemme de Gronwall-Bellman pour la stabilisation des systèmes fractionnaires." Phd thesis, Université Henri Poincaré - Nancy I, 2011. http://tel.archives-ouvertes.fr/tel-00584402.

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Dans ce mémoire, nous avons proposé une méthode basée sur l'utilisation de la généralisation du lemme de Gronwall-Bellman pour garantir des conditions suffisantes de stabilisation asymptotique pour une classe de systèmes non linéaires fractionnaires. Nous avons étendu ces résultats dans la stabilisation asymptotique des systèmes non linéaires singuliers fractionnaires et proposé des conditions suffisantes de stabilité asymptotique de l'erreur d'observation dans le cas de l'étude des observateurs pour les systèmes non linéaires fractionnaires et singuliers fractionnaires. Pour les systèmes non linéaires à dérivée d'ordre entier, nous avons proposé par l'application de la généralisation du lemme de Gronwall-Bellman des conditions suffisantes pour : - la stabilisation exponentielle par retour d'état statique et par retour de sortie statique, - la stabilisation exponentielle robuste en présence d'incertitudes paramétriques, - la commande basée sur un observateur. Nous avons étudié la stabilisation des systèmes linéaires fractionnaires avec les lois de commande suivantes~: retour d'état statique, retour de sortie statique et retour de sortie basé sur un observateur. Puis, nous avons proposé des conditions suffisantes de stabilisation lorsque le système linéaire fractionnaire est affecté par des incertitudes non linéaires paramétriques. Enfin, nous avons traité la synthèse d'un observateur pour ces systèmes. Les résultats proposés pour les systèmes linéaires fractionnaires ont été étendus au cas où ces systèmes fractionnaires sont singuliers. La technique de stabilisation basée sur l'utilisation de la généralisation du lemme de Gronwall-Bellman est étendue aux systèmes non linéaires fractionnaires et aux systèmes non linéaires singuliers fractionnaires. Des conditions suffisantes de stabilisation asymptotique, de stabilisation asymptotique robuste et de commande basée sur un observateur ont été obtenues pour les classes de systèmes non linéaires fractionnaires et non linéaires singuliers fractionnaires. Par ailleurs, une méthode de synthèse d'observateurs pour ces systèmes non linéaires fractionnaires et non linéaires singuliers fractionnaires est proposée. Cette approche est basée sur la résolution d'un système d'équations de Sylvester. L'avantage de cette méthode est que, d'une part, l'erreur d'observation ne dépend pas explicitement de l'état et de la commande du système et, d'autre part, qu'elle unifie la synthèse d'observateurs de différents ordres (observateurs d'ordre réduit, d'ordre plein et d'ordre minimal).
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Arman, Andrii. "Generalizations of Ahlfors lemma and boundary behavior of analytic functions." 2013. http://hdl.handle.net/1993/22095.

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In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.
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SARFATTI, GIULIA. "Elements of function theory in the unit ball of quaternions." Doctoral thesis, 2013. http://hdl.handle.net/2158/806320.

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The theory of regular functions over the quaternions introduced by Gentili and Struppa in 2006, already quite rich, is in continuous development. Despite their diverse peculiarities, regular functions reproduce numerous properties of holomorphic functions of one complex variable. This Thesis is devoted to investigate properties of regular functions defined on the unit ball B of the quaternions H. As it happens in the complex case, this particular subset of H represents a special domain for the class of regular function. It is the simplest example of the most natural set of definition for a regular function, namely of a "symmetric slice domain". Furthermore, on open balls centred at the origin, regular functions are characterized by having a power series expansion, hence they behave very nicely. The first Chapter, starting from the very first definitions, includes all the preliminary results that will be used in the sequel. The second Chapter discusses some properties of the modulus of regular functions, in particular how it is related with the modulus of the "regular conjugate" of a regular function. The main result presented is an analogue of the Borel-Carathéodory Theorem, a tool useful to bound the modulus of a regular function by means of the modulus of its real part. The central part of the Thesis contains geometric theory results. The third Chapter contains the analogue of the Bohr Theorem concerning power series, together with a weaker version, that follows as in the complex case from the Borel-Carathéodory Theorem. In the fourth Chapter we prove a Bloch-Landau type theorem, showing that in some sense the image of a ball under a regular function can not be too much thin. The fifth Chapter is dedicated to Landau-Toeplitz type theorems, that study the possible shapes that the image of a regular function can assume. The last Chapter is devoted to the study of the quaternionic Hardy spaces. We begin by the definition of the spaces H^p(B) and H^{\infty}(B), then we prove some of their basic properties. We introduce in conclusion the Corona Problem in the quaternionic setting, proving a partial statement of the Corona Theorem.
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Lin, Cheng-Tsai, and 林成財. "Schwarz Lemma on Symmetrized Bidisc." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/05462082649779495998.

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碩士
東海大學
數學系
89
Let $\Gamma$ denote the set of symmetrized bidisc. In this thesis we discuss the Schwarz lemma on $\Gamma$ also known as the special flat problem on $\Gamma$ as: Given $\alpha_{2}\in\mathbb{D},~\alpha_{2}\neq0~$ and $(s_{2},p_{2})\in\Gamma$, find an analytic function $\varphi:\mathbb{D}\rightarrow\Gamma$with $\varphi(\lambda)=(s(\lambda),p(\lambda))$ satisfies $$\varphi(0)=(0,0),~\varphi(\alpha_{2})=(s_{2},p_{2})$$ Based on the equality of Carath\'odory and Kobayashi distances, and the Schur's theorem, we construct an analytic function $\varphi$ to solve this problem. Keywords: Spectral Nevanlinna-Pick interpolation, Poincar\'{e} distance, Carath\'odory distance, Kobayashi distance, Symmetrized bidisc, Schwarz lemma.
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Erickson, John D. Ph D. "Generalization, lemma generation, and induction in ACL2." Thesis, 2008. http://hdl.handle.net/2152/4004.

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Formal verification is becoming a critical tool for designing software and hardware today. Rising complexity, along with software's pervasiveness in the global economy have meant that errors are becoming more difficult to find and more costly to fix. Among the formal verification tools available today, theorem provers offer the ability to do the most complete verification of the most complex systems. However, theorem proving requires expert guidance and typically is too costly to be economical for all but the most mission critical systems. Three major challenges to using a theorem prover are: finding generalizations, choosing the right induction scheme, and generating lemmas. In this dissertation we study all three of these in the context of the ACL2 theorem prover.
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SAMBUSETTI, Andrea. "Aspetti topologici e geometrici di un lemma di Schwarz in geometria riemanniana." Doctoral thesis, 1998. http://hdl.handle.net/11573/221025.

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Chandel, Vikramjeet Singh. "The Pick-Nevanlinna Interpolation Problem : Complex-analytic Methods in Special Domains." Thesis, 2017. http://etd.iisc.ernet.in/2005/3700.

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The Pick–Nevanlinna interpolation problem, in its fullest generality, is as follows: Given domains D1, D2 in complex Euclidean spaces, and a set f¹ zi; wiº : 1 i N g D1 D2, where zi are distinct and N 2 š+, N 2, find necessary and sufficient conditions for the existence of a holomorphic map F : D1 ! D2 such that F¹ziº = wi, 1 i N. When such a map F exists, we say that F is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem — which we shall study in this thesis — have been of lasting interest: Interpolation from the polydisc to the unit disc. This is the case D1 = „n and D2 = „, where „ denotes the open unit disc in the complex plane and n 2 š+. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case n = 1. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for n 2, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur– Agler class. This is notable because, when n = 2, the latter result completely solves the problem for the case D1 = „2; D2 = „. However, Pick’s approach can also be effective for n 2. In this thesis, we give an alternative characterization for the existence of a 3-point interpolant based on Pick’s approach and involving the study of rational inner functions. Cole–Lewis–Wermer lifted Sarason’s approach to uniform algebras — leading to a char-acterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of N N matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a „n-to-„ interpolant in terms of the positivity of a family of N N matrices parametrized by a class of polynomials. Interpolation from the unit disc to the spectral unit ball. This is the case D1 = „ and D2 = n , where n denotes the set of all n n matrices with spectral radius less than 1. The interest in this arises from problems in Control Theory. Bercovici–Foias–Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc — leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any n and N = 2. In this thesis, we shall present a necessary condition for the existence of an interpolant in the case when N = 3. This we shall achieve by adapting Pick’s approach and applying the aforementioned result of Bharali.
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Books on the topic "Schwarz Lemma and generalization"

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The Schwarz lemma. Oxford: Clarendon Press, 1989.

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Dineen, Seán. The Schwarz lemma. Mineola, New York: Dover Publications, 2016.

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Kim, Kang-Tae. Schwarz's lemma from a differential geometric viewpoint. Singapore: World Scientific, 2011.

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Shapiro, Harold S. The Schwarz function and its generalization to higher dimensions. New York: Wiley, 1992.

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Dineen, Seán. Schwarz Lemma. Dover Publications, Incorporated, 2016.

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Sogge, Christopher D. Improved spectral asymptotics and periodic geodesics. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160757.003.0005.

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This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral bands, details and proves a lemma, and gives a related generalization of the Weyl formula from Chapter 3 that involves pseudodifferential operators. The chapter then proves its main result by using a version of the Duistermaat-Guillemin theorem, which allows the use of the Hadamard parametrix and the arguments from Chapter 3. To conclude, the chapter shows that one can improve the sup-norm estimates from Chapter 3 if one assumes a condition on the geodesic flow that is similar to a hypothesis laid out in the Duistermaat-Guillemin theorem.
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Book chapters on the topic "Schwarz Lemma and generalization"

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Kodaira, Kunihiko. "Schwarz–Kobayashi Lemma." In SpringerBriefs in Mathematics, 19–38. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-6787-7_2.

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Kobayashi, Shoshichi. "Schwarz Lemma and Negative Curvature." In Grundlehren der mathematischen Wissenschaften, 19–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/978-3-662-03582-5_2.

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Elin, Mark, Fiana Jacobzon, Marina Levenshtein, and David Shoikhet. "The Schwarz Lemma: Rigidity and Dynamics." In Harmonic and Complex Analysis and its Applications, 135–230. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-01806-5_3.

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Gamelin, Theodore W. "The Schwarz Lemma and Hyperbolic Geometry." In Undergraduate Texts in Mathematics, 260–73. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-0-387-21607-2_9.

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Burckel, Robert B. "Schwarz’ Lemma and its Many Applications." In Classical Analysis in the Complex Plane, 397–456. New York, NY: Springer US, 2021. http://dx.doi.org/10.1007/978-1-0716-1965-0_7.

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Moriya, Katsuhiro. "The Schwarz Lemma for Super-Conformal Maps." In Hermitian–Grassmannian Submanifolds, 59–68. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-5556-0_6.

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Iwanuma, Koji, and Kenichi Kishino. "Lemma Generalization and Non-unit Lemma Matching for Model Elimination." In Advances in Computing Science — ASIAN’99, 163–76. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-46674-6_15.

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Balakrishnan, A. V. "A Generalization of the Kalman-Yakubovic Lemma." In Systems, Models and Feedback: Theory and Applications, 59. Boston, MA: Birkhäuser Boston, 1992. http://dx.doi.org/10.1007/978-1-4757-2204-8_5.

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Burgeth, Bernhard. "Schwarz Lemma Type Inequalities for Harmonic Functions in the Ball." In Classical and Modern Potential Theory and Applications, 133–47. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-1138-6_13.

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Wang, Wu-Sheng, Zhengfang Mo, and Zongyi Hou. "Generalization of Lemma Gronwall–Bellman on Retarded Integral Inequality." In Lecture Notes in Electrical Engineering, 749–56. London: Springer London, 2012. http://dx.doi.org/10.1007/978-1-4471-4790-9_98.

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Conference papers on the topic "Schwarz Lemma and generalization"

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Akyel, Tuğba, and Bülent Nafi Örnek. "On the rigidity part of Schwarz Lemma." In THIRD INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2019). AIP Publishing, 2019. http://dx.doi.org/10.1063/1.5136123.

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Örnek, Bülent Nafi, and Tuğba Akyel. "An application of Schwarz Lemma for analytic functions in the unit disc." In 10TH INTERNATIONAL CONFERENCE ON APPLIED SCIENCE AND TECHNOLOGY. AIP Publishing, 2022. http://dx.doi.org/10.1063/5.0117524.

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Sun, Ningxin, and Haiyan Wang. "A version of Schwarz lemma based on Cauchy integral formula in octonionic analysis." In 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2022), edited by Chi-Hua Chen, Xuexia Ye, and Hari Mohan Srivastava. SPIE, 2022. http://dx.doi.org/10.1117/12.2638800.

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Имомкулов, Севдиёр, and Усмон Собиров. "Some generalization of Hartogs's lemma about analytic extension of functions of several complex variables." In International scientific conference "Ufa autumn mathematical school - 2021". Baskir State University, 2021. http://dx.doi.org/10.33184/mnkuomsh1t-2021-10-06.44.

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Wang, Di, and Jinhui Xu. "Lower Bound of Locally Differentially Private Sparse Covariance Matrix Estimation." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/665.

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In this paper, we study the sparse covariance matrix estimation problem in the local differential privacy model, and give a non-trivial lower bound on the non-interactive private minimax risk in the metric of squared spectral norm. We show that the lower bound is actually tight, as it matches a previous upper bound. Our main technique for achieving this lower bound is a general framework, called General Private Assouad Lemma, which is a considerable generalization of the previous private Assouad lemma and can be used as a general method for bounding the private minimax risk of matrix-related estimation problems.
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Monchiet, V., T. H. Tran, and G. Bonnet. "Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-82060.

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A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a generalization of the Hill-Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. It is rigorously proved that the macroscopic quantities derived from this homogenization procedure comply with the equations of strain gradient elasticity. As an illustration, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed whose efficiency is shown in the particular case of composites reinforced by long fibers.
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