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Doctoral Thesis
DOI
10.11606/T.82.2014.tde-17042015-153207
Document
Author
Full name
Uziel Paulo da Silva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2014
Supervisor
Committee
Aguiar, Adair Roberto (President)
Carvalho, Jonas de
Proença, Sergio Persival Baroncini
Savassi, Walter
Silva, Orivaldo Lopes da
Title in Portuguese
Emprego do método de homogeneização assintótica no cálculo das propriedades efetivas de estruturas ósseas
Keywords in Portuguese
Estrutura óssea
Método de homogeneização assintótica
Modelagem multiescala
Propriedades efetivas
Abstract in Portuguese
Ossos são sólidos não homogêneos com estruturas altamente complexas que requerem uma modelagem multiescala para entender seu comportamento eletromecânico e seus mecanismos de remodelamento. O objetivo deste trabalho é encontrar expressões analíticas para as propriedades elástica, piezoelétrica e dielétrica efetivas de osso cortical modelando-o em duas escalas: microscópica e macroscópica. Utiliza-se o Método de Homogeneização Assintótica (MHA) para calcular as constantes eletromecânicas efetivas deste material. O MHA produz um procedimento em duas escalas que permite obter as propriedades efetivas de um material compósito contendo uma distribuição periódica de furos cilíndricos circulares unidirecionais em uma matriz piezoelétrica linear e transversalmente isotrópica. O material da matriz pertence à classe de simetria cristalina 622. Os furos estão centrados em células de uma matriz periódica de secções transversais quadradas e a periodicidade é a mesma em duas direções perpendiculares. O compósito piezoelétrico está sob cisalhamento antiplano acoplado a um campo elétrico plano. Os problemas locais que surgem da análise em duas escalas usando o MHA são resolvidos por meio de um método da teoria de variáveis complexas, o qual permite expandir as soluções correspondentes em séries de potências de funções elípticas de Weierstrass. Os coeficientes das séries são determinados das soluções de sistemas lineares infinitos de equações algébricas. Truncando estes sistemas infinitos até uma ordem finita de aproximação, obtêm-se fórmulas analíticas para as constantes efetivas elástica, piezoelétrica e dielétrica, que dependem da fração de volume dos furos e de um fator de acoplamento eletromecânico da matriz. Os resultados numéricos obtidos a partir destas fórmulas são comparados com resultados obtidos pelas fórmulas calculadas via método de Mori-Tanaka e apresentam boa concordância. A boa concordância entre todas as curvas obtidas via MHA sugere que a expressão correspondente da primeira aproximação fornece uma fórmula muito simples para calcular o fator de acoplamento efetivo do compósito. Os resultados são úteis na mecânica de osso.
Title in English
Using the asymptotic homogenization method to evaluate the effective properties of bone structures
Keywords in English
Asymptotic homogenization method
Bone structure
Effective properties
Multiscale modeling
Abstract in English
Bones are inhomogeneous solids with highly complex structures that require multiscale modeling to understand its electromechanical behavior and its remodeling mechanisms. The objective of this work is to find analytical expressions for the effective elastic, piezoelectric, and dielectric properties of cortical bone by modeling it on two scales: microscopic and macroscopic. We use Asymptotic Homogenization Method (AHM) to calculate the effective electromechanical constants of this material. The AHM yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linear transversely isotropic piezoelectric matrix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The piezoelectric composite is under antiplane shear deformation together with in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method, which allows us to expand the corresponding solutions in power series of Weierstrass elliptic functions. The coefficients of these series are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical formulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori-Tanaka approach and show good agreement. The good agreement between all curves obtained via AHM suggests that the corresponding expression of first approximation provides a very simple formula to calculate the effective coupling factor of the composite. The results are useful in bone mechanics.
 
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Publishing Date
2015-07-15
 
WARNING: The material described below relates to works resulting from this thesis or dissertation. The contents of these works are the author's responsibility.
  • AGUIAR, A. R., et al. EFFECTIVE ELECTROMECHANICAL PROPERTIES OF 622 PIEZOELECTRIC MEDIUM WITH UNIDIRECTIONAL CYLINDRICAL HOLES [doi:10.1115/1.4023475]. Journal of Applied Mechanics [online], 2013, p. 1-11.
  • SEVOSTIANOV, IGOR, DA SILVA, UZIEL PAULO, and AGUIAR, ADAIR ROBERTO. Green-s function for piezoelectric 622 hexagonal crystals [doi:10.1016/j.ijengsci.2014.06.001]. International Journal of Engineering Science [online], 2014, vol. 84, p. 18-28.
  • AGUIAR, A. R., et al. Effective Electromechanical Properties Of 622 Piezoelectric Medium with Unidirectional Cylindrical Holes. In 12th Pan American Congress of Applied Mechanics, Port of Spain, 2012. APPLIED MECHANICS IN THE AMERICAS 2012., 2012. Abstract. Available from: http://www.pacamxii.org.
  • AGUIAR, A. R., et al. Effective Electromechanical Properties of a Medium Containing Unidirectional Cylindrical Holes Embedded in a 622 Piezoelectric Matrix. In 49th Annual Technical Meeting of the Society of Engineering Science (SES 2012), Atlanta, GA. Society of Engineering Science - 49th Annual Technical Meeting.Atlanta, GA : Georgia Institute of Technology, 2012. Resumo. Dispon?vel em: http://ses2012.gatech.edu/sites/ses2012.org/files/images/SES_Abstract_Book_FINAL_lores.pdf.
  • Silva, U. P, and AGUIAR, A. R. Multiscale Modeling to Assess Effective Electromechanical Properties of Bone - See more at: http://www.proceedings.blucher.com.br/article-details/multiscale-modeling-to-assess-effective-electromechanical-properties-of-bone-10770#sthash.9SwaXUsF.dpuf. In 13th International Symposium on Multiscale, Multifunctional and Functionally Graded Materials - MM&FGM 2014, Atibaia, SP, 2014. 13th International Symposium on Multiscale, Multifunctional and Functionally Graded Materials.São Paulo : Blucher, 2014. Abstract. Available from: http://www.proceedings.blucher.com.br/article-details/multiscale-modeling-to-assess-effective-electromechanical-properties-of-bone-10770.
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