Homogenization and Asymptotic Expansions for Solutions of the Elasticity System with Rapidly Oscillating Periodic Coefficients

Oleinik, A.; Panasenko, Grigory; Yosifian, G. A.

doi:10.1080/00036818308839437

Cited by 15 publications

(9 citation statements)

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“…Since the 1970s, the mathematical theory of homogenization has developed and is used as an alternative approach to find the effective properties of the equivalent homogenized material [24][25][26][27]. It is applicable to all kinds of processes that might occur in periodic media, such as elastic vibrations, heat propagation, diffusion, fluid percolation, electromagnetic oscillations and radiation.…”

Section: Two-scale Asymptotic Homogenization Methods (Ahm)mentioning

confidence: 99%

“…It is applicable to all kinds of processes that might occur in periodic media, such as elastic vibrations, heat propagation, diffusion, fluid percolation, electromagnetic oscillations and radiation. From a mathematical point of view, the theory of homogenization is a limit theory which uses the asymptotic expansion and the assumption of periodicity to substitute the differential equations with rapidly oscillating coefficients, with differential equations whose coefficients are constant or slowly varying in such a way that the solutions are close to the initial equations [26].…”

Section: Two-scale Asymptotic Homogenization Methods (Ahm)mentioning

confidence: 99%

“…Using the double-scale asymptotic expansion, the displacements (u ε i (x)) are expressed as [25][26][27]:…”

Section: Two-scale Asymptotic Homogenization Methods (Ahm)mentioning

confidence: 99%

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Delamination influence on elastic properties of laminated composites

et al. 2018

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The present work aims to predict the behavior of effective elastic properties for laminated composites, considering localized damage in the interface between two layers. In practical terms, the damage in the adhesion, which influences the effective elastic properties of a laminate, is evaluated like a delamination between adjacent layers. Thus, the effective properties of laminated composites with different delamination extensions are calculated via finite element method and two-scale asymptotic homogenization method. It is investigated how the properties of the laminated composites are affected by the delamination extension and the thickness of the interface between layers. It is possible to conclude that the effective coefficient values decrease as the damage extension increases due to the fact that the delamination area increases. Besides, for all effective coefficients, except the effective coefficients C * 12 , C * 13 , and C * 23 , in the case without delamination, the coefficients decrease as the adhesive region thickness increases, and almost all coefficients decrease for complete separation of the interface. Numerical and analytical results are compared in order to show the potentialities and limitations of the proposed approaches. Finally, a numerical approach is used to simulate a specific case, where the interface is considered a functionally graded material.

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Section: Two-scale Asymptotic Homogenization Methods (Ahm)mentioning

confidence: 99%

Section: Two-scale Asymptotic Homogenization Methods (Ahm)mentioning

confidence: 99%

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Delamination influence on elastic properties of laminated composites

et al. 2018

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“…Different techniques have been used to estimate the effective properties of composites materials; the twoscale asymptotic expansion method [19,20] was applied by Galka et al [21] to compute macro behavior in thermo-piezoelectric composites. Further research activities have focused on studies on the micro-scale, where different approaches [22][23][24][25][26][27][28][29][30] have been considered for describing perfect and imperfect adhesion with a uniform interface between the constituents. A mathematical structure was developed to calculate the mechanical behavior of inhomogeneous media under the statement of an ordered microstructure with perfect contact.…”

Section: Introductionmentioning

confidence: 99%

“…The theoretical details of AHM have been rigorously developed in previous studies, e.g., Refs. [19,20,22,23,30]. The general method to calculate the effective properties is performed assuming the point group 2 mm for material symmetry.…”

Section: Introductionmentioning

confidence: 99%

Behavior of piezoelectric layered composites with mechanical and electrical non-uniform imperfect contacts

et al. 2020

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In this work, the two scales asymptotic homogenization method (AHM) is applied for determining the effective coefficients of laminated piezoelectric composite with periodic structure under nonuniform electrical and mechanical imperfect contact conditions. The analytical expressions of the local problems and the effective coefficients as result of the AHM are explicitly described. The constituent materials have properties belonging to 2 mm symmetry point group. Numerical values of the effective coefficients are reported and compared with limit cases, where perfect and uniform imperfect contact conditions are considered. Good agreements are found for these comparisons. Hence, the effect of the nonuniform imperfect contact conditions on the effective coefficients can be analyzed.

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On homogenization problems

Oleinik

1984

Lecture Notes in Physics

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The theory of homogenization for ordinary differential equations has been developed in connection with problems in mechanics by N.N.Bogo- In the theory of elasticity, the theory of composite and perfora- We also consider in detail the system of linear elasticity.Let V be a real reflexive separable Banach space and V' its dual.The value of a functional f ~ V' at an element v ~ V is denoted by < f,v > . We use llull E to denote the norm of an element u in a Banach space E

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Homogenization and Asymptotic Expansions for Solutions of the Elasticity System with Rapidly Oscillating Periodic Coefficients

Cited by 15 publications

References 1 publication

Delamination influence on elastic properties of laminated composites

Delamination influence on elastic properties of laminated composites

Behavior of piezoelectric layered composites with mechanical and electrical non-uniform imperfect contacts

On homogenization problems

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