Effective properties of periodic fibrous electro-elastic composites with mechanic imperfect contact condition

International audienceIn this work, two-phase parallel fiber-reinforced periodic piezoelectric composites are considered wherein the constituents exhibit transverse isotropy and the cells have different configurations. Mechanical imperfect contact at the interface of the composites is studied via linear spring model. The statement of the problem for two phase piezoelectric composites with mechanical imperfect contact is given. The local problems are formulated by means of the asymptotic homogenization method (AHM) and their solutions are found using complex variable theory. Analytical formulae are obtained for the effective properties of the composites with spring imperfect type of contact and different parallelogram cells. Some numerical examples and comparisons with other theoretical results illustrate that the model is efficient for the analysis of composites with presence of parallelogram cells and the aforementioned imperfect contact

Homogeneizacion reiterada de un problema de contorno unidimensional

The asymptotic homogenization method is applied to homogenize a one-dimensional family of elliptic boundary value problems with periodic and rapidly oscillating coefficients which depend on two fast variables. The homogenized problem, the local problems and the corresponding effective coefficient are obtained. A necessary and sufficient condition for constructing an asymptotic solution with periodic terms is demonstrated. Based on a Maximum Principle the proximity between the solutions of the homogenized and original problems is proved. Some numerical computations are used to illustrate the mathematical justificationEl metodo de homogeneizacion asintotica es aplicado para homogeneizar una familia unidimensional de problemas elıpticos, con coeficientes periodicos y rapidamente oscilantes que dependen de dos variables rapidas. El problema homogeneizado, los problemas locales y los correspondientes coeficientes efectivos son obtenidos. Una condicion necesaria y suficiente para la construccion de una solucion asintotica con terminos periodicos es demostrada. Basados en el principio del maximo, se demuestra la proximidad entre la soluci´on del problema homogeneizado y la del problema original. Se propone un ejemplo numerico para ilustrar la justificacion matematica

The Alestle - Vol. 64 No. 16 - 12/13/2011

Vol. 64 No. 1

A hierarchical asymptotic homogenization approach for viscoelastic composites

Effective properties of non-aging linear viscoelastic and hierarchical composites are investigated via a three-scale asymptotic homogenization method. In this approach, we consider the assumption of a generalized periodicity in the different structural levels and their characterization through the so-called stratified functions. The expressions for the associated local and homogenized problems, and the effective coefficients are derived at each level of organization by using the correspondence principle and the Laplace-Carson transform. Considering isotropic components and a perfect contact at the interfaces between the constituents, analytical solutions, in the Laplace-Carson space, are found for the local problems and the effective coefficients are computed. An interconversion procedure between the effective relaxation modulus and the effective creep compliance is carried out for obtaining information about both viscoelastic properties. The numerical inversion to the original temporal space is also performed. Finally, we exploit the potential of the approach and study the overall properties of a hierarchical viscoelastic composite structure representing the dermis

Comparação caso contínuo com o caso contínuo por partes com contato perfeito para a equação elíptica via método de homogeneização assintótica

The methods of mathematical homogenization allow the effective properties of heterogeneous media to be found with great precision and rigor based on the physical and geometric properties of their components. In particular, the asymptotic homogenization method is used to find the coefficients that represent the effective properties of a medium with a periodic structure. The present work aims to study this mathematical homogenization technique to obtain the effective behavior of micro-heterogeneous media, and to apply mathematical formalism to build a formal asymptotic solution of a one-dimensional linear problem with continuous and constant coefficients by parts. Still, the proximity between the solutions of the original and homogenized problems will be mathematically justified. In order to illustrate the theoretical results, an example is presented considering both types of heterogeneity in a case that presents the same effective behavior. Os métodos de homogeneização matemática permitem encontrar com grande precisão e rigor as propriedades efetivas de meios heterogêneos a partir das propriedades físicas e geométricas de seus componentes. Em particular, o método de homogeneização assintótica, é utilizado para encontrar os coeficientes que representam as propriedades efetivas de um meio com estrutura periódica. O presente trabalho tem como objetivo o estudo desta técnica matemática de homogeneização para obtenção do comportamento efetivo de meios micro-heterogêneos, e aplicar o formalismo matemático para construir uma solução assintótica formal de um problema unidimensional linear com coeficientes contínuos e constante por partes. Ainda, justificar-se-á matematicamente a proximidade entre as soluções dos problemas original e homogeneizado. A fim de ilustrar os resultados teóricos, apresenta-se um exemplo considerando ambos os tipos de heterogeneidade em um caso que apresenta o mesmo comportamento efetivo

Reiterated homogenization of a two-point boundary value problem

El metodo de homogeneizacion asintotica es aplicado para homogeneizar una familia unidimensional de problemas elıpticos, con coeficientes periodicos y rapidamente oscilantes que dependen de dos variables rapidas. El problema homogeneizado, los problemas locales y los correspondientes coeficientes efectivos son obtenidos. Una condicion necesaria y suficiente para la construccion de una solucion asintotica con terminos periodicos es demostrada. Basados en el principio del maximo, se demuestra la proximidad entre la soluci´on del problema homogeneizado y la del problema original. Se propone un ejemplo numerico para ilustrar la justificacion matematica.The asymptotic homogenization method is applied to homogenize a one-dimensional family of elliptic boundary value problems with periodic and rapidly oscillating coefficients which depend on two fast variables. The homogenized problem, the local problems and the corresponding effective coefficient are obtained. A necessary and sufficient condition for constructing an asymptotic solution with periodic terms is demonstrated. Based on a Maximum Principle the proximity between the solutions of the homogenized and original problems is proved. Some numerical computations are used to illustrate the mathematical justificatio

Mathematical modeling of non-ageing linear viscoelastic composites with general periodicity

International audienc

Average of the elastic properties of bio-heterogeneous structures

International audienc

Soft and hard anisotropic interface in composite materials

International audienceFor a large class of composites, the adhesion at the fiber–matrix interface is imperfect i.e. the continuity conditions for displacements and often for stresses is not satisfied. In the present contribution, effective elastic moduli for this kind of composites are obtained by means of the Asymptotic Homogenization Method (AHM). Interaction between fiber and matrix is considered for linear elastic fibrous composites with parallelogram periodic cell. In this case, the contrast or jump in the displacements on the boundary of each phase is proportional to the corresponding component of the tension on the interface. A general anisotropic behavior of the interphase is assumed and the interface stiffnesses are explicitly given in terms of the elastic constants of the interphase. The constituents of the composites exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions is considered. Comparisons with theoretical and experimental results verified that the present model is efficient for the analysis of composites with presence of imperfect interface and parallelogram cell. The present method can provide benchmark results for other numerical and approximate methods