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A novel spectral framework for second-order homogenization theories
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|Title: ||A novel spectral framework for second-order homogenization theories|
|Authors: ||Binci, Massimiliano|
|Keywords: ||Materials science|
Homogenization (Differential equations)
|Issue Date: ||11-Feb-2008 |
|Abstract: ||In the modern scenario of materials engineering, composite systems are emerging as leading materials in many technological applications. The increasing popularity depends on the specific capacity for these materials to tailor their proprieties by varying their internal structure. Composites have unique combinations of properties that are not achievable by traditional materials. The synergy between the different constituents results in improved performance. Composites can reach very high specific strength and be extremely anisotropic. The anisotropic character can be enormously enhanced by optimizing the morphology of the microstructure. The unique capabilities and the competitive edge of composites in all fields of engineering have steered our curiosity toward the study their microstructure-property relationships.
Recently we have developed a novel mathematical framework, called the Microstructure Sensitive Design (MSD), specifically introduced to address to the microstructure-property-processing paradigm of materials. The main feature of MSD is the simplification of the linkages in an appropriate spectral (Fourier) space and identification of the microstructure hull, i.e. the spectral domain of all possible microstructural realizations of a given material system. In this work we have extended the MSD framework to microstructure-property relationships of composites that utilize second-order homogenization theories. These latter are a formidable advancement over the previous first-order theories that accounted only for the volume fraction information of the material constituents. Second-order theories seize the morphological details of the microstructure and capture the anisotropy of materials. Using this new spectral framework we have successfully established macroscopic elastic properties of composites and delineated for the first time in literature the second-order property closures, i.e. the ensemble of feasible microstructures of a given material system satisfying prescribed performance conditions. Next, we shifted our attention to an important class of scalebridging relations, called localization linkages, which represent relevant phenomena, such as fracture and creep. The localization linkages connect the microscopic stresses or strains to the macroscopic loading conditions and they are described by fourth-rank tensors. The localization tensors are efficiently recast in the Fourier space as algebraic expressions whose coefficients do not depend on the microstructural details. These parameters, called influence coefficients, are constant for a given material system and define the local response of a composite with any morphology, however, are difficult to evaluate. In this work we have developed two different strategies to calculate the influence coefficients: the first consists in the direct numerical integration of the homogenization theories; the second approach is based on the calibration of these coefficients to the results of finite element micromechanical models. We show that that the spectral method is a useful tool to predict local properties of composite materials with weak and moderate contrasts of the material moduli. Finally, we discuss some ways to improve the current methodology.|
|Appears in Collections:||Drexel Theses and Dissertations|
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