Asymptotic homogenisation in linear elasticity. Part I: Mathematical formulation and finite element modelling

Pinho-da-Cruz, J.; Dias-de-Oliveira, João; Teixeira-Dias, Filipe

doi:10.1016/j.commatsci.2009.02.025

Cited by 100 publications

(37 citation statements)

References 4 publications

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“…Using the standard method of asymptotic expansion 8 applied to the linear elastic problem, we get a microscopic equation formulated in terms of the characteristic response function and a macroscopic equation that involves the homogenized elasticity parameters. 6,8 The problem at the microscopic level can be formulated to find the characteristic response function c …”

Section: Assymptotic Homogenizationmentioning

confidence: 99%

“…[1][2][3][4] Nevertheless, in some cases, it is advantageous to support experimental measurements with the knowledge of the behavior of the considered material at the micro-or meso-scale level (low scale). [5][6][7] A precisely calibrated low-scale model can be helpful in the cases where all the necessary material parameters for a macro-scale model are impossible to be characterized or in the case when the initiation or propagation of the processes observed on the micro-scale are not clear.…”

Section: Introductionmentioning

confidence: 99%

“…The first approach uses the advantages of commercial software Abaqus 6.14-2 and its wide range of pre-and post-processing tools including the possibility of an automatic model construction using the Python software. 5 The second approach uses the well established theory of asymptotic homogenization, 6,7 which is implemented in the SfePy finite-element software. [8][9][10] …”

Section: Introductionmentioning

confidence: 99%

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Comparison of homogenization approaches used for the identification of the material parameters of unidirectional composites

Srbová¹,

Kroupa²,

Lukeš³

2017

Mater. Tehnol.

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The paper is aimed at the identification of the material parameters of the constituents of unidirectional long-fiber carbon/epoxy composites. The ratios between the real fiber and the matrix volume of unidirectional fiber composites were identified from the images obtained with scanning electron microscopy (SEM). Fiber and matrix areas were analyzed using Matlab and its image-processing toolbox. Determined volume ratios were used to propose the geometry of a micromechanical representative volume element. An algorithm ensuring an irregular fiber distribution in the representative volume element cross-section and periodicity in the cross-section plane were used. A single-layer representative volume element with a mesh was built in Abaqus/CAE. Experimentally obtained force-elongation dependencies for the tensile tests of the specimens with different fiber orientations were processed and effective moduli were determined. In the first approach, the material parameters of phases were identified using the OptiSlang optimization software and finite-element code Abaqus. A periodically constrained representative volume element was loaded according to the experiments where specimens with different fiber orientations were loaded by tensile force. In the second approach, the SfePy software and asymptotic homogenization were used for the numerical computation on the microscopic scale, resulting in the homogenized material parameters that were afterwards used at the macroscopic level. The optimization function characterized the sum of the differences of the effective moduli and the difference of Poisson's ratios of the whole composite. Keywords: composite, unidirectional, micro-model, constituents, material parameters, cross-section, homogenization Namen~lanka je ugotavljanje materialnih parametrov sestavin epoksi kompozitov z enosmernim dolgimi ogljikovimi vlakni. Razmerje realnih vlaken in volumna osnove kompozitov z usmerjenimi vlakni je bilo dolo~eno iz slik, dobljenih na vrsti~nem elektronskem mikroskopu (SEM). Vlakna in podro~ja osnove so bili analizirani s pomo~jo Matlaba in njegovega orodja za obdelavo slik. Dolo~ena volumska razmerja so bila uporabljena za predlog geometrije reprezentativnih mikromehanskih volumskih elementov. Uporabljen je bil algoritem, ki zagotavlja neenakomerno razporeditev vlaken na preseku reprezentativnega volumskega elementa in periodi~nost na ravnini preseka. Plast reprezentativnega volumskega elementa z mre`o, je bila postavljena z Abaqus/CAE. Izvedeni so bili eksperimenti za dolo~anje odvisnosti sila-raztezek pri nateznih preizku{ancih z razli~no orientacijo vlaken in dolo~eni so bili efektivni moduli. V prvem pribli`ku so bili ugotavljani materialni parametri faz s pomo~jo OptiSlang programske opreme za optimizacijo in kodo kon~nih elementov Abaqus. Periodi~no omejen reprezentan~ni volumski element je bil obremenjen, skladno s preizkusi. Pri~emer so bili vzorci z razli~no orientacijo vlaken, obremenjeni z natezno silo. Pri drugem pribli`ku je uporabljena programska oprema SfePy in asimptot...

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Section: Assymptotic Homogenizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comparison of homogenization approaches used for the identification of the material parameters of unidirectional composites

Srbová¹,

Kroupa²,

Lukeš³

2017

Mater. Tehnol.

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“…Their computational issues and applications were overviewed and a relationship between macro and micro-scale properties were developed [8,41]. Mathematical expansions for modeling physical phenomena on inhomogeneous materials with periodic microstructure and also the explicit mathematical equations which describes the local stress and strain fields associated with a given global domain were derived [50]. Performance of such formulations were tested on composite structures and effect of reinforcement volume fraction and their geometry and distribution in the matrix on overall material properties were studied [47].…”

Section: Literature Reviewmentioning

confidence: 99%

“…(2.8) has to be valid for any given ǫ → 0 + , it is necessary that coefficients of any power of ǫ be zero [50]. Therefore:…”

Section: Homogenizationmentioning

confidence: 99%

Characterization of Homogenized Mechanical Properties of Porous Ceramic Materials Based on Their Realistic Microstructure

Rastkar¹

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The recent advances in the Materials Engineering have led to the development of new materials with customized microstructure in which the properties of its constituents and their geometric distribution have a considerable effect on determination of the macroscopic properties of the substance. Direct inclusion of the material microstructure in the analysis on a macro level is challenging since spatial meshes created for the analysis should have enough resolution to be able to accurately capture the geometry of the microstructure. In most cases this leads to a huge finite element model which requires a substantial amount of computational resources.To circumvent this limitation a number of homogenization techniques were developed. By considering a small element of the material, referred to as Representative Volume Element (RVE), homogenization methods make it possible to include the effects of a materials microstructure on the overall properties at the macro level.However, complexity of the microstructure geometry and the necessity of satisfying periodic boundary conditions introduce additional difficulties into the analysis procedure.In this dissertation we propose a hybrid homogenization method that combines Asymptotic homogenization with MeshFree Solution Structures Method (SSM). Our vi approach allows realistic inclusion of complex geometry of the microstructure that can be captured from micrographs or micro CT scans. In addition to unprecedented flexibility in handling complex geometries, this method also provides a completely automatic analysis procedure. Using meshfree solution structures simplifies meshing to creating a simple cartesian grid which only needs to contain the domain. This also eliminates manual modifications which usually needs to be performed on meshes created from image data.A computational platform is developed in C++ based on meshfree/asymptotic method. In this platform also a novel meshfree solution structure is designed to provide exact satisfaction of periodic boundary conditions for boundary value problems such as homogenization. Performance of the developed platform is tested over 2D and 3D domains against previously published data and/or conventional finite element methods. After getting satisfactory results, homogenized properties are used to compute localized stress and strain distributions over inhomogeneous structures.Furthermore, effects of geometric features of pores/inclusions on homogenized mechanical properties is investigated and it is demonstrated that the developed platform could provide an automated quantitative analysis tool for studying effects of different design parameters on homogenized properties.

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Physics‐Informed and Thermodynamics‐Based Neural Networks

MASI,

STEFANOU

2024

Machine Learning in Geomechanics 2

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Asymptotic homogenisation in linear elasticity. Part I: Mathematical formulation and finite element modelling

Cited by 100 publications

References 4 publications

Comparison of homogenization approaches used for the identification of the material parameters of unidirectional composites

Comparison of homogenization approaches used for the identification of the material parameters of unidirectional composites

Characterization of Homogenized Mechanical Properties of Porous Ceramic Materials Based on Their Realistic Microstructure

Physics‐Informed and Thermodynamics‐Based Neural Networks

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