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1

A 3D trabecular bone homogenization technique

Marques Marco Costa, Belinha J., Oliveira A. F., Manzanares Cespedes M.C., Natal Jorge R.M.

Acta of Bioengineering and Biomechanics

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2020

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Vol. 22, no. 3

139--152

EN

Purpose: Bone is a hierarchical material that can be characterized from the microscale to macroscale. Multiscale models make it possible to study bone remodeling, inducing bone adaptation by using information of bone multiple scales. This work proposes a computationally efficient homogenization methodology useful for multiscale analysis. This technique is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium. Methods: In this work, a morphology-based fabric tensor and a set of anisotropic phenomenological laws for bone tissue was used, in order to define the bone micro-scale mechanical properties. To validate the developed methodology, several examples were performed in order to analyze its numerical behavior. Thus, trabecular bone and fabricated benchmarks patches (representing special cases of trabecular bone morphologies) were analyzed under compression. Results: The results show that the developed technique is robust and capable to provide a consistent material homogenization, indicating that the homogeneous models were capable to accurately reproduce the micro-scale patch mechanical behavior. Conclusions: The developed method has shown to be robust, computationally less demanding and enabling the authors to obtain close results when comparing the heterogeneous models with equivalent homogenized models. Therefore, it is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.

2

Combining radial point interpolation meshless method with a new homogenization technique for trabecular bone multiscale structural analyses

Marques Marco, Belinha Jorge, Oliveira António, Manzanares Céspedes, Jorge Renato Natal

Acta of Bioengineering and Biomechanics

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2019

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Vol. 21, no. 2

101--113

EN

Bone tissue is a dynamic tissue, possessing different functional requirements at different scales. This layered organization indicates the existence of a hierarchical structure, which can be characterized to distinguish macro-scale from micro-scale levels. Structurally, both scales can be linked by the use of classic multiscale homogenization techniques. Since in bone tissue each micro-scale domain is distinct form its neighbour, applying a classic multiscale homogenization technique to a complete bone structure could represent an inadmissible computational cost. Thus, this work proposes a homogenization methodology that is computationally efficient, presenting a reduced computational cost, and is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium. Methods: The methodology uses the fabric tensor concept in order to define the material principal directions. Then, using an anisotropic phenomenological law for bone tissue correlating the local apparent density with directional elasticity moduli, the anisotropic homogenized material properties of the micro-scale patch are fully defined. To validate the developed methodology, several numerical tests were performed, measuring the sensitivity of the technique to changes in the micro-patch size and preferential orientation. Results: The results show that the developed technique is robust and capable to provide a consistent material homogenization. Additionally, the technique was combined with two discrete numerical techniques: the finite element method and radial point interpolation meshless method. Conclusions: Structural analyses were performed using real trabecular patches, showing that the proposed methodology is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.

3

Homogenization techniques of unidirectional composite material

Kormaníková E.

Roczniki Inżynierii Budowlanej

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2015

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z. 15

79--83

EN

The paper deals with analytical and numerical homogenization of unidirectional fiber matrix composite. There are described the Mori-Tanaka method, periodic analytical model and numerical periodic and hexagonal models. The example of homogenization is solved by programs Heat and Elasticity Properties, MATLAB and ANSYS. The obtained results are summarized in the table and compared to each other.

4

A two-scale numerical approach to granular systems

Nitka M., Tejchman J.

Archives of Civil Engineering

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2011

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Vol. 57, nr 3

313-330

EN

A two-scale numerical homogenization approach was used for granular materials. At small-scale level, granular micro-structure was simulated using the discrete element method. At macroscopic level, the finite element method was applied. An up-scaling technique took into account a discrete model at each Gauss integration point of the FEM mesh to derive numerically an overall constitutive response of the material. In this process, a tangent operator was generated with the stress increment corresponding to the given strain increment at the Gauss point. In order to detect a loss of the solution uniqueness, a determinant of the acoustic tensor associated with the tangent operator was calculated. Some elementary geotechnical tests were numerically calculated using a combined DEM-FEM technique.

PL

Zastosowano dwuskalowe numeryczne podejście homogenizacyjne do materiałów granulowanych. Na poziomie małej skali symulowano granulowaną mikrostrukturę przy zastosowaniu metody elementów dyskretnych. Na poziome dużej skali zastosowano metodę elementów skończonych. Technika przechodzenia do wyższej skali uwzględniła dyskretny model w każdym punkcie całkowania Gaussa siatki MES w celu wyprowadzenia numerycznego obliczenia wynikowej konstytutywnej odpowiedzi materiału. W tym procesie, operator styczny został obliczony za pomocą przyrostu naprężenia odpowiadającego danemu przyrostowi odkształcenia w punkcie Gaussa. W celu wykrycia utraty jednoznaczności rozwiązania, określono wyznacznik tensor akustycznego związanego z operatorem stycznym. Obliczono numerycznie kilka podstawowych testów geotechnicznych stosując połączoną technikę MED-MES.

5

Numerical homogenization of elastic brick masonry

Kuczma M., Wybranowska K.

Civil and Environmental Engineering Reports

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2005

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No. 1

135-152

EN

The paper is concerned with a numerical homogenization technique for determination of effective material properties of brick masonry in the elastic range. The homogenization problem is posed in the plane state of stress. The corresponding boundary value problem on a representative cell is discretized by the finite element method. The quadrilateral finite element with four nodes and eight degrees of freedom is applied and our own computer program is developed. The homogenization technique allows one to determine for masonry, which is an inhomogeneous two-phase composite medium, an equivalent homogeneous orthotropic material characterized by five material constants. The homogenized material constants can further be used in an analysis of large-scale masonry structures. The obtained results of numerical simulations are compared with predictions of the value of elastic modulus for masonry by other researches, and qualitative agreement can be observed.

PL

Praca dotyczy numerycznego sposobu homogenizacji muru ceglanego w zakresie sprężystym. Problem homogenizacji postawiono w płaskim stanie naprężenia. Odpowiednie zagadnienie brzegowe na reprezentatywnej komórce zdyskretyzowano metodą elementów skończonych wykorzystując czterowęzłowy element skończony o ośmiu stopniach swobody i opracowany własny program komputerowy. Zastosowana metoda homogenizacji pozwala wyznaczyć dla muru, który jest niejednorodnym dwuskładnikowym materiałem kompozytowym, wartości pięciu efektywnych parametrów materiałowych dla jednorodnego materiału ortotropowego. Wyznaczone parametry mogą następnie być użyte w analizach całych, dużych konstrukcji murowych. Otrzymane wyniki analiz numerycznych porównano z propozycjami obliczania modułu sprężystości muru według innych badaczy, uzyskując dobrą zgodność jakościową.