A Computational Modeling Approach for Homogenization Techniques Based on the Finite Element Method

1
A Computational Modeling Approach for
Homogenization Techniques Based on the Finite
Element Method

• Francisco Evangelista Junior
• Flávio Vasconcelos de Souza
• Joaquim Bento Cavalcante Neto
• Jorge Barbosa Soares

2
Introduction

• - Composite materials - heterogeneous media with
  dissimilar constituents and properties
• For design purposes, these media have been
  approximated as homogeneous even though they are
  heterogeneous
• Example asphalt mixture (binder and aggregates).

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Introduction

• - Material heterogeneity has a significant impact
  in the macroscopic behavior of the composite
  materials
• The analysis at heterogeneous smaller scales
  (microscale, µ) explains some phenomena observed
  at a larger scale (macroscale, µ1)
• Developments have been undertaken with
  micromechanical techniques aiming the
  understanding of these phenomena
• (i) detailed modeling
• entire heterogeneous structure and its details
• (ii) average modeling
• predict macro (homogeneous) by micro
  (heterogeneous).

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Objectives

• - Formulate a micromechanical-based approach for
  composite materials using the FEM
• (i) heterogeneous microscale (µ) -
  performed with different components.
• (ii) homogeneous macroscale (µ1) - by means
  of homogenization.
• Computational modeling approach for
  homogenization techniques geometric modeling and
  mesh generation
• - Case study composite formed by aggregates into
  a binder media.

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Background

• Material is heterogenous in microscale (µ) and
  homogeneous in macroscale (µ1)
• -The constitutive behavior of the macrostructure
  can be microscopically represented by a
  Representative Volume Element (RVE)
• - Especially desirable when numerical methods are
  used - reduce the computational effort
• - Analysis steps
• (i) RVE selection and IBVP formulation
• (ii) perform the microscale analysis of the
  RVE
• (iii) homogenize the results (microscale)
  through average theorems
• (iv) solve the macroscale with the results
  by step (iii) .

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Background
- Depiction of the homogenization techniques
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Background
- Pavement multi-scale analysis
Microscale
Mesoscale
Macroscale
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Formulation
- RVE (interior V and boundary ?V )
Micromechanical problem

• Uniform tractions - stress formulation
• Uniform displacements at the boundary - strain
  formulation
• Elastic formulation

linear displacements u in V
uniform boundary averaged strain tensor
constitutive relationship
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Formulation
Homogenization of the Micromechanics Solution

• To obtain locally averaged (macroscopic)
  constitutive equations for the global structure
  domain (interior VG and boundary ?VG )
• Average theorems

averaged stress tensor
averaged strain tensor
constitutive relationship
10
Modeling
Geometry definition

• Domain of RVE and overall structure - composite
  material formed by aggregates into a binder
  media
• Geometry of RVE - obtained from image processing
  (scanning)
• Distinguish geometry from topology - regions are
  modeled as subdomains that have some sort of
  hierarchy.

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Mesh Generation
Mesh generation

• Mesh is generated for each subdivision - each
  subdivison knows its boundary and discretization
• The geometry is very irregular - meshing is
  based on Delaunay and Advancing Front techniques
  (Cavalcante-Neto et al., 1993).
• Meshing also uses a quadtree - important to
  speed up the process because there are a great
  number of subdivisions in the model.

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Case Study
Modeling

• Macrostructure a cylindrical specimen of
  sand-asphalt (SA)
• Domain 2D cross-section
• Materials asphalt binder (91.27) and
  sand-aggregates (8.73)
• Elastic properties

• RVE selection and microscopic IBVP

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Case Study
Modeling

• Macroscopic IBVP simulation of diametral
  compression test (IDT)
• The IDT test was numerically simulated in both
  loccally averaged model (homogenized) and
  detailed model (heterogeneous)

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Case Study
Results

• The homogenized elastic modulus obtained from RVE
  was 1,570 MPa
• The IDT test was numerically simulated
  (0.8mm/sec) in both loccally averaged model
  (homogenized) and detailed model (heterogeneous)
• Force-displacement curves (top) and left/right
  displacements

maximum error 4.42
maximum error 4.80
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Case Study
Computational aspects

• Time for homogenized model heterogeneous model
• The number of nodes and elements is reduced by
  the homogenization procedures - analysis is
  divided into two parts

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Conclusions

• - Homogenization procedure is accurate enough for
  predicting microscopic behavior
• The details in µ scale can be passed to µ1
• The process can be automatized and the same
  algorithm can be used for µ and µ1 scales but a
  proper linking should be accomplished via
  homogenization principle.
• Void contents, crack growth and damage can be
  considered in µ
• The computational modeling approach separates
  geometry and topology, ensuring that models can
  be generated for any domain
• The computational time to perform the FE analysis
  is reduced many orders of magnitude - d.o.f. are
  drastically reduced.

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Acknowledgements

• CNPq Brazil for research support
• Dr. David H. Allen University of Nebraska at
  Lincoln
• Eng. Felipe A. Freitas University of Nebraska
  at Lincoln.

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A Computational Modeling Approach for
Homogenization Techniques Based on the Finite
Element Method

• Francisco Evangelista Junior
• fejr_at_det.ufc.br
• Flávio Vasconcelos de Souza
• flaviovs_at_det.ufc.br
• Joaquim Bento Cavalcante Neto
• joaquimb_at_lia.ufc.br
• Jorge Barbosa Soares
• jsoares_at_det.ufc.br