On overall properties of elastic heterogeneous bodies smaller than the representative volume

Hazanov, S.; AMIEUR, Mohamed

doi:10.1016/0020-7225(94)00129-8

Cited by 92 publications

(42 citation statements)

References 8 publications

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“…In fact, periodic boundary conditions have been shown to provide better approximation to the actual behavior than either direct Dirichlet or Neumann conditions (Amieur et al, 1993 ;Hazanov and Amieur, 1995;Terada et al, 2000;van der Sluis et al, 2000) for first order simulations; however, this has not been quantitatively studied for second order simulations. The disadvantage of such boundary conditions is that they require, at least conceptually, a periodic rSVE geometry.…”

Section: Discussionmentioning

confidence: 99%

A hierarchical framework for the multiscale modeling of microstructure evolution in heterogeneous materials.

Luscher¹

2010

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ACKNOWLEDGEMENTSAs far as I am concerned, there is no better way to celebrate a success than to share the credit with those that have contributed to the effort. Here I share credit with those people that have had tremendous impact to the successful completion of requirements for my PhD degree at Georgia Tech, including this dissertation.Of primary importance in this acknowledgement is to convey how substantial I consider my wife's sacrifices over the last five years. With the sole exception of unmarried PhD candidates, Ido not imagine that there exists a PhD thesis whose acknowledgement section does not make reference to the author's spouse. So I strive to make this acknowledgement of my wife's contributions to my success so compelling and unique that it escapes the status of 'formality'. To attempt this, I will provide a single anecdote selected from five years of worthy material.Stephanie Luscher endured months alone keeping our house in Los Alamos while I was in Atlanta attending classes and fulfilling residency requirements. During that time, she continued working as an engineering design review coordinator at Los Alamos National Lab. While in Oregon to visit family during the winter of 2004-5, a water pipe inside our Los Alamos house froze and burst. Over the next week, 90,000 gallons of water flowed throughout our house ruining just about everything. We returned home and opened soggy Christmas presents from the relative comfort of our car parked in the driveway while contemplating our next course of action.Our house required extensive repairs including a new heating system, kitchen cabinets, flooring, and complete refinishing of the interior walls. We lived in a motor home, a rented house, and a hotel over the course of six months of these repairs. Within two weeks of this happening, I was back in Atlanta and Stephanie was left to manage the reconstruction effort by herself. Of course, this is a single example of how Stephanie has sacrificed, and tended to the many details of life, in order to allow me to focus attention on the work presented within this thesis. Stephanie, I thank iv you for the support and sacrifice you have made to help me accomplish this goal. I'm glad you embraced "my goal" as "our goal."Well in advance of thoughts regarding graduate school, my parents instilled a few helpful traits in me. As a young boy, during hot summers spent roofing with my Dad, Mark Luscher, I learned to work hard and to play hard. My Dad taught me that sometimes "getting the job done" is not elegant; it simply requires you to roll up your sleeves and get to work. As a complement to this, my Mom taught me the importance of precision and attention to detail. Together, my parents have taught me the importance of family, which supersedes all else. Engineering division leader (W-DO), gave me opportunities to excel within the weapons program. Ryan Maupin, group leader for Advanced Engineering Analysis (W-13) has enthusiastically supported and trusted me, both as my manager and friend, to pursue a course of research th...

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Section: Discussionmentioning

confidence: 99%

A hierarchical framework for the multiscale modeling of microstructure evolution in heterogeneous materials.

Luscher¹

2010

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“…Xia et al [234] reported that the homogeneous boundary conditions, when applied on periodic microstructures, "are not only overconstrained, but may also violate the boundary traction periodicity conditions" under loading types with shear components. Hazanov and Huet [235], Hazanov and Amieur [236], and Pahr and Zysset [237] proposed uniform mixedtype boundary conditions that consider applying constant traction boundary conditions on some parts of the boundary and linear displacement boundary conditions to the other parts such that the apparent elasticity tensor for this boundary condition lies between the apparent tensors obtained with homogeneous boundary conditions. Mesarovic and Padbidri [238] argued that there is no reason to assume that an RVE with random microstructure behaves as a periodic unit cell and suggested the use of minimal kinematic boundary conditions.…”

Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…That Θ( χ,ω) is mean-ergodic means the spatial average equals the ensemble average 5 1 A Figure 1 shows the proposed homogenization methodology. The checkerboard is composed of randomly arranged isotropic phases with differing phase conductivities.…”

Section: A the Hill-mandel Condition In Electrical Conductivitymentioning

confidence: 99%

Electrical properties of random checkerboards at finite scales

2015

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Under investigation is the scale dependent electrical conductivity (and resistivity) of two-phase random checkerboards at arbitrary volume fractions and phase contrasts. Using variational principles, rigorous mesoscale bounds are obtained on the electrical properties at finite scales by imposing a boundary condition that is either uniform electric potential or uniform current density. We demonstrate the convergence of these bounds to the effective properties with increasing length scales. This convergence gives rise to the notion of a scalar-valued scaling function that accounts for the statistical nature of the mesoscale responses. A semi-analytical closed form solution for the scaling function is obtained as a function of phase contrast, volume fraction and the mesoscale. Finally, a material scaling diagram is constructed with which the convergence to the effective properties can be assessed for any random checkerboard with arbitrary phase contrast and volume fraction.

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On overall properties of elastic heterogeneous bodies smaller than the representative volume

Cited by 92 publications

References 8 publications

A hierarchical framework for the multiscale modeling of microstructure evolution in heterogeneous materials.

A hierarchical framework for the multiscale modeling of microstructure evolution in heterogeneous materials.

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Electrical properties of random checkerboards at finite scales

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