Scale-dependent homogenization of random composites as micropolar continua

Trovalusci, Patrizia; Ostoja–Starzewski, Martin; Bellis, Maria Laura De; Murrali, Agnese

doi:10.1016/j.euromechsol.2014.08.010

Cited by 137 publications

(119 citation statements)

References 37 publications

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“…Since an SVE is not statistically representative by definition, the homogenized meso-scale responses change with the SVE size, with the applied Boundary Conditions (BCs) on the SVE, but also for different SVE realizations of the same size. This last property of SVEs has been used to up-scale the uncertainties at the micro-scale to the meso-scale, for example to define the probability convergence criterion of RVE for masonry [10], to study the scale-dependency of homogenization for random composite materials [42], to obtain the property variations of poly-silicon film [25], to extract effective properties of random two-phase composites [39], or again to capture the stochastic properties of the parameters in a constitutive model [47]. The problem of finite elasticity was also considered through the resolution of composite material elementary cells in [4,24], which allows defining a meso-scale potential as proposed in [4].…”

Section: Introductionmentioning

confidence: 99%

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

Lucas

Nguyễn

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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The aim of this work is to study the thermo-elastic quality factor (Q) of micro-resonators with a stochastic multi-scale approach. In the design of high-Q micro-resonators, thermo-elastic damping is one of the major dissipation mechanisms, which may have detrimental effects on the quality factor, and has to be predicted accurately. Since material uncertainties are inherent to and unavoidable in micro-electromechanical systems (MEMS), the effects of those variations have to be considered in the modeling in order to ensure the required MEMS performance. To this end, a coupled thermo-mechanical stochastic multi-scale approach is developed in this paper. Thermo-mechanical micro-models of polycrystalline materials are used to represent micro-structure realizations. A computational homogenization procedure is then applied on these statistical volume elements to obtain the stochastic characterizations of the elasticity tensor, thermal expansion, and conductivity tensors at the meso-scale. Spatially correlated meso-scale random fields can thus be generated to represent the stochastic behavior of the homogenized material properties. Finally, the distribution of the thermo-elastic quality factor of MEMS resonators is studied through a stochastic finite element method using as input the generated stochastic random field.

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Section: Introductionmentioning

confidence: 99%

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

Lucas

Nguyễn

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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“…To this end, guided by the analogy to upscaling of a spatially random micropolar elastic continuum [30], we set up a homogenization condition of Hill-Mandel type for a micropolar fluid medium [a generalization of (11) for Cauchy media]:…”

Section: Upscaling From Micropolar To Classical Fluid Mechanicsmentioning

confidence: 99%

Second law violations, continuum mechanics, and permeability

Ostoja–Starzewski

2015

Continuum Mech. Thermodyn.

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The violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem and mathematically, the irreversible entropy is a submartingale. First, we discuss the consequences of these results for the axioms of continuum mechanics, arguing in favor of a framework relying on stochastic functionals of energy and entropy. We next determine a Lyapunov function for diffusion-type problems governed by stochastic rather than deterministic functionals of internal energy and entropy, where the random field coefficients of diffusion are not required to satisfy the positive definiteness everywhere. Next, a formulation of micropolar fluid mechanics is developed, accounting for the lack of symmetry of stress tensor on molecular scales. This framework is then applied to employed to show that spontaneous random fluctuations of the microrotation field will arise in Couette-and Poiseuille-type flows in the absence of random (turbulence-like) fluctuations of the classical velocity field. Finally, while the permeability is classically modeled by the Darcy law or its modifications, besides considering the violations of the second law, one also needs to account for the spatial randomness of the channel network, implying a modification of the hierarchy of scale-dependent bounds on the macroscopic property of the network.Keywords Continuum mechanics · Second law violations · Fluctuation theorem · Submartingale · Micropolar fluid · Permeability · Darcy law MotivationAs shown by theory, simulations, and experiments, the violations of the second law of thermodynamics are relevant where/when the length and/or time scales become very small. The second law must then be replaced by the fluctuation theorem [10,12,16,29,31]. [Strictly speaking, the fluctuation theorem stands for a group of such theorems (see e.g. Table 4.1 in [13])]. In effect, the second law holds on average, be it an ensemble average, or a spatial average over a sufficiently large domain, or a temporal average over a sufficiently large time interval. Interestingly, the second law violations may occur for up to several seconds (!) in, say, cholesteric liquids. While the work by physicists focuses on stochastic thermodynamics, our interest is in building these results into a continuum mechanics theory, i.e., in formulating stochastic continuum thermomechanics.From the standpoint of mathematics, the irreversible entropy is a submartingale [23]: a stochastic process conditioned by the entire past history, while it fluctuates randomly but grows on average; this is reviewed in Sect. 2. All these concepts and results have consequences for the foundations of continuum mechanics: since the fluctuation theorem is derived from the Axiom of Causality in statistical mechanics, this suggests Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.M. Ostoja-Starzewski (B)

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“…In order to determine the constitutive tensors components of the homogenized continuum and to detect the RVE size, we follow the statistical procedure presented in [26] and briefly recalled in the following steps:…”

Section: Computational Homogenization For Random Compositesmentioning

confidence: 99%

“…In this paper we adopt the statistically-based scaledependent homogenization procedure developed in [26], in order to reproduce the actual microstructure of two-phase random media and to estimate the constitutive moduli of energy-equivalent continuum with rigid local structure (micropolar) [1,13]. This procedure applies to random composites perceived micropolar continua both at the micro and macro level.…”

Section: Introductionmentioning

confidence: 99%

Particulate random composites homogenized as micropolar materials

et al. 2014

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Many composite materials, widely used in different engineering fields, are characterized by random distributions of the constituents. Examples range from polycrystals to concrete and masonry-like materials. In this work we propose a statistically-based scale-dependent multiscale procedure aimed at the simulation of the mechanical behavior of a two-phase\ud particle random medium and at the estimation of the elastic moduli of the energy-equivalent homogeneous\ud micropolar continuum. The key idea of the procedure is to approach the so-called Representative Volume Element (RVE) using finite-size scaling of Statistical Volume Elements (SVEs). To this end properly defined Dirichlet, Neumann, and periodic-type non-classical boundary value problems are numerically solved on the SVEs defining hierarchies of constitutive bounds.\ud The results of the performed numerical simulations point out the importance of accounting for spatial randomness as well as the additional degrees of freedom of the continuum with rigid local structure

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Scale-dependent homogenization of random composites as micropolar continua

Cited by 137 publications

References 37 publications

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators

Second law violations, continuum mechanics, and permeability

Particulate random composites homogenized as micropolar materials

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