Electrostatic and magnetostatic properties of random materials

Abstract

Scale dependence of electrostatic and magnetostatic properties is investigated in the setting of spatially random linear lossless materials with statistically homogeneous and spatially ergodic random microstructures. First, from the Hill-Mandel homogenization conditions adapted to electric and magnetic fields, uniform boundary conditions are formulated for a statistical volume element (SVE). From these conditions, there follow upper and lower mesoscale bounds on the macroscale (effective) electrical permittivity and magnetic permeability. Using computational electromagnetism methods, these bounds are obtained through numerical simulations for composites of two types: (i) 2D random checkerboard (two-phase) microstructures and (ii) analogous 3D random (three-phase) media. The simulation results demonstrate a scale-dependent trend of these bounds towards the properties of a representative volume element (RVE). This transition from SVE to RVE is described using a scaling function dependent on the mesoscale δ, the volume fraction v f , and the property contrast k between two phases. The scaling function is calibrated through fitting the data obtained from extensive simulations (∼10,000) conducted over the aforementioned parameter space. The RVE size of a given microstructure can be estimated down to within any desired accuracy using this scaling function as parametrized by the contrast and the volume fraction of two phases.

Multiscale modeling of cancellous bone considering full coupling of mechanical, electric and magnetic effects

“…To connect the macro-and microscale in FE 2 , it is important to discuss the transition between the scales. The Hill-Mandel conditions (Hill 1963(Hill , 1972Schröder 2009;Schröder et al 2016;Labusch et al 2019;Karimi et al 2019) have to be fulfilled, guaranteeing energy conservation during the scale transition. Thus, the virtual work on the macroscale has to be equal to the virtual work on the microscale:…”

“…2019 ; Karimi et al. 2019 ) have to be fulfilled, guaranteeing energy conservation during the scale transition. Thus, the virtual work on the macroscale has to be equal to the virtual work on the microscale: For the macro-to-micro-transition, these conditions can be fulfilled by three different types of boundary conditions on the microscale: Dirichlet, Neumann and periodic boundary conditions (Ilic et al.…”

Multiscale modeling of cancellous bone considering full coupling of mechanical, electrical and magnetic effects

“…To connect the macro-and microscale in FE 2 , it is important to discuss the transition between the scales. The Hill-Mandel conditions [33,34,35,36,37,38] have to be fulfilled, guaranteeing energy conservation during the scale transition. Thus, the virtual work on the macroscale has to be equal to the virtual work on the microscale:…”

Violations of the Clausius–Duhem inequality in Couette flows of granular media

“…In general, the SVE plays the same role for a statistical (or stochastic) continuum as the RVE for a deterministic continuum, e.g. [6][7][8]. Continuing in the vein of [9], the fundamental goal of this paper is to study the scale dependence of SVE to establish the demarcation in the parameter space between violations and non-violations of the CD inequality.…”

“…This type of homogenization has several advantages: (i) it provides a handshake between the interpretations of a theoretician and an experimentalist; (ii) it avoids intuitively postulating more-or-less correct effective medium theories; (iii) it naturally grasps the statistical scale effects (including the upscaling from SVE to RVE); (iv) it has been employed in a wide range of random media problems: conductivity, mechanics, electromagnetism and coupled-field phenomena, e.g. [6][7][8].…”