Electrostatic and magnetostatic properties of random materials.

Area of Science:

• Computational Electromagnetics
• Materials Science
• Statistical Physics

Background:

• Investigating scale dependence of material properties is crucial for accurate modeling.
• Homogenization theories provide bounds but often assume scale independence.
• Understanding the transition from statistical volume element (SVE) to representative volume element (RVE) is key.

Purpose of the Study:

• To investigate the scale dependence of electrostatic and magnetostatic properties in random linear lossless materials.
• To derive and validate mesoscale bounds on effective electrical permittivity and magnetic permeability.
• To develop a predictive scaling function for estimating the RVE size.

Main Methods:

• Adapted Hill-Mandel homogenization conditions for SVE analysis.
• Formulated uniform boundary conditions for SVE.
• Employed computational electromagnetics for numerical simulations of 2D and 3D random composites.
• Developed and calibrated a scaling function based on extensive simulations.

Main Results:

• Established upper and lower mesoscale bounds on effective permittivity and permeability.
• Demonstrated scale-dependent trends of these bounds towards RVE properties.
• Quantified the transition from SVE to RVE using a scaling function.
• Validated the scaling function's accuracy across various mesoscales, volume fractions, and property contrasts.

Conclusions:

• The RVE size of random microstructures is accurately predictable using the derived scaling function.
• The scaling function provides a method to estimate RVE size to any desired accuracy.
• This work offers a robust framework for understanding and predicting scale-dependent properties in composite materials.