Variationally consistent homogenization of diffusion in particle composites with material interfaces using dual macroscale chemical potentials

Abstract Computational homogenization of particle-matrix composites with material interfaces is considered, whereby linear transient diffusion driven by a chemical potential is used as a model problem. Due to the models linearity, it is beneficial to assume micro-stationarity in order to provide direct upscaling and avoid excessive computational cost. First order homogenization with a single macroscale chemical potential is taken as the most basic approach; however, the accuracy is negatively affected whenever relevant micro-transient effects can not be captured using the stationary sub-scale problem. To improve the accuracy while still upscaling from a stationary sub-scale problem, different formulations based on dual macroscale potentials, one for each phase, are proposed and investigated in this paper. As to the prolongation order within the particles and matrix phase, respectively, two types are considered: constant-linear and linear-linear. Most importantly, for the case of linear prolongation, different ways of defining the macroscale variables (acting as loading on the RVE-problem) in terms of suitable measures of the chemical potential can be envisioned: (1) averaging of $$0\text {th}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>th</mml:mtext> </mml:mrow> </mml:math> and $$1\text {st}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mtext>st</mml:mtext> </mml:mrow> </mml:math> gradient of the potential, (2) averaging of $$0\text {th}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>th</mml:mtext> </mml:mrow> </mml:math> and $$1\text {st}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mtext>st</mml:mtext> </mml:mrow> </mml:math> moment of the potential. The pros and cons of the different approaches were assessed in a numerical study and compared to a reference solution from Direct Numerical Simulation (DNS) for an example problem. It was concluded that the moment-based linear-linear method was the only one that could match the DNS solution for all considered material parameters. However, for sufficiently large interface resistance, leading to a more pronounced potential jump across the interfaces, the constant-linear prolongation gave comparable results.