Abstract
The work develops a universal approach to accounting for imperfect contacts in determining effective properties of various nature, namely, effective diffusivity and thermal and electrical conductivity. Imperfect contacts appear when fields at the microlevel are not continuous. The possibility of creating a unified approach is due to the similarity of the governing equations. At the same time, the appearance of imperfect contacts may be caused by microstructural features and by the specifics of the process itself. For concreteness, the effective diffusion permeability is determined, since various reasons for the appearance of imperfect contacts can be considered. The reasons can be associated both with the formation of structural defects and with the presence of the specific segregation effect. The paper generalizes and compares two approaches to accounting for imperfect contacts. In the first case, a field jump is set. In the second case, an inhomogeneity with a thin coating possessing extreme properties is introduced. A comprehensive analysis is carried out on the example of a material with spherical inhomogeneities. Analytical expressions for the contribution tensor of the equivalent inhomogeneity are obtained, which results in simplification of the generalization of various homogenization methods.
REFERENCES
S. K. Kanaun and V. M. Levin, “Effective field method in mechanics of matrix composite materials,” in Advances in Mathematical Modelling of Composite Materials (World Scientific, Singapore, 1994), in Ser.: Series on Advances in Mathematics for Applied Sciences, Vol. 15, pp. 1–58. https://doi.org/10.1142/9789814354219_0001
M. Kachanov and I. Sevostianov, Micromechanics of Materials, with Applications (Springer-Verlag, Cham, 2018).
Y. Zhang and L. Liu, “On diffusion in heterogeneous media,” Am. J. Sci. 312, 1028–1047 (2012). https://doi.org/10.2475/09.2012.03
B. S. Bokshtein, I. A. Magidson, and I. L. Svetlov, “On diffusion in volume and along grain boundaries,” Fiz. Met. Metalloved. 6, 1040–1052 (1958).
I. Kaur, Y. Mishin, and W. Gust, Fundamentals of Grain and Interphase Boundary Diffusion (Wiley, Chichester, 1995).
J. R. Kalnin, E. A. Kotomin, and J. Maier, “Calculations of the effective diffusion coefficient for inhomogeneous media,” J. Phys. Chem. Solids 63, 449–456 (2002). https://doi.org/10.1016/S0022-3697(01)00159-7
I. V. Belova and G. E. Murch, “Calculation of the effective conductivity and diffusivity in composite solid electrolytes,” J. Phys. Chem. Solids 66, 722–728 (2005). https://doi.org/10.1016/j.jpcs.2004.09.009
A. G. Knyazeva, G. P. Grabovetskaya, I. P. Mishin, and I. Sevostianov, “On the micromechanical modelling of the effective diffusion coefficient of a polycrystalline material,” Philos. Mag. 95, 2046–2066 (2015). https://doi.org/10.1080/14786435.2015.1046965
K. P. Frolova and E. N. Vilchevskaya, “Effective diffusivity of transversely isotropic material with embedded pores,” Mater. Phys. Mech. 47, 937–950 (2021). https://doi.org/10.18149/MPM.4762021_12
T. Miloh and Y. Benveniste, “On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces,” Proc. R. Soc. London, Ser. A 455, 2687–2706 (1999). https://doi.org/10.1098/rspa.1999.0422
H. L. Duan and B. L. Karihaloo, “Effective thermal conductivities of heterogeneous media containing multiple imperfectly bonded inclusions,” Phys. Rev. B 75, 064206 (2007). https://doi.org/10.1103/PhysRevB.75.064206
V. I. Kushch, I. Sevostianov, and A. S. Belyaev, “Effective conductivity of spheroidal particle composite with imperfect interfaces: Complete solutions for periodic and random micro structures,” Mech. Mater. 89, 1–11 (2015). https://doi.org/10.1016/j.mechmat.2015.05.010
A. L. Endres and R. J. Knight, “A model for incorporating surface phenomena into the dielectric response of a heterogeneous medium,” J. Colloid Interface Sci. 157, 418–425 (1993). https://doi.org/10.1006/jcis.1993.1204
V. Levin and M. Markov, “Effective thermal conductivity of micro-inhomogeneous media containing imperfectly bonded ellipsoidal inclusions,” Int. J. Eng. Sci. 109, 202–215 (2016). https://doi.org/10.1016/j.ijengsci.2016.09.012
R. Hill, “Elastic properties of reinforced solids: Some theoretical principles,” J. Mech. Phys. Solids 11, 357–372 (1963). https://doi.org/10.1016/0022-5096(63)90036-X
H. Fricke, “A mathematical treatment of the electric conductivity and capacity of disperse systems I. The electric conductivity of a suspension of homogeneous spheroids,” Phys. Rev. 24, 575 (1924). https://doi.org/10.1103/PhysRev.24.575
K. Z. Markov, Elementary Micromechanics of Heterogeneous Media (Birkhäuser, Boston, Mass., 2000).
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflict of interest.
Additional information
Translated by K. Gumerov
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Frolova, K.P., Vilchevskaya, E.N. & Polyanskiy, V.A. Modeling of Imperfect Contacts in Determining the Effective Diffusion Permeability. Vestnik St.Petersb. Univ.Math. 56, 459–469 (2023). https://doi.org/10.1134/S1063454123040088
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063454123040088