×

Properties of a composite material with mixed imperfect contact conditions. (English) Zbl 1488.30226

Summary: We present an analytical solution of a mixed boundary value problem for an unbounded 2D doubly periodic domain which is a model of a composite material with mixed imperfect interface conditions. We find the effective conductivity of the composite material with mixed imperfect interface conditions, and also give numerical analysis of several of their properties such as temperature and flux.

MSC:

30E25 Boundary value problems in the complex plane
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
39B32 Functional equations for complex functions
74E30 Composite and mixture properties
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74M15 Contact in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
74S70 Complex-variable methods applied to problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. Allaire. Shape Optimization by the Homogenization Method. Springer Verlag, New York, 2002. http://dx.doi.org/10.1007/978-1-4684-9286-6. · Zbl 0990.35001 · doi:10.1007/978-1-4684-9286-6
[2] I.V. Andrianov, V. I. Bolshakov, V.V. Danishevskyy, and D. Weichert.: Asymptotic study of imperfect interfaces in conduction through a granular composite material. Proceedings of the Royal Society A, 466(2121):2707-2725, 2010. http://dx.doi.org/10.1098/rspa.2010.0052. · Zbl 1211.74056 · doi:10.1098/rspa.2010.0052
[3] Y. Benveniste and T. Miloh. Imperfect soft and stiff interfaces in two– dimensional elasticity. Mechanics of Materials, 33(6):309-323, 2001. · doi:10.1016/S0167-6636(01)00055-2
[4] L. Berlyand and V. Mityushev. Generalized Clausius-Mossotti formula for random composite with circular fibers. Journal of Statistical Physics, 102(1):115-145, 2001. http://dx.doi.org/10.1023/A:1026512725967. · Zbl 1072.82586 · doi:10.1023/A:1026512725967
[5] L.P. Castro, D. Kapanadze and E. Pesetskaya. Effective conductivity of a composite material with stiff imperfect contact conditions. Mathematical Methods in the Applied Sciences, 38(18):4638-4649, 2015. http://dx.doi.org/10.1002/mma.3423. · Zbl 1345.31005 · doi:10.1002/mma.3423
[6] L.P. Castro, D. Kapanadze and E. Pesetskaya. A heat conduction problem of 2D unbounded composites with imperfect contact conditions. Zeitschrift fuer Angewandte Mathematik und Mechanik, 95(9):952-965, 2015. http://dx.doi.org/10.1002/zamm.201400067. · Zbl 1326.74037 · doi:10.1002/zamm.201400067
[7] L.P. Castro, E. Pesetskaya and S. Rogosin. Effective conductivity of a composite material with non-ideal contact conditions. Complex Variables and Elliptic Equations, 54(12):1085-1100, 2009. http://dx.doi.org/10.1080/17476930903275995. · Zbl 1184.30029
[8] P. Drygas. A functional-differential equation in a class of analytic functions and its application. Aequationes Mathematicae, 73(3):222-232, 2007. http://dx.doi.org/10.1007/s00010-006-2865-3. · Zbl 1125.30018 · doi:10.1007/s00010-006-2865-3
[9] P. Drygas. Functional-differential equations in Hardy-type classes. Trudy Instituta Matematiki, 15(1):105-110, 2007. · Zbl 1497.31001
[10] P. Drygas and V. Mityushev. Effective conductivity of arrays of unidirectional cylinders with interfacial resistance. The Quarterly Journal of Mechanics and Applied Mathematics, 62(3):235-262, 2009. http://dx.doi.org/10.1093/qjmam/hbp010. · Zbl 1170.74043 · doi:10.1093/qjmam/hbp010
[11] R. Guinovart-Díaz, R. Rodŕıguez-Ramos, J. Bravo-Castillero, F.J. Sabina, R. Dario Santiago and R. Martínez Rosado.: Asymptotic analysis of linear thermoelastic properties of fiber composites. Journal of Thermoplastic Composite Materials, 20(4):389-410, 2007. http://dx.doi.org/10.1177/0892705707079607. · doi:10.1177/0892705707079607
[12] V.V. Jikov, S.M. Kozlov and O.A. Olejnik. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994. http://dx.doi.org/10.1007/978-3-642-84659-5. · Zbl 0838.35001 · doi:10.1007/978-3-642-84659-5
[13] D. Kapanadze, G. Mishuris and E. Pesetskaya. Improved algorithm for analytical solution of the heat conduction problem in composites. Complex Variables and Elliptic Equations, 60(1):1-23, 2015. http://dx.doi.org/10.1080/17476933.2013.876418. · Zbl 1316.30041
[14] V. Mityushev. Transport properties of doubly periodic arrays of circular cylinders and optimal design problems. Applied Mathematics and Optimization, 44(1):17-31, 2001. http://dx.doi.org/10.1007/s00245-001-0013-y. · Zbl 0982.30019 · doi:10.1007/s00245-001-0013-y
[15] V. Mityushev. Exact solution of the R-linear problem for a disk in a class of doubly periodic functions. Journal of Applied Functional Analysis, 2:115-127, 2007. · Zbl 1113.30035
[16] B. Noetinger. An explicit formula for computing the sensitivity of the effective conductivity of heterogeneous composite materials to local inclusion transport properties and geometry. Multiscale Modeling and Simulation, 11(3):907-924, 2013. http://dx.doi.org/10.1137/120884961. · Zbl 1280.35115 · doi:10.1137/120884961
[17] N. Rylko. Transport properties of a rectangular array of highly conducting cylinders. Journal of Engineering Mathematics, 38(1):1-12, 2000. http://dx.doi.org/10.1023/A:1004669705627. · Zbl 0963.78023 · doi:10.1023/A:1004669705627
[18] N. Rylko. Structure of the scalar field around unidirectional circular cylinders. Proceedings R. Soc. London A, 464(2090):391-407, 2008. http://dx.doi.org/10.1098/rspa.2007.0114. · Zbl 1143.82338 · doi:10.1098/rspa.2007.0114
[19] A. Weil. Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer, Berlin, 1999. · Zbl 0955.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.