December  2020, 15(4): 581-603. doi: 10.3934/nhm.2020015

Perturbation analysis of the effective conductivity of a periodic composite

1. 

Dipartimento di Matematica 'Tullio Levi-Civita', Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy

2. 

Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia, via Torino 155, 30172 Venezia Mestre, Italy

* Corresponding author: Paolo Musolino

Received  March 2020 Revised  June 2020 Published  December 2020 Early access  July 2020

We consider the effective conductivity $ \lambda^{\mathrm{eff}} $ of a periodic two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. Then we study the behavior of $ \lambda^{\mathrm{eff}} $ upon perturbation of the shape of the inclusions, of the periodicity structure, and of the conductivity of each material.

Citation: Paolo Luzzini, Paolo Musolino. Perturbation analysis of the effective conductivity of a periodic composite. Networks and Heterogeneous Media, 2020, 15 (4) : 581-603. doi: 10.3934/nhm.2020015
References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, 162, Springer, New York, 2007.

[3]

H. AmmariH. Kang and K. Kim, Polarization tensors and effective properties of anisotropic composite materials, J. Differential Equations, 215 (2005), 401-428.  doi: 10.1016/j.jde.2004.09.010.

[4]

H. AmmariH. Kang and M. Lim, Effective parameters of elastic composites, Indiana Univ. Math. J., 55 (2006), 903-922.  doi: 10.1512/iumj.2006.55.2681.

[5]

H. AmmariH. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41 (2005), 119-140. 

[6]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.

[7]

L. BerlyandD. GolovatyA. Movchan and J. Phillips, Transport properties of densely packed composites. Effect of shapes and spacings of inclusions, Quart. J. Mech. Appl. Math., 57 (2004), 495-528.  doi: 10.1093/qjmam/57.4.495.

[8]

L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase composites due to polydispersity, J. Stat. Phys., 118 (2005), 481-509.  doi: 10.1007/s10955-004-8818-0.

[9]

L. P. CastroD. Kapanadze and E. Pesetskaya, A heat conduction problem of 2D unbounded composites with imperfect contact conditions, ZAMM Z. Angew. Math. Mech., 95 (2015), 952-965.  doi: 10.1002/zamm.201400067.

[10]

L. P. CastroD. Kapanadze and E. Pesetskaya, Effective conductivity of a composite material with stiff imperfect contact conditions, Math. Methods Appl. Sci., 38 (2015), 4638-4649.  doi: 10.1002/mma.3423.

[11]

L. P. Castro and E. Pesetskaya, A composite material with inextensible-membrane-type interface, Math. Mech. Solids, 24 (2019), 499-510.  doi: 10.1177/1081286517746717.

[12]

G. P. Cherepanov, Inverse problems of the plane theory of elasticity, J. Appl. Math. Mech., 38 (1974), 915–931 (1975); translated from Prikl. Mat. Meh., 38 (1974), 963–979 (Russian). doi: 10.1016/0021-8928(75)90085-4.

[13]

A. Cherkaev, Variational Methods for Structural Optimization, Applied Mathematical Sciences, 140. Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1188-4.

[14]

M. Dalla Riva, Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ., 58 (2013), 231-257.  doi: 10.1080/17476933.2011.575462.

[15]

M. Dalla Riva and M. Lanza de Cristoforis, Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: A functional analytic approach, Complex Anal. Oper. Theory, 5 (2011), 811-833.  doi: 10.1007/s11785-010-0109-y.

[16]

M. Dalla Riva and P. Musolino, A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite, SIAM J Appl Math., 73 (2013), 24-46.  doi: 10.1137/120886637.

[17]

M. Dalla RivaP. Musolino and R. Pukhtaievych, Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface, Asymptot. Anal., 111 (2019), 217-250.  doi: 10.3233/ASY-181495.

[18]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[19]

P. Drygaś, S. Gluzman, V. Mityushev and W. Nawalaniec, Applied Analysis of Composite Media. Analytical and Computational Results for Materials Scientists and Engineers, , Woodhead Publishing Series in Composites Science and Engineering. Woodhead Publishing, 2020.

[20]

P. Drygaś and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance, Quart. J. Mech. Appl. Math., 62 (2009), 235-262.  doi: 10.1093/qjmam/hbp010.

[21]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

[22]

L. V. Gibiansky and A. V. Cherkaev, Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in Topics in the mathematical modelling of composite materials, 273–317, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[24]

S. Gluzman, V. Mityushev and W. Nawalaniec, Computational Analysis of Structured Media, Mathematical Analysis and Its Applications. Academic Press, London, 2018.

[25]

Y. Gorb and L. Berlyand, Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape, Quart. J. Mech. Appl. Math., 58 (2005), 84-106.  doi: 10.1093/qjmamj/hbh022.

[26]

S. Gryshchuk and S. Rogosin, Effective conductivity of 2D disk-ring composite material, Math. Model. Anal., 18 (2013), 386-394.  doi: 10.3846/13926292.2013.804890.

[27]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[28]

D. KapanadzeG. Mishuris and E. Pesetskaya, Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials, Complex Var. Elliptic Equ., 60 (2015), 1-23.  doi: 10.1080/17476933.2013.876418.

[29]

S. M. Kozlov, Geometric aspects of averaging, Russian Math. Surveys, 44 (1989), 91-144.  doi: 10.1070/RM1989v044n02ABEH002039.

[30]

P. Kurtyka and N. Rylko, Quantitative analysis of the particles distributions in reinforced composites, Composite Structures, 182 (2017), 412-419.  doi: 10.1016/j.compstruct.2017.09.048.

[31]

M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces, Comput. Methods Funct. Theory, 2 (2002), 1-27.  doi: 10.1007/BF03321008.

[32]

M. Lanza de Cristoforis, A domain perturbation problem for the Poisson equation, Complex Var. Theory Appl., 50 (2005), 851-867.  doi: 10.1080/02781070500136993.

[33]

M. Lanza de Cristoforis, Perturbation problems in potential theory, a functional analytic approach, J. Appl. Funct. Anal., 2 (2007), 197-222. 

[34]

M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. (FJMS), 52 (2011) 75–120.

[35]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl., 16 (2004), 137-174.  doi: 10.1216/jiea/1181075272.

[36]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density, in Analytic methods of analysis and differential equations: AMADE 2006, Camb. Sci. Publ., Cambridge, 2008,193–220.

[37]

H. Lee and J. Lee, Array dependence of effective parameters of dilute periodic elastic composite, in Imaging, multi-scale and high contrast partial differential equations, 59–71, Contemp. Math., 660, Amer. Math. Soc., Providence, RI, 2016.

[38]

K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1984), 71-87.  doi: 10.1017/S030821050002597X.

[39]

P. LuzziniP. Musolino and R. Pukhtaievych, Shape analysis of the longitudinal flow along a periodic array of cylinders, J. Math. Anal. Appl., 477 (2019), 1369-1395.  doi: 10.1016/j.jmaa.2019.05.017.

[40]

P. Luzzini, P. Musolino and R. Pukhtaievych, Real analyticity of periodic layer potentials upon perturbation of the periodicity parameters and of the support, in Proceedings of the 12th ISAAC congress (Aveiro, 2019). Research Perspectives. Birkhuser, accepted.

[41]

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613357.

[42]

V. MityushevW. NawalaniecD. Nosov and E. Pesetskaya, Schwarz's alternating method in a matrix form and its applications to composites, Appl. Math. Comput., 356 (2019), 144-156.  doi: 10.1016/j.amc.2019.03.032.

[43]

V. MityushevYu. ObnosovE. Pesetskaya and S. Rogosin, Analytical methods for heat conduction in composites, Math. Model. Anal., 13 (2008), 67-78.  doi: 10.3846/1392-6292.2008.13.67-78.

[44]

V. Mityushev and N. Rylko, Optimal distribution of the nonoverlapping conducting disks, Multiscale Model. Simul., 10 (2012), 180-190.  doi: 10.1137/110823225.

[45]

P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach, Math. Methods Appl. Sci., 35 (2012), 334-349.  doi: 10.1002/mma.1575.

[46]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35245-4.

[47]

R. Pukhtaievych, Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain, Math. Methods Appl. Sci., 41 (2018), 3392-3413.  doi: 10.1002/mma.4832.

[48]

R. Pukhtaievych, Effective conductivity of a periodic dilute composite with perfect contact and its series expansion, Z. Angew. Math. Phys., 69 (2018), Paper no. 83, 22 pp. doi: 10.1007/s00033-018-0976-z.

[49]

N. Rylko, Dipole matrix for the 2D inclusions close to circular, ZAMM Z. Angew. Math. Mech., 88 (2008), 993-999.  doi: 10.1002/zamm.200700114.

[50]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, volume 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, 162, Springer, New York, 2007.

[3]

H. AmmariH. Kang and K. Kim, Polarization tensors and effective properties of anisotropic composite materials, J. Differential Equations, 215 (2005), 401-428.  doi: 10.1016/j.jde.2004.09.010.

[4]

H. AmmariH. Kang and M. Lim, Effective parameters of elastic composites, Indiana Univ. Math. J., 55 (2006), 903-922.  doi: 10.1512/iumj.2006.55.2681.

[5]

H. AmmariH. Kang and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41 (2005), 119-140. 

[6]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.

[7]

L. BerlyandD. GolovatyA. Movchan and J. Phillips, Transport properties of densely packed composites. Effect of shapes and spacings of inclusions, Quart. J. Mech. Appl. Math., 57 (2004), 495-528.  doi: 10.1093/qjmam/57.4.495.

[8]

L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase composites due to polydispersity, J. Stat. Phys., 118 (2005), 481-509.  doi: 10.1007/s10955-004-8818-0.

[9]

L. P. CastroD. Kapanadze and E. Pesetskaya, A heat conduction problem of 2D unbounded composites with imperfect contact conditions, ZAMM Z. Angew. Math. Mech., 95 (2015), 952-965.  doi: 10.1002/zamm.201400067.

[10]

L. P. CastroD. Kapanadze and E. Pesetskaya, Effective conductivity of a composite material with stiff imperfect contact conditions, Math. Methods Appl. Sci., 38 (2015), 4638-4649.  doi: 10.1002/mma.3423.

[11]

L. P. Castro and E. Pesetskaya, A composite material with inextensible-membrane-type interface, Math. Mech. Solids, 24 (2019), 499-510.  doi: 10.1177/1081286517746717.

[12]

G. P. Cherepanov, Inverse problems of the plane theory of elasticity, J. Appl. Math. Mech., 38 (1974), 915–931 (1975); translated from Prikl. Mat. Meh., 38 (1974), 963–979 (Russian). doi: 10.1016/0021-8928(75)90085-4.

[13]

A. Cherkaev, Variational Methods for Structural Optimization, Applied Mathematical Sciences, 140. Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1188-4.

[14]

M. Dalla Riva, Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ., 58 (2013), 231-257.  doi: 10.1080/17476933.2011.575462.

[15]

M. Dalla Riva and M. Lanza de Cristoforis, Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: A functional analytic approach, Complex Anal. Oper. Theory, 5 (2011), 811-833.  doi: 10.1007/s11785-010-0109-y.

[16]

M. Dalla Riva and P. Musolino, A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite, SIAM J Appl Math., 73 (2013), 24-46.  doi: 10.1137/120886637.

[17]

M. Dalla RivaP. Musolino and R. Pukhtaievych, Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface, Asymptot. Anal., 111 (2019), 217-250.  doi: 10.3233/ASY-181495.

[18]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[19]

P. Drygaś, S. Gluzman, V. Mityushev and W. Nawalaniec, Applied Analysis of Composite Media. Analytical and Computational Results for Materials Scientists and Engineers, , Woodhead Publishing Series in Composites Science and Engineering. Woodhead Publishing, 2020.

[20]

P. Drygaś and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance, Quart. J. Mech. Appl. Math., 62 (2009), 235-262.  doi: 10.1093/qjmam/hbp010.

[21]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

[22]

L. V. Gibiansky and A. V. Cherkaev, Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in Topics in the mathematical modelling of composite materials, 273–317, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997.

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[24]

S. Gluzman, V. Mityushev and W. Nawalaniec, Computational Analysis of Structured Media, Mathematical Analysis and Its Applications. Academic Press, London, 2018.

[25]

Y. Gorb and L. Berlyand, Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape, Quart. J. Mech. Appl. Math., 58 (2005), 84-106.  doi: 10.1093/qjmamj/hbh022.

[26]

S. Gryshchuk and S. Rogosin, Effective conductivity of 2D disk-ring composite material, Math. Model. Anal., 18 (2013), 386-394.  doi: 10.3846/13926292.2013.804890.

[27]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[28]

D. KapanadzeG. Mishuris and E. Pesetskaya, Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials, Complex Var. Elliptic Equ., 60 (2015), 1-23.  doi: 10.1080/17476933.2013.876418.

[29]

S. M. Kozlov, Geometric aspects of averaging, Russian Math. Surveys, 44 (1989), 91-144.  doi: 10.1070/RM1989v044n02ABEH002039.

[30]

P. Kurtyka and N. Rylko, Quantitative analysis of the particles distributions in reinforced composites, Composite Structures, 182 (2017), 412-419.  doi: 10.1016/j.compstruct.2017.09.048.

[31]

M. Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces, Comput. Methods Funct. Theory, 2 (2002), 1-27.  doi: 10.1007/BF03321008.

[32]

M. Lanza de Cristoforis, A domain perturbation problem for the Poisson equation, Complex Var. Theory Appl., 50 (2005), 851-867.  doi: 10.1080/02781070500136993.

[33]

M. Lanza de Cristoforis, Perturbation problems in potential theory, a functional analytic approach, J. Appl. Funct. Anal., 2 (2007), 197-222. 

[34]

M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. (FJMS), 52 (2011) 75–120.

[35]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl., 16 (2004), 137-174.  doi: 10.1216/jiea/1181075272.

[36]

M. Lanza de Cristoforis and L. Rossi, Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density, in Analytic methods of analysis and differential equations: AMADE 2006, Camb. Sci. Publ., Cambridge, 2008,193–220.

[37]

H. Lee and J. Lee, Array dependence of effective parameters of dilute periodic elastic composite, in Imaging, multi-scale and high contrast partial differential equations, 59–71, Contemp. Math., 660, Amer. Math. Soc., Providence, RI, 2016.

[38]

K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1984), 71-87.  doi: 10.1017/S030821050002597X.

[39]

P. LuzziniP. Musolino and R. Pukhtaievych, Shape analysis of the longitudinal flow along a periodic array of cylinders, J. Math. Anal. Appl., 477 (2019), 1369-1395.  doi: 10.1016/j.jmaa.2019.05.017.

[40]

P. Luzzini, P. Musolino and R. Pukhtaievych, Real analyticity of periodic layer potentials upon perturbation of the periodicity parameters and of the support, in Proceedings of the 12th ISAAC congress (Aveiro, 2019). Research Perspectives. Birkhuser, accepted.

[41]

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613357.

[42]

V. MityushevW. NawalaniecD. Nosov and E. Pesetskaya, Schwarz's alternating method in a matrix form and its applications to composites, Appl. Math. Comput., 356 (2019), 144-156.  doi: 10.1016/j.amc.2019.03.032.

[43]

V. MityushevYu. ObnosovE. Pesetskaya and S. Rogosin, Analytical methods for heat conduction in composites, Math. Model. Anal., 13 (2008), 67-78.  doi: 10.3846/1392-6292.2008.13.67-78.

[44]

V. Mityushev and N. Rylko, Optimal distribution of the nonoverlapping conducting disks, Multiscale Model. Simul., 10 (2012), 180-190.  doi: 10.1137/110823225.

[45]

P. Musolino, A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach, Math. Methods Appl. Sci., 35 (2012), 334-349.  doi: 10.1002/mma.1575.

[46]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35245-4.

[47]

R. Pukhtaievych, Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain, Math. Methods Appl. Sci., 41 (2018), 3392-3413.  doi: 10.1002/mma.4832.

[48]

R. Pukhtaievych, Effective conductivity of a periodic dilute composite with perfect contact and its series expansion, Z. Angew. Math. Phys., 69 (2018), Paper no. 83, 22 pp. doi: 10.1007/s00033-018-0976-z.

[49]

N. Rylko, Dipole matrix for the 2D inclusions close to circular, ZAMM Z. Angew. Math. Mech., 88 (2008), 993-999.  doi: 10.1002/zamm.200700114.

[50]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, volume 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

Figure 1.  The sets $ \mathbb{S}_{q}[q\mathbb{I}[\phi]]^- $, $ \mathbb{S}_{q}[q\mathbb{I}[\phi]] $, and $ q\phi(\partial\Omega) $ in case $ n = 2 $
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