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Small volume asymptotics for anisotropic elastic inclusions
1. | Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5 - 00185 Roma, Italy |
2. | Laboratoire Jean Kuntzmann, Université de Joseph Fourier, CNRS, 38041 Grenoble Cedex 9, France |
3. | Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67A - 50134 Firenze, Italy |
4. | Mathematics Department, Penn State University, University Park, PA, 16802, United States |
References:
[1] |
H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements,'' Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004. |
[2] |
H. Ammari, H. Kang, G. Nakamura and K. Tanuma, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity, 67 (2002), 97-129.
doi: 10.1023/A:1023940025757. |
[3] |
E. Beretta and E. Francini, An asymptotic formula for the displacement field in the presence of thin elastic inhomogeneities, SIAM J. Math. Anal., 38 (2006), 1249-1261.
doi: 10.1137/050648596. |
[4] |
S. Campanato, "Sistemi ellittici in forma divergenza. Regolaritá all'interno,'' (Italian) Quaderni, Scuola Normale Superiore Pisa, Pisa, 1980. |
[5] |
Y. Capdeboscq and H. Kang, Improved Hashin-Shtrikman bounds for elastic moment tensors and an application, Appl. Math. Optim., 57 (2008), 263-288.
doi: 10.1007/s00245-007-9022-9. |
[6] |
Y. Capdeboscq and M. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Modelling Num. Anal., 37 (2003), 159-173.
doi: 10.1051/m2an:2003014. |
[7] |
Y. Capdeboscq and M. Vogelius, A review of some recent work on impedance imaging for inhomogeneities of low volume fraction, in "Partial Differential Equations and Inverse Problems," Contemp. Math., 362, Amer. Math. Soc., Providence, RI, (2004), 69-87. |
[8] |
Y. Capdeboscq and M. Vogelius, Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities, Asymptot. Anal., 50 (2006), 175-204. |
[9] |
G. Fichera, Existence theorems in elasticity, in "Handbuch der Physik,'' Vol. VI, Springer-Verlag, Berlin, Heidelberg, New York, (1972), 347-389. |
[10] |
G. A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity, Arch. Rational Mech. Anal., 94 (1986), 307-334.
doi: 10.1007/BF00280908. |
[11] |
M. Fuchs, The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition, Manuscripta Math., 46 (1984), 97-115.
doi: 10.1007/BF01185197. |
[12] |
Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892-925.
doi: 10.1002/cpa.10079. |
[13] |
Y. Y. Li and M. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rat. Mech. Anal., 153 (2000), 91-151.
doi: 10.1007/s002050000082. |
[14] |
R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites, J. Mech. Phys. Solids, 41 (1993), 809-833.
doi: 10.1016/0022-5096(93)90001-V. |
[15] |
G. W. Milton, "The Theory of Composites,'' Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002. |
[16] |
O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization,'' Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992. |
[17] |
M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal., 34 (2000), 732-748. |
show all references
References:
[1] |
H. Ammari and H. Kang, "Reconstruction of Small Inhomogeneities from Boundary Measurements,'' Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004. |
[2] |
H. Ammari, H. Kang, G. Nakamura and K. Tanuma, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity, 67 (2002), 97-129.
doi: 10.1023/A:1023940025757. |
[3] |
E. Beretta and E. Francini, An asymptotic formula for the displacement field in the presence of thin elastic inhomogeneities, SIAM J. Math. Anal., 38 (2006), 1249-1261.
doi: 10.1137/050648596. |
[4] |
S. Campanato, "Sistemi ellittici in forma divergenza. Regolaritá all'interno,'' (Italian) Quaderni, Scuola Normale Superiore Pisa, Pisa, 1980. |
[5] |
Y. Capdeboscq and H. Kang, Improved Hashin-Shtrikman bounds for elastic moment tensors and an application, Appl. Math. Optim., 57 (2008), 263-288.
doi: 10.1007/s00245-007-9022-9. |
[6] |
Y. Capdeboscq and M. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Modelling Num. Anal., 37 (2003), 159-173.
doi: 10.1051/m2an:2003014. |
[7] |
Y. Capdeboscq and M. Vogelius, A review of some recent work on impedance imaging for inhomogeneities of low volume fraction, in "Partial Differential Equations and Inverse Problems," Contemp. Math., 362, Amer. Math. Soc., Providence, RI, (2004), 69-87. |
[8] |
Y. Capdeboscq and M. Vogelius, Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities, Asymptot. Anal., 50 (2006), 175-204. |
[9] |
G. Fichera, Existence theorems in elasticity, in "Handbuch der Physik,'' Vol. VI, Springer-Verlag, Berlin, Heidelberg, New York, (1972), 347-389. |
[10] |
G. A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity, Arch. Rational Mech. Anal., 94 (1986), 307-334.
doi: 10.1007/BF00280908. |
[11] |
M. Fuchs, The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition, Manuscripta Math., 46 (1984), 97-115.
doi: 10.1007/BF01185197. |
[12] |
Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892-925.
doi: 10.1002/cpa.10079. |
[13] |
Y. Y. Li and M. Vogelius, Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rat. Mech. Anal., 153 (2000), 91-151.
doi: 10.1007/s002050000082. |
[14] |
R. Lipton, Inequalities for electric and elastic polarization tensors with applications to random composites, J. Mech. Phys. Solids, 41 (1993), 809-833.
doi: 10.1016/0022-5096(93)90001-V. |
[15] |
G. W. Milton, "The Theory of Composites,'' Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002. |
[16] |
O. A. Oleĭnik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization,'' Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992. |
[17] |
M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal., 34 (2000), 732-748. |
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