# American Institute of Mathematical Sciences

February  2012, 6(1): 1-23. doi: 10.3934/ipi.2012.6.1

## 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

Received  May 2011 Revised  November 2011 Published  February 2012

We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of Capdeboscq and Vogelius (Math. Modeling Num. Anal. 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains a moment or polarization tensor $\mathbb{M}$ that encodes the effect of the inclusions. We also derive some basic properties of this tensor $\mathbb{M}$. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for $\mathbb{M}$ only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of $\mathbb{M}$ in this setting and recover the formula previously obtained by Beretta and Francini (SIAM J. Math. Anal., 38, 2006).
Citation: Elena Beretta, Eric Bonnetier, Elisa Francini, Anna L. Mazzucato. Small volume asymptotics for anisotropic elastic inclusions. Inverse Problems and Imaging, 2012, 6 (1) : 1-23. doi: 10.3934/ipi.2012.6.1
##### 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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##### 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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