On sets of multiple equally strong holes in an infinite elastic plate: parameterization and existence. (English) Zbl 1425.74206
Summary: This paper addresses the inverse problem of determining sets of any finite number of “equally strong” holes (with smooth boundaries) in an infinite, homogeneous, isotropic elastic plate that is subjected to uniform in-plane loadings along the holes’ boundaries and at infinity. Our two main results are (i) we construct an exact, explicit parameterization that describes all such sets of holes in terms of a one-to-one conformal map from a circular domain; (ii) we derive a condition on the loading parameters that is both necessary and sufficient for such a set of holes to exist. We derive result (i) by considering the Schwarz functions of the holes’ boundaries, introducing a Schottky group that is generated from our preimage circular domain, and making use of the theory of automorphic functions. We derive result (ii) by making use of our parameterization and exploiting existing results relating to univalent conformal maps. In addition to presenting results (i) and (ii), we discuss a connection to
quadrature domain theory.
MSC:
| 74G75 | Inverse problems in equilibrium solid mechanics |
| 74G05 | Explicit solutions of equilibrium problems in solid mechanics |
| 74E05 | Inhomogeneity in solid mechanics |
| 74P10 | Optimization of other properties in solid mechanics |
| 30E25 | Boundary value problems in the complex plane |
| 30C20 | Conformal mappings of special domains |
| 30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
Keywords:
equally strong holes; multiply connected domains; Schwarz function; conformal mapping; automorphic functions; null quadrature domainsReferences:
| [1] | SKPrime, 2015, https://github.com/ehkropf/SKPrime. |
| [2] | Y. A. Antipov, Slit maps in the study of equal-strength cavities in \(n\)-connected elastic planar domains, SIAM J. Appl. Math., 78 (2018), pp. 320-342. · Zbl 1387.74009 |
| [3] | Y. A. Antipov, Method of automorphic functions for an inverse problem of antiplane elasticity, Quart. J. Mech. Appl. Math., 72 (2019), pp. 213-234. · Zbl 1425.30012 |
| [4] | H. F. Baker, Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions, 2nd ed., Cambridge University Press, Cambridge, UK, 1995. · Zbl 0848.14012 |
| [5] | N. V. Banichuk, Optimality conditions in the problem of seeking the hole shapes in elastic bodies, J. Appl. Math. Mech., 41 (1977), pp. 946-951. · Zbl 0401.73030 |
| [6] | A. F. Beardon, A Primer on Riemann Surfaces, London Math. Soc. Lecture Note Ser. 78, Cambridge University Press, Cambridge, UK, 1984. · Zbl 0546.30001 |
| [7] | E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Evolution Equations, Springer-Verlag, Berlin, 1994. · Zbl 0809.35001 |
| [8] | G. Cherepanov, Inverse problems of the plane theory of elasticity, J. Appl. Math. Mech., 38 (1974), pp. 915-931. · Zbl 0315.73106 |
| [9] | D. Crowdy and J. Marshall, Conformal maps to generalized quadrature domains, in Special Functions and Orthogonal Polynomials, Contemp. Math. 471, AMS, Providence, RI, 2008, pp. 105-116. · Zbl 1188.30052 |
| [10] | D. G. Crowdy, Quadrature domains and fluid dynamics, in Quadrature Domains and Their Applications, Oper. Theory Adv. Appl. 156, Birkhäuser, Basel, 2005, pp. 113-129. · Zbl 1329.76290 |
| [11] | D. G. Crowdy, Stress fields around two pores in an elastic body: Exact quadrature domain solutions, Proc. A, 471 (2015), 20150240. · Zbl 1371.74026 |
| [12] | D. G. Crowdy, private communication, 2018. |
| [13] | D. G. Crowdy, E. H. Kropf, C. C. Green, and M. M. S. Nasser, The Schottky-Klein prime function: A theoretical and computational tool for applications, IMA J. Appl. Math., 81 (2016), pp. 589-628. · Zbl 1408.30035 |
| [14] | D. G. Crowdy and J. S. Marshall, Conformal mappings between canonical multiply-connected domains, Comput. Methods Funct. Theory, 6 (2006), pp. 59-76. · Zbl 1101.30010 |
| [15] | D. G. Crowdy and J. S. Marshall, Computing the Schottky-Klein prime function on the Schottky double of planar domains, Comput. Methods Funct. Theory, 7 (2007), pp. 293-308. · Zbl 1148.30023 |
| [16] | D. G. Crowdy and J. S. Marshall, Multiply-connected quadrature domains and the Bergman kernel function, Complex Anal. Oper. Theory, 3 (2009), pp. 379-397. · Zbl 1186.30006 |
| [17] | P. J. Davis, The Schwarz Function and Its Applications, Mathematics Association of America, Washington, DC, 1974. · Zbl 0293.30001 |
| [18] | L. R. Ford, Automorphic Functions, 2nd ed., Chelsea, New York, 1972. · JFM 55.0810.04 |
| [19] | P. R. Garabedian and M. Schiffer, Identities in the theory of conformal mapping, Trans. Amer. Math. Soc., 65 (1949), pp. 187-238. · Zbl 0035.05204 |
| [20] | G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, AMS, Providence, RI, 1969. · Zbl 0183.07502 |
| [21] | Y. Grabovsky and R. V. Kohn, Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: The Vigdergauz microstructure, J. Mech. Phys. Solids, 43 (1995), pp. 949-972. · Zbl 0877.73041 |
| [22] | H. Grötzsch, Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche, Ber. Verh. sächs Akad. Wiss. Leipzig, Math.-phys. Kl., 84 (1932), pp. 15-36. · Zbl 0005.06802 |
| [23] | H. Grunsky, Lectures on Theory of Functions in Multiply Connected Domains, Vandenhoeck & Ruprecht, Göttingen, 1978. · Zbl 0371.30015 |
| [24] | B. Gustafsson, C. He, P. Milanfar, and M. Putinar, Reconstructing planar domains from their moments, Inverse Problems, 16 (2000), pp. 1053-1070. · Zbl 0959.44010 |
| [25] | B. Gustafsson and H. S. Shapiro, What is a quadrature domain?, in Quadrature Domains and Their Applications, Oper. Theory Adv. Appl. 156, Birkhäuser, Basel, 2005, pp. 1-25. · Zbl 1086.30002 |
| [26] | D. A. Hejhal, Theta Functions, Kernel Functions and Abelian Integrals, Mem. Amer. Math. Soc. 129, AMS, Providence, RI, 1972. · Zbl 0244.30016 |
| [27] | H. Kang, Conjectures of Pólya-Szegö and Eshelby, and the Newtonian potential problem: A review, Mech. Mater., 41 (2009), pp. 405-410. |
| [28] | H. Kang, E. Kim, and G. Milton, Inclusion pairs satisfying Eshelby’s uniformity property, SIAM J. Appl. Math., 69 (2008), pp. 577-595. · Zbl 1159.74387 |
| [29] | L. P. Liu, Solutions to the Eshelby’s conjectures, Proc. A, 464 (2008), pp. 573-594. · Zbl 1132.74010 |
| [30] | J. S. Marshall, Steady uniform vortex patches around an assembly of walls or flat plates, Quart. J. Mech. Appl. Math., 65 (2011), pp. 27-60. · Zbl 1248.76027 |
| [31] | J. S. Marshall, Exact solutions describing the injection-driven growth of a doubly-periodic fluid region in a Hele-Shaw cell, IMA J. Appl. Math., 81 (2016), pp. 723-749. · Zbl 1408.76230 |
| [32] | D. Mumford, C. Series, and D. Wright, Indra’s Pearls: The Vision of Felix Klein, Cambridge University Press, Cambridge, UK, 2002. · Zbl 1141.00002 |
| [33] | N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity: Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending, 2nd ed., Noordhoff, Leiden, 1975. · Zbl 0297.73008 |
| [34] | M. Sakai, Null quadrature domains, J. Anal. Math., 40 (1981), pp. 144-154. · Zbl 0483.30002 |
| [35] | M. Schiffer, The span of multiply connected domains, Duke Math. J., 10 (1943), pp. 209-216. · Zbl 0060.23704 |
| [36] | M. Schiffer, Some recent developments in the theory of conformal mapping, in Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces, R. Courant, Springer, New York, 1950. |
| [37] | M. Schiffer and D. C. Spencer, Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton, NJ, 1954. · Zbl 0059.06901 |
| [38] | M. Shiba, Univalence of a complex linear combination of two extremal parallel slit mappings, Analysis, 27 (2007), pp. 301-310. · Zbl 1132.30306 |
| [39] | G. L. Vasconcelos, J. S. Marshall, and D. G. Crowdy, Secondary Schottky-Klein prime functions associated with multiply connected planar domains, Proc. A, 471 (2015), 20140688. · Zbl 1371.30012 |
| [40] | S. B. Vigdergauz, Integral equation of the inverse problem of the plane theory of elasticity, J. Appl. Math. Mech., 40 (1976), pp. 518-522. · Zbl 0402.73026 |
| [41] | S. B. Vigdergauz, On a case of the inverse problem of two-dimensional theory of elasticity, J. Appl. Math. Mech., 41 (1977), pp. 927-933. · Zbl 0401.73038 |
| [42] | S. B. Vigdergauz, Effective elastic parameters of a plate with a regular system of equal strength holes, Mech. Solids, 21 (1986), pp. 162-166. |
| [43] | X. Wang, Uniform fields inside two non-elliptical inclusions, Math. Mech. Solids, 17 (2012), pp. 736-761. · Zbl 07278887 |
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.
