Vigdergauz, Shmuel Planar grained structures with multiple inclusions in a periodic cell: elastostatic solution and its potential applications. (English) Zbl 1299.74150 Math. Mech. Solids 19, No. 7, 805-820 (2014). Summary: The simple analytical expressions for the effective moduli and related quantities of interest in a planar doubly periodic matrix-inclusion structure previously obtained for only one inclusion in a cell are generalized on a finite number of non-intersecting inclusions each having its own local properties. As before, all phases are treated as linear and isotropic with perfectly bonding along smooth material interfaces. The derivations are performed by the complex variable technique applied to the quasi-periodic Weierstrassian zeta-function. Special attention is given to the equi-stress inclusion shapes (ESSs) where the analytical development can be fully completed. In particular, they are proved to saturate the multi-phase Hashin-Shtrikman bounds on the effective bulk modulus. The necessary condition of the ESSs existence is also found, although the question of whether they really exist is left aside. The results obtained form a basis for further numerical analysis of the attendant direct and optimization problems which are briefly discussed. MSC: 74Q15 Effective constitutive equations in solid mechanics 74Q20 Bounds on effective properties in solid mechanics 74S70 Complex-variable methods applied to problems in solid mechanics Keywords:plane elasticity problem; multi-phase lattices; shape optimization; Kolosov-Muskhelishvili potentials; hoop stresses; extremal elastic structures; Hashin-Shtrikman bounds PDFBibTeX XMLCite \textit{S. Vigdergauz}, Math. Mech. Solids 19, No. 7, 805--820 (2014; Zbl 1299.74150) Full Text: DOI References: [1] DOI: 10.1017/CBO9780511613357 · Zbl 0993.74002 [2] Whittaker ET, A Course of Modern Analysis, 4. ed. (1996) [3] Vigdergauz S, Math Mech Solids 4 pp 407– (1999) · Zbl 1001.74550 [4] Cherepanov GP, J Appl Math Mech 38 pp 913– (1974) · Zbl 0315.73106 [5] Hashin Z, J Mech Phys Solids 11 pp 127– (1963) · Zbl 0108.36902 [6] Allaire G, Struct Opt 17 pp 86– (1999) [7] Muskhelishvili NI, Some Basic Problems of the Mathematical Theory of Elasticity, 2. ed. (1975) [8] Gakhov FD, Boundary Value Problems (1990) [9] Abramowitz M, Handbook of Mathematical Functions (1965) [10] Ahlfors L, Complex Analysis, 3. ed. (1979) [11] Vigdergauz S, Math Mech Solids 18 (4) pp 431– (2013) [12] Natanson IP, Theory of Functions of a Real Variable (1956) [13] Gibiyansky LV, J Mech Phys Solids 48 pp 461– (2000) · Zbl 0989.74060 [14] Grabovsky Y, J Mech Phys Solids 43 pp 949– (1994) · Zbl 0877.73041 [15] Vigdergauz SB, Q J Mech Appl Maths 49 pp 565– (1996) · Zbl 0874.73045 [16] Liu LP, Proc R Soc A: Math Phys Eng Sci 466 pp 3693– (2010) · Zbl 1211.74178 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.