×

zbMATH — the first resource for mathematics

On the homogenization of problems in the theory of elasticity on composite structures. (Russian, English) Zbl 1085.35027
J. Math. Sci., New York 132, No. 3, 313-330 (2006); translation from Zap. Nauchn. Semin. POMI 310, 114-144, 227-228 (2004).
The paper deals with the elastic behaviour of 2-dimensional periodic structures, with a reinforced periodic thin network. The underlying structure is described by means of the two-dimensional Lebesgue measure, while the reinforcement is represented by a normalised measure, proportional to the two-dimensional Lebesgue measure, and concentrated on a network \(F^h_\varepsilon\), with period \(\varepsilon\) and thickness \(h\). The combined structure is obtained by the normalised sum \(\mu^h_\varepsilon\) of the two measures. The main problem is the study of the limit behaviour of the solutions to a boundary-value problem for the system of linear elasticity related to such structure, and expressed in a variational form by means of the measures \(\mu^h_\varepsilon\).
The main result states that, as the thickness \(h\) and the size of the period \(\varepsilon\) tend to zero, then the solutions converge strongly in \(L^2\) with respect to measures, to a solution of a homogenised problem, where the effective tensor can be expressed by a homogenisation formula of classical type. In particular, the result is independent of the choice of the function \(h=h(\varepsilon)\). A crucial technical point is the proof of a Korn’s inequality with respect to measures. Moreover, the proof of the main result uses the theory of two-scale convergence with respect to varying measures, developed by V. V. Zhikov. The author remarks that this result is different from what happens in homogenisation of thin elastic structures without reinforcement, where “non-classical” homogenisation results have been proved, and indicates some reasons for these phenomena.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74Q15 Effective constitutive equations in solid mechanics
74R05 Brittle damage
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] V. V. Zhikov and S. E. Pastukhova, ”Homogenization of problems of elasticity theory on periodic grids of critical thickness,” Dokl. Ros. Akad. Nauk, 385, 590–595 (2002). · Zbl 1146.35310
[2] V. V. Zhikov and S. E. Pastukhova, ”Homogenization of problems of elasticity theory on periodic grids of critical thickness,” Mat. Sb., 194, 61–95 (2003). · Zbl 1077.35023
[3] S. E. Pastukhova, ”Homogenization of problems of elasticity theory on periodic box structures of critical thickness,” Dokl. Ros. Akad. Nauk, 387, 447–451 (2002). · Zbl 1197.35032
[4] S. E. Pastukhova, ”Homogenization of problems of elasticity theory on periodic rod frames of critical thickness,” Dokl. Ros. Akad. Nauk, 394, 26–31 (2003).
[5] S. E. Pastukhova, ”Homogenization of problems of elasticity theory on periodic box and rod frames of critical thickness,” Contemp. Math. Appl., 12, 51–98 (2004). · Zbl 1085.35028
[6] V. V. Zhikov, ”One generalization and application of the method of two-scale convergence,” Mat. Sb., 191, 31–72 (2000). · Zbl 0969.35048
[7] V. V. Zhikov, ”Homogenization of problems of elasticity theory on singular structures,” Izv. RAN, Ser. Mat., 66, 81–148 (2002). · Zbl 1043.35031
[8] V. V. Zhikov, ”On two-scale convergence,” Trudy Semin. Petrovskii, 23, 149–187 (2003). · Zbl 1284.35054
[9] G. Nguetseng, ”A general convergence result for a functional related to the theory of homogenization,” SIAM J. Math. Anal., 20, 608–623 (1989). · Zbl 0688.35007 · doi:10.1137/0520043
[10] G. Allaire, ”Homogenization and two-scale convergence,” SIAM J. Math. Anal., 23, 1482–1518 (1992). · Zbl 0770.35005 · doi:10.1137/0523084
[11] S. E. Pastukhova, ”Homogenization of problems of elasticity theory for a periodic combined structure,” Dokl. Ros. Akad. Nauk, 395, 316–321 (2004).
[12] S. E. Pastukhova, ”Approximative properties of the Sobolev spaces of elasticity theory on thin rod structures,” Contemp. Math. Appl., 12, 99–106 (2004). · Zbl 1099.46025
[13] V. V. Zhikov, ”On the homogenization technique for variational problems,” Funkts. Anal. Prilozh., 33, 14–29 (1999). · Zbl 0959.49016
[14] V. V. Zhikov, ”On weighted Sobolev spaces,” Mat. Sb., 189, 27–58 (1998). · Zbl 0919.46026
[15] V. V. Zhikov, ”Note on Sobolev spaces,” Contemp. Math. Appl., 10, 54–58 (2003). · Zbl 1098.46028
[16] T. Kato, Theory of Perturbations of Linear Operators [Russian translation], Moscow (1979).
[17] S. E. Pastukhova, ”Convergence of hyperbolic semigroups in a variable Hilbert space,” Trudy Semin. Petrovskii, 24, 216–241 (2004).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.