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.

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.

Reviewer: Marco Codegone (Torino)

##### MSC:

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

74Q15 | Effective constitutive equations in solid mechanics |

74R05 | Brittle damage |

##### References:

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