Abstract
The paper is dedicated to the asymptotic behavior of varepsilon-periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as varepsilon to 0. In case of plate-like or beam-like structures the asymptotic reduction of dimension from 3D to 2D or 1D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as varepsilon to 0 we use the periodic unfolding method.
Highlights
This paper deals with the linearized elasticity problem posed in different periodic domains
These domains are obtained by reproducing a representative cell of size ε in such a way that one can get beam-like, plate-like or N -dimensional structures
We summarize the result of this section: for ε-periodic porous materials with a known structure, for e.g. structures made of beams whose thicknesses are of order ε, or dense packages of small compressed balls, the solution to the linearized elasticity problem (2.5)-(2.6) in a heterogeneous 3D domain is approximated by uε(x) ≈ u(x) + ε enp(u)(x)χnp x ε n,p=1 for x ∈ Ωε∗, (2.17)
Summary
This paper deals with the linearized elasticity problem posed in different periodic domains. These domains are obtained by reproducing a representative cell of size ε in such a way that one can get beam-like, plate-like or N -dimensional structures. Note that in case of beam-like and plate-like domains the derivation of Korn’s inequalities is based on the decomposition of beam or plate displacements. These decompositions have been introduced in [2, 18]. The proofs of Korn’s inequalities for different types of domains, namely N -dimensional, plate-like and beam-like, are given in Appendix A. In all the estimates the constants do not depend on ε
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