On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions

@article{Mazya2020OnMA,
  title={On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions},
  author={Vladimir Maz'ya and Alexander B. Movchan and Michael J. Nieves},
  journal={ArXiv},
  year={2020},
  volume={abs/2002.02810}
}
In this paper we consider formal asymptotic algorithms for a class of meso-scale approximations for problems of vibration of elastic membranes, which contain clusters of small inertial inclusions distributed along contours of pre-defined smooth shapes. Effective transmission conditions have been identified for inertial structured interfaces, and approximations to solutions of eigenvalue problems have been derived for domains containing lower-dimensional clusters of inclusions. 

Figures from this paper

Asymptotic behavior of integral functionals for a two‐parameter singularly perturbed nonlinear traction problem
We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive
On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions
TLDR
The proposed homogenization framework provides a convenient platform for the synthesis of a wide range of wave phenomena in metamaterials and phononic crystals by approximating asymptotically the dispersion relationships for Kagome lattice featuring hexagonal Neumann exclusions, and "pinned" square lattice with circular Dirichlet exclusions.
A Dirichlet Problem in a Domain with a Small Hole
In this chapter we introduce the Functional Analytic Approach for the analysis of elliptic boundary value problems in singularly perturbed domains. To illustrate how the method works, we closely
Localized waves in elastic plates with perturbed honeycomb arrays of constraints
In this paper, we study wave propagation in elastic plates incorporating honeycomb arrays of rigid pins. In particular, we demonstrate that topologically non-trivial band-gaps are obtained by
Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach
In this paper, a coherent boundary value problem to model metamaterials' behaviour based on the relaxed micromorphic model is established. This boundary value problem includes well-posed boundary
Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem
: We study the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in R n , n ≥ 3, with a (nonlinear) Robin boundary condition on the

References

SHOWING 1-10 OF 16 REFERENCES
Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis
We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of
Green's Kernels and Meso-Scale Approximations in Perforated Domains
Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale
Mesoscale Asymptotic Approximations to Solutions of Mixed Boundary Value Problems in Perforated Domains
We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the
Asymptotic treatment of perforated domains without homogenization
As a main result of the paper, we construct and justify an asymptotic approximation of Green's function in a domain with many small inclusions. Periodicity ofthe array ofinclusions is not required.
Green's kernels for transmission problems in bodies with small inclusions
The uniform asymptotic approximation of Green's kernel for the transmission problem of antiplane shear is obtained for domains with small inclusions. The remainder estimates are provided. Numerical
The Green's Function for the Two-Dimensional Helmholtz Equation in Periodic Domains
Analytical techniques are described for transforming the Green's function for the two-dimensional Helmholtz equation in periodic domains from the slowly convergent representation as a series of
Homogenization of Partial Differential Equations
* Preface * Introduction * The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Fine-Grained Boundary * The Dirichlet Boundary Value Problem in Strongly Perforated Domains with
Periodic Integral and Pseudodifferential Equations with Numerical Approximation
1 Preliminaries.- 2 Single Layer and Double Layer Potentials.- 3 Solution of Boundary Value Problems by Integral Equations.- 4 Singular Integral Equations.- 5 Boundary Integral Operators in Periodic
Bloch Waves in an Arbitrary Two-Dimensional Lattice of Subwavelength Dirichlet Scatterers
TLDR
The method presented, that simplifies and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of the operating wavelength.
Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach
TLDR
This work solves the two-dimensional problem of acoustic scattering by a semi-infinite array of identical isotropic point scatterers and confirms that a number of phenomena reported for specific geometries are in fact present in the general case.
...
...