Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation
- PDF / 1,551,424 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 101 Downloads / 203 Views
Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation Delfina Gómez1 · Sergei A. Nazarov2,3 · María-Eugenia Pérez-Martínez1
Received: 25 March 2020 / Published online: 23 September 2020 © Springer Nature B.V. 2020
Abstract We consider a spectral homogenization problem for the linear elasticity system posed in a domain Ω of the upper half-space R3+ , a part of its boundary Σ being in contact with the plane {x3 = 0}. We assume that the surface Σ is traction-free out of small regions T ε , where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function M(x) and a reaction parameter β(ε) that can be very large when ε → 0. The size of the regions T ε is O(rε ), where rε ε, and they are placed at a distance ε between them. We provide all the possible spectral homogenized problems depending on the relations between ε, rε and β(ε), while we address the convergence, as ε → 0, of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on Σ. New capacity matrices are introduced to define these strange terms. Keywords Boundary homogenization · Spectral perturbations · Elasticity · Winkler foundation · Capacity matrices · Critical relations Mathematics Subject Classification (2010) 35B27 · 74B05 · 74Q20 · 35P05 · 35B25 · 35J25 · 35P15
B M.-E. Pérez-Martínez [email protected]
D. Gómez [email protected] S.A. Nazarov [email protected] 1
Universidad de Cantabria, Avenida Los Castros s/n, Santander, 39005, Spain
2
St-Petersburg State University, Universitetskaya nab., 7-9, St. Petersburg, 199034, Russia
3
Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia
90
D. Gómez et al.
1 Introduction In this paper, we address the asymptotic behavior of a spectral problem associated with the vibrations of a deformable elastic solid Ω ⊂ R3+ = {x : x3 > 0} whose boundary ∂Ω has a part clamped to an absolutely rigid profile ΓΩ and the other part Σ ⊂ {x : x3 = 0} in contact with a strainer Winkler foundation which can be modeled by a series of small springs periodically placed along Σ , the reaction regions T ε . On these small regions, the boundary conditions are of Winkler-Robin type, also so-called of spring type, while outside, they are traction-free. The small regions T ε have diameter O(rε ) and are at a distance ε between them, where ε measures the period of the structure. Here ε and rε are two small parameters rε ε 1; see Fig. 1. The elastic coefficients of the small springs are defined through the so-called Robin reaction matrix, which we denote by β(ε)M(x). Matrix M(x) depends on the point where the reaction regions T ε are placed, while the parameter β(ε), which is referred to as the reaction parameter, can range from very small to very large. Each T ε is assumed to be a domain of the plane R2 homothetic to a fixed domain T , with a Lipschitz boundary.
Data Loading...
