TY - JOUR
T1 - A singularly perturbed nonlinear traction problem in a periodically perforated domain
T2 - A functional analytic approach
AU - Riva, M. Dalla
AU - Musolino, P.
PY - 2014/1/15
Y1 - 2014/1/15
N2 - We consider a periodically perforated domain obtained by making in Rn a periodic set of holes, each of them of size proportional to ε. Then, we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector-valued function u, which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ε contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then, our aim is to describe what happens to the displacement vector function u when ε tends to 0. Under suitable assumptions, we prove the existence of a family of solutions {u(ε, ×)} ε â̂̂ ]0,ε ′ [ with a prescribed limiting behavior when ε approaches 0. Moreover, the family {u(ε, ×)}ε â̂̂ ]0,ε ′ [ is in a sense locally unique and can be continued real analytically for negative values of ε.
AB - We consider a periodically perforated domain obtained by making in Rn a periodic set of holes, each of them of size proportional to ε. Then, we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector-valued function u, which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ε contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then, our aim is to describe what happens to the displacement vector function u when ε tends to 0. Under suitable assumptions, we prove the existence of a family of solutions {u(ε, ×)} ε â̂̂ ]0,ε ′ [ with a prescribed limiting behavior when ε approaches 0. Moreover, the family {u(ε, ×)}ε â̂̂ ]0,ε ′ [ is in a sense locally unique and can be continued real analytically for negative values of ε.
KW - integral representations, integral operators, integral equation methods
KW - linearized elastostatics
KW - nonlinear boundary value problems for linear elliptic equations
KW - periodically perforated domain
KW - real analytic continuation in Banach space
KW - singularly perturbed domain
UR - http://www.scopus.com/inward/record.url?scp=84890552427&partnerID=8YFLogxK
U2 - 10.1002/mma.2788
DO - 10.1002/mma.2788
M3 - Article
AN - SCOPUS:84890552427
SN - 0170-4214
VL - 37
SP - 106
EP - 122
JO - Mathematical Methods in the Applied Sciences
JF - Mathematical Methods in the Applied Sciences
IS - 1
ER -