VISCOELASTIC PROPERTIES AND THEIR IMPACT ON THE DYNAMICS OF FLEXIBLE STRUCTURES
Keywords:
Elastic structures, viscoelastic boundary conditions, thin plate, partial differential equations, memory effects.Abstract
This article delves into the study of elastic structures with viscoelastic boundary conditions, focusing on an elastic thin plate in a bounded domain Ω ⊂ ℝ² with ????²-smooth boundary Γ. The plate is clamped, and memory effects are considered on a subset Γ₀ with positive boundary measure. The vertical deflection ????(????, ????) of this thin elastic plate is governed by a partial differential equation involving wave equations and memory effects. The specific problem can be described by the following equations: ????????????(????, ????) + Δ²????(????, ????) = 0, in Ω × ℝ⁺, (1.1a) ????(????, ????) = ∂????????(????, ????) = 0, on Γ₀ × ℝ⁺, (1.1b) ℬ₁????(????, ????) − ∫₀⁺∞ ????′(????) ∂????[????(????, ????) − ????(????, ???? − ????)]d???? = 0, on Γ₀ × ℝ⁺, (1.1c) ℬ₂????(????, ????) + ∫₀⁺∞ ????′(????)[????(????, ????) − ????(????, ???? − ????)]d???? = ????(????, ????), on Γ₁ × ℝ⁺, (1.1d) ????(????, 0⁺) = ????₀(????), ????????(????, 0⁺) = ????₁(????), (1.1e) ????(????, −????) = ????(????, ????), for 0 < ???? < ∞. (1.1f) This study explores the interplay between elastic structures and viscoelastic boundary conditions, examining the behavior of the thin plate under these specified conditions.