VISCOELASTICITY AND FLEXIBLE STRUCTURES: EXPLORING THEIR INTERPLAY IN DYNAMIC SYSTEMS

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 𝐶²-smooth boundary Γ. The plate is clamped, and memory effects are considered on a subset Γ₀ with positive boundary measure. The vertical deflection 𝑦(𝑥, 𝑡) 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: 𝑦𝑡𝑡(𝑥, 𝑡) + Δ²𝑦(𝑥, 𝑡) = 0, in Ω × ℝ⁺, (1.1a) 𝑦(𝑥, 𝑡) = ∂𝑣𝑦(𝑥, 𝑡) = 0, on Γ₀ × ℝ⁺, (1.1b) ℬ₁𝑦(𝑥, 𝑡) − ∫₀⁺∞ 𝑔′(𝑠) ∂𝑣[𝑦(𝑥, 𝑡) − 𝑦(𝑥, 𝑡 − 𝑠)]d𝑠 = 0, on Γ₀ × ℝ⁺, (1.1c) ℬ₂𝑦(𝑥, 𝑡) + ∫₀⁺∞ 𝑔′(𝑠)[𝑦(𝑥, 𝑡) − 𝑦(𝑥, 𝑡 − 𝑠)]d𝑠 = 𝑢(𝑥, 𝑡), on Γ₁ × ℝ⁺, (1.1d) 𝑦(𝑥, 0⁺) = 𝑦₀(𝑥), 𝑦𝑡(𝑥, 0⁺) = 𝑦₁(𝑥), (1.1e) 𝑦(𝑥, −𝑠) = 𝜗(𝑥, 𝑡), for 0 < 𝑠 < ∞. (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.