Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Abstract

An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch.

A Numerical Method for Simulating Viscoelastic Plates Based on Fractional Order Model

“…Three properties of Caputo fractional differential are as follows, for λ, µ ∈ R, 0 < α ≤ 1, C is a constant, and f (t) ∈ C 1 (R) [26,32] (1)…”

“…Hashim et al [25] proposed shifted Chebyshev polynomials of the second kind to solve approximate solutions of time-delay variable fractional differential equations. Cao et al [26] studied a significant method based on fractional rheological model to solve viscoelastic column problems. The fractional differential equation was solved by shifted Chebyshev wavelet function.…”

Analysis of the variable‐order fractional viscoelastic modeling with application to polymer materials

“…Polymers and polymer composites display apparent viscoelastic property, whose deformation behaviors exhibit both elastic and viscous characteristics, and it has been attracted increased attention due to its wide range of engineering application from aerospace to transportation, electrical, and construction. 1 The viscoelastic behavior is much more complicated than the ordinary elasticity, which brings considerable difficulty to engineering design and safety applications. To effectively trace the evolution of viscoelastic behavior and facilitate engineering applications, rheological models are the most widely used, which include the elastic components modeled as springs and the viscous components modeled as dashpots.…”

“…Polymers and polymer composites display apparent viscoelastic property, whose deformation behaviors exhibit both elastic and viscous characteristics, and it has been attracted increased attention due to its wide range of engineering application from aerospace to transportation, electrical, and construction 1 . The viscoelastic behavior is much more complicated than the ordinary elasticity, which brings considerable difficulty to engineering design and safety applications.…”

Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

“…Such derivatives are called variable‐order (VO) fractional derivatives 10 . Recently, special consideration has been paid to the application of VO fractional derivatives to model several problems, such as viscoelastic constitutive models, 11 dual phase lag bioheat problems, 12 and ultra‐short pulsed laser therapy 13 . Since it is often impossible to solve VO fractional problems, analytically, varied numerical techniques have been used for such problems.…”