Generalized solutions for the Euler-Bernoulli model with Zener viscoelastic foundations and distributional forces

TL;DR

This paper develops a mathematical framework to analyze a complex Euler-Bernoulli beam model with discontinuous, distributional coefficients and a Zener viscoelastic foundation, proving existence and uniqueness of generalized solutions.

Contribution

It introduces a novel combination of functional analytic and regularization methods to handle the distributional coefficients and fractional viscoelasticity in the model.

Findings

Existence and uniqueness of generalized solutions established.

Effective handling of distributional and discontinuous coefficients.

Application of fractional differential equations to viscoelastic foundation.

Abstract

We study the initial-boundary value problem for an Euler-Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is fourth order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.