Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

Abstract

The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method. The material behaviour is modelled by using a general Kelvin-Voigt constitutive law. Mathematics Subject Classification (2000). Primary 74K10 · Secondary 35C20 · 74A05 · 74A10 · 74D05 · 41A60.

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

“…The most remarkable feature was that from the asymptotic analysis of the three-dimensional problems which included a short term memory represented by a time 50 derivative, a long term memory arised in the two-dimensional limit problems, represented by an integral with respect to the time variable. This fact, agreed with previous asymptotic analysis of viscoelastic rods in [32,33] where an analogous behaviour was presented as well.…”

“…In [34], we justified the two-dimensional equations of the viscoelastic membrane shells, where the middle surface S is elliptic and the boundary condition of place is considered on the whole lateral face of the shell. These assumptions lead to V F (ω) = {0} (see [34] for details).…”

“…Also, in the reference [27], the authors study the effects of great deflections in thin plates covering both short and long term memory cases. Concerning viscoelastic shell problems, in [28] we 35 can find different kind of studies where the authors also remark the viscoelastic property of the material of a shell. For the problems dealing with the shell-type equations, there exists a very limited amount of results available, for instance, [29] where the authors present a model for a dynamic contact problem where a short memory (Kelvin-Voigt) material is considered.…”

“…In [34,35] we justified the equations of a viscoelastic membrane shell where the surface S is elliptic and the boundary condition of place is considered in the whole lateral face of 55 the shell. Therefore, in this paper the main aim is to justify the remaining cases in the group of viscoelastic membrane cases, known as the viscoelastic generalized membrane shell equations.…”

A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis

“…Using asymptotic analysis, several classical reduced models have been mathematically developed over the years. The asymptotic method was originally introduced by Lions [22] and since then it has been extensively used to derive and justify reduced models for elastic plates and shells [10][11][12], elastic beams [4,19,21,31,[40][41][42][43], viscoelastic beams [28,29] and also for elastic beams in contact with a foundation (see [20,30,44,47], the last two to justify and generalize contact models found in [17,37]). The success of this method is due to the inherent small geometrical parameters involved (thickness of plates and shells and diameter of the cross-section in beams).…”