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A Bermúdez–Moreno algorithm adapted to solve a viscoplastic problem in alloy solidification processes

P. Barral, P. Quintela, M. T. Sánchez (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The aim of this work is to present a computationally efficient algorithm to simulate the deformations suffered by a viscoplastic body in a solidification process. This type of problems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. In our previous papers, this difficulty has been solved by means of a duality method, known as Bermúdez–Moreno algorithm, involving a multiplier which was computed with a fixed point algorithm or a Newton method. In this paper, we will...

A general decay estimate for a finite memory thermoelastic Bresse system

Cyril Dennis Enyi, Soh Edwin Mukiawa (2022)

Applications of Mathematics

This work considers a Bresse system with viscoelastic damping on the vertical displacement and heat conduction effect on the shear angle displacement. A general stability result with minimal condition on the relaxation function is obtained. The system under investigation, to the best of our knowledge, is new and has not been studied before in the literature. What is more interesting is the fact that our result holds without the imposition of the equal speed of wave propagation condition, and differentiation...

A viscoelastic contact problem with normal damped response and friction

B. Awbi, El H. Essoufi, M. Sofonea (2000)

Annales Polonici Mathematici

We study an evolution problem which describes the quasistatic contact of a viscoelastic body with a foundation. We model the contact with normal damped response and a local friction law. We derive a variational formulation of the model and we establish the existence of a unique weak solution to the problem. The proof is based on monotone operators and fixed point arguments. We also establish the continuous dependence of the solution on the contact boundary conditions.

Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue

Freed, Alan, Diethelm, Kai (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 74B20, 74D10, 74L15The popular elastic law of Fung that describes the non-linear stress- strain behavior of soft biological tissues is extended into a viscoelastic material model that incorporates fractional derivatives in the sense of Caputo. This one-dimensional material model is then transformed into a three-dimensional constitutive model that is suitable for general analysis. The model is derived in a configuration that differs from the current, or spatial,...

Development of three dimensional constitutive theories based on lower dimensional experimental data

Satish Karra, Kumbakonam R. Rajagopal (2009)

Applications of Mathematics

Most three dimensional constitutive relations that have been developed to describe the behavior of bodies are correlated against one dimensional and two dimensional experiments. What is usually lost sight of is the fact that infinity of such three dimensional models may be able to explain these experiments that are lower dimensional. Recently, the notion of maximization of the rate of entropy production has been used to obtain constitutive relations based on the choice of the stored energy and rate...

Dissipatività e unicità per il problema dinamico unidimensionale della viscoelasticità lineare

Giorgio Vergara Caffarelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Fissato lo spazio di Sobolev H 1 , 2 come ambiente del problema dinamico per un corpo viscoelastico unidimensionale si dimostra un teorema di unicità per la classe delle funzioni di rilassamento convesse. Si fa inoltre vedere come tale unicità sia strettamente legata allo spazio ambiente considerato.

Dissipatività ed esistenza per il problema dinamico unidimensionale della viscoelasticità lineare

Giorgio Vergara Caffarelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questa nota si completa la studio (iniziato in [1]) della caratterizzazione delle funzioni di rilassamento per le quali il problema dinamico della viscoelasticità lineare, con condizioni di spostamento nullo agli estremi, risulta ben posto nello spazio di Sobolev H 1 , 2 . Precisamente, per un'opportuna classe di sollecitazioni esterne, si dimostra l'esistenza della soluzione, se le funzioni di rilassamento sono positive, convesse ed hanno il modulo di elasticità all'equilibrio strettamente maggiore...

Dynamic frictional contact of a viscoelastic beam

Marco Campo, José R. Fernández, Georgios E. Stavroulakis, Juan M. Viaño (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect...

Effects of In-plane Elastic Stress and Normal External Stress on Viscoelastic Thin Film Stability

F. Closa, F. Ziebert, E. Raphaël (2012)

Mathematical Modelling of Natural Phenomena

Motivated by recent experiments on the electro-hydrodynamic instability of spin-cast polymer films, we study the undulation instability of a thin viscoelastic polymer film under in-plane stress and in the presence of either a close by contactor or an electric field, both inducing a normal stress on the film surface. We find that the in-plane stress affects both the typical timescale of the instability and the unstable wavelengths. The film stability...

Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation

Jong Yeoul Park, Sun Hye Park (2006)

Czechoslovak Mathematical Journal

We consider the damped semilinear viscoelastic wave equation u ' ' - Δ u + ∫ 0 t h ( t - τ ) div { a ∇ u ( τ ) } d τ + g ( u ' ) = 0 in Ω × ( 0 , ∞ ) with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.

Free energy and internal variables in linear viscoelasticity

Angelo Morro, Maurizio Vianello (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In connection with the determination of the free energy functional for the viscoelastic stress tensor, a viscoelastic material is considered as described by a material with internal variables. In this framework the free energy is uniquely determined. It proves to be the minimal one in the class of thermodynamically admissible free energies.

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