On the mixtures of MGT viscoelastic solids

Noelia Bazarra; José R. Fernández; Ramón Quintanilla; Noelia Bazarra; José R. Fernández; Ramón Quintanilla

doi:10.3934/era.2022219

Electronic Research Archive

2022, Volume 30, Issue 12: 4318-4340. doi: 10.3934/era.2022219

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On the mixtures of MGT viscoelastic solids

1.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, Vigo 36310, Spain
2.
Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, Terrassa 08222, Barcelona, Spain

Received: 08 July 2022 Revised: 14 September 2022 Accepted: 21 September 2022 Published: 28 September 2022

In this paper, we study, from both analytical and numerical points of view, a problem involving a mixture of two viscoelastic solids. An existence and uniqueness result is proved using the theory of linear semigroups. Exponential decay is shown for the one-dimensional case. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained and the linear convergence is derived under suitable regularity conditions. Finally, one- and two-dimensional numerical simulations are presented to demonstrate the convergence, the discrete energy decay and the behavior of the solution.
- mixtures,
- viscoelasticity,
- linear semigroups,
- existence and uniqueness,
- energy decay,
- finite elements
Citation: Noelia Bazarra, José R. Fernández, Ramón Quintanilla. On the mixtures of MGT viscoelastic solids[J]. Electronic Research Archive, 2022, 30(12): 4318-4340. doi: 10.3934/era.2022219

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Abstract

In this paper, we study, from both analytical and numerical points of view, a problem involving a mixture of two viscoelastic solids. An existence and uniqueness result is proved using the theory of linear semigroups. Exponential decay is shown for the one-dimensional case. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained and the linear convergence is derived under suitable regularity conditions. Finally, one- and two-dimensional numerical simulations are presented to demonstrate the convergence, the discrete energy decay and the behavior of the solution.

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