Variational formulation and asymptotic analysis of viscoelastic problem with Riemann‐Liouville fractional derivatives

We consider two mathematical models which describe the antiplane shear deformation of a piezoelectric cylinder in adhesive contact with a rigid foundation. The material is assumed to be electro-viscoelastic in the first model and electro-elastic in the second one. In both models the process is quasistatic, the foundation is electrically conductive and the adhesion is described with a surface variable, the bonding field. We derive a variational formulation of the models which is given by a system coupling two variational equations for the displacement and the electric potential fields, respectively, and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to each model. We also investigate the behavior of the solution of the electro-viscoelastic problem as the viscosity converges to zero and prove that it converges to the solution of the corresponding electro-elastic problem.

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Asymptotic analysis of quasistatic electro‐viscoelastic problem with Tresca's friction Law

Computational and Mathematical Methods ◽

10.1002/cmm4.1028 ◽

2019 ◽

Vol 1 (3) ◽

pp. e1028

Author(s):

Mohamed Dilmi ◽

Mourad Dilmi ◽

Hamid Benseridi

Keyword(s):

Asymptotic Analysis ◽

Viscoelastic Problem ◽

Friction Law ◽

Tresca’S Friction

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MODELING OF THE INCLUSION OF A REINFORCING SHEET WITHIN A 3D MEDIUM

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202503002635 ◽

2003 ◽

Vol 13 (04) ◽

pp. 573-595 ◽

Cited By ~ 14

Author(s):

D. CHAPELLE ◽

A. FERENT

Keyword(s):

Asymptotic Analysis ◽

Variational Formulation ◽

Degrees Of Freedom ◽

Coupled Model ◽

Numerical Results ◽

Crucial Issue ◽

Coupling Conditions ◽

Limit Solutions

We propose a variational formulation suitable for coupling a shell and a surrounding softer 3D medium. This is intended to model the behavior of a reinforcing sheet as frequently occurs in applications. The crucial issue of the kinematical coupling conditions is addressed by taking into account the rotation degrees of freedom of the shell. We then perform an asymptotic analysis of the coupled model in which the distinction between bending-dominated and membrane-dominated shells arises. For the limit solutions we identify coupling kinematical conditions that do not involve rotations. Finally we present some numerical results that illustrate our discussions.

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