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Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin’s strain gradient porous elasticity

Abstract

In this paper, we derive a nonlocal theory for porous elastic materials in the context of Mindlin’s strain gradient model. The second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. The obtained system of equations is a coupling of a two hyperbolic equations with higher gradients terms due to the strain gradient length scale parameter l and the elastic nonlocal parameter \(\varpi \). This poses some new mathematical difficulties due to the lack of regularity. Under quite general assumptions on nonlinear sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the one dimensional nonlinear problem. By an approach based on the Gearhart–Herbst–Prüss–Huang Theorem, we prove that the semigroup associated with the derived model is not analytic in general (\(\varpi =0\) or not). A frictional damping for the elastic component, whose form depends on the elastic nonlocal parameter (\(\varpi =0\) or not), is shown to lead to exponential stability at a rate of decay determined explicitly. Without frictional damping, the derived system can be exponentially stable only in the absence of body forces and under the condition of equal wave speeds.

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Acknowledgements

The author would like to thank the Editor Prof. Laurent Chupin and the anonymous reviewers for their recommendations and remarks aiming at improving the manuscript in terms of clarity.

ORCID Moncef Aouadi http://orcid.org/0000-0003-3400-5670

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Aouadi, M. Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin’s strain gradient porous elasticity. Z. Angew. Math. Phys. 73, 111 (2022). https://doi.org/10.1007/s00033-022-01764-2

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  • DOI: https://doi.org/10.1007/s00033-022-01764-2

Keywords

  • Nonlocality
  • Strain gradient
  • Porous elasticity
  • Well-posedness
  • Lack of analyticity
  • Exponential stability

Mathematics Subject Classification

  • Primary 35B40
  • 74H40
  • Secondary 35Q74
  • 74F05