Decay for strain gradient porous elastic waves. (English) Zbl 1505.74096
Summary: We study the one-dimensional problem for the linear strain gradient porous elasticity. Our aim is to analyze the behavior of the solutions with respect to the time variable when a dissipative structural mechanism is introduced in the system. We consider five different scenarios: hyperviscosity and viscosity for the displacement component and hyperviscoporosity, viscoporosity and weak viscoporosity for the porous component. We only apply one of these mechanisms at a time. We obtain the exponential decay of the solutions in the case of viscosity and a similar result for the viscoporosity. Nevertheless, in the hyperviscosity case (respectively hyperviscoporosity) the decay is slow and it can be controlled at least by \(t^{-1/2}\). Slow decay is also expected for the weak viscoporosity in the generic case, although a particular combination of the constitutive parameters leads to the exponential decay. We want to emphasize the fact that the hyperviscosity (respectively hyperviscoporosity) is a stronger dissipative mechanism than the viscosity (respectively viscoporosity); however, in this situation, the second mechanism seems to be more “efficient” than the first one in order to pull along the solutions rapidly to zero. This is a striking fact that we have not seen previously at any other linear coupling system. Finally, we also present some numerical simulations by using the finite element method and the Newmark-\(\beta\) scheme to show the behavior of the energy decay of the solutions to the above problems, including a comparison between the hyperviscosity and the viscosity cases.
MSC:
| 74J05 | Linear waves in solid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74D99 | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |
| 74H40 | Long-time behavior of solutions for dynamical problems in solid mechanics |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
| 74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
hyperviscoelasticity; viscoporosity; exponential decay; polynomial decay; finite element method; Newmark schemeReferences:
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