We consider a recent plate model obtained as a scaled limit of the three dimensional Biot system of poro-elasticity. The result is a "2.5" dimensional linear system that couples traditional Euler-Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We alow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions to implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate, and exposits a variety of interesting related models and associated analytical investigations.
The paper addresses the homogenization of a micro-model of poroelasticity coupled with thermal effects for two-constituent media and with imperfect interfacial contact.The homogenized model is obtained by means of the two-scale convergence technique. It is shown that the macro-thermo-poro-elasticity model with double porosity/diffusivity contains in particular some extra terms accounting for the micro-heterogeneities and imperfect contact at the local scale. Finally, a corrector result is given under more regularity assumptions on the data.
In this paper we investigate the interaction of fluid flow with a thin porous elastic layer. We consider two fluid-filled bulk domains which are separated by a thin periodically perforated layer consisting of a fluid and an elastic solid part. Thickness and periodicity of the layer are of order $ε$, where $ε$ is small compared to the size of the bulk domains. The fluid flow is described by an instationary Stokes equation and the solid via linear elasticity. The main contribution of this paper is the rigorous homogenization of the porous structure in the layer and the reduction of the layer to …
Modelling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits. Models for inclusions larger or comparable to the thickness of the plate
In this work we derive by Gamma-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $ρ$ and study the limit as $ρ$ tends to 0. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.
We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.
A quasistatic model for a horizontally loaded thin elastic composite at small strains is studied. The composite consists of two adjacent plates whose interface behaves in a cohesive fashion with respect to the slip of the two layers. We allow for different loading-unloading regimes, distinguished by the presence of an irreversible variable describing the maximal slip reached during the evolution. Existence of energetic solutions, characterized by equilibrium conditions together with energy balance, is proved by means of a suitable version of the Minimizing Movements scheme. A crucial tool to a…
This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biot's model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in…
In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as generalized gradient flows requiring choices of physical states, corresponding energies, dissipation potentials and external work rates. It is demonstrated that various existing models can be in fact written within this framework. Ultimately, the particular structure allows for a unified well-posedness analysis performed for different classes of linear and non-l…
This paper investigates the effects of finite flat porous extensions to semi-infinite impermeable flat plates in an attempt to control trailing-edge noise through bio-inspired adaptations. Specifically the problem of sound generated by a gust convecting in uniform mean steady flow scattering off the trailing edge and permeable-impermeable junction is considered. This setup supposes that any realistic trailing-edge adaptation to a blade would be sufficiently small so that the turbulent boundary layer encapsulates both the porous edge and the permeable-impermeable junction, and therefore the int…
We rigorously derive a Kirchhoff plate theory, via $Γ$-convergence, from a three-di\-men\-sio\-nal model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables $x'$. At the same time, the spontaneous strain is $h$-close to the identity, where $h$ is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into $\mathbb…
We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resol…
We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under…
In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis funct…
This paper investigates the optimal distribution of hard and soft material on elastic plates. In the class of isometric deformations stationary points of a Kirchhoff plate functional with incorporated material hardness function are investigated and a compliance cost functional is taken into account. Under symmetry assumptions on the material distribution and the load it is shown that cylindrical solutions are stationary points. Furthermore, it is demonstrated that the optimal design of cylindrically deforming, clamped rectangular plates is non trivial, i.e. with a material distribution which i…
The aim of this work is to present theoretical tools to study wave propagation in elastic waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects in a 2D and 3D elastic plate. Given surface multi-frequency wavefield measurements, we use a Born approximation to reconstruct localized defect in the geometry of the plate. To justify this approximation, we introduce a rigorous framework to study the propagation of elastic wavefield generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous Lamb mode, we prove the well-posedn…
