Decay of waves in strain gradient porous elasticity with Moore-Gibson-Thompson dissipation

We study a one-dimensional problem arising in strain gradient porous-elasticity. Three different Moore–Gibson–Thompson dissipation mechanisms are considered: viscosity and hyperviscosity on the displacements, and weak viscoporosity. The existence and uniqueness of solutions are proved. The energy decay is also shown, being polynomial for the two first situations, unless a particular choice of the constitutive parameters is made in the hyperviscosity case. Finally, for the weak viscoporosity, only the slow decay can be expectedPeer ReviewedPostprint (author's final draft

On the stability of a double porous elastic system with visco-porous dampings

In this paper we consider a one dimensional elastic system with double porosity structure and with frictional damping in both porous equations. We introduce two stability numbers $\chi_{0}$ and $\chi_{1}$ and prove that the solution of the system decays exponentially provided that $\chi_{0}=0$ and $\chi_{1}\neq0.$ Otherwise, we prove the lack of exponential decay. Our results improve the results of \cite{Bazarra} and \cite{Nemsi}

A unifying perspective: the relaxed linear micromorphic continuum

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict non-polar size-effects. It is intended for the homogenized description of highly heterogeneous, but non polar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models

Scaling laws in permeability and thermoelasticity of random media

Under consideration is the finite-size scaling of two thermomechanical responses of random heterogeneous materials. Stochastic mechanics is applied here to the modeling of heterogeneous materials in order to construct the constitutive relations. Such relations (e.g. Hooke's Law in elasticity or Fourier's Law in heat transfer) are well-established under spatial homogeneity assumption of continuum mechanics, where the Representative Volume Element (RVE) is the fundamental concept. The key question is what is the size L of RVE? According to the separation of scales assumption, L must be bounded according to d<L<<LMacro where d is the microscale (or average size of heterogeneity), and LMacro is the macroscale of a continuum mechanics problem. Statistically, for spatially ergodic heterogeneous materials, when the mesoscale is equal to or bigger than the scale of the RVE, the elements of the material can be considered homogenized. In order to attain the said homogenization, two conditions must be satisfied: (a) the microstructure's statistics must be spatially homogeneous and ergodic; and (b) the material's effective constitutive response must be the same under uniform boundary conditions of essential (Dirichlet) and natural (Neumann) types.In the first part of this work, the finite-size scaling trend to RVE of the Darcy law for Stokesian flow is studied for the case of random porous media, without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. By analogy to the existing methodology in thermomechanics of solid random media, the Hill-Mandel condition for the Darcy flow velocity and pressure gradient fields was first formulated. Under uniform essential and natural boundary conditions, two variational principles are developed based on minimum potential energy and complementary energy. Then, the partitioning method was applied, leading to scale dependent hierarchies on effective (RVE level) permeability. The proof shows that the ensemble average of permeability has an upper bound under essential boundary conditions and a lower bound under uniform natural boundary conditions.To quantitatively assess the scaling convergence towards the RVE, these hierarchical trends were numerically obtained for various porosities of random disk systems, where the disk centers were generated by a planar Poisson process with inhibition. Overall, the results showed that the higher the density of random disks---or, equivalently, the narrower the micro-channels in the system---the smaller the size of RVE pertaining to the Darcy law.In the second part of this work, the finite-size scaling of effective thermoelastic properties of random microstructures were considered from Statistical to Representative Volume Element (RVE). Similarly, under the assumption that the microstructure's statistics are spatially homogeneous and ergodic, the SVE is set-up on a mesoscale, i.e. any scale finite relative to the microstructural length scale. The Hill condition generalized to thermoelasticity dictates uniform essential and natural boundary conditions, which, with the help of two variational principles, led to scale dependent hierarchies of mesoscale bounds on effective (RVE level) properties: thermal expansion strain coefficient and stress coefficient, effective stiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds for the thermal expansion are more complicated than those for the stiffness tensor and the heat capacity. To quantitatively assess the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of a planar, hard-core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to microscale, depending on the microstructural parameters, the random fluctuations in the SVE response become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE.Based on the above studies, further work on homogenization of heterogeneous materials is outlined at the end of the thesis.Keywords: Representative Volume Element (RVE), heterogeneous media, permeability, thermal expansion, mesoscale, microstructure

Scale effects in finite elasticity and thermoelasticity

The main focus of this thesis is on investigating the minimum size of the Representative Volume Element (RVE) and finite-size scaling of properties of random linear and nonlinear elastic composites. The RVE is a material volume which accurately describes the overall behavior of a heterogeneous solid, and is the core assumption of continuum mechanics theory. If the composite microstructure admits the assumption of spatial homogeneity and ergodicity, the RVE can be attained within a specific accuracy on a finite length-scale. Determining this scale is the key objective of this thesis.In order to theoretically analyze the scale-dependence of the apparent response of random microstructures, essential and natural boundary conditions which satisfy Hill's averaging theorem in finite deformation elasticity are first considered. It is shown that the application of the partitioning method and variational principles in nonlinear elasticity and thermoelasticity, under the two above-mentioned boundary conditions, leads to the hierarchy of mesoscale bounds on the effective strain- and free-energy functions, respectively. These theoretical derivations lay the ground for the quantitative estimation of the scale-dependence of nonlinear composite responses and their RVE size.The hierarchies were computed for planar matrix-inclusion composites with the microstructure modeled by a homogeneous Poisson point field. Various nonlinear composites with Ogden-type strain-energy function are considered. The obtained results are compared with those where both matrix and inclusions are described by a neo-Hookean strain-energy function as well as with the results obtained from the linear elasticity theory. The trends toward the RVE are also computed for nonlinear elastic composites subjected to non-isothermal loading. The accuracy of the RVE size estimation is calculated in terms of the discrepancy between responses under essential and natural boundary conditions. Overall, the results show that the trends toward the RVE as well as its minimum size are functions of the deformation, deformation mode, temperature, and the mismatch between material properties of the phases.The last part of the thesis presents an investigation of the size effect on thermoelastic damping of a micro-/nanobeam resonator. It does not follow the framework described above. The main concern here is the size and the vibration frequency, at which the classical Fourier law of heat conduction is no longer valid, and the finite speed of heat propagation has to be taken into account

Thermodynamische Materialtheorien

The meeting was focused on research in the broad field of thermodynamic constitutive theories. It provided a contact between physicists, engineers and mathematicians, whose talks led to lively and interesting discussions. The debate concentrated on the physical motivation of the models subjected to mathematical analysis