Eduard Rohan

Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps

We consider a three-dimensional composite material made of small inclusions periodically embedded into an elastic matrix; the whole structure presents strong heterogeneities between its different components. In the general framework of linearized elasticity we show that, when the size of the microstructures tends to zero, the limit homogeneous structure presents, for some wavelengths, a negative “mass density” tensor. Hence we are able to rigorously justify the existence of forbidden bands, i.e., intervals of frequencies in which there is no propagation of elastic waves. In particular, we show how to compute these band gaps and illustrate the theoretical results with some numerical simulations.

A multiscale theoretical investigation of electric measurements in living bone : piezoelectricity and electrokinetics.

This paper presents a theoretical investigation of the multiphysical phenomena that govern cortical bone behaviour. Taking into account the piezoelectricity of the collagen–apatite matrix and the electrokinetics governing the interstitial fluid movement, we adopt a multiscale approach to derive a coupled poroelastic model of cortical tissue. Following how the phenomena propagate from the microscale to the tissue scale, we are able to determine the nature of macroscopically observed electric phenomena in bone.

Sensitivity strategies in modelling heterogeneous media undergoing finite deformation

This paper deals with homogenization of microscopically heterogeneous media which are subjected to finite deformations. The updated Lagrangian scheme is applied to obtain linear subproblems which can be homogenized using the two-scale convergence. Microscopic equations and homogenized stiffness coefficients are derived for the hyperelastic material with incompressible inclusions. A sensitivity analysis of homogenized coefficients is proposed to study their dependence on local deformations of the microstructure. This approach can assist in reducing the number of the local microscopic equations that have to be solved in each iteration of macroscopic problems.

Shape optimization of elasto-plastic structures and continua

This article is concerned with the sensitivity analysis and optimization of elasto-plastic bodies when isotropic strain hardening takes place. The elasto-plastic behaviour of the material is governed by a nonlinear complementarity problem. After discretization the evolutionary state problem, associated with the optimal shape problem, is formulated as a sequence of nonsmooth equations and a general form of the optimal design problem is treated by using a nonsmooth approach. The sensitivity analysis based on the adjoint-variable technique is derived for a history-dependent problem. As applications optimal design problems are solved for an elasto-plastic truss structure and for an elasto-plastic continuum.

Homogenization of the acoustic transmission through a perforated layer

The paper deals with homogenized transmission conditions imposed on an interface plane separating two halfspaces occupied by an acoustic medium. The conditions are obtained as the two-scale homogenization limit of a standard acoustic problem imposed on the layer perforated by a sieve-like obstacle with a periodic structure. Both the characteristic scale of the perforations and the layer thickness are parametrized by @e->0. The limit model involving some homogenized coefficients governs the interface discontinuity of the acoustic pressure associated with the two halfspaces and the magnitude of the transverse acoustic velocity. This novel approach allows for the treatment of complicated designs of perforations and presents an alternative to the usual description of acoustic impedance, which relies on a rough averaging of the quasi-experimental data.

Homogenization and Shape Sensitivity of Microstructures for Design of Piezoelectric Bio-Materials

Shape sensitivity of effective constitutive parameters is studied for homogenized piezoelectric composite which formerly was intended for bio-material application. It consists of the piezoelectric matrix in which elastic inclusions are distributed periodically. The microstructures are assumed to be parametrized in terms of the shape of the inclusions. Microstructures with elliptic inclusions are considered in numerical examples which indicate strong influence of the geometry on the homogenized piezoelectric properties. As the main theoretical result of the paper, the shape sensitivity formulae of the homogenized elastic, dielectric and piezoelectric coefficients are derived using the domain method of the material derivative.

Homogenization of elastic waves in fluid-saturated porous media using the Biot model

We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method, we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated from the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

On modelling the parallel diffusion flow in deforming porous media

We present a macroscopic model of the fluid diffusion in deformable porous media, motivated by diffusion-deformation phenomena influencing the heart muscle blood perfusion, or the mechanical properties of kidneys. The problems are described by the displacement field and by several fluid pressure fields associated with parallel porosities interpenetrating the material matrix and mutually separated by interface sectors. The model consists of the equilibrium equation, and a number of mass conservation equations, each incorporating the Darcy law of fluid diffusion. The steady state problem attains the form of the Barenblatt model of parallel flows, while, in the non-steady regime, the coupled diffusion and deformation phenomena induce the apparent viscoelastic behaviour of the bulk material. Numerical examples are given to illustrate some features of the finite element model.

Poro-viscoelasticity modelling based on upscaling quasistatic fluid-saturated solids

In this paper, quasistatic models are developed for the slow flow of compressible fluids through porous solids, where the solid exhibits fading memory viscoelasticity. Problems of this type are important in practical geomechanics contexts, for example, in the context of fluid flow through unconsolidated reservoir sands and of wellbore deformation behaviour in gas and oil shale reservoirs, all of which have been studied extensively. For slow viscous fluid flow in the poro-viscoelastic media we are able to neglect the dynamic effects related to inertia forces, as well as the dissipation associated with the viscous flows. This is in contrast to the vast body of work in the poro-elastic context, where much faster flow of the viscous fluids may give rise to memory effects associated with microflows in pores of the solid media. Such problems have been treated extensively in both the dynamic and quasistatic cases. We are taking a specific type of the porous medium subject to slow deformation processes possibly inducing moderate pressure gradients and flow rates characterised by negligible inertia effects. As the result of homogenisation of such a two-phase medium, we observe the fading memory behaviour in the Biot modulus which controls the pressure increase due to skeleton macroscopic deformation and pore fluid content. Although our derivation leads to a result which is consistent with the formal phenomenological approach proposed by Biot (J Appl Phys 23:1482–1498, 1962), we offer the reader more insight into the structure of the poro-viscoelastic constitutive relations obtained; in particular, we can show that the Biot compressibility evolves in time according to the creep function while the skeleton stiffness is driven by the relaxation function.

Multiscale FE simulation of diffusion-deformation processes in homogenized dual-porous media

AbstractThe paper deals with a model of the homogenized fluid saturated porous material which recently was obtained by the authors using the asymptotic analysis of the Biot type medium characterized by the double porosity. The homogenized macroscopic model is featured by the fading memory effects arising from the microflow in the dual porosity. We derive the steady state formulations and discuss several topics related to the numerical implementation of the model, namely the solution procedure of the discretized microscopic problems, evaluation of the homogenized coefficients and an approximation of the convolution integrals of the macroscopic model, so that the fading memory effects are computationally tractable. Numerical examples are presented to illustrate the approximation schemes discussed in the paper. All computations were performed using the in-house developed finite element code SfePy allowing the multiscale simulations. Besides various potential engineering applications, the present model is intended for simulations of compact bone poroelasticity.