A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation

In this note we formulate a sufficient condition for the quasiconvexity at $x \mapsto \lambda x$ of certain functionals $I(u)$ which model the stored-energy of elastic materials subject to a deformation $u$. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to M\"{u}ller and Spector, on admissible deformations. Deformations obey the condition $u(x)= \lambda x$ whenever $x$ belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit upper bound on those $\lambda>0$ such that $I(u) \geq I(u_{\lambda})$ for all admissible $u$, where $u_{\lambda}$ is the linear map $x \mapsto \lambda x$ applied across the entire domain. This is the quasiconvexity condition referred to above

Line-tension model for plasticity as the Gamma-limit of a nonlinear dislocation energy

In this paper we rigorously derive a line-tension model for plasticity as the Gamma-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The Gamma-limit we obtain as the length of the Burgers vector tends to zero has the same form as the Gamma-limit obtained by starting from a linear, semi-discrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration

The Neumann sieve problem and dimensional reduction: a multiscale approach

We perform a multiscale analysis for the elastic energy of a $n$-dimensional bilayer thin film of thickness $2\delta$ whose layers are connected through an $\epsilon$-periodically distributed contact zone. Describing the contact zone as a union of $(n-1)$-dimensional balls of radius $r\ll \epsilon$ (the holes of the sieve) and assuming that $\delta \ll \epsilon$, we show that the asymptotic memory of the sieve (as $\epsilon \to 0$) is witnessed by the presence of an extra interfacial energy term. Moreover we find three different limit behaviors (or regimes) depending on the mutual vanishing rate of $\delta$ and $r$. We also give an explicit nonlinear capacitary-type formula for the interfacial energy density in each regime.Comment: 43 pages, 4 figure

Geometric rigidity for incompatible fields and an application to strain-gradient plasticity

<p>In this paper we show that a strain-gradient plasticity model arises as the Gamma-limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so that edge dislocations can be modelled as point singularities of the strain field.</p> <p>A key ingredient in the derivation is the extension of the rigidity estimate proved by Friesecke, James and Mueller to the case of fields with nonzero curl.</p&gt

Γ-Convergence of free discontinuity problems

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper

Evaluation of Interphase Drag Models for the Determination of Gas Hold-Up of an Air-Water System in a Spouted Bed using CFD

Abstract: The hydrodynamics of a dispersed air-water system within a spouted column with a concentric draft tube and a conical base is simulated using CFD based on a two–fluid Euler–Euler (E-E) modeling framework and k-ε two-equation turbulence closures. The interaction between the dispersed gas phase and the continuous liquid phase is characterized by bubble–liquid interphase forces (drag, turbulent dispersion and lift forces). The Ishii-Zuber drag model [1] and Grace adjusted drag model [2], the latter represented by: GraceDpg Grace dense D C C , , are compared for their capability to match experimental gas hold- up. Numerical results of Reynolds-averaged Navier-Stokes equations with k-ε two-equation turbulence closures models when compared with Pironti experimental data [3] indicated that both drag models, predicted the air hold-up within experimental error. Furthermore, Ishii-Zuber liquid-gas drag model consistently provided better agreement of experimental results; it correctly determines the hold-up within 0.14%. Numerical agreement with adjusted Grace liquid-gas drag model, is exponent dependent 4 p 0.5, turning down that the best computed hold-up is within 0.44%. for p 0.5