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We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing thin infinite elastic rods with material coefficients that rapidly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the rod thickness and the period of material oscillations, which are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under the assumption on material symmetries as well as in the general case when these two types of displacements are intertwined. [ABSTRACT FROM AUTHOR]

For arbitrarily small values of ε > 0 , we formulate and analyse the Maxwell system of equations of electromagnetism on ε -periodic sets S ε ⊂ R 3. Assuming that a family of Borel measures μ ε , such that supp (μ ε) = S ε , is obtained by ε -contraction of a fixed 1-periodic measure μ , and for right-hand sides f ε ∈ L 2 (R 3 , d μ ε) , we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic "singular structures", when μ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for L 2 functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method. [ABSTRACT FROM AUTHOR]

New boron-rich sulfide B6S and selenide B6Se have been discovered by combination of high pressure – high temperature synthesis and ab initio evolutionary crystal structure prediction, and studied by synchrotron X-ray diffraction and Raman spectroscopy at ambient conditions. As it follows from Rietveld refinement of powder X-ray diffraction data, both chalcogenides have orthorhombic symmetry and belong to Pmna space group. All experimentally observed Raman bands have been attributed to the theoretically calculated phonon modes, and the mode assignment has been performed. Prediction of mechanical properties (hardness and elastic moduli) of new boron-rich chalcogenides has been made using ab initio calculations, and both compounds were found to be members of a family of hard phases. [ABSTRACT FROM AUTHOR]

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media. [ABSTRACT FROM AUTHOR]

We prove operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the M-matrix of an associated boundary triple ('Krein resolvent formula'). The resulting asymptotic behaviour is shown to be described, up to a unitary transformation, by a non-standard version of the Kronig-Penney model on $${\mathbb{R}}$$ . [ABSTRACT FROM AUTHOR]

We show that nonlinearly elastic plates of thickness $$h\rightarrow 0$$ with an $$\varepsilon $$ -periodic structure such that $$\varepsilon ^{-2}h\rightarrow 0$$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is 'discontinuously anisotropic' in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale. [ABSTRACT FROM AUTHOR]

We consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations, which may be a model of, e.g., nonlinear conducting, dielectric, or deforming in a restricted way hardening elastic-plastic composites, with “outer” periodicity conditions of a fixed large period. Under some natural growth assumptions on the stored-energy function, we construct for uniformly elliptic problems a full two-scale asymptotic expansion, which has a precise “double-series” structure, separating the slow and the fast variables “in all orders”, so that its “slowly varying” part solves asymptotically an “infinite-order homogenised equation” (cf. Bakhvalov, N.S., Panasenko, G.P.:Homogenisation: Averaging Processes in Periodic Media. Nauka, Moscow, 1984 (in Russian); English translation: Kluwer, 1989), and whose higher-order terms depend on the higher gradients of the slowly varying part. We prove the error bound, i.e., that the truncated asymptotic expansion is “higher-order” close to the actual solution in appropriate norms. The approach is extended to a non-uniformly elliptic case: for two-dimensional power-law potentials we prove the “non-degeneracy” using topological index methods. Examples and explicit formulae for the higher-order terms are given. In particular, we prove that the first term in the higher-order homogenised equations is related to the first-order corrector to the “mean” flux, and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest (second) derivative being a linear combination of the second minors of the Hessian with coefficients depending on the first gradient, and in dimension two is of Monge-Ampère type. We show that this term is present at least for some examples (three-phase power-law laminates).In the second part of the paper we extend to this nonlinear context some of the results previously developed by us in the linear case (Smyshlyaev, V.P., Cherednichenko, K.D.J. Mech. Phys. Solids48, 1325-1357, 2000). In particular, we prove that the slowly varying part of the full asymptotic expansion is the rigorous asymptotics “in all orders” for the “translationally averaged” actual solution and flux, or in the sense of a higher-order version of the weak convergence. We then explore to what extent the method of variational truncation of the infinite-order homogenised equation, successfully implemented by us in the linear context in the previous work for constructing explicit “higher-order” homogenised equations, is extendable to the nonlinear regime. We propose a natural extension and prove that at least under some further natural “non-degeneracy” assumptions it has a solution (the existence), and that any such solution is close to the actual solution in appropriate norms. [ABSTRACT FROM AUTHOR]

This study investigated the use of cellulose nanocrystals (CNC)/chitosan (Chit) polyelectrolyte complex as a stabilizing agent for Pickering emulsions. We demonstrated that chitosan reduces the surface charge of CNC improving the emulsification process. An optimal stabilizing complex containing 1% chitosan results in emulsions with minimal zeta potential (3.2 ± 0.3 mV), droplet size (2.8 ± 0.8 μm), and creaming index (19.8 ± 1.0%) values, along with high stability during storage, a change in pH, and high centrifugal forces (up to 2000 g). The study also showed that the maximum neutralized surface charge of the CNC in the CNC-Chit complex allows for effective adsorption on the surface of sunflower oil droplets, producing a denser stabilizing layer with a smaller droplet size. Additionally, chitosan addition is linked to improved stability and higher viscosity, with little dependence on ionic strength and temperature. Potentiometric titration revealed that compared with sulfated CNCs, five times less chitosan is needed to neutralize the negative surface charge of acetylated CNC. The wettability of a hydrophilic surface depends on the surface charge of the complex, and the wettability and adhesion performance increase with increasing chitosan content. Additionally, we showed that tuning the stabilizer composition can change the bioaccessibility of lipophilic compounds during oral administration.Graphical Abstract: This study investigated the use of cellulose nanocrystals (CNC)/chitosan (Chit) polyelectrolyte complex as a stabilizing agent for Pickering emulsions. We demonstrated that chitosan reduces the surface charge of CNC improving the emulsification process. An optimal stabilizing complex containing 1% chitosan results in emulsions with minimal zeta potential (3.2 ± 0.3 mV), droplet size (2.8 ± 0.8 μm), and creaming index (19.8 ± 1.0%) values, along with high stability during storage, a change in pH, and high centrifugal forces (up to 2000 g). The study also showed that the maximum neutralized surface charge of the CNC in the CNC-Chit complex allows for effective adsorption on the surface of sunflower oil droplets, producing a denser stabilizing layer with a smaller droplet size. Additionally, chitosan addition is linked to improved stability and higher viscosity, with little dependence on ionic strength and temperature. Potentiometric titration revealed that compared with sulfated CNCs, five times less chitosan is needed to neutralize the negative surface charge of acetylated CNC. The wettability of a hydrophilic surface depends on the surface charge of the complex, and the wettability and adhesion performance increase with increasing chitosan content. Additionally, we showed that tuning the stabilizer composition can change the bioaccessibility of lipophilic compounds during oral administration. [ABSTRACT FROM AUTHOR]