Calculation of stability of laminated composite coatings in tribotechnics

The stability of laminated coatings in tribotechnics under compression loads is investigated on the basis of the three-dimensional linearized theory of stability within the framework of a model of piecewise-homogeneous medium. The general formulation of the problem is given, and the solution of a particular example is considered. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 229–236, 2000.     

Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: Instabilities and post-bifurcation response

This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.     

Cyclic density functional theory: A route to the first principles simulation of bending in nanostructures

We formulate and implement Cyclic Density Functional Theory (Cyclic DFT) — a self-consistent first principles simulation method for nanostructures with cyclic symmetries. Using arguments based on Group Representation Theory, we rigorously demonstrate that the Kohn-Sham eigenvalue problem for such systems can be reduced to a fundamental domain (or cyclic unit cell) augmented with cyclic-Bloch boundary conditions. Analogously, the equations of electrostatics appearing in Kohn-Sham theory can be reduced to the fundamental domain augmented with cyclic boundary conditions. By making use of this symmetry cell reduction, we show that the electronic ground-state energy and the Hellmann-Feynman forces on the atoms can be calculated using quantities defined over the fundamental domain. We develop a symmetry-adapted finite-difference discretization scheme to obtain a fully functional numerical realization of the proposed approach. We verify that our formulation and implementation of Cyclic DFT is both accurate and efficient through selected examples.The connection of cyclic symmetries with uniform bending deformations provides an elegant route to the ab-initio study of bending in nanostructures using Cyclic DFT. As a demonstration of this capability, we simulate the uniform bending of a silicene nanoribbon and obtain its energy-curvature relationship from first principles. A self-consistent ab-initio simulation of this nature is unprecedented and well outside the scope of any other systematic first principles method in existence. Our simulations reveal that the bending stiffness of the silicene nanoribbon is intermediate between that of graphene and molybdenum disulphide — a trend which can be ascribed to the variation in effective thickness of these materials. We describe several future avenues and applications of Cyclic DFT, including its extension to the study of non-uniform bending deformations and its possible use in the study of the nanoscale flexoelectric effect.     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.

     

Thermodynamically based fictitious crack/interface model for general normal and shear loading

For small deformations a crack/interface model that considers general 3D normal and shear loading is proposed. It involves elasticity, plasticity and damage and it is thermodynamically based. An essential feature of the model is its consistency with the concepts behind the fictitious crack model. In particular, no crack deformation occurs before the crack is initiated and when a crack has just been initiated the proposed model provides an unloading stiffness that is infinitely large. For the same set of parameters, it is demonstrated that the proposed model is able to provide predictions that are in close agreement with experimental data for concrete for a wide range of loading situations.     

Modelling of contact deformation for a pinch gripper in automated material handling

Grasping analysis is concerned with relating the characteristics of the grasped object to the requirements of the gripper structure. When a gripper presses a flexible specimen with a size larger than that of the gripping surface, complex deformations are produced (compression, shearing, bending and tension). The boundary conditions are complex too because of the changing compressed region and, for many grippers, a changeable contact area. In this paper a mathematical predictive model is presented for simulating these complex deformations, relating a flat gripper’s performance to the properties of flexible gripped materials using elasticity theory. In particular, the distributions of stress and strain within the materials are derived when the gripper presses them. The main variables in the picking-up action have been identified. Moreover, a contact solution has been derived for calculating the contact length between a sample and a rigid support table. The change in the compressed region within a sample as a function of external load has been calculated. In order to match experimental behaviour the non-linear elastic response of the flexible material and the large deformations have to be incorporated in the model. The model predictions have been tested against experimental data and the results of finite element analysis and reasonably good agreement has been obtained.     

Non-resonant light-induced chirality in non-chiral nematic liquid crystals

We propose a simple experimental technique to carry out the observation of light-induced stationary, precessing and oscillating quasi-macroscopic chiral structures of molecular orientation in non-chiral nematics. Moreover, the optical control of these chiral structures is demonstrated both in time and space, using two incoherent counter-propagating circularly polarized excitation beams. Finally, the relaxation of those orientational structures is also investigated and experimental results are shown to be in good agreement with the theory.     

Numerical investigation of influence of ionic space charge and flexoelectric polarization on measurements of elastic constants in nematic liquid crystals

Deformations of nematic layers caused by magnetic field allow determination of the elastic constants of liquid crystal. In this paper, we simulated numerically the deformations of planar and homeotropic nematic layers. The flexoelectric properties of the nematic and presence of ions were taken into account. Our aim was to show the influence of flexoelectricity on the results of the real measurement of the elastic constants k₃₃ and k₁₁. In these simulations, we calculated the optical phase difference ΔΦ between the ordinary and extraordinary rays of light passing through the layer placed between crossed polarizers as a function of the magnetic field induction B. One of the elastic constants can be calculated from the magnetic field threshold for deformation. The ratio k₃₃/k₁₁ can be found by means of fitting theoretical ΔΦ(B) dependence to the experimental results. The calculations reveal that the flexoelectric properties influence the deformations induced by the external magnetic field. In the case of highly pure samples, this may lead to false results of measurement of the elastic constants ratio k₃₃/k₁₁. This influence can be reduced if the nematic material contains ions of sufficiently high concentration. These results show that the flexoelectric properties may play an important role, especially in well purified samples.