Atomistic simulation of free transverse vibration of graphene,hexagonal SiC,and BN nanosheets

Free transverse vibration of monolayer graphene, boron nitride (BN), and silicon carbide (SiC) sheets is inves-tigated by using molecular dynamics finite element method. Eigenfrequencies and eigenmodes of these three sheets in rectangular shape are studied with different aspect ratios with respect to various boundary conditions. It is found that aspect ratios and boundary conditions affect in a similar way on nat-ural frequencies of graphene, BN, and SiC sheets. Natural frequencies in all modes decrease with an increase of the sheet's size. Graphene exhibits the highest natural frequen-cies, and SiC sheet possesses the lowest ones. Missing atoms have minor effects on natural frequencies in this study.     

Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization

The determination of the effective mechanical moduli of textiles from mechanical measurements is usually difficult due to their discrete architecture, which makes micromechanical analyses a relevant alternative to access those properties. Micropolar continuum models describing the effective mechanical behavior of woven fabric monolayers are constructed from the homogenization of an identified repetitive pattern of the textile within a representative unit cell. The interwoven yarns within the textile are represented as a network of trusses connected by nodes at their crossover points. These trusses have extensional and bending rigidities to allow for yarn stretching and flexion, and a transverse shear deformation is additionally considered. Interactions between yarns at the crossover points are captured by beam segments connecting the nodes. The woven fabric is modeled after homogenization as an anisotropic planar continuum with two preferred material directions in the mean plane of the textile. Based on the developed methodology, the effective mechanical properties of plain weave and twill are evaluated, including their bending moduli and characteristic flexural lengths. A satisfactory agreement is obtained between the effective moduli obtained by homogenization and numerical values obtained by finite element simulations performed over periodic unit cells.     

Asymptotic expansion homogenization of discrete fine-scale models with rotational degrees of freedom for the simulation of quasi-brittle materials

Discrete fine-scale models, in the form of either particle or lattice models, have been formulated successfully to simulate the behavior of quasi-brittle materials whose mechanical behavior is inherently connected to fracture processes occurring in the internal heterogeneous structure. These models tend to be intensive from the computational point of view as they adopt an “a priori” discretization anchored to the major material heterogeneities (e.g. grains in particulate materials and aggregate pieces in cementitious composites) and this hampers their use in the numerical simulations of large systems. In this work, this problem is addressed by formulating a general multiple scale computational framework based on classical asymptotic analysis and that (1) is applicable to any discrete model with rotational degrees of freedom; and (2) gives rise to an equivalent Cosserat continuum. The developed theory is applied to the upscaling of the Lattice Discrete Particle Model (LDPM), a recently formulated discrete model for concrete and other quasi-brittle materials, and the properties of the homogenized model are analyzed thoroughly in both the elastic and the inelastic regime. The analysis shows that the homogenized micropolar elastic properties are size-dependent, and they are functions of the RVE size and the size of the material heterogeneity. Furthermore, the analysis of the homogenized inelastic behavior highlights issues associated with the homogenization of fine-scale models featuring strain-softening and the related damage localization. Finally, nonlinear simulations of the RVE behavior subject to curvature components causing bending and torsional effects demonstrate, contrarily to typical Cosserat formulations, a significant coupling between the homogenized stress–strain and couple-curvature constitutive equations.     

Third-order elastic,piezoelectric, and dielectric constants

The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry.These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.     

Antiplane elastic wave cloaking using metamaterials,homogenization and hyperelasticity

We consider the problem of how to cloak objects from antiplane elastic waves using two alternative techniques. The first is the use of a layered metamaterial in the spirit of the work of Torrent and Sanchez-Dehesa (2008) who considered acoustic cloaks, motivated by homogenization theories, whilst the second is the use of a hyperelastic cloak in the spirit of the work of Parnell et al. (2012). We extend the hyperelastic cloaking theory to the case of a Mooney–Rivlin material since this is often considered to be a more realistic constitutive model of rubber-like media than the neo-Hookean case studied by Parnell et al. (2012), certainly at the deformations required to produce a significant cloaking effect. Although not perfect, the Mooney–Rivlin material appears to be a reasonable hyperelastic cloak. This is clearly encouraging for applications. We quantify the effectiveness of the various cloaks considered by plotting the scattering cross section as a function of frequency, noting that this would be zero for a perfect cloak.     

Acoustic cloaking: Geometric transform,homogenization and a genetic algorithm

A general process is proposed to experimentally design anisotropic inhomogeneous metamaterials obtained through a change of coordinates in the Helmholtz equation. The method is applied to the case of a cylindrical transformation that allows cloaking to be performed. To approximate such complex metamaterials we apply results of the theory of homogenization and combine them with a genetic algorithm. To illustrate the power of our approach, we design three types of cloaks composed of isotropic concentric layers structured with three types of perforations: curved rectangles, split rings and crosses. These cloaks have parameters compatible with existing technology and they mimic the behavior of the transformed material. Numerical simulations have been performed to qualitatively and quantitatively study the cloaking efficiency of these metamaterials.     

Combination of the homogenization and effective medium methods for flow through porous media in a nonperiodic external perturbation field

Models described by parabolic equations with a rapidly oscillating nonperiodic right side are investigated by means of averaging theory methods. For a nonperiodic perturbation field a combined homogenization and effective medium method is developed. This method makes it possible to obtain the solution of the cell problems in a finite form correct to the second order in the inhomogeneity parameter. The method is applied to problems of single-phase and two-phase flow through porous media. The technique of the method is outlined and explicit solutions of cell problems are constructed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–115, July–August, 1995.     

A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams

A rigorous method for the homogenization of general elastoplastic periodic lattices is presented. A discrete unit cell problem with finite number of degrees of freedom is solved for the determination of the overall elastic stiffness and ultimate strength of the lattice. Both static and kinematic methods are developed. It is shown that the overall yield strength domain of a large specimen, subjected to the so-called kinematically uniform boundary conditions, is asymptotically equal to the homogenized yield strength domain, as the size of the specimen goes to infinity. The method is applied to metallic honeycomb materials with arbitrary non-uniform cell wall thickness. New results concerning non-symmetric material distribution in the cell edges of the honeycomb are obtained. The model shows that the effects of this type of defect on the overall properties are less important than the already known effects of symmetric non-uniform cell wall thickness. Good agreement is observed between the proposed analytical beam model predictions and the finite element computations.     

Mechanical response of fabric sheets to three-dimensional bending,twisting, and stretching

A model for the mechanics of woven fabrics is developed in the framework of two-dimensional elastic surface theory. Thickness effects are modeled indirectly in terms of appropriate constitutive equations. The model accounts for the strain of the fabric and additional effects associated with the normal bending, geodesic bending, and twisting of the constituent fibers.     

Formation and interaction of intense vortex sheets in 3-dimensional,incompressible, hydrodynamics

Various questions related to the physics of inviscid flows are reviewed. The emergence of strong vortex sheets has repeatedly been observed in the simulation of the 3-dimensional equations, with a variety of initial conditions. In the case of axisymmetric Euler flows, the origin of these sheets can be easily understood with the help of an analogy with thermally driven flows. A more general mechanism to explain these sheets is proposed. Questions of singularities are briefly reviewed. Lastly, preliminary results on the connection between the sheets forming in inviscid flows, and the vortex tubes observed in high Reynolds number flows are presented.

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Sommario Sono considerate varie questioni correlate con la fisica dei flussi non-viscosi. La nascita di strati di forti vortici è stata ripetutamente osservata nella simulazione delle equazioni tridimensionali, per diverse condizioni iniziali. Nel caso di flussi di Eulero assialsimmetrici, l'origine di questi piani può essere facilmente compresa con l'aiuto di una analogia con i flussi guidati termicamente. Viene inoltre proposto un più generale meccanismo per giustificare questi strati e si passano in rassegna brevemente questioni riguardanti le singolarità. Infine, vengono presentati alcuni risultati preliminari sulla connessione tra i piani formantisi in flussi non viscosi ed i vortici tubolari osservati nei flussi ad alti numeri di Reynolds.

     

On the limit and applicability of dynamic homogenization

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. Through arguments of calculated reflected energy we note that infinite-domain homogenization is much more applicable to finite cases of the 3-phase composite than it is to the 2-phase composite. In fact, the applicability of dynamic homogenization extends to most of the first branch (negligible reflection) for the 3-phase composite. This is in contrast with a periodic composite without local resonance where the approximation of homogenization worsens with increasing frequency over the first branch and is demonstrably bad on the second branch. We also study the effect of the interface location on the applicability of homogenization. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction.     

Displacement reconstruction and strain refinement of clustering-based homogenization

Recently proposed clustering-based methods provide an efficient way for homogenizing heterogeneous materials, yet without concerning the detailed distribution of the mechanical responses. With coarse fields of the clustering-based methods as an initial guess, we develop an iteration strategy to fastly and accurately resolve the displacement, strain and stress based on the Lippmann-Schwinger equation, thereby benefiting the local mechanical analysis such as the detection of the stress concentration. From a simple elastic case, we explore the convergence of the method and give an instruction for the selection of the reference material. Numerical tests show the efficiency and fast convergence of the reconstruction method in both elastic and hyper-elastic materials.     

Numerical estimation of heterogeneous reaction constants in the flow of rarefied gas through a cylindrical channel

The Direct Simulation Monte Carlo method is used to study the influence of the coefficients of heterogeneous dissociation and recombination reactions on the rarefied gas flow through a cylindrical channel. It is established that the degree of dissociation of the flow coming out of the channel is significantly dependent on the relationship between the dissociation and recombination coefficients. The technique for determining the dissociation and recombination coefficients on the basis of the experimental data is proposed.     

Decay of shear layers and vortex sheets

An incompressible fluid is contained in the domain between two stationary infinited parallel rigid plates. It is assumed that for shear flows, the shear stress in an element of the fluid depends linearly on history of the velocity gradient in that element. It is supposed that initially two steady shear layers exist in the fluid and are symmetrically disposed with respect to the mid-plane. The time-dependent velocity field which results from the removal of the forces maintaining this steady flow is calculated in the cases when the fluid is Newtonian and when it is Maxwellian. The limiting cases when the shear layers reside in an unbounded space of the fluid and when they further become vortex sheets are discussed.     

Force and Torque Measurement,a Technology Overview

Experimental Techniques -     

Stability and capillary dynamics of circular vortex sheets

We investigate the motion of circular vortex sheets with surface tension. A linear stability analysis shows that high modes of the circular vortex sheet are stabilized by surface tension, and the sheet is stable if surface tension is larger than a critical value. The modes of perturbations, n = 1 and 2, are always stable, regardless of surface tension, and the mode n = 3 is also stable for large surface tension. The numerical results show that a stable vortex sheet rotates and oscillates weakly. The oscillations of each mode of the interface mainly consist of two travelling waves of different frequencies in time. The amplitude and the period of the oscillation depend on the mode of the perturbation and surface tension. We also perform long-time computations for the unstable evolution of circular sheets. For a high Weber number, ripples are produced on the sheets, as well as pinching and self-intersection. It is found that the appearance of ripples is associated with the growth of noise. For an intermediate Weber number, the sheet evolves to an exotic structure with small spikes on the fingers, while for a low Weber number, it is nonlinearly stable.