Elastic Term Research Articles

Exact formulations of non-linear planar and spatial Euler–Bernoulli beams with finite strains

Using Hamilton’s principle, exact equations of motion for non-linear planar and spatial Euler–Bernoulli beams are derived. In the existing non-linear Euler–Bernoulli beam formulations, some elastic terms are dropped by differentiation from the incomplete Green–Lagrange strain tensor followed by negligible elastic deformations of cross-sectional frame. On the other hand, in this article, the exact strain field concerning considerable elastic deformations of cross-sectional frame is used as a source in differentiations. As a result, the achieved closed-form equations are exact and more accurate than formerly reported equations in the literature. Moreover, the applicable dynamic model of inextensional beams which is fully accurate, yet simple has been shown. The planar and inextensional dynamic models have been compared with the existing dynamic models in the literature, and the proposed dynamic models demonstrate significant improvements in the numerical results. Finally, experiments on the carbon fibre rods verify the model presented for inextensional beams.

First-principles-based mesoscale modeling of the solute-induced stabilization of 〈1 0 0〉 tilt grain boundaries in an Al–Pb alloy

A first-principles disclination structural units (DSUM) model was used to calculate the energies of 12 〈0 0 1〉 symmetric tilt grain boundaries in Al for a pure system and a system in which Pb atoms substitute for Al at one-half of the lattices sites along the interfacial plane. The grain boundaries are modeled as disclination dipole walls with energies given as a weighted sum of individual disclination energies determined from special low-sigma structures, elastic terms and disclination core energies. Predictions of the DSUM using the bulk Al shear modulus and Poisson’s ratio in the elastic and core energy terms are found to be comparable to energies for fully atomistic simulations calculated using a modified embedded-atom method (MEAM) potential. No relation between grain boundary energies in pure Al and the degree of stabilization due to Pb doping was found. The DSUM parameterized to density functional theory calculations predicts a ∼50% reduction in the energy anisotropy with respect to angle due to doping compared to the pure system, while MEAM calculations yield no appreciable reduction in the energy anisotropy.

Predicting average velocity of a non-linear sliding microrobot

Among the locomotion concepts employed in the microrobotics, friction-based locomotion principles are of considerable importance. In this study, the dynamic modelling of friction drive microrobots subjected to the tangential and normal excitations is investigated, which have the same frequency, but are shifted in phase. The motion equation of the microrobot reveals a strongly non-linear differential equation with discontinuity for which the elastic force term is proportional to a signum function. Using the homotopy perturbation method, simple expressions are derived for predicting average velocity of the slider. The obtained results are in good agreement with those achieved from numerical integration of the full equation and experimental results, reported in the literature. Using the concept of step efficiency, some useful guidelines for the design and control of this type of microrobots are provided.

Attenuation tomography of the western United States from ambient seismic noise

[1] We show that the spatial coherency of the ambient seismic field can be used for attenuation tomography in the western United States. We evaluate the real portion of the spatial coherency with an elastic geometric spreading term (a Bessel function) and a distance dependent decay (an attenuation coefficient). In order to invert the spatial coherency, a weight stack inversion technique is applied. We recover phase velocity and attenuation coefficient maps at periods of 8–32s, which correspond to the elastic and anelastic structure at crustal and upper mantle depths. The phase velocity maps obtained by this method are of similar resolution to more standard two‐station methods. The attenuation results provide an important complement to the information extracted from earthquake‐based tomography. Several geological features are easily identifiable in the attenuation coefficient maps, such as the highly attenuating sedimentary basins along the West Coast of the United States, and the highly attenuating Yellowstone region, and the boundaries of the Snake River Plains.

A periodic solution for friction drive microrobots based on the iteration perturbation method

The friction drive principle, which is based on the superposition of two synchronized perpendicular vibrations at the interface of the robot and the work floor, plays a fundamental role in the locomotion of miniaturized robots. In this paper, the iteration perturbation method proposed by He is used to generate a periodic solution for this type of friction drive microrobot. The equation of motion for the system reveals a parametrically excited oscillator with discontinuity, the elastic force term for which is proportional to a signum function. The obtained solutions are in excellent agreement with those achieved from numerical integration and experiments reported in the literature. Results show that for a wide range of variation of system parameters, a phase shift near 90°between the perpendicular vibrations leads to the maximum achievable velocity of the slider. Besides, we have shown that driving frequency and the relative amplitude of vibrations are the best parameters for the control purposes of this type of microrobot.

Dynamic analysis of microrobots with Coulomb friction using harmonic balance method

In this paper, we investigate the dynamic analysis of a strongly nonlinear microrobot using a three-term harmonic balance method. The employed locomotion concept, namely “friction drive principle,” is based on the superposition of a horizontal vibration at the interface between the robot and work floor and an active variation of friction force, obtained by the vertical vibration of the base at the same interface. The equation of motion for the system reveals a parametrically excited oscillator with discontinuity for which the elastic force term is proportional to a signum function. The obtained periodic solution not only is of high accuracy, but also can predict the contribution of the friction coefficient in the average velocity of the slider. Results show that the velocity and the step efficiency of motion depend almost sinusoidally on the phase shift between the horizontal and the vertical vibration. Unlike traditional analytical techniques and in agreement with both numerical simulations and experimental results reported in the literature, the utilized method demonstrates that the maximum average velocity occurs at a phase shift that varies with respect to system’s configuration parameters. Besides, the effect of variation of different configuration parameters on the behavior of this type of microrobots has been studied and the maximum achievable performance in terms of velocity and the step efficiency has been evaluated.

Effects of weak anchoring on the azimuthal anchoring energy of a nematic liquid crystal at a grooved interface

Fukuda et al. re-examined Berreman's model on the azimuthal anchoring energy at a grooved interface with infinite polar anchoring. We investigate the effects of finite polar anchoring on the azimuthal anchoring energy in the frame of Fukuda et al.’s theory and propose a more general expression for the azimuthal anchoring energy. We show that finite polar anchoring can provide an important contribution to the azimuthal anchoring energy and that the surface-like elastic term still leads to a non-zero contribution to the Rapini–Papoular anchoring energy.

Efficient formulation for dynamics of corotational 2D beams

The corotational method is an attractive approach to derive non-linear finite beam elements. In a number of papers, this method was employed to investigate the non-linear dynamic analysis of 2D beams. However, most of the approaches found in the literature adopted either a lumped mass matrix or linear local interpolations to derive the inertia terms (which gives the classical linear and constant Timoshenko mass matrix), although local cubic interpolations were used to derive the elastic force vector and the tangent stiffness matrix. In this paper, a new corotational formulation for dynamic nonlinear analysis is presented. Cubic interpolations are used to derive both the inertia and elastic terms. Numerical examples show that the proposed approach is more efficient than using lumped or Timoshenko mass matrices.

Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms

In this paper, Hamiltonian Approach (HA) is applied to obtain the analytical approximate solution of the nonlinear oscillators with Rational and Irrational Elastic Terms. Periodic solutions are analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. A comparison of the period of oscillation and obtained solutions with the exact results illustrates that the Hamiltonian approach is very effective and quite accurate for nonlinear equations.

A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces

A Mixed Finite Element (MFE) method for 3D non-steady flow of a viscoelastic compressible fluid is presented. It was used to compute polymer injection flows in a complex mold cavity, which involves moving free surfaces. The flow equations were derived from the Navier-Stokes incompressible equations, and we extended a mixed finite element method for incompressible viscous flow to account for compressibility (using the Tait model) and viscoelasticity (using a Pom-Pom like model). The flow solver uses tetrahedral elements and a mixed velocity/pressure/extra-stress/density formulation, where elastic terms are solved by decoupling our system and density variation is implicitly considered. A new DEVSS-like method is also introduced naturally from the MINI-element formulation. This method has the great advantage of a low memory requirement. At each time slab, once the velocity has been calculated, all evolution equations (free surface and material evolution) are solved by a space-time finite element method. This method is a generalization of the discontinuous Galerkin method, that shows a strong robustness with respect to both re-entrant corners and flow front singularities. Validation tests of the viscoelastic and free surface models implementation are shown, using literature benchmark examples. Results obtained in industrial 3D geometries underline the robustness and the efficiency of the proposed methods.

An Intrusive Entropic Barrier Induced by Force

Single-molecule force spectroscopy has opened up new approaches to the study of protein dynamics. For example, an extended protein folding after an abrupt quench in the pulling force was shown to follow variable collapse trajectories marked by well-defined stages that departed from the expected two-state folding behavior that is commonly observed in bulk. Here, we explain these observations by developing a simple approach that models the free energy of a mechanically extended protein as a combination of an entropic elasticity term and a short-range potential representing enthalpic hydrophobic interactions. The resulting free energy of the molecule shows a force-dependent energy barrier of magnitude, ΔE = e(F - Fc)3/2, separating the collapsed state of the molecule from a force-driven extended conformation, that vanishes at a critical force Fc. By solving the Langevin equation under force quench conditions, we generate folding trajectories corresponding to the diffusional collapse of an extended polypeptide that mimics those observed experimentally. Further we apply this model to force extension conditions in order to investigate the role played by the force-induced energy barrier on the two-state hopping phenomena that has been observed in single protein molecules placed under a stretching force. Langevin dynamics across such force induced barrier readily demonstrates the hopping behavior observed for a variety of single molecules placed under force. Our model interprets AFM force-clamp data and accounts as well for force-extension and hopping observed in optical tweezers, thus unifying the field of protein force spectroscopy. Moreover, given that this barrier does not exist at zero force, extrapolating hopping trajectories to zero force could not be compared to bulk measurements.

Multiple‐Parameter Hamiltonian Approach for Higher Accurate Approximations of a Nonlinear Oscillator with Discontinuity

We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinuity. He′s Hamiltonian approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). We employed this method for higher‐order approximate solution of the nonlinear oscillator equation. This property is used to obtain approximate frequency‐amplitude relationship of a nonlinear oscillator with high accuracy. Many numerical results are given to prove the efficiency of the suggested technique.

Berreman近似下任意形状表面沟槽对双轴向列相方位锚定的影响

Under the approximation of Berreman,any grooved surface with arbitrary shape can be described in terms of Fourier series.Based on the theory of Fukuda et al,the expression of anchoring energy with an additional term was obtained,taking into the surfacelike elastic term in biaxial nematic liquid crystal.The figures of reduced anchoring energies versus the angle between the direction of the grooves and main director were simulated under the different values of surfacelike elastic constant in certain cases and the effect of biaxial elastic constant on easy axes was discussed.

Polymer-Grafted-Nanoparticles Nanocomposites: Dispersion, Grafted Chain Conformation, and Rheological Behavior

We investigate the dispersion mechanisms of nanocomposites made of well-defined polymer (polystyrene, PS) grafted-nanoparticles (silica) mixed with free chains of the same polymer using a combination of scattering (SAXS/USAXS) and imaging (TEM) techniques. We show that the relevant parameter of the dispersion, the grafted/free chains mass ratio R tuned with specific synthesis process, enables to manage the arrangement of the grafted nanoparticles inside the matrix either as large and compact aggregates (R < 0.24) or as individual nanoparticles dispersion (R > 0.24). From the analysis of the interparticles structure factor, we can extract the thickness of the spherical corona of grafted brushes and correlate it with the dispersion: aggregation of the particles is associated with a significant collapse of the grafted chains, in agreement with the theoretical models describing the free energy as a combination of a mixing entropy term between the free and the grafted chains and an elastic term of deformation of the grafted brushes. At fixed grafting density, the individual dispersion of particles below the theoretical limit of R = 1 can be observed, due to interdiffusion between the grafted and the free chains but also to processing kinetics effects, surface curvature and chains poly dispersity. Mechanical analysis of nanocomposites show the appearance of a longer relaxation time at low frequencies, more pronounced in the aggregated case even without direct connectivity between the aggregates. Correlation between the local structure and the rheological behavior suggests that the macroscopic elastic modulus of the nanocomposite could be described mainly by a short-range contribution, at the scale of the interactions between grafted particles, without significant effect of larger scale organizations.

On the intensity of linear elastic high order singularities ahead of cracks and re-entrant corners

The paper deals with high order elastic singular terms at cracks and re-entrant corners (sharp V-notches), which are commonly omitted in linear elastic analyses by the argument that the strain energy and displacements in the near-tip region should be bounded. The present analysis proves that these terms are fully included in the elastic part of complete elastic–plastic stress and strain solutions. The intensities of high order singular terms are found to be linked to the linear elastic stress intensity factor and the extension of the plastic zone along the crack bisector line. The smaller the plastic radius, the smaller the intensities of high order singular terms are. A physical justification of the existence of high order singular terms is provided on the basis of the strain energy density distribution detected along the crack bisector line. Finally, the influence of the V-notch opening angle is made explicit, discussing also the relationship between the singularity orders and the solution of a Williams’ type sinusoidal eigen-equation.

The generation of tetrahedral mesh models for neuroanatomical MRI

In this article, we describe a detailed method for automatically generating tetrahedral meshes from 3D images having multiple region labels. An adaptively sized tetrahedral mesh modeling approach is described that is capable of producing meshes conforming precisely to the voxelized regions in the image. Efficient tetrahedral mesh improvement is then performed minimizing an energy function containing three terms: a smoothing term to remove the voxelization, a fidelity term to maintain continuity with the image data, and a novel elasticity term to prevent the tetrahedra from becoming flattened or inverted as the mesh deforms while allowing the voxelization to be removed entirely. The meshing algorithm is applied to structural MR image data that has been automatically segmented into 56 neuroanatomical sub-divisions as well as on two other examples. The resulting tetrahedral representation has several desirable properties such as tetrahedra with dihedral angles away from 0 and 180 degrees, smoothness, and a high resolution. Tetrahedral modeling via the approach described here has applications in modeling brain structure in normal as well as diseased brain in human and non-human data and facilitates examination of 3D object deformations resulting from neurological illness (e.g. Alzheimer's disease), development, and/or aging.

Rheo-NMR studies of the behavior of a nematic liquid crystal in a low-shear-rate regime: The transition from director alignment to reorientation

Deuterium NMR spectroscopy has been used to study the director dynamics of the nematic liquid-crystal system cetyl trimethylammonium bromide/D2O under the action of applied viscous torques. Shear forces were applied using a custom-built Couette cell that was introduced into an NMR superconducting magnet, so that its rotational axis was parallel to the magnetic field direction, along which the liquid-crystal director originally aligned. Subsequently, the inner cylinder of the cell was rotated continuously at different rates using a stepper motor. The resulting time evolution and ultimate steady-state orientation of the director, governed by the competition between the applied viscous torque with elastic and magnetic terms, was measured via observed changes in the deuterium spectrum. Using a simple gearbox allowed unprecedented access to a low-shear-rate regime in which, above a threshold shear rate, the director of part of the sample was observed to reorient, while the remaining part still aligned with the magnetic field. Subsequent increases in the applied rotational rate were found to increase the relative proportion of the orienting fraction. Spatially resolved NMR spectra showed that the orienting and field-aligned fractions formed separated bands across the gap of the Couette cell, with director reorientation being initiated at the moving inner wall. The behavior was found to be consistent with the often ignored variation in velocity gradient manifest across the gap of a cylindrical cell, so that as the angular frequency of the inner cylinder was increased the radial location of the critical shear rate required for reorientation traversed the gap. Once the applied rotational rate was sufficient to reorient the director of the entire sample, the dependence of the exhibited steady-state orientation on the average applied shear rate was measured. These results could be fitted to an analytical solution of the force-balance equation, made tractable by the assumption that the elasticity term was of minor significance and could be ignored. Additionally, the use of a numerical solution of the full force-balance equation, which explicitly includes elasticity and secondary flow and additionally allows the time evolution of the director orientation to be calculated, was investigated.

A simple thermodynamic description of the combined Einstein and elastic models

A simple application of the Einstein model combined with an elastic description of the solidstate is developed. The frequency of quantum oscillators has been assumed as volumedependent and elastic energy terms with a static character have been included tocomplete the description. Such an extension enables us to construct the completethermodynamics. In particular, the model yields a practical equation of state anddescribes the thermal expansion coefficient as well as the isothermal compressibilityof solids. The thermodynamic properties resulting from the Gibbs free-energyanalysis have been calculated and are illustrated in figures. Some comparisons ofthe theoretical results with experimental data for solid argon have been made.

Smart metering: what potential for householder engagement?

The argument for the implementation of ‘smart’ metering, which is an elastic term, varies according to circumstance and place. In some countries, the business case for establishing an advanced metering infrastructure (AMI) relies in part on improving consumption feedback to customers and assisting in the transition to lower-impact energy systems. There is an expectation that AMI will lead to reductions in both the demand and the cost to serve customers through improved communication, but little evidence exists to show overall demand reduction. To what extent might smart meters improve the prospects for customer engagement? To assess this question, end-user perceptions and practices must be considered along with metering hardware and economics. Using the theory of affordances, qualitative research is examined to understand how householders have used consumption feedback, with and without smart meters. Although AMI offers possibilities for household energy management and customer–utility relations, there is little evidence to suggest it will automatically achieve a significant reduction in energy demand. For that, there has to be a determined focus on overall demand reduction (rather than on peak electricity demand reduction), on designing customer interfaces for ease of understanding, and on guiding occupants towards appropriate action. Appropriate forms of interface, feedback, narrative, and support will be needed to reach diverse populations. Les arguments en faveur de la mise en œuvre d'un comptage «intelligent», qui est un terme souple, varient en fonction des circonstances et du pays concerné. Dans certains pays, le business case pour mettre en place une infrastructure de comptage avancée (AMI) s'appuie en partie sur une amélioration du retour d'information fourni aux clients sur leur consommation, et sur l'aide apportée pour passer à des systèmes énergétiques présentant un impact moindre. Il est attendu d'une AMI qu'elle conduise à la fois à une réduction de la demande et à une réduction du coût de desserte des clients grâce à des améliorations en terme de communication, mais il existe peu de données probantes à l'appui d'une réduction globale de la demande. Dans quelle mesure des compteurs intelligents pourraient-ils améliorer les chances de voir le client s'impliquer? Afin d'évaluer cette question, il est nécessaire d'examiner les perceptions et les pratiques des utilisateurs finaux en même temps que le matériel de comptage et les aspects économiques. En utilisant la théorie des affordances, l'étude qualitative est examinée de façon à comprendre comment les occupants ont utilisé le retour d'information sur leur consommation, avec et sans compteurs intelligents. Bien qu'une AMI offre des possibilités en matière de gestion énergétique des ménages et de relations client-fournisseur d'énergie, il y a peu de données probantes permettant de suggérer que cela assurera automatiquement une réduction sensible de la demande énergétique. Pour ce faire, il importe de mettre l'accent sur une réduction globale de la demande (plutôt que sur une réduction des pics de consommation électrique), sur la conception d'interfaces client faciles à comprendre et sur la fourniture de conseils aux occupants pour qu'ils prennent les mesures appropriées. Il sera nécessaire de disposer d'interfaces, d'un retour d'information, de descriptifs et d'une assistance aux formes adaptées pour pouvoir toucher des populations diverses. Mots clés: affordances, comportement, demande énergétique, gestion énergétique, retour d'information, ménages, logement, human agency, comptage intelligent

Scaling of viscoelastic shell properties of lipid encapsulated microbubbles with frequency.

While it has been established that nonlinear oscillations of lipid encapsulated microbubbles occur at frequencies of &amp;gt;10 MHz, the understanding of bubble behavior in this frequency range is limited and is not predicted by bubble models using shell properties derived at &amp;lt;5 MHz. Here we investigate the scaling of the viscoelastic shell properties of lipid encapsulated contrast agents with frequency and bubble size. Attenuation measurements were performed over 2–30 MHz and, using size distribution measurements and the linearized Rayleigh–Plesset equation, shell elasticity and viscosity terms were estimated. Experiments were performed on a series of narrow size distribution in-house agents with progressively smaller diameters, as well as the high frequency agent Micromarker. In-house agents peaked in attenuation at 4.2, 8.5, 13, and 22 MHz, and Micromarker rose until 21 MHz. The elasticity for in-house agents was comparable to other agents at lower frequencies, while the viscosity rapidly decreased above 10 MHz. For Micromarker, which has a different fabrication process, the elasticity was substantially higher than the in-house agent and over 15–30 MHz its viscosity was far lower viscosity than those reported at lower frequencies. The use of low shell viscosities in bubble models enables the prediction of nonlinear oscillations at higher frequencies.