Elastic Sphere Research Articles

Depth-sensing spherical indentation of an elastic sphere on an elastic substrate

The axisymmetric three-body, double-unilateral contact problem for an elastic sphere is treated by analytical techniques. It is assumed that the elastic sphere is put on the surface of an elastic layer of finite thickness and, afterwards, is indented at the upper pole of the sphere with a rigid spherical punch. Friction is neglected at both contact interfaces. A first-order asymptotic solution for the displacement–force relation, which generalizes the corresponding Hertzian formula, is derived in explicit form in terms of dimensionless asymptotic constants that account for the finite sizes of the elastic sphere and layer. Under the assumption that the sphere/substrate contact diameter is less than the substrate thickness, the presented results can cover a wide range of parameter combinations, which may be used for the purpose of benchmark assessments in the finite element analysis. The constructed asymptotic model is remarkably simple and elegant, and yet can be applied to various contact problems of practical importance (including a very timely problem of indentation of spherical viruses). The underlying theoretical framework is versatile and can be further extended for analysis of multiple contacts involving elastic spheres and elastic layers.

Failure in granular materials based on acoustic tensor: a numerical analysis

We investigate localization in granular material with the support of numerical simulations based upon DEM (Distinct Element Method). Localization is associated with a discontinuity in a component of the incremental strain over a plane surface through the condition of the determinant of the acoustic tensor to be zero. DEM simulations are carried out on an aggregate of elastic frictional spheres, initially isotropically compressed and then sheared at constant pressure p0. The components of the stiffness tensor are evaluated numerically in stressed states along the triaxial test and employed to evaluate the acoustic tensor in order to predict localization. This occurs in the pre-peak region, where the aggregate hardens under the circumstance to be incrementally frictionless: it is a regime in which the tangential force does not change as the deformation proceedes and, consequently, the deviatoric stress varies only with the normal component of the contact force.

Silo discharge of mixtures of soft and rigid grains.

We study the outflow dynamics and clogging phenomena of mixtures of soft, elastic low-friction spherical grains and hard frictional spheres of similar size in a quasi-two-dimensional (2D) silo with narrow orifice at the bottom. Previous work has demonstrated the crucial influence of elasticity and friction on silo discharge. We show that the addition of small amounts, even as low as 5%, of hard grains to an ensemble of soft, low-friction grains already has significant consequences. The mixtures allow a direct comparison of the probabilities of the different types of particles to clog the orifice. We analyze these probabilities for the hard, frictional and the soft, slippery grains on the basis of their participation in the blocking arches, and compare outflow velocities and durations of non-permanent clogs for different compositions of the mixtures. Experimental results are compared with numerical simulations. The latter strongly suggest a significant influence of the inter-species particle friction.

Of balls, bladders, and balloons: The time required to deflate an elastic sphere

Experience teaches us that a large balloon takes longer to deflate than a small one of the same kind. But what is the quantitative relation between the deflation time τ and the radius R of a balloon? A simple analysis, depending only upon elementary physics, shows that τ∼R7/2—a prediction that is surprisingly easy to illustrate with a party balloon, a tape measure, and a smart-phone app.

Влияние распределенных дислокаций на устойчивость полого нелинейно-упругого шара

The phenomenon of stability loss of a hollow elastic sphere containing distributed dislocations and loaded with external hydrostatic pressure is studied. The study was carried out in the framework of the nonlinear elasticity theory and the continuum theory of continuously distributed dislocations. An exact statement and solution of the stability problem for a three-dimensional elastic body with distributed dislocations are given. The static problem of nonlinear elasticity theory for a body with distributed dislocations is reduced to a system of equations consisting of equilibrium equations, incompatibility equations with a given dislocation density tensor, and constitutive equations of the material. The unperturbed state is caused by external pressure and a spherically symmet-ric distribution of dislocations. For distributed edge dislocations in the framework of a harmonic (semi-linear) mate-rial model, the unperturbed state is defined as an exact spherically symmetric solution to a nonlinear boundary value problem. This solution is valid for any function that characterizes the density of edge dislocations. The perturbed equilibrium state is described by a boundary value problem linearized in the neighborhood of the equilibrium. The analysis of the axisymmetric buckling of the sphere was performed using the bifurcation method. It consists in determining the equilibrium positions of an elastic body, which differ little from the unperturbed state. By solving the linearized problem, the value of the external pressure at which the sphere first loses stability is found. The effect of dislocations on the buckling of thin and thick spherical shells is analyzed.

An energy-landscape-based crossover temperature in glass-forming liquids.

The systematic identification of temperature scales in supercooled liquids that are key to understanding those liquids' underlying glass properties, and their formation-history dependence, is a challenging task. Here, we study the statistics of particles' squared displacements δr2 between equilibrium liquid configurations at temperature T and their underlying inherent states, using computer simulations of 11 different computer glass formers. We show that the relative fluctuations of δr2 are nonmonotonic in T, exhibiting a maximum whose location defines the crossover temperature TX. Therefore, TX marks the point of maximal heterogeneity during the process of tumbling down the energy landscape, starting from an equilibrium liquid state at temperature T down to its underlying inherent state. We extract TX for the 11 employed computer glasses, ranging from tetrahedral glasses to packs of soft elastic spheres, and demonstrate its usefulness in putting the elastic properties of different glasses on the same footing. Interestingly, we further show that TX marks the crossover between two distinct regimes of the mean ⟨δr2⟩: a high temperature regime in which ⟨δr2⟩ scales approximately as T0.5 and a deeply supercooled regime in which ⟨δr2⟩ scales approximately as T1.3. Further research directions are discussed.

On modeling of airflow in human lungs: constitutive relations to describe deformation of porous medium

Within the framework of a multilevel mathematical model to describe the evolution of functional disorders of the human organism under the influence of environment factors, a mathematical model of the "meso-level" of the human respiratory system is developed. The article is deals with the development of the meso-level model - the formulation of a constitutive model to describe the airflow in a porous lung medium. Human lungs filled with small airways and alveoli, with air contained in them, are modeled by an elastically deformable saturated porous medium enclosed in an internal chamber with varying volume (movable walls). Conceptual and mathematical statements are presented. Air movement in the deformable porous medium of lungs is described by ratios of the mechanics of deformable solid body and filtration theory. As an element of this sub-model an analytical solution is obtained for an auxiliary geometrically linear problem of the all-round compression of an elastic thin-walled hollow sphere filled with air to determine the rate of mean stress of the two-phase medium of the lungs, taking into account the interaction between the lung tissue and the air contained in the lungs. To confirm the hypothesis on the acceptability of a linear solution of an auxiliary problem for large deformations, a similar problem was numerically solved in a geometrically nonlinear formulation. The results show that the obtained analytical solution is in satisfactory agreement with the solution of a similar problem in a nonlinear formulation both for calm and deep breathing, which indicates the possibility of using the former in the construction of the considered sub-model.

Atomistic tribological investigation of ultrathin layer of carbon disulfide between diamond surfaces

Using the method of classical molecular dynamics the tribological properties are studied of one molecular layer of liquid carbon disulfide, pressed by two atomically smooth diamond surfaces. We treat the models of elastic spheres for carbon disulfide molecules and absolutely rigid diamond plates. The found time series of atomic level friction force at the contact is heterogeneous. An increase of the external normal pressure leads to the transition of the film to a solidlike state. This is manifested in a decrease of the diffusion constant, in the behavior of velocity autocorrelation function of liquid molecules and average friction force.

Generation of spherical vortex beams to trap large particles for enhanced axial force

Recent studies have shown the possibility to manipulate small elastic spheres in 3D with a single-sided beam. Acoustical tweezers are very appealing because they provide a fine spatial control of the motion of a single particle in space. Their main limitations are due to the weak restoring axial force and improving this force is still a challenge. We show theoretically that the spherical vortex beams can trap large particles and enhance the axial force. Indeed, the special features of these unusual beams look like the bottle beams in optics. Nevertheless, their spatial complexity presupposes that they can be produced with sufficient precision. Therefore, attention is paid to the synthesis of the spherical vortices. A method based on the inverse filter method is proposed. It allows to synthesize them with a very good precision since the theoretical force is recovered experimentally with an error smaller than 10%. Then, the spherical vortices are used to trap polyethylene beads with radii between 500 and 590µm. Experiments show that the radial trap is working while no beads have been trapped in the axial direction. This failure is analyzed in detail and is shown to be mainly due to sensitivity to the properties of the materials which influences the resonance modes of the elastic sphere. This study paves the way to the optimization of acoustical tweezers for the manipulation of large objects.

Numerical study on the deformation behaviors of elastic spheres during water entry

When elastic spheres enter the water from air, the fluid–structure interaction becomes a complex problem characterized by large deformations. This paper focuses on the deformation behaviors of elastic spheres during water entry. Based on a strong coupling method, a numerical approach to the water entry of elastic spheres is established and validated using the experimental data. In addition to the descriptions of the free surface flows under impacting elastic spheres, the physical mechanism underlying the continuous deformation behaviors of elastic spheres during water entry is also identified. The results show that the magnitude of the prolate deformation is smaller than that of the oblate deformation under the action of hydrodynamic pressure. Additionally, the deformation period of the elastic spheres is related to the material shear modulus, but not to the impact velocity. Previous experimental results describing a linear relationship between the deformation period and the combination parameter for time, D∕(μ0∕ρS)1∕2, are confirmed. The magnitude of sphere deformation is affected by the material shear modulus and impact velocity. The results also show the change characteristics for the kinetic and strain energies of elastic spheres with respect to the water entry time.

Acoustic radiation force on a compressible spheroid

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions. There is no restriction on the size or acoustic impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The solution is the same as that developed previously for the radiation force on an elastic sphere, which is expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii etal., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Spheroidal wave expansions are used to satisfy the boundary conditions on the compressible spheroid, and far-field asymptotes are used to relate the spheroidal and spherical wave scattering coefficients analytically. Explicit expressions for the scattering coefficients are available for spheroids with rigid or free surfaces. Results are compared with available analytical solutions for various limiting cases. The theoretical framework may be employed for objects of arbitrary shape provided the spherical wave expansion of the scattered field is known. [T.S.J. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.] a)Deceased.

Acoustic radiation force on a compressible spheroid.

The acoustic radiation force on a compressible spheroid is calculated using expansions of the scattered field in terms of both spherical and spheroidal wave functions that are matched analytically in the far field. There is no restriction on the size or impedance of the spheroid, the structure of the incident field, or the orientation of the spheroid with respect to the incident field. The form of the solution is the same as that developed previously for the radiation force on an elastic sphere, which is a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields. Spheroidal wave expansions are employed to satisfy the boundary conditions and obtain the scattering coefficients. While the scattering coefficients must be obtained numerically for compressible spheroids, explicit expressions in terms of radial wave functions are available for spheroids with rigid or free surfaces. Results are compared with available analytical expressions for various limiting cases. The theoretical framework may be extended to objects of arbitrary shape.

DEM simulation of anisotropic granular materials: elastic and inelastic behavior

In this work, Discrete Elements Method simulations are carried out to investigate the effective stiffness of an assembly of frictional, elastic spheres under anisotropic loading. Strain probes, following both forward and backward paths, are performed at several anisotropic levels and the corresponding stress is measured. For very small strain perturbations, we retrieve the linear elastic regime where the same response is measured when incremental loading and unloading are applied. Differently, for a greater magnitude of the incremental strain a different stress is measured, depending on the direction of the perturbation. In the case of unloading probes, the behavior stays elastic until non-linearity is reached.Under forward perturbations, the aggregate shows an intermediate inelastic stiffness, in which the main contribution comes from the normal contact forces. That is, when forward incremental probes are applied the behavior of anisotropic aggregates is an incremental frictionless behavior. In this regime we show that contacts roll or slide so the incremental tangential contact forces are zero.Graphical

Modeling thermal-visco-elastohydrodynamic lubrication (TVEHL) interfaces of polymer-based materials

This paper reports a novel thermal-visco-elasohydrodynamic lubrication (TVEHL) model for analyzing the lubrication behavior of the interface formed by an elastic sphere and a polymer half-space. The temperature-dependent viscoelastic displacement of the polymer surface is calculated through the elastic-viscoelastic correspondence theory and frequency-temperature superposition. The discrete convolution and fast Fourier transform (DC-FFT) algorithm is used for efficient solution computation. The model is verified by comparing results from its degenerated forms with those from thermal-viscoelastic (TVE) contact and thermal-elastohydrodynamic lubrication (TEHL) theories. The results from the current model with and without considering temperature effect are also compared. The new TVEHL model is explored to study the viscoelastic material property, entraining speed, sliding-to-rolling ratio, and the coupled thermal-viscoelasticity effects.

An analytical prediction model for residual stress distribution and plastic deformation depth in ultrasonic-assisted single ball burnishing process

Mechanical and metallurgical characteristics of the surface layers are modified as the material is subjected to the burnishing process. Plastic deformation is known as a major reason for property enhancement of the surface and subsurface layers. Residual stress distribution and influenced depth of plastic deformation provide useful information regarding the functionality and life cycles of the burnished sample. In the present study, a novel analytical approach was presented to predict residual stress distribution and the depth of plastic defamation in the ultrasonic-assisted ball burnishing process. The burnishing process was firstly analyzed using the contact mechanic of an elastic sphere with semi-infinite body theorem. Then, the plastic deformation and residual stress were modeled using the McDowell algorithm. The model could incorporate effects of vibration amplitude and frequency, static pressure, feed rate, and ball dimensions. A series of ultrasonic-assisted ball burnishing experiments have been carried out on aluminum 6061-T6 and AISI 1045 steel to confirm the proposed model prediction results. The prediction accuracy of the proposed model was further verified by residual stress distributions of AISI 304, Ti-6Al-4V, and Inconel 718 from other literatures. The research findings in this study indicated that the developed model could be used for a variety of engineering materials in the prediction of residual stress with adequate precision.

Collapse modes in simple cubic and body-centered cubic arrangements of elastic beads.

Collapse modes in compressed simple cubic (SC) and body-centered cubic (BCC) periodic arrangements of elastic frictionless beads were studied numerically using the discrete element method. Under pure hydrostatic compression, the SC arrangement tends to transform into a defective hexagonal close-packed or amorphous structure. The BCC assembly exhibits several modes of collapse, one of which, identified as cI16 structure, is consistent with the behavior of BCC metals Li and Na under high pressure. The presence of a deviatoric stress leads to the transformation of the BCC structure into face-centered cubic (FCC) one via the Bain path. The observed effects expand the knowledge on possible packings of soft elastic spheres and transformations between them, while providing an unexpected link with the mechanical behavior of certain atomic systems.

Elastohydrodynamic Dewetting of Thin Liquid Films: Elucidating Underwater Adhesion of Topographically Patterned Surfaces.

In underwater adhesion of a topographically patterned surface with a very soft material such as human skin, the elastic deformation can be large enough to achieve solid-on-solid contact not only on top of the hills but also in the valleys of the substrate topography. In this context, we have studied the dynamics of dewetting of a thin liquid film confined between a rigid, periodic micropillar array and a soft, elastic sphere. In our experiments, we observed two very distinct dewetting morphologies. For large ratios of array period to micropillar height and width, the dewetted areas tend to have a diamond-like shape and expand with a rate similar to a flat, unpatterned substrate. When the array period is reduced, the morphology of the dry spot becomes irregular and its expansion rate is significantly reduced. We developed a fully coupled numerical model of the dewetting process that reproduces the key features observed in experiments. Moreover, we performed contact mechanics simulations to characterize the deformation of the elastomer and the shape of the dewetted area in a unit cell of the micropillar array.

A massively parallel and scalable multi-GPU material point method

Harnessing the power of modern multi-GPU architectures, we present a massively parallel simulation system based on the Material Point Method (MPM) for simulating physical behaviors of materials undergoing complex topological changes, self-collision, and large deformations. Our system makes three critical contributions. First, we introduce a new particle data structure that promotes coalesced memory access patterns on the GPU and eliminates the need for complex atomic operations on the memory hierarchy when writing particle data to the grid. Second, we propose a kernel fusion approach using a new Grid-to-Particles-to-Grid ( G2P2G ) scheme, which efficiently reduces GPU kernel launches, improves latency, and significantly reduces the amount of global memory needed to store particle data. Finally, we introduce optimized algorithmic designs that allow for efficient sparse grids in a shared memory context, enabling us to best utilize modern multi-GPU computational platforms for hybrid Lagrangian-Eulerian computational patterns. We demonstrate the effectiveness of our method with extensive benchmarks, evaluations, and dynamic simulations with elastoplasticity, granular media, and fluid dynamics. In comparisons against an open-source and heavily optimized CPU-based MPM codebase [Fang et al. 2019] on an elastic sphere colliding scene with particle counts ranging from 5 to 40 million, our GPU MPM achieves over 100x per-time-step speedup on a workstation with an Intel 8086K CPU and a single Quadro P6000 GPU, exposing exciting possibilities for future MPM simulations in computer graphics and computational science. Moreover, compared to the state-of-the-art GPU MPM method [Hu et al. 2019a], we not only achieve 2x acceleration on a single GPU but our kernel fusion strategy and Array-of-Structs-of-Array ( AoSoA ) data structure design also generalizes to multi-GPU systems. Our multi-GPU MPM exhibits near-perfect weak and strong scaling with 4 GPUs, enabling performant and large-scale simulations on a 1024 3 grid with close to 100 million particles with less than 4 minutes per frame on a single 4-GPU workstation and 134 million particles with less than 1 minute per frame on an 8-GPU workstation.

Modeling of many-body interactions between elastic spheres through symmetry functions.

Simple models for spherical particles with a soft shell have been shown to self-assemble into numerous crystal phases and even quasicrystals. However, most of these models rely on a simple pairwise interaction, which is usually a valid approximation only in the limit of small deformations, i.e., low densities. In this work, we consider a many-body yet simple model for the evaluation of the elastic energy associated with the deformation of a spherical shell. The resulting energy evaluation, however, is relatively expensive for direct use in simulations. We significantly reduce the associated numerical cost by fitting the potential using a set of symmetry functions. We propose a method for selecting a suitable set of symmetry functions that capture the most relevant features of the particle's environment in a systematic manner. The fitted interaction potential is then used in Monte Carlo simulations to draw the phase diagram of the system in two dimensions. The system is found to form both a fluid and a hexagonal crystal phase.

Performances of the Lamb Model to Describe the Vibrations of Gold Quantum-Sized Clusters

Lamb modes describe the vibrations of an object as a whole from the stellar scale to the nanometer one. Lamb description has been built from the linear elasticity theory and considers a homogeneous elastic sphere. Our work tries to determine the minimum scale where this description remains valid by studying the vibration of quantum-sized gold clusters (Au6, Au9, and Au25) stabilized by organic molecules. First, our work shows that experimental frequencies of small-functionalized gold clusters obtained by low-frequency Raman spectroscopy can be interpreted with density functional theory calculations. Moreover, the Lamb model broadly succeeds in predicting these Raman acoustic modes only if a correction considering the mass of the surrounding ligands is added. Ligands affect vibrational modes of the core by their mass but also by their covalent bond with the core. The unexpected consequence of this electronic stabilization by the ligands is the sustainability of the Lamb description for clusters as small as six atoms. Finally, the limit of the Lamb model can be reached out at low temperature where the vibration mode spectrum presents a substructuration that the Lamb description, developed for a homogeneous sphere, is unable to predict.