Elastic Sphere Research Articles

Depth-sensing indentation of spherical particles on corrugated substrates — An asymptotic model

An AFM-based nanoindentation of a spherical particle absorbed on a corrugated surface of rigid substrate is considered in the framework of continuum mechanics. The corresponding problem of unilateral frictionless contact with multiple contact zones is formulated using isotropic infinitesimal elasticity. Both the AFM probe surface and the surfaces of rigid supports, which are exposed to contact with the elastic sphere, are assumed to be spherical. The method of matched asymptotic expansions is utilized for constructing the leading order asymptotic model of multiple contact. Explicit formulas are derived for the force-displacement relationship and the incremental indentation stiffness. It is shown that the difference between the one-point contact and the multi-point contact (e.g., with two and three local supports) is found to be significant. The case of tight hexagonal packing of the support spheres is considered in detail and illustrated by a comparison with the Hertzian theory of local contact.

Research of Contact Response of an Elastic Sphere Impacting a Rigid Plate Based on the Mass-Spring-Damped Oscillator Model and Dimensionless Parameter \omega_\Omega

The contact theory, such as the Hertz theory and Hunt-Crossley model can just solve the contact response when the sphere goes to the maximum compression but can't solve the contact process. In traditional research the impact process is considered as a single degree of freedom dynamical spring-damping system, however, the calculated contact force is much greater than the actual one. Based on the shortage of the conducted studies, a new dimensionless parameter Omega is proposed in this paper, the mass-spring-damped oscillator model (MSDOM) and the new contact theory with the consideration of the sphere mass and Omega are established. The correctness of the theoretical model is verified by 3D simulation, and the optimal contact force model is obtained. Furthermore, influence rules of material properties and contact parameters on contact response were studied. The results show that the contact deformation, velocity, piecewise nonlinear contact force and acceleration based on MSDOM and Omega are effective to calculate the contact response. The contact stiffness, damping, material properties of sphere and elastic modulus of the plate have different influence on the contact response while the Poisson's ratio nu_2 has little effect on it.

Stress-induced ordering of two-dimensional packings of elastic spheres.

Packing of particles in confined environments is a common problem in multiple fields. Here, based on the two-dimensional Hertzian particle model, we study the packing of deformable spherical particles under compression and reveal the crucial role of stress as an ordering field in regulating particle arrangement. Specifically, under increasing compression, the squeezed particles spontaneously organize into regular packings in the sequence of triangular and square lattices, pentagonal tessellation, and the reentrant triangular lattice. The rich ordered patterns and complex structures revealed in this work suggest a fruitful organizational strategy based on the interplay of external stress and intrinsic elastic instability of particle arrays.

Thermoelastic contact of layered materials with interfacial imperfection

This paper reports an investigation of the thermoelastic frictional contact responses of an elastic sphere sliding on an elastic layer-substrate system involving imperfect interfaces. The theoretical solution and numerical modeling of the thermoelastic contact of such material systems include: (1) the mathematical derivation of the closed-form frequency response functions (FRFs) of the thermoelastic field in terms of temperature, thermoelastic displacements and stresses in the layer and the substrate, (2) the numerical contact pressure determination and heat partition evaluation by employing the conjugate gradient method (CGM), and (3) stress and temperature calculations by utilizing the discrete convolution-fast Fourier transform algorithm (DC-FFT). Numerical results from the proposed model indicate that, a stronger discontinuity of interface displacements causes the contact pressure, temperature, and in-plane compressive stress to decrease, but the in-plane tensile stress, out-of-plane (compressive and tensile) stress, and von Mises stress to increase. The results also reveal the behavior transition of the out-of-plane shear stress state due to the change of the interface discontinuity status.

Unloading behavior of low velocity impact between elastic and elastic–plastic bodies

In this paper, the unloading behavior is investigated by finite element (FE) analysis for low-velocity frictionless impact between elastic sphere and elastic-perfectly plastic half-space. The material property and impact velocity are varied to cover the elastic-plastic and fully plastic unloading regimes by 301 FE simulations. Based on the FE results, the yield strain is a crucial material property to characterize the unloading law, and spherical residual cavity and Hertzian-type unloading assumptions are validated. By introducing a scalar factor to revise Stronge's geometric similarity, a new unloading model is presented to fast and accurate precut the unloading behavior. The unloading model is validated numerically, experimentally and compared with other unloading models. Its applications combining with loading models are also validated experimentally.

Towards a soft magnetoelastic twist actuator

The authors propose the realization of a torsional actuator based on a soft magnetoelastic material. The paper investigates the response of different spatial arrangements of magnetizable particles enclosed by a soft elastic sphere and identifies those structures showing a maximized twist deformation under saturating external magnetic fields.

On the Analysis of Free Vibrations of Nonlocal Elastic Sphere of FGM Type in Generalized Thermoelasticity

Free vibrations of non-homogenous (functionally graded material (FGM)) axisymmetric nonlocal thermoelastic hollow sphere has been taken into consideration for investigation. The material of nonlocal thermoelastic sphere is supposed to be graded using power law in radial direction. The solution of continued power series is employed to resolve the differential equations and to investigate the analytical solutions for field functions i.e. temperature, stress and displacement. The analytical results have been authenticated using numerical computations with computer based software like MATLAB. The numerical results have been shown graphically for the comparison of frequency shift and thermoelastic damping for local and nonlocal elastic materials. Deduction of results has been validated with the already published literature.

The tensile strength of dust aggregates consisting of small elastic grains: constraints on the size of condensates in protoplanetary discs

ABSTRACT A consensus view on the formation of planetesimals is now exposed to a threat, since recent numerical studies on the mechanical properties of dust aggregates tend to dispute the conceptual picture that submicrometer-sized grains conglomerate into planetesimals in protoplanetary discs. With the advent of precise laboratory experiments and extensive computer simulations on the interaction between elastic spheres comprising dust aggregates, we revisit a model for the tensile strength of dust aggregates consisting of small elastic grains. In the framework of contact mechanics and fracture mechanics, we examine outcomes of computer simulations and laboratory experiments on the tensile strength of dust aggregates. We provide a novel analytical formula that explicitly incorporates the volume effect on the tensile strength, namely, the dependence of tensile strength on the volume of dust aggregates. We find that our model for the tensile strength of dust aggregates well reproduces results of computer simulations and laboratory experiments, if appropriate values are adopted for the elastic parameters used in the model. Moreover, the model with dust aggregates of submicrometer-sized grains is in good harmony with the tensile strength of cometary dust and meteoroids derived from astronomical observations. Therefore, we reaffirm the commonly believed idea that the formation of planetesimals begins with conglomeration of submicrometer-sized grains condensed in protoplanetary discs.

Scattering coefficients for a sphere in a visco-acoustic medium for arbitrary partial wave order

Analytical solutions are reported for the scattering coefficients of a solid elastic sphere suspended in a viscous fluid for arbitrary partial wave order. Expressions are derived for incident compressional and shear wave modes, taking into account the viscosity of the surrounding fluid and resultant wave mode conversion. The long compressional wavelength limit is employed to simplify the derivation, whereas no restriction is placed on the shear wavelength in the fluid compared to the particle dimension. The analytical approximations are compared with numerical results obtained from matrix inversion of the boundary equations and agree within the validity domain of the solutions.

A note on the spheroidal modes vibration of an elastic sphere in linear viscoelastic fluid

Vibration characteristics of elastic nanostructures embedded in fluid medium have been used for biological and mechanical sensing, and also to investigate materials mechanical properties. An analytical approach based on the exact theory has been developed in this paper, to establish a new accurate and simple generalized frequency equation to predict spheroidal vibration of an elastic nanosphere, in a compressible viscoelastic fluid using linear Maxwell fluid model. To demonstrate the accuracy of the present approach, a comparison is made with the published theoretical results in the literature in some particular cases, which shows a very good agreement. Thus, the obtained frequency equation can be very useful to interpret the experimental measurements of vibrational dynamics of nanospheres and can serve as benchmark solution in design of liquid sensors.

Lumped parameter models for adhesive contact mechanics

Adhesion is very common when the nanoscale or microscale solids come into contact, thus playing an important role in a variety of situations of interest. Due to the special nature of these problems, computational methods are often superior to analytical and experimental techniques for problems in adhesive contact mechanics. Finite element method is probably the most commonly used tool for such computational studies, due to its versatility and robustness. A major difficulty with these studies is the extremely high computational resource requirement. In this work, we propose two lumped parameter algorithms, which can significantly alleviate this issue. These algorithms provide results which approach the correct solution from either side, thereby providing upper and lower bounds to the correct solution, and thus, an estimate of the error involved in the analysis. Further, these algorithms can easily be incorporated into commercial finite element software and hence can be used more widely. Applicability of these algorithms is illustrated by studying the adhesive interactions between an elastic sphere and a rigid half-space, including the jump-to-contact instability caused by adhesion at small separations.

Forced vibrations and wave propagation in multilayered solid spheres using a one-dimensional semi-analytical finite element method

A numerical model is proposed to compute the eigenmodes and the forced response of multilayered elastic spheres. The main idea is to describe analytically the problem along the angular coordinates with spherical harmonics and to discretize the radial direction with one-dimensional finite elements. The proper test function must be carefully chosen so that both vector and tensor spherical harmonics orthogonality relationships can be used. The proposed approach yields a general one-dimensional formulation with a fully analytical description of the angular behaviour, suitable for any interpolating technique. A linear eigenvalue problem, simple and fast to solve, is then obtained. The eigensolutions are the spheroidal and torsional modes. They are favourably compared with literature results for a homogeneous sphere. The eigensolutions are superposed to compute explicitly the forced response. The latter is used to reconstruct the propagation of surfaces waves. In particular, the collimation of a Rayleigh wave (non-diffracted surface wave propagating with a quasi-constant width) excited by a line source in a homogeneous sphere is recovered with the model. Based on the vibration eigenmodes, a modal analysis shows that such a wave is a superposition of fundamental spheroidal modes with a displacement confined at the equator of the sphere. These modes are the so-called Rayleigh modes, of sectoral type and high polar wavenumbers. When a thin viscoelastic coating is added to the sphere, the Rayleigh mode behaviour is recovered in a limited frequency range, allowing the generation of a collimating wave at the interface between the sphere and the coating.

Pulling force of acoustic-vortex beams on centered elastic spheres based on the annular transducer model**Project supported by the National Natural Science Foundation of China (Grant Nos. 11934009, 11974187, and 11604156).

To solve the difficulty of generating an ideal Bessel beam, an simplified annular transducer model is proposed to study the axial acoustic radiation force (ARF) and the corresponding negative ARF (pulling force) exerted on centered elastic spheres for acoustic-vortex (AV) beams of arbitrary orders. Based on the theory of acoustic scattering, the axial distributions of the velocity potential and the ARF for AV beams of different orders generated by the annular transducers with different physical sizes are simulated. It is proved that the pulling force can be generated by AV beams of arbitrary orders with multiple axial regions. The pulling force is more likely to exert on the sphere with a smaller k0a (product of the wave number and the radius) for the AV beam with a bigger topological charge due to the strengthened off-axis acoustic scattering. The pulling force decreases with the increase of the axial distance for the sphere with a bigger k0a. More pulling force areas with wider axial regions can be formed by AV beams using a bigger-sized annular transducer. The theoretical results demonstrate the feasibility of generating the pulling force along the axes of AV beams using the experimentally applicable circular array of planar transducers, and suggest application potentials for multi-position stable object manipulations in biomedical engineering.

Equilibrium stability of nonlinear elastic sphere with distributed dislocations

An exact formulation and a solution of the stability problem for a three-dimensional elastic body containing distributed dislocations are given. The buckling phenomenon for a hollow nonlinear elastic sphere of the semi-linear (harmonic) material with edge dislocations is studied. The study is carried out within the framework of the continuum theory of continuously distributed dislocations using the bifurcation method of buckling analysis. The bifurcation method is to find the equilibrium positions of an elastic body, which differ little from the subcritical (unperturbed) state. The perturbed equilibrium state is described by a linearized boundary value problem. By solving a homogeneous linear boundary value problem, the minimum critical value of the external pressure at which the sphere loses stability is found. The influence of dislocations on the buckling of both thin and thick spherical shells is analyzed.

Likely cavitation and radial motion of stochastic elastic spheres

The cavitation of solid elastic spheres is a classical problem of continuum mechanics. Here, we study this problem within the context of ‘stochastic elasticity’ where the constitutive parameters are characterised by probability density functions. We consider homogeneous spheres of stochastic neo-Hookean material, composites with two concentric stochastic neo-Hookean phases, and inhomogeneous spheres of locally neo-Hookean material with a radially varying parameter. In all cases, we show that the material at the centre determines the critical load at which a spherical cavity forms there. However, while under dead-load traction, a supercritical bifurcation, with stable cavitation, is obtained in a static sphere of stochastic neo-Hookean material, for the composite and radially inhomogeneous spheres, a subcritical bifurcation, with snap cavitation, is also possible. For the dynamic spheres, oscillatory motions are produced under suitable dead-load traction, such that a cavity forms and expands to a maximum radius, then collapses again to zero periodically, but not under impulse traction. Under a surface impulse, a subcritical bifurcation is found in a static sphere of stochastic neo-Hookean material and also in an inhomogeneous sphere, whereas in composite spheres, supercritical bifurcations can occur as well. Given the non-deterministic material parameters, the results can be characterised in terms of probability distributions.

Adhesion of an elastic sphere on a tensioned membrane

Inspirited by the fact that classical models of adhesion contact (such as the Johnson–Kendall–Roberts model, Young’s equation or the Neumann equation) cannot be directly applied to adhesion of an elastic sphere on a membrane, the present work aims to develop an explicit general model for axisymmetric adhesion mechanics of an elastic sphere on a tensioned circular membrane. An explicit expression for the potential energy of the sphere–membrane system is derived, and explicit equations are given to determine the adhesion equilibrium state. The validity and accuracy of the proposed model are verified by good agreement between the predicted results and known results on both adhesion of a rigid sphere on a membrane and the critical condition for full wrapping of a rigid sphere by a membrane of non-zero bending rigidity.

Understanding transport of an elastic, spherical particle through a confining channel

The transport of soft particles through narrow channels or pores is ubiquitous in biological systems and industrial processes. On many occasions, the particles deform and temporarily block the channel, inducing a built-up pressure. This pressure buildup often has a profound effect on the behavior of the respective system; yet, it is difficult to be characterized. In this work, we establish a quantitative correlation between the built-up pressure and the material and geometry properties through experiments and mechanics analysis. We fabricate microgels with a controlled diameter and elastic modulus by microfluidics. We then force them to individually pass through a constrictive or straight confining channel and monitor the pressure variation across the channel. To interpret the pressure measurement, we develop an analytical model based on the Neo-Hookean material law to quantify the dependence of the maximum built-up pressure on the radius ratio of the elastic sphere to the channel, the elastic modulus of the sphere, and two constant parameters in the friction constitutive law between the sphere and the channel wall. This model not only agrees very well with the experimental measurement conducted at large microgel deformation but also recovers the classical theory of contact at small deformation. Featuring a balance between accuracy and simplicity, our result could shed light on understanding various biological and engineering processes involving the passage of elastic particles through narrow channels or pores.

Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres

The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic equation, recently derived for this system. As a consequence of the confinement, the pressure tensor and the heat flux contain, in addition to the terms associated to the motion of the particles, collisional transfer contributions, similar to those that appear beyond the dilute limit. The complexity of these terms, and of the kinetic equation itself, compromise the potential of the equation to describe the rich phenomenology observed in this kind of systems. For this reason, a simpler model equation based on the Boltzmann equation is proposed. The model is formulated to keep the main properties of the underlying equation, and it is expected to provide relevant information in more general states than the original equation. As an illustration, the solution describing a macroscopic state with uniform temperature, but a density gradient perpendicular to the plates is considered. This is the equilibrium state for an elastic system, and the inhomogeneous cooling state for the case of inelastic hard spheres. The results are in good agreement with previous results obtained directly from the Boltzmann equation.

The influence of friction on behavior of a hyperelastic body in a conic channel

The model problem about the axisymmetric quasistatic motion of an elastic sphere considering the interaction with inner surface of a rigid cone die is investigated. The constitutive model is represented by the physically nonlinear generalization of the linear model of Hencky’s elastic material. The tensor of logarithmic strains is used by way of strain measure due to described below special properties. The influence of friction on macro characteristics of a process is studied using proposed numerical model of accounting of mixed boundary conditions on time-changing contact surface.

A Finite Element Analysis of The Indentation of an Elastic-Work Hardening Double Layered Half-Space by a Rigid Sphere

In this work, the stresses caused by an elastic sphere indenting double layered coating on an elastic-work hardening substrate with a relatively low elastic modulus regime have been examined. An axisymmetric finite element mesh was set up for the stress analysis. The contact stress arising on the free surface, the stresses at both interface sides of the first and second coating layers and at the symmetry axis, and the stresses within the coating layers have been examined depending on the first coating layer (interlayer) thickness using ANSYS. It was obtained that the thickness of the first coating layer affected considerably the stresses formed at the contact surface, the interfaces, the symmetry axis and within the coating layers.