Hertzian Contact Law Research Articles

Justification of leading order quasicontinuum approximations of strongly nonlinear lattices

We consider the leading order quasicontinuum limits of a one-dimensionalgranular medium governed by the Hertz contact law under precompression.The approximate model which is derived in this limit is justifiedby establishing asymptotic bounds for the error with the help of energy estimates.The continuum model predicts the development of shock waves, which are also studiedin the full system with the aid of numerical simulations.We also show that existing resultsconcerning the Nonlinear Schrödinger (NLS) and Korteweg de-Vries (KdV) approximation of FPU models apply directly to a precompressed granular medium in the weakly nonlinear regime.

Study on mechanical behavior of the transverse processing on a granular matter layer

When the granular matter is being pushed a certain point can be found, before arriving at which the force increases linearly, and after that it increases exponentially. The granular matter is a large assemblage of solid grains, which are fundamentally different from any other type of matters, such as solid blue and liquids; they have different physical properties. The meso-scale of force chain bridges single particles and granular matter, and leads to unique properties and behaviours of granular matters, thus acts as the key issue in the study of granular matter. In this work, we first find the variation of different forces by simulation. We then propose a discrete element model based on rigorous Hertzian contact law and Mindlin-Deresiewicz contact theory for normal and tangential contact forces, respectively, and then point out that the change of force chain is the key to the granular dynamics. The force distribution and the force chain length distribution are calculated and analyzed to show the change rule and the origin of the drag force.

Effect of Twist and Striker Velocity of Torsion Stiff Composite Conical Shells under Multiple Delaminations

The low velocity impact response of torsion stiff composite pretwisted conical shells is portrayed in the present study. The formulation is based on Mindlin's theory incorporating rotary inertia and effects of transverse shear deformation. An eight noded isoparametric plate bending element is employed to satisfy the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. A multipoint constraint algorithm is incorporated which leads to unsymmetric stiffness matrices. The modified Hertzian contact law is employed to simulate the contact force and the time dependent equations are solved by Newmark's time integration algorithm. Parametric studies are conducted with respect to triggering parameters like twist angle, target displacement, initial velocity of striker considering low velocity centrally impacted spherical mass.

Contact Behaviour of Composite Laminate under Quasi-Static Indentation Load

A viable solution to the ever-demanding weight-saving target in aerospace industry is the replacement of conventional engineering alloys with composite materials in primary structures. A major concern to the effective use of composite laminates is the substantial reduction in the compressive strength when the contact force has exceeded the delamination threshold load (DTL). This paper focuses on the study of the contact behavior of composite laminates under quasi-static indentation (QSI) forces. The effect of damage initiation and growth on contact behavior has been investigated via detailed assessment of the relation between the indentation force and the dent depth. Different phases corresponding to undamaged, local damage, global damage, and final failure of the laminate have been identified. A modification to the classical Hertz contact law has been proposed to account for the matrix material resistance to plastic deformation.

Contact-coupled impact of slender rods: analysis and experimental validation.

To validate models of contact mechanics in low speed structural impact, slender rods were impacted in a drop tower, and measurements of the contact and vibration were compared to analytical and finite element (FE) models. The contact area was recorded using a novel thin-film transfer technique, and the contact duration was measured using electrical continuity. Strain gages recorded the vibratory strain in one rod, and a laser Doppler vibrometer measured speed. The experiment was modeled analytically on a one-dimensional spatial domain using a quasi-static Hertzian contact law and a system of delay differential equations. The three-dimensional FE model used hexahedral elements, a penalty contact algorithm, and explicit time integration. A small submodel taken from the initial global FE model economically refined the analysis in the small contact region. Measured contact areas were within 6% of both models' predictions, peak speeds within 2%, cyclic strains within 12 με (RMS value), and contact durations within 2 μs. The global FE model and the measurements revealed small disturbances, not predicted by the analytical model, believed to be caused by interactions of the non-planar stress wavefront with the rod's ends. The accuracy of the predictions for this simple test, as well as the versatility of the diagnostic tools, validates the theoretical and computational models, corroborates instrument calibration, and establishes confidence that the same methods may be used in experimental and computational study of contact mechanics during impact of more complicated structures. Recommendations are made for applying the methods to a particular biomechanical problem: the edge-loading of a loose prosthetic hip joint which can lead to premature wear and prosthesis failure.

The effect of base roughness on the development of a dense granular flow down an inclined plane

The development of the flow of a granular material down an inclined plane starting from rest is studied as a function of the base roughness. In the simulations, the particles are rough frictional spheres interacting via the Hertz contact law. The rough base is made of a random configuration of fixed spheres with diameter different from the flowing particles, and the base roughness is decreased by decreasing the diameter of the base particles. The transition from an ordered to a disordered flowing state at a critical value of the base particle diameter, first reported by Kumaran and Maheshwari [Phys. Fluids 24, 053302 (2012)]10.1063/1.4710543 for particles with the linear contact model, is observed for the Hertzian contact model as well. The flow development for the ordered and disordered flows is very different. During the development of the disordered flow for the rougher base, there is shearing throughout the height. During the development of the ordered flow for the smoother base, there is a shear layer at the bottom and a plug region with no internal shearing above. In the shear layer, the particles are layered and hexagonally ordered in the plane parallel to the base, and the velocity profile is well approximated by Bagnold law. The flow develops in two phases. In the first phase, the thickness of the shear layer and the maximum velocity increase linearly in time till the shear front reaches the top. In the second phase, after the shear layer encompasses the entire flow, there is a much slower increase in the maximum velocity until the steady state is reached.

Low velocity impact response of thick FGM beams with general boundary conditions in thermal field

Low velocity impact response of an FGM beam with through-the-thickness distributed properties is studied in this paper. Third order shear deformation beam theory is implemented to derive the governing equations. A modification of the Hertz contact law which is suitable for graded materials is adopted. A Ritz-based solution in space domain appropriate for various in-plane and out-of-plane boundary conditions is developed. Solution of the resulted system of equations in time domain is accomplished via the fourth-order Runge–Kutta method. The developed method is validated with the results of commercial finite elements software. The effects of various involved parameters, such as graded property profile, impactor velocity, impactor mass, boundary conditions, thermal environment and impact position are studied in detail.

Contact forces of polyhedral particles in discrete element method

A general contact force law for arbitrarily shaped bodies is presented. At first an advanced contact force law is derived from the well know Hertz contact law. The obtained formulation of the Hertz contact law can be applied to the contact of arbitrarily shaped bodies. In a second step this contact model is applied to the contacts among polyhedral particles. The results are compared to finite element simulations. The model is extended by terms for damping and friction. The behaviour of the damping and friction model are demonstrated with simple examples. The force law is then implemented in the discrete element method (DEM). The application of this DEM is demonstrated by a simulation of the particle movement in a mixer.

Time dependent response of low velocity impact induced composite conical shells under multiple delamination

This paper presents the time dependent response of multiple delaminated angle-ply composite pretwisted conical shells subjected to low velocity normal impact. The finite element formulation is based on Mindlin’s theory incorporating rotary inertia and effects of transverse shear deformation. An eight-noded isoparametric plate bending element is employed to satisfy the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. A multipoint constraint algorithm is incorporated which leads to asymmetric stiffness matrices. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the contact force, and the time dependent equations are solved by Newmark’s time integration algorithm. Parametric studies are conducted with respect to triggering parameters like laminate configuration, location of delamination, angle of twist, velocity of impactor, and impactor’s displacement for centrally impacted shells.

A new finite element model for low-velocity impact analysis of sandwich beams subjected to multiple projectiles

In this paper, a new finite element formulation is presented for high-order impact analysis of sandwich beams with any boundary conditions subjected to simultaneous multiple small projectiles. The contact of the projectiles may occur on arbitrary places on both faces of the beam. Hamilton’s principle is used for driving the governing partial differential equations of motion. The effect of each projectile is modeled by a one degree of freedom spring-mass system with variable stiffness by assuming the Hertzian contact law. The appropriate elemental matrices are derived for a sandwich beam element that carrying four sprung masses on both face sheets. Based on the presented formulation, an FE code is developed using the Newmark method with direct iteration process. The various cases are used to validate the predictions of the presented higher order theory and FE formulation shows good agreement with available results in the open literature and exact solution. Furthermore, numerical examples are presented to investigate effects of various parameters on the response of the structure.

Nonlinear resonances in a general class of granular dimers with no pre-compression

In this work we study the existence of nonlinear resonances in a general 1: $$N$$ one-dimensional granular dimer chains, i.e., granular chains consisting of periodic sets of ‘heavy’ beads followed and preceded by $$N$$ ‘light’ beads. Each bead is assumed to be spherical and purely elastic and to interact with its neighbors through Hertzian contact law. In a previous work (Jayaprakash et al. in 112(3):1–17, 2012) we proved the existence of countably infinite families of solitary waves in these systems; these are localized pulses that propagate without distortion of their waveforms through these highly inhomogeneous nonlinear media. We attributed these waves to nonlinear anti-resonance that led to complete elimination of radiating waves in the trail of the propagating localized pulse. Anti-resonances were associated with certain symmetries of the velocity waveforms of the beads of the dimer. In this work we report on the opposite phenomenon, that is, of the breakup of waveform symmetries of the bead responses leading to drastic attenuation of propagating pulses due to energy radiation to the far field by means of nonlinear traveling waves. We use the connotation of resonance to describe this dynamical phenomenon resulting in the maximum amplification of the amplitudes of radiated waves that emanate from the propagating pulse. We study numerically and analytically the nonlinear resonance mechanism in this class of strongly nonlinear periodic media, and demonstrate that it can lead to drastic attenuation of shock-induced pulses propagating in the dimers.

Study of Dynamic Behavior of Multilayered Clamped Composite Skewed Hypar Shell Roofs under Impact Load

With advancement in the field of structural engineering, hunt for smarter materials has channelised the research towards the application of composite material. It is the high specific weight and specific stiffness of this material that have drawn the interest of different industrial sectors. Civil engineers also picked up composites to use it as a roofing material. Laminated composite shells, which can cover large column-free area and reduces dead weight of structure, show vulnerability under sudden impact due to their low transverse shear resistances. This study utilises finite element tool to investigate the dynamic response of a multilayered laminated composite hypar shells for fully clamped boundary condition. This class of shells is unique in a sense that the curvature has only the radius of cross curvature and these shells do not admit easy closed form solution particularly when the boundary conditions are complicated. Contact behavior of impactor and impacted mass has been modeled by modified Hertzian contact law and time-dependent equations are solved using Newmark’s time integration technique. Basic aim is to analyse the shell for symmetrically placed multilayered angle and cross ply lamination under different impact velocities.

Seismic Control of Power Transmission Tower Using Pounding TMD

Lattice transmission towers are vital components of transmission line systems, which play an important role in the operation of electrical power systems. This paper proposes a new type of tuned mass damper (TMD), the pounding tuned mass damper (PTMD), to upgrade the seismic resistant performance of a transmission tower. In the PTMD, a limiting collar with viscoelastic material laced on the inner rim is installed to restrict the stroke of the TMD and to dissipate energy through collision. The pounding force is modeled based on the Hertz contact law, whereas the pounding stiffness β is estimated in a small-scale test. A multimass model of a 55-m tower is established to verify the effectiveness of the PTMD numerically. Harmonic excitation and time-history analysis demonstrate the PTMD’s superiority over the traditional TMD. Finally, a parametric study is performed for the optimal design.

Dynamic analysis of delaminated composite conical shells under low velocity impact

The dynamic response of graphite-epoxy delaminated cross-ply composite shallow conical shells subjected to low velocity normal impact is investigated using the finite element method. An eight-noded isoparametric plate bending element is employed in the formulation incorporating rotary inertia and effects of transverse shear deformation based on Mindlin’s theory. In the time integration, Newmark constant average acceleration method in conjunction with an efficient iteration scheme is used. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the time-dependent equations. Numerical results present the histories of contact force, deflection/thickness and impactor’s displacement for centrally impacted delaminated cross-ply composite cantilevered conical shells.

Site-Specific Quantification of Bone Quality Using Highly Nonlinear Solitary Waves

Osteoporosis is a well recognized problem affecting millions of individuals worldwide. The ability to diagnose problems in an effective, efficient, and affordable manner and identify individuals at risk is essential. Site-specific assessment of bone mechanical properties is necessary, not only in the process of fracture risk assessment, but may also be desirable for other applications, such as making intraoperative decisions during spine and joint replacement surgeries. The present study evaluates the use of a one-dimensional granular crystal sensor to measure the elastic properties of bone at selected locations via direct mechanical contact. The granular crystal is composed of a tightly packed chain of particles that interact according to the Hertzian contact law. Such chains represent one of the simplest systems to generate and propagate highly nonlinear acoustic signals in the form of compact solitary waves. First, we investigated the sensitivity of the sensor to known variations in bone density using a synthetic cancellous bone substitute, representing clinical bone quality ranging from healthy to osteoporotic. Once the relationship between the signal response and known bone properties was established, the sensor was used to assess the bone quality of ten human cadaveric specimens. The efficacy and accuracy of the sensor was then investigated by comparing the sensor measurements with the bone mineral density (BMD) obtained using dual-energy x-ray absorptiometry (DEXA). The results indicate that the proposed technique is capable of detecting differences in bone quality. The ability to measure site-specific properties without exposure to radiation has the potential to be further developed for clinical applications.

Wave propagation in random granular chains

The influence of randomness on wave propagation in one-dimensional chains of spherical granular media is investigated. The interaction between the elastic spheres is modeled using the classical Hertzian contact law. Randomness is introduced in the discrete model using random distributions of particle mass, Young's modulus, or radius. Of particular interest in this study is the quantification of the attenuation in the amplitude of the impulse associated with various levels of randomness: two distinct regimes of decay are observed, characterized by an exponential or a power law, respectively. The responses are normalized to represent a vast array of material parameters and impact conditions. The virial theorem is applied to investigate the transfer from potential to kinetic energy components in the system for different levels of randomness. The level of attenuation in the two decay regimes is compared for the three different sources of randomness and it is found that randomness in radius leads to the maximum rate of decay in the exponential regime of wave propagation.

Effect of the Impact Velocity on the Dynamic Response of E-Glass/Epoxy Laminated Composites

The present paper describes the dynamic response of E-glass/epoxy laminated composite structures under the variation of the aluminum projectile velocity from 0.54 to 3.1 m/s. The results are obtained using a Hertzian contact law and a drop weight impact machine and they are presented for three different cross-ply laminates ([02/906/02], [03/904/03] and [04/902/04]). The objective of this research is to determine the influence of projectile velocities on the impact response of the clamped laminated plates. The contact law used in the theoretical analysis is based on the combination of the dynamic and static solutions of the impact problem of two elastic bodies. In order to determine the stiffness of the composite plates, the system was represented by a spring–mass model. Particular attention was given during the comparison of the theoretical and experimental results. The difference between the two results was observed higher every time that the impact velocity increased. However, the application of the theoretical model is found to be an efficient guide for the experimental results validation.

Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach

This paper presents a general harmonic balance method for studying plane wave propagation in strongly nonlinear periodic media. The proposed approach starts by assuming a multi-wavenumber and frequency solution for the unit cell degrees of freedom. A Galerkin projection then transforms the nonlinear differential equations of motion into a set of nonlinear algebraic equations, which are subsequently solved numerically through a Newton-like iteration scheme. These solutions reveal amplitude-dependent dispersion behavior and group velocities. Specific example systems studied include one-dimensional chains and two-dimensional lattices, both formed by a periodic arrangement of spheres interacting under a Hertzian contact law. Amplitude-dependent dispersion is noted in monatomic and diatomic chains, and in hexagonally close-packed two-dimensional lattices. The validity of the presented technique is assessed through direct numerical simulation of the equations governing finite-extent lattices. Strong agreement is documented for results calculated using the harmonic balance approach and the direct numerical simulations.

Impact response of simply supported skewed hypar shell roofs by finite element

Use of laminated composite in the different weight-sensitive branches of engineering has gained immense popularity due to its high stiffness-to-weight ratio. In spite of this great advantage, this material is vulnerable to damage under impact loading due to its low transverse shear strength. A look through the literature on shell reveals that the impact response of civil engineering shell structures has not received proper attention. This article aims to carry out such a study of composite civil engineering shell considering skewed hypar as the geometric configuration. The modified Hertzian contact law which accounts for the permanent indentations is utilized to compute the contact force and the time-dependent equations are solved by Newmark’s time integration algorithm. Parametric studies are performed for the centrally impacted cross and angle ply skewed hypar shells modeled with eight noded isoparametric doubly curved shell elements. Simply supported boundary condition and different impact velocities a...

A piezoelectric device for impact energy harvesting

This paper studies a piezoelectric impact energy harvesting device consisting of twopiezoelectric beams and a seismic mass. The aim of this work is to find the influence ofseveral mechanical design parameters on the output power of such a harvester so as tooptimize its performance; the electrical design parameters were not studied. Toaccount for the dynamics of the beams, a model including the mechanical andpiezoelectric properties of the system is proposed. The impacts involved in theenergy harvesting process are described through a Hertzian contact law thatrequires a time domain simulation to solve the nonlinear equations. A transientregime and a steady-state regime have been identified and the performance ofthe device is characterized by the steady-state mean electrical power and thetransient electrical power. The time simulations have been used to study theinfluence of various mechanical design parameters (seismic mass, beam length, gap,gliding length, impact location) on the performance of the system. It has beenshown that the impact location is an important parameter and may be optimizedonly through simulation. The models and the simulation technique used in thiswork are general and may be used to assess any other impact energy harvestingdevice.