Elastic-plastic Sphere Research Articles

Study on characterization of moderate energy impact response of elastic-plastic plate structure

This paper presents a study on the characterization of the moderate energy impact response between the elastic-plastic plate and the elastic-plastic sphere. An equivalent contact stiffness is proposed to substitute the linear contact stiffness, and a universal analytical expression is presented according to the intensive FE simulations for different material combinations under moderate impact energy. Then, a modified characterization diagram is suggested using the analytical expression. The modified diagram is validated by the FE simulations and experimental results. The results show the modified diagram can predict the maximum impact force and the response type in a fast and simple way.

Role of intermolecular potential in adhesive wear behaviors for elastic-plastic spherical microcontacts

This paper aims to investigate the effects of intermolecular potential on adhesive wear behaviors of microcontacts between a rigid flat and an elastic-plastic sphere, using a Finite Element (FE) model. In this model, a separation-dependent adhesive pressure, governed by the Lennard-Jones (LJ) potential, is applied to the surfaces of both the rigid flat and the sphere under combined normal loading and tangential displacement. To assess material damage, the Johnson–Cook (JC) failure and fracture energy criteria are utilized, allowing for element deletion upon reaching the damage threshold. The role of intermolecular potential in the adhesive wear behaviors of spherical microcontacts is evaluated, focusing on wear volume, wear rate, wear morphology and fracture propagation mode under various dimensionless normal loads and adhesive energies. Two fracture propagation modes, namely bidirectional-like and unidirectional-like modes, dependent on normal load and adhesive energy, have been identified. The transition from mild to severe wear regime is found to be closely related to the fracture propagation mode. Additionally, an interesting observation is that relatively high adhesive energy could result in decreased wear volume and wear rate under certain normal loads, which can be attributed to the change in fracture propagation mode caused by the adhesive pressure.

Linear Contact Load Law of an Elastic–Perfectly Plastic Half-Space vs. Sphere under Low Velocity Impact

The impact of contact between two elastic–plastic bodies is highly complex, with no established theoretical contact model currently available. This study investigates the problem of an elastic–plastic sphere impacting an elastic–plastic half-space at low speed and low energy using the finite element method (FEM). Existing linear contact loading laws exhibit significant discrepancies as they fail to consider the impact of elasticity and yield strength on the elastic–plastic sphere. To address this limitation, a novel linear contact loading law is proposed in this research, which utilizes the concept of equivalent contact stiffness rather than the conventional linear contact stiffness. The theoretical expressions of this new linear contact loading law are derived through FEM simulations of 150 sphere and half-space impact cases. The segmental linear characteristics of the equivalent contact stiffness are identified and fitted to establish the segmental expressions of the equivalent contact stiffness. The new linear contact loading law is dependent on various factors, including the yield strain of the half-space, the ratio of elastic moduli between the half-space and sphere, and the ratio of yield strengths between the half-space and sphere. The accuracy of the proposed linear contact loading law is validated through extensive Finite Element Method simulations, which involve an elastic–plastic half-space being struck by elastic–plastic spheres with varying impact energies, sizes, and material combinations.

Numerical modelling of contact adhesion in a random assembly of elastic–plastic particles

The prediction of tensile strength properties of powder compacts remains an important industrial issue. In particular, one of the main problems of the powder compaction process is the failure of compacts. Indeed, some powder compacts exhibit cracks which appear during compaction. Such defects occur due to localised tensile or shear stresses, for example close to geometrical singularities. They are also related to the ability of a powder to create enough adhesion at contacts between particles to withstand tensile stresses. Therefore, cracks are a consequence of phenomena occurring at the particle scale and below, down to the molecular scale. To help understanding this mechanism, a particle-scale, numerical method called the multi-particle finite-element method was developed using the finite-element software suite Abaqus (Abaqus 6.14 Documentation, 2016). Such a method allowed to explicitly model the microstructure of a granular media idealised as an assembly of elastic–plastic spheres. The particles were meshed such that their deformations were fully taken into account, using a continuum-mechanics-based material model. The interactions between particles were managed using finite-element contact formulations. A multi-scale, adhesive contact model was developed based on the literature and implemented into the multi-particle finite-element code. The contact model was based on a surface energy formulation weighted by the roughness model developed by Pullen and Williamson (1972). It introduced a novel aspect, the development of adhesion under the effect of external mechanical loads, which is consistent with the cold compaction process. This model was then applied to predict the mesoscopic properties of a packing of spheres, i.e. its response to mechanical stresses of any type, in particular strongly deviatoric stress paths. Such a method intends to be a help toward the development of an efficient continuum model for the modelling of the powder compaction process.

Effect of mixed plastic hardening on the cyclic contact between a sphere and a rigid flat

This paper studies the effect of mixed plasticity mode (combined with isotropic and kinematic hardening law) on the cyclic contact between an elastic-plastic sphere and a rigid flat. Assuming power-law hardening with different levels of mixed plasticity for the sphere, we derived a semi-analytical expression of load versus interference during the first loading and unloading process. During cyclic loading, our results indicate that the isotropic plasticity model shows no variation of residual interference, while kinematic plasticity has the cyclic effect on the residual interference, and this effect is bigger for the material with a higher hardening exponent. In addition, we provided the semi-analytical expression for the evolution of residual interference, which is accurate for the strain hardening exponent from 0.1 to 0.5.

Characterization of Impacts of Elastic-plastic Spheres

This work presents explicit dynamic finite element simulations of various impacts of elastic-plastic solid spheres with flat walls. Different analytical models describing the mechanics of the impact phenomenon are also presented. Elastic and elastic-plastic material models for the sphere and the wall are considered during the analyses. The applicability of these different models is demonstrated and their accuracies are investigated. Closed-form analytical functions are proposed to describe the relationship between the initial velocity of the sphere and the investigated contact characteristics for the given material models. Analysis is carried out to study the effect of the friction coefficient as well as the angle of impact for various cases.

Cyclic tangential loading of a power-law hardening elastic–plastic spherical contact in pre-sliding stage

The cyclic tangential loading of the contact between a power-law hardening elastic–plastic sphere and a rigid flat under full stick condition is studied. The contact behavior in pre-sliding stage is investigated with a developed finite element model, and the contact parameters are obtained. Under the same maximum load in the cyclic tangential loading process, the tangential load–displacement relation is significantly affected by hardening exponents. Meanwhile, the dissipated energy obtained from the tangential load–displacement loops is larger for the smaller hardening exponent cases because the plastic deformation is more severe for these cases. The contact area increases mainly during the first tangential load cycle, and the increase is larger for materials with smaller hardening exponents.

The Relationship between the Deformation of Spherical Indentation and Tensile Deformation

The paper is devoted to the definition of the deformation during indentation of the sphere and its relationship with the tensile deformation. Proposed by different authors methods of determining the deformation of the contact are considered. The results of their researches may vary significantly. It is shown that in the last decade to determine the deformation, the finite element analysis taking into account the "sink-in/pile-up", i.e. an elastic sinking in and plastic piling up of the material on the edges of the indent during the indentation process is widely used. The purpose of this research is to determine the relationship between tension deformation and sphere indentation deformation with taking into account the last achievements in the field of finite-element modeling of elastic-plastic sphere indentation. It is considered two methods of determining of deformation. One uses the equation proposed by S.I. Bulychev, in which the Mayer’s index is determined from the results of finite element analysis. The second method use the energy concept of hardness. It is based on the assumption that within the range of uniform deformation during uniaxial tension and during sphere indentation, the same energy is consumed to the plastic displacement of the part of the material volume out of limits of initial volume. They have close results. The corresponding graphic relations are shown.

A new mathematical model to calculate sand-packed fracture conductivity

The conductivity of sand-packed fractures is influenced by the fracture width and permeability. Proppants become rearranged, embedded in formations and even deformed under certain pressures. Thus, the fracture width, the porosity and the radius of seepage channels decrease with increasing pressure. Currently, sand-packed fracture conductivity is measured in the laboratory but rarely calculated by mathematical methods or based on ideal conditions, which may not always be true in practice. This paper demonstrates proppant rearrangement under low and high pressure and presents a rhombohedron model to calculate sand-packed fracture conductivity that considers the looseness coefficient, deformations of rock and proppants, particle size, crushing rate, number of proppant layers, closure pressure and so on. The mathematical model is based on the principle of elastic-plastic sphere contacts, the capillary bundle model and the Carman-Kozeny equation. In this case, the changing width and permeability of sand-packed fractures during compression can be calculated. Verification experiments and comparisons with existing models indicate that the mathematical models can reasonably predict the sand-packed fracture width and conductivity under real conditions.

Spherical Elastic–Plastic Contact Model for Power-Law Hardening Materials Under Combined Normal and Tangential Loads

The contact between a power-law hardening elastic–plastic sphere and a rigid flat under combined normal and tangential loads in full stick is studied in this work. The displacement-driven loading is used since the frictional contact problems under the displacement-driven loading are widespread in the fields of metal forming and orthogonal cutting. The loading process is as follows: First, a normal displacement-driven loading is imposed on the rigid flat and kept constant; then, an additional tangential displacement-driven loading is applied to the rigid flat. The elastic–plastic contact behavior in presliding is investigated with a proposed finite element (FE) model, including the tangential force, the von Mises stress, the normal force, the contact pressure, and the contact area. The effect of the strain-hardening exponent on contact behavior is considered. It is seen that the tangential force increases nonlinearly with the increase of the tangential displacement, exhibiting gradual stiffness reduction which implies that the junction becomes more plastic. The von Mises stresses moves along the direction of the tangential load, while the maximum stress moves to the contact surface from the below. The normal force diminishes as the tangential load increases, and more obviously for the lower hardening exponent cases. The contact pressure also decreases more significantly for the lower hardening exponent cases. In addition, smaller exponents result in a greater increase of the contact area. The empirical expressions of the tangential force and the contact area in the tangential loading process are also proposed by fitting to the FE results.

Thermo-mechanical contact between a rigid sphere and an elastic–plastic sphere

The mechanical and thermal response of a rigid sphere sliding over an elastic–plastic sphere with radius larger than that of the rigid sphere, namely, plowing contact model, are investigated using a finite element method. Dimensionless solutions for the maximum values of contact force, friction force, contact area, residual interference, and interfacial temperature rise are obtained as a function of dimensionless contact time. A detailed comparison is performed with the results between the plowing contact model and the flattening contact model in which an elastic–plastic sphere flattened by a rigid sphere with radius larger than that of the elastic–plastic sphere.

A unified model for the contact behaviour between equal and dissimilar elastic–plastic spherical bodies

A unified method for calculating the contact force and the contact area between two dissimilar elastic-plastic spheres is presented with the aim of simulating granular materials using particle methods. Explicit equations are presented for the case when the plastic behaviour of the spheres is described with three material parameters. This makes the analysis applicable for a wide range of materials. The model is partly based on dimensionless quantities emerging from the Brinell hardness test. Large deformation of the contact is accounted for in the analysis, which allows for accurate contact relations up to indentation depths relevant for powder compaction. The presented model shows excellent agreement with finite element simulations of two spheres in contact and the results found in literature. An implementation of the contact model in Python is provided together with the online version of this paper.

Loading and unloading of a power-law hardening spherical contact under stick contact condition

The loading and unloading process of a deformable sphere pressed by a rigid flat is a primary problem in contact mechanics. This work studies the contact of an elastic–plastic sphere with a rigid flat under stick condition during loading and unloading process by the finite element method. The sphere material is assumed isotropic with power-law hardening. The contact load and contact area are calculated at various Poisson׳s ratios and strain hardening exponents, with the sphere material properties varying from purely elastic to elastic-perfectly plastic. Analytical dimensionless expressions of the contact load, contact area and the residual interference after fully unloading for the power-law hardening materials in the elastic–plastic case are proposed for a wide range of interferences.

An accurate force–displacement law for the modelling of elastic–plastic contacts in discrete element simulations

This paper presents an accurate model for the normal force–displacement relationship between elastic–plastic spheres in contact for use in discrete element method (DEM) simulations. The model has been developed by analysing the normal force–displacement relationship between an elastic–perfectly plastic sphere and a rigid surface using the finite element method (FEM). Empirical relationships are found that relate the parameters of the new model to material properties. This allows the model to be used in the DEM for direct simulation of well characterised elastic–plastic materials without fitting parameters to experimental results. This gives the model an advantage over models in the literature for which fitting to experimental results is required. The implementation of the model into an existing DEM code is discussed and validated against the results from FEM simulations. The new model shows a good match to the FEM results and the DEM implementation correctly distinguishes between the loading, unloading and re-loading phases of contact between two spheres.

Guided Impact Mitigation in 2D and 3D Granular Crystals

We simulate the dynamics of impacts on 1D, 2D and 3D arrays of metallic spheres in order to design novel granular protection systems. The dynamics of these highly organized systems of spheres, commonly called granular crystals, are governed by the contact law that describes how each particle interacts with the others. We use our recently developed force-displacement model of the dynamic compression of elastic-plastic spheres as the building block to investigate the response of systems comprised of metallic spheres to an impact. We first provide preliminary experimental results using a drop tower as validation of our numerical approach for 2D and 3D systems. We then use simulations of large periodic granular crystals in order to determine which particle properties govern the velocity of stress waves in these materials. We show that the properties of 1D systems can be scaled to predict the behavior of more complex 2D and 3D granular crystals. Because we can choose the material properties of each of the constituent particles and design how the particles are geometrically packed, we can leverage the heterogeneity of the system to create materials with unique properties such as anisotropic local stiffnesses and wave propagation velocities. We show that these materials allow us to design the dispersion and dissipation properties within the material in order to influence the propagation of a stress wave. Using these materials, we can therefore design protection systems or armor that directs damage away from sensitive parts or localizes damage to an unimportant area after impact from a projectile or a blast.

Thermomechanical modeling and transient analysis of sliding contacts between an elastic–plastic asperity and a rigid isothermal flat

To investigate thermomechanical contacts between an elastic–plastic sphere and a rigid flat, simulations with slip rates ranging from 0.1m/s to 10m/s were performed. As interfaces with strong interfacial bonding but weak substrate were specifically targeted, slip initiation was treated as shear failure of the softer material in numerical simulations. The simulations show that both sliding friction coefficient and friction stress are significantly dependent on slip rate while the maximum static friction coefficient is independent of that. Moreover, the energy release during the transition from full stick to full slip is comparable to the shear fracture energy of the material.

On the tangential contact behavior at elastic–plastic spherical contact problems

The problem of tangential contact between an elastic–plastic sphere and a rigid plane is studied analytically and numerically with the specific aim to derive force–displacement relations to be used in numerical simulations of granular materials. The simulations are performed for both ideal-plastic and strain hardening materials with different yield stresses and including large deformation effects in order to draw general conclusions. The results are correlated using normalized quantities pertinent to the correlation of indentation testing experiments leading to a general description of the tangential contact problem. Explicit formulas for the normal and tangential forces are presented as a function of the tangential displacement using data that are easily available from axi-symmetric analyses of spherical contact. The proposed model shows very good agreement when compared with the FE-simulations.

Analysis of Thick-Walled Elastic-Plastic Sphere Subjected to Temperature Gradient

The aim of this work is the study of full thermal cycle (thermal loading and unloading) and the influence of linear hardening coefficients of the material on behavior of thick-walled elastic-plastic sphere subjected to thermal gradient. The influence of thermal cycle and hardening coefficients on the distribution of temperature and strains, on the position of elastic-plastic zone and current stresses, and in particular residual (inner stresses) arisen after the process of elastic unloading is analyzed. The role of residual stresses can be important in estimation of exploitation lifetime of loaded constructions and in analysis of thermal fatigue in loading-unloading thermal cycle. In analysis of the thick-walled sphere, the critical values of sphere geometric parameter β are also determined in the moment of formation of the second plastic zone on outer radius of sphere.

Flattening of a deformable sphere by a rigid sphere during transient thermo-mechanical contact

A finite element model is developed to study the transient thermo-mechanical contact between a rigid sphere and an elastic–plastic sphere. Interference and tangential displacement are applied to the rigid sphere while the elastic–plastic sphere is stationary. The radius of the rigid sphere is larger than that of the elastic–plastic sphere to model flattening. Universal solutions are obtained for the maximum contact force, the maximum contact area, the maximum residual interference, and the maximum temperature rise independent of sphere radius, material properties and loading parameters. The effects of various parameters (sphere radius, interference, coefficient of friction, sliding velocity and material properties) on the contact characteristics of the deformable sphere are discussed.

Effects of load configuration on partial slip contact between an elasticplastic sphere and a rigid flat

A model utilizing finite-element method (FEM) was introduced to investigate the dependency of partial slip behavior on the loading configuration for a deformable sphere and a rigid flat in contact. It was found that the maximum static friction under tangential displacement control is lower than that under tangential force control with an equivalent normal preload. Elastic shakedown was observed under cyclic tangential force loading while plastic shakedown was observed under cyclic tangential displacement loading. As a fully coupled contact configuration with alternating loading, the properties dissimilarity of two contacting bodies plays a boosting role on the displacement and force coupling between normal and tangential.