Hertz-Mindlin Theory Research Articles

Study on Compaction Characteristics and Compaction Process of an Unsaturated Silt Based on PFC3D

In order to reduce the construction cost of excessively wet silt roadbeds and improve the compaction quality, this paper studies the compaction characteristics and compaction technology of excessively wet silts through theoretical derivation, indoor experiments, numerical simulation, engineering verification, and other research methods. Based on the pendulum-type liquid bridge structure, this paper uses the Hertz–Mindlin theory, particle motion equation of state, and other theories to modify the force between particles of an unsaturated soil. Through the contact angle, which is a medium, the matrix suction is linked to the water content and is used to approximate the water content state of soil samples. Using PFC3D (particle flow code 3D) numerical simulation software, an unsaturated silt model was established based on Hill contact, and the model parameters were calibrated through basic geotechnical tests and triaxial compression tests. The correctness of the theory is verified by comparing the simulated triaxial compression test of wet silts with the indoor test. By simulating subgrade rolling, considering factors such as the distribution of contact force chain and the variation of compaction degree, the effects of rolling times, strong vibration, and weak vibration on the compaction effect were compared and analyzed. Finally, the optimal rolling and compaction process under the optimal water content is obtained through on-site engineering tests. The results show that: (1) The Hill contact model is reasonable for simulating the wet silt. (2) During the simulation of the roadbed compaction process using PFC3D software, it was found that the compaction degree change during the entire compaction process can be roughly divided into the initial compaction stage and the re-compaction stage. The reasonable number of compaction times was determined to be five through discrete element simulation. (3) It is found from the numerical simulation results that compared to static compaction, both strong and weak vibration compaction can effectively improve the compaction effect of subgrade compaction. The larger the vibration amplitude, the more obvious the improvement of the compaction effect. The compaction effect of strong vibration followed by weak vibration is stronger than that of weak vibration followed by strong vibration. (4) The optimal compaction process obtained under the optimal moisture content of 15% is static pressure once + strong vibration twice + weak vibration twice.

Axial Symmetric Granular Flow Due to Gravity in a Circular Pipe

Axisymmetric granular flows in vertical cylindrical pipes under action of gravity are studied using mathematical particle–particle models based on the Hertz–Mindlin theory. By and large, in granular flows, the density field and the pressure are unknown scalar functions. A well-known relationship between these fields gives the pressure field a power law of the density. The aim of this paper was to study unsteady, axisymmetric, fully developed granular flow under gravity action in a vertical cylindrical pipe, under the assumptions that the density field is constant and the velocity on the pipe’s wall is time-dependent. Using integral transforms method and appropriate initial-boundary conditions, the analytical solution for axial velocity is determined. The obtained analytical solution is used to determine the steady-state solution (the solution for large values of the time). The properties of the flow in some particular cases of the velocity on the pipe’s surface are analyzed and the transient flow is compared with the stationary one.

Electromagnetic piezoelectric hybrid driven multi-DOF motor contact model considering drive and load state.

This paper proposes a new research method in view of the rare research on the contact state of hybrid drive multi-degree-of-freedom motor. Firstly, the basic principle and structure of hybrid drive motor are introduced. The air gap magnetic field model and electromagnetic torque model of permanent magnet rotor are obtained by analytical method. Static analysis of piezoelectric stators are carried out by stress-strain relationship. According to the analytical method and the spatial geometric relationship, the piezoelectric driving torque under different driving conditions is obtained. Introducing Hertz contact theory and Mindlin theory, combined with piezoelectric driving torque, electromagnetic driving torque and load torque to analyze the friction situation. A friction interface model considering the dynamic nonlinearity of friction coefficient and the nonlinear change of friction force distribution is established. Finally, using Matlab software, the friction distribution diagram of the contact surface of the hybrid drive three-degree-of-freedom motor under different driving and motion states is obtained. The analysis results verify the rationality of the electromagnetic piezoelectric hybrid driven three-degree-of-freedom motor. It shows that the contact state of the three sets of piezoelectric stators are determined according to the driving condition and load, the relative speed of the piezoelectric stators and the spherical rotor is proportional to the frictional force. It provides an idea for the research of contact state of hybrid drive multi-degree-of-freedom motor, and also lays a basis for further optimizing the friction interface of this kind of motors or actuators.

Estimation of grain elasticity properties from ultrasonic measurements on dry granular pack

ABSTRACTThe elastic properties of crystalline rocks can be estimated from ultrasonic measurements on the powders of crushed rocks produced by the drilling process. To determine the elastic properties of grains from properties of powder packs, we study the dependence of their ultrasonic wave velocities on pressure. From this dependency, using the Hertz–Mindlin theory, we can calculate the effective ratio of the grain shear modulus to one minus the Poisson ratio. The Hertz–Mindlin theory requires the knowledge of grain coordination number as a function of pressure, which can be obtained using an empirical relation based on published numerical simulations. Previous work has shown that this approach gives an accurate prediction of the effective bulk modulus of glass beads but produces a significant discrepancy for sand. This discrepancy may be attributed to the angularity of sand grains. To overcome this problem, we introduce a shape factor into the empirical relation for the coordination number. This new shape factor allows us to reconcile the rock physics model with laboratory measurements. We show that the shape factor varies from 1 to 2.5 for different grain shape angularity and sorting (grain size distributions). The modified theory allows us to estimate a combination of elasticity parameters of the grains from the measured dependence of P‐wave velocity in the pack on the pressure.

Seismic Velocity Prediction in Shallow (<30 m) Partially Saturated, Unconsolidated Sediments Using Effective Medium Theory

Seismic velocity models of the near-surface (<30 m) better explain seismic velocities when all elements of total effective stress are considered, especially in materials with large cohesive and soil suction stress such as clays. Traditional constitutive elastic models that predict velocities in granular materials simplify the effect of total effective stress by equating it to net overburden stress, while excluding interparticle stresses and soil suction stress. A new proposed methodology calculates elastic moduli of granular matrices in near-surface environments by incorporating an updated definition of total effective stress into Hertz-Mindlin theory and calculates the elastic moduli of granular materials by extending Biot-Gassmann theory to include pressure effects induced by water saturation changes and cohesion. At shallow depths, when water saturation increases, theoretically calculated seismic velocities decrease in clay and increase in sand because interparticle stresses suppress the Biot-Gassmann effect. For standard sand and clay properties, net overburden stress becomes more influential than interparticle stresses at depths greater than 10 cm in sand and 100 m in clay. Pore pressure in the new model also incorporates the effect of layer thickness and pore size variation. Traditional calculation of pore pressure assumes a constant pore size medium, but may lead to an under- or overestimation of velocity by up to 20%. In clays, the variation of seismic velocity with water saturation is almost double the range predicted when only net overburden stress is considered to influence stress at the grain contacts. The proposed model predicts seismic velocities that compare well with measured field velocities from the literature.

固有振動数を用いた表面性状パラメータの抽出に関する研究

The surface texture describes the surface topography formed by micro asperities on a surface of a solid member. The Hertz-Mindlin theory is one of contact theories based on the mechanics of elasticity, and the main parameters for the theory are the standard deviation of height, mean radius, and density of asperity summits on the surface. The conventional approach to obtain these parameters is curve fittings applied on the surface texture. However, the 3D measurement with a spatial resolution of a micron is required in the measurement of surface texture and the measureable area is limited. This study has developed a novel method to extract the surface texture parameters based on the natural frequency of the target. In the proposed method, the natural frequency is obtained by the hammering test and the finite element analysis. Then applying the data assimilation on the test result and the analysis result, the most likelihood set of the surface texture parameters is derived. In order to investigate the basic behavior of the proposed method, a series of hammering tests was carried out and the surface texture parameters of specimens were estimated. The surface texture parameters extracted by the proposed method were compared with those directly measured by 3D surface measuring instruments.

Analysis of rigid tire traction performance on a sandy soil by 3D finite element–discrete element method

A 3D simulation method was developed to investigate the tire–sand interactions, where the discrete element (DE) and finite element (FE) methods were coupled. An algorithm for the contact of FE and DE was introduced. The Hertz–Mindlin theory was applied to calculate the contact force among elements. A 3D numerical model was established based on the soil-bin experiment to validate the feasibility of the method in the analysis of the tire–sand interactions. The traveling process of the smooth tire on the sandy soil was simulated and the traction performance parameters of the tire under different slip values (0%, 10%, 20%, 30%, 40%, 50% and 60%) were obtained. The overall trend of the drawbar pull versus the slip ratio is qualitatively in agreement with current experimental results. The result also proved the feasibility of the 3D FE–DEM in the simulation of the tire–sand interactions.

Modeling the pressure sensitivity of uncemented sediments using a modified grain contact theory: Incorporating grain relaxation and porosity effects

We develop a modified grain contact theory to better describe the pressure-dependent elastic properties of uncemented sediments. The Hertz-Mindlin (HM) theory typically predicts shear moduli that are much higher than observed laboratory measurements, resulting in inaccurate estimates of the dry bulk to shear modulus ratio [Formula: see text]. The HM theory further predicts that the [Formula: see text] ratio is constant with pressure, whereas ultrasonic core measurements typically show an increasing [Formula: see text] ratio as effective pressure decreases. Laboratory data also suggest that the dry bulk and shear moduli variation with effective pressure is greater than the cube-root power law predicted by the HM theory. We introduce two new pressure-dependent calibration parameters to account for the shortcomings in effective medium theory, and we develop a new method to predict pressure-dependent elastic properties. Our calibration parameters agree with the results of published granular dynamics simulations, and they incorporate grain relaxation and porosity effects not included in existing effective medium theories. Our new model provides improved fits to laboratory data when compared with existing models, and it can be used for improved prediction of elastic properties as a function of effective pressure. Our new theory can also be used to model uncemented sediments with values of Poisson’s ratio [Formula: see text], where many existing grain contact and effective medium theories currently fail.

Molecular Dynamics simulation for PBR pebble tracking simulation via a random walk approach using Monte Carlo simulation

Using a Monte Carlo (MC) simulation, random walks were used for pebble tracking in a two-dimensional geometry in the presence of a biased gravity field. We investigated the effect of viscosity damping in the presence of random Gaussian fluctuations. The particle tracks were generated by Molecular Dynamics (MD) simulation for a Pebble Bed Reactor. The MD simulations were conducted in the interaction of noncohesive Hertz–Mindlin theory where the random walk MC simulation has a correlation with the MD simulation. This treatment can easily be extended to include the generation of transient gamma-ray spectra from a single pebble that contains a radioactive tracer. Then the inverse analysis thereof could be made to determine the uncertainty of the realistic measurement of transient positions of that pebble by any given radiation detection system designed for that purpose.

Dispersion of elastic waves in three-dimensional noncohesive granular phononic crystals: Properties of rotational modes

The dispersion relations of bulk modes propagating within a hexagonal close-packed structure of noncohesive monodisperse spherical elastic beads are derived. The contacts are modeled by two springs with stiffnesses given by the Hertz-Mindlin theory, one for normal interactions and one for transverse interactions. The existence of the transverse interaction requires to take into account the rotational degrees of freedom of the beads in the analysis. This leads to the prediction of translational modes and, due to the rotational degrees of freedom, of rotational modes and coupled rotational and translational modes. The study of the dispersion relations in a direction of high symmetry allows to identify the different modes and the influence of the rotational degrees of freedom on the bulk mode propagation. The evaluated dispersion relations provide guidelines for the experimental observation of rotational modes. Opportunities for controlling the dispersion laws of the modes by an external loading on the granular structure are discussed.

Preisach analysis of the Hertz–Mindlin system

In this paper we apply the Preisach hysteretic formalism to the Hertz–Mindlin system dealing with frictional and elastic interactions of two spheres in contact subjected to a varying oblique force. The classical solution for this fundamental mechanical problem, obtained by Mindlin and Deresiewicz over 50 years ago, has become the foundation for many applications, including friction of rough surfaces, physics of granular materials, rock mechanics, nonlinear acoustics, seismology, geological exploration, monitoring of oil reservoirs and mining processes, etc. However, due to the complex nature of the general case of a two-sphere interaction subjected to an arbitrary force protocol, researchers have commonly utilized only the basic results of the Hertz–Mindlin theory, i.e. the formulations of the normal and tangential stiffnesses and of the hysteretic loss of energy. The full description with arbitrarily varying forces has been possible only on a case-to-case basis. To overcome this hurdle, we propose to link the Hertz–Mindlin system to the fundamental theory introduced by Preisach over 70 years ago, which provides the tools to mathematically describe arbitrary hysteretic functions, i.e. non-unique and history-dependent functional relations with irreversible behavior. We show how the Preisach formalism simplifies the mathematical complexity of the Hertz–Mindlin problem and makes it possible to consider complicated force protocols in a universal manner. We prove that the Hertz–Mindlin interaction with dN/ dT=const is a particular case of a Preisach system and we derive the essential parameters of the corresponding Preisach formalism. Finally, we demonstrate the analogy between force-driven systems and displacement-driven systems, and we propose a new solution for the displacement-driven Hertz–Mindlin friction.

The apparent failure of effective medium theory in granular materials

The linear and nonlinear elastic properties of granular media are analyzed within the context of effective medium theories, as well as with numerical molecular dynamic simulations, assuming the validity of the Hertz-Mindlin theory at the single contact level. There is a crucial distinction between force laws which are path independent, leading to a hyper-elastic effective medium theory, and those which are path dependent, for which the deformation history must be followed explicitly. The effective medium theories provide a reasonable description of existing experimental data, considered as a function of applied stress, but there are significant discrepancies. Numerical simulations resolve the question as to whether the problem lies with the treatment of the individual grain-grain contact or with the effective medium approximation (ema). We find that the problem lies principally with the latter: The bulk modulus is well-described by the ema but the shear modulus is not, principally because the ema does not correctly allow for the grains to relax from the affine motion assumed by the ema.

Nonlinear elasticity of granular media

The linear and nonlinear elastic properties of granular media are analyzed within the context of effective medium theories, as well as with numerical molecular dynamic simulations, assuming the validity of the Hertz–Mindlin theory at the single contact level. There is a crucial distinction between force laws which are path independent, leading to a hyper-elastic effective medium theory, and those which are path dependent, for which the deformation history must be followed explicitly. The effective medium theories provide a reasonable description of existing experimental data, considered as a function of applied stress, but there are significant discrepancies. Numerical simulations resolve the question as to whether the problem lies with the treatment of the individual grain–grain contact or with the effective medium approximation (ema). We find that the problem lies principally with the latter: The bulk modulus is well described by the ema but the shear modulus is not, principally because the ema does not correctly allow for the grains to relax from the affine motion assumed by the ema.

Linear and Nonlinear Elasticity of Granular Media: Stress-Induced Anisotropy of a Random Sphere Pack

We develop an effective medium theory of the nonlinear elasticity of a random sphere pack based upon the underlying Hertz-Mindlin theory of grain-grain contacts. We compare our predictions for the stress-dependent sound speeds against new experimental data taken on samples with stress-induced uniaxial anistropy. We show that the second-order elastic moduli, Cijkl, and therefore the sound speeds, can be calculated as unique path-independent functions of an arbitrary strain environment, {εkl}, thus generalizing earlier results due to Walton. However, the elements of the stress tensor, σij, are not unique functions of {εkl} and their values depend on the strain path. Consequently, the sound speeds, considered as functions of the applied stresses, are path dependent. Illustrative calculations for three cases of combined hydrostatic and uniaxial strain are presented. We show further, that, even when the additional applied uniaxial strain is small, these equations are not consistent with the usual equations of third-order hyperelasticity. Nor should they be, for the simple reason that there does not exist an underlying energy function which is simply a function of the current state of the strain. Our theory provides a good understanding of our new data on sound speeds as a function of uniaxial stress.

Assessment of tangential and normal stiffness of contact between rough surfaces using ultrasonic method

The acoustic model of contact, previously proposed for longitudinal ultrasonic waves, was adopted for shear waves. It was found that the measurements of the phase shift of coefficient of reflection from the contact interface, taken both for shear and longitudinal ultrasonic waves of the same frequency, allows us to find the tangential to normal contact stiffness ratio for the contact tested. However, an analysis based on the Greenwood-Williamson model and the Hertz-Mindlin theory revealed that for a contact between rough surfaces with spherical asperities, the ratio of the tangential to the normal contact stiffness is independent of surface height distribution and is, moreover, equal to that ratio for an elementary contact. In order to verify the theoretical predictions, measurements of the phase shift of longitudinal and shear ultrasonic waves of frequency 10 MHz were carried out for the loaded contact (up to 300 MPa) between ground quartz samples. It appeared that the value of the tangential to normal contact stiffness ratio resulting from the ultrasonic measurements is fairly close to that predicted by the Hertz-Mindlin theory.

Properties of granular packings under low amplitude cyclic loading

The initial tangent and the secant moduli for granular materials under low amplitude cyclic loading is obtained from an analytical model considering particle interactions. The Hertz-Mindlin theory of contact of two elastic spheres are used to describe the contact displacement behavior under oscillating forces. The analytical model includes the effect of the void ratio, the coordination number, the constituent particle properties, and the structural anisotrophy. Results computed from the derived relations are compared with the experimental measurements for initial tangent and secant moduli.

Stochastic analysis of non-top-top contacting rough surfaces

Within the framework of the Hertz-Mindlin theory of contact the dependence of the mean values of the resulting bending moments, the torsion moment and the shear and vertical forces of stochastically rough surfaces in elastic contact on the shape of the asperities, and the effect of lateral contact of the asperities on the vertical force in the slip-free and slip-afflicted cases are investigated.