Granular Solid Hydrodynamics (GSH): a broad-ranged macroscopic theory of granular media

A unified continuum-mechanical theory has been until now lacking for granular media, some believe it could not exist. Derived employing the hydrodynamic approach, Granular Solid Hydrodynamic is such a theory, though as yet a qualitative one. The behavior being accounted for includes static stress distribution, elastic wave, elasto-plastic motion, the critical state and rapid dense flow. The equations and application to a few typical experiments are presented here.     

A macroscopic description of the quasi-static behavior of granular materials based on the theory of porous media

Granular materials fall into the class of porous media. But in contrast to materials like foams and sponges their structure is discontinous on a microscopic level. For this reason the particles may undergo independent displacements and rotations. This is the classical kinematics which may be captured by a micropolar or Cosserat theory on the macroscopic level. The goal of this paper is to combine the theory of porous media as a macroscopic theory dealing with multi-phase systems and the micropolar theory describing extended kinematics and taking care of the discountinous structure of granular media on the micro scale. The resulting micropolar theory of porous media may be used to describe the quasistatic behavior of granular materials. In the present contribution thermodynamically consistent constitutive relations for the elastic response of a dense granular matrix material saturated by a viscous pore fluid are given and applied to some boundary value problems which demonstrate the physical relevance of the proposed model. Received: 9 June 1999     

Elastic energy and relaxation in triaxial compressions

For a quasi-statically sheared granular system, the deformation of individual particles leads to reversible energy storage that sustains elastic stress. But, the system would subsequently relax because particles jiggle and slide. By employing the complete continuum mechanical theory, also known as Granular Solid Hydrodynamics (GSH), the elastic energy and its relaxation (denoted by granular temperature) are both calculated and explained. For a dense assembly, it is found that the elastic energy and energy dissipation rate reach peak values simultaneously, as it reaches peak strength. To observe the mesoscale characteristics, a two-dimension biaxial test is simulated with a discrete element method. The motion of particles and the evolution of force networks are exhibited at different strain values. The discrete element simulations results are helpful to understand GSH results.     

Parametric hypoplasticity as continuum model for granular media: from Stokesium to Mohr-Coulombium and beyond

Fluid-particle systems, in which internal forces arise only from viscosity or intergranular friction, represent an important special case of strictly dissipative materials defined by a history-dependent 4th-rank viscosity tensor. In a recently proposed simplification, this history dependence is represented by a symmetric 2nd-rank fabric tensor with evolution determined by a given homogeneous deformation. That work suggests an essential physical link between idealized suspensions (“Stokesium”) and granular media (“Mohr-Coulombium”) along with possible models for the visco-plasticity of fluid-saturated and dry granular media. The present paper deals with the elastoplasticity of dilatant non-cohesive granular media composed of nearly rigid, frictional particles. Based on the underlying physics and past modeling by others, a continuum model based on parametric hypoplasticity is proposed, which involves a set of rate-independent ODEs in the state-space of stress, void ratio and fabric. As with the standard theory of hypoplasticity, the present model does not rely on plastic potentials but, in contrast to that theory, it is based explicitly on positive-definite elastic and plastic moduli. The present model allows for elastic loading or unloading within a dissipative yield surface and also provides a systematic treatment of Reynolds dilatancy as a kinematic constraint. Some explicit forms are proposed and comparisons are made to previous hypoplastic models of granular media.     

Stress softening of elastomers in hydrostatic tension

Summary. This study is concerned with inelastic effects of non-reinforcing carbon-black filled elastomers when subjected to periodic hydrostatic loading-unloading cycles in tension. During cyclic testing of sufficient magnitude, a critical state may be reached where microcavities suddenly grow inside the rubber, possibly initiated at sites of internal imperfections. As a result of cavitation damage the tensile bulk modulus in the natural configuration is reduced. A series of hydrostatic tension tests are performed at room temperature to provide new insight into the progressive deterioration of the bulk stiffness. We define dilatational stress softening as a phenomenon where the hydrostatic stress on unloading and subsequent submaximal reloading is significantly less than that on primary loading for the same volumetric strain. Dilatational stress softening during initial loading cycles and the permanent volumetric change upon unloading are not accounted for when the mechanical properties are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper a constitutive model is derived to include the progressive reduction of the bulk stiffness and the permanent volumetric change of carbon-black filled elastomers subjected to quasi–static loading. The basis of the model is the theory of pseudo-elasticity, which including a softening variable modifies the dilatational strain energy function. An acceptable correspondence between the theory and the data is obtained.     

How to handle the inelastic collapse of a dissipative hard-sphere gas with the TC model

The inelastic hard sphere model of granular material is simple, easily accessible to theory and simulation, and captures much of the physics of granular media. It has three drawbacks, all related to the approximation that collisions are instantaneous: 1) The number of collisions per unit time can diverge, i.e. the “inelastic collapse” can occur. 2) All interactions are binary; multiparticle contacts cannot occur and 3) no static limit exists. We extend the inelastic hard sphere model by defining a duration of contact t _c such that dissipation is allowed only if the time between contacts is larger than t _c . We name this generalized model the TC model and discuss it using examples of dynamic and static systems. The contact duration used here does not change the instantaneous nature of the hard sphere contacts, but accounts for a reduced dissipation during “multiparticle contacts”. Kinetic and elastic energies are defined as well as forces and stresses in the system. Finally, we present event-driven numerical simulations of situations far beyond the inelastic collapse, possible only with the TC model.     

Proportional paths, Barodesy, and granular solid hydrodynamics

Proportional paths as summed up by the Goldscheider Rule (GR)—stating that given a constant strain rate, the evolution of the stress maintains the ratios of its components—is a characteristics of elasto-plastic motion in granular media. Barodesy, a constitutive relation proposed recently by Kolymbas, is a model that, with GR as input, successfully accounts for data from soil mechanical experiments. Granular solid hydrodynamics (GSH), a theory derived from general principles of physics and two assumptions about the basic behavior of granular media, is constructed to qualitatively account for a wide range of observation – from elastic waves over elasto-plastic deformation to rapid dense flow. In this paper, showing the close resemblance of results from Barodesy and GSH, we further validate GSH and provide an understanding for GR.     

On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

One-dimensional selfsimilar problems for waves in an elastic half-space generated by a sudden change of the boundary stress (the “piston” problem) and problems of disintegration of an arbitrary discontinuity are considered. For the case when small-amplitude waves are generated in a medium with small anisotropy a qualitative analysis shows that these problems have nonunique solutions when it is assumed that the solutions involve Riemann waves and evolutionary discontinuities. The above-mentioned problems are considered as limits of properly formulated problems for visco-elastic media when the viscosity tends to zero or (what is the same) that time tends to infinity. It is numerically found that all above-mentioned inviscid solutions can represent the asymptotics of visco-elastic solutions. The type of asymptotics depends on those details of the visco-elastic problem formulation which are absent when formulating inviscid selfsimilar problems. Similar considerations are made for elastic media with dispersion along with dissipation which are manifested in small-scale processes. In such media the number of available asymptotics (as t→∞) for the above-mentioned solutions depends on a relation between dispersion and dissipation and can be large. Thus, two possible causes for the nonuniqueness of solutions to the equations of elasticity theory are investigated.     

Dynamic cylindrical cavity expansion in an incompressible elastoplastic medium

Summary Self-similar dynamic expansion of a pressurized circular cylindrical cavity, embedded in an infinite elastoplastic incompressible medium, is here investigated with the large strain J₂ flow theory. Assuming steady-state conditions, thus bypassing the initial loading history, it is shown that plane-strain fields are sustained with no diverging logarithmic stress appearing in the remote elastic field. Yet, even in the absence of remotely applied stress, the appearance of small stresses at infinity is unavoidable. The present solution is exact but limited to relatively low cavity expansion velocities. A closed form expression is given for the cavitation pressure with elastic/perfectly-plastic response. A fairly general result is derived for the cavitation pressure in hardening media with a definite yield point and in linear-hardening solids as a special case. Contact is made with earlier results of quasi-static cavity expansion along with a comparison to the self-similar dynamic expansion of a spherical cavity in an incompressible Mises solid. Upper and lower bounds for penetration depth tests are suggested by using the present cylindrical cavitation model and the incompressible spherical cavitation model.     

A theory for the fracture of thin plates subjected to bending and twisting moments

Stress fields near the tip of a through crack in an elastic plate under bending and twisting moments are reviewed assuming both Kirchhoff and Reissner plate theories. The crack tip displacement and rotation fields based on the Reissner theory are calculated here for the first time. These results are used to calculate the J-integral (energy release rate) for both Kirchhoff and Reissner plate theories. Invoking Simmonds and Duva's [16] result that the value of the J-integral based on either theory is the same for thin plates, a universal relationship between the Kirchhoff theory stress intensity factors and the Reissner theory stress intensity factors is obtained for thin plates. Calculation of Kirchhoff theory stress intensity factors from finite elements based on energy release rate is illustrated. A small scale yielding like model of the crack tip fields is discussed, where the Kirchhoff theory fields are considered to be the far field conditions for the Reissner theory fields. It is proposed that, for thin plates, fracture toughness and crack growth rates be correlated with the Kirchhoff theory stress intensity factors.     

Thermal fracture of bonded dissimilar media containing a penny-shaped crack

The problem of uniform heat flow disturbed by an insulated penny-shaped crack along the common plane between two semi-infinite elastic media with different thermo-mechanical properties is formulated in terms of two potential functions in a half-space which in turn is reduced to a plane problem solvable by Muskhelishvili's method in complex function theory. Explicit expressions for the stress-intensity factors and the local stress field are derived and used in conjunction with Griffith's energy criterion to obtain the critical temperature gradient which motivates and produces initial crack extension along the bonding surface. The stress analysis involved is also applicable to a penny-shaped crack between two dissimilar solids under shear loadings.

     

Energy characteristics of simple shear granular flows

This work uses a 3-D discrete element simulation to calculate the elastic and kinetic energy for a nonuniform granular shear flow to determine whether the ratio of these energies is sufficient to identify specific flow regimes of granular materials in a fashion to other dimensionless parameters such as inertial number and dimensionless stiffness. We first obtain the critical packing fraction under isostatic compression, then analyze the mean and fluctuating parts of the elastic and kinetic energy as the granular flow reaches a steady state. External work performed on a system during granular flow partially dissipates into heat, while the remaining work is stored in particles as elastic and kinetic energy; thus processes occurring at a particle level not only control the energy transformation, but also affect the bulk behavior of a granular flow. The effective frictions are correlated with the mean elastic energy to mean kinetic energy ratio and it is interesting to find a power law function with an index of $-0.16$ for the systems used in this work. Analysis of this ratio’s ability to classify flow shows that its determination is quite sufficient to identify specific flow regimes of granular materials, even though energy has a scalar expression. Therefore, these energetics studies can provide a theoretical basis for unifying the mechanics of granular flows over the entire range of regimes.     

Coherent elastic energy calculation of ω particles in β matrix for Zr–Nb alloys

The misfit and coherent elastic energy caused by ω particles in β matrix is quantitatively calculated in this study. First, the coherent strain matrixes for four ω variants are established including the misfit parameters based on Khachaturyan’s theory. Then, the misfit and coherent elastic energy in athermal β → ω transition, and isothermal β → ω transition are calculated, respectively. The calculation results indicate that the coherent elastic energy gets maximum value when x _Nb = 0.08 (Nb content) and gets minimum value when x _Nb = 0.1518 in quenching Zr–Nb alloys, which are in fair agreement with experimental results. For isothermal β → ω transition, the misfit and coherent elastic energy depend on composition and aging temperature. The misfit caused by isothermal ω phase is much larger than the one caused by athermal ω phase. This results in larger coherent elastic energy in isothermal β → ω transition. In addition, the misfit is found as an approximate linear function of temperature and composition for Zr–Nb alloys, and the coherent elastic energy is revealed as an increasing function of |v _F –v _S | for the two kinds of transition.     

Effect of uniaxial stress on the electromechanical properties in ferroelectric thin films under combined loadings

The electromechanical properties of ferroelectric thin films under an alternating electric field and a static uniaxial compressive stress are investigated using the modified planar four-state Potts model. To implement the electromechanical properties and the coupling of the electrical and mechanical response, the mechanical energy density as well as the energy due to anisotropic switching between a-domain and c-domain are incorporated in the Hamiltonian. Besides, there are two contributions to the strain at each cell: eigenstrain and elastic strain. Our simulation results show that the longitudinal strain-electric field butterfly loop shifts downward along strain axis and that for the transverse strain shifts upward as the stress magnitude is increased. Moreover, the polarization-electric field hysteresis loop becomes a double-loop under a large compressive stress. The piezoelectric coefficient increases with the stress magnitude and reaches a maximum value at a critical stress level. It then gradually decreases to a small value at large stress magnitudes. Our results qualitatively agree with experimental ones.     

Jammed systems in slow flow need a new statistical mechanics

The slow dynamics of granular flow is studied as an extension of static granular problems, which, as a consequence of shaking or related regimes, can be studied by the methods of statistical mechanics. For packed (i.e. 'jammed'), hard and rough objects, kinetic energy is a minor and ignorable quantity, as is strain. Hence, in the static case, the stress equations need supplementing by 'missing equations' depending solely on configurations. These are in the literature; this paper extends the equilibrium studies to slow dynamics, claiming that the strain rate (which is a consequence of flow, not of elastic strain) takes the place of stress, and as before, the analogue of Stokes's equation has to be supplemented by new 'missing equations' which are derived and which depend only on configurations.     

Strain rate concentration and dynamic stress concentration for double‐edge‐notched specimens subjected to high‐speed tensile loads

Engineering plastics provide superior performance to ordinary plastics for wide range of the use. For polymer materials, dynamic stress and strain rate may be major factors to be considered when the strength is evaluated. Recently, high‐speed tensile test is being recognized as a standard testing method to confirm the strength under dynamic loads. In this study, therefore, high‐speed tensile test is analysed by the finite element method; then, the maximum dynamic stress and strain rate are discussed with varying the tensile speed and maximum forced displacement. The maximum strain rate increases with increasing the tensile speed u/t, but the strain rate concentration factor is found to be constant independent of tensile speed, which is defined as the maximum strain rate appearing at the notch root over the average nominal strain rate at the minimum section . It is found that the strain rate at the notch root depends on the dynamic stress rate at the notch root and independent of the notch root radius ρ. It is found that the difference between the static and dynamic maximum stress concentration (σ_yA,max ? σ_yA,st) at the notch root is proportional to the tensile speed when u/t = 5000 mm/s. Strain rate concentration factors are also discussed with varying the notch depth and specimen length. Based on the elastic strain rate concentration factor, the master curve is obtained useful for understanding the impact fracture of polycarbonate for the wide range of temperature and impact speed.     

Internal wave—granular temperature interaction: an energy balance study on granular flow

Summary Recent work demonstrated that under a certain resonance condition a granular flow becomes unstable to internal wave oscillations. The generated oscillation may result in an optimization of the flow by reducing the net power loss to frictional drag. The present paper examines possible interaction between ordered oscillations and granular temperature within a high-speed granular flow. It will be shown that one consequence of the initial generation of the internal waves is that there will be a net reduction in the granular temperature. In effect, the instability amounts to a conversion from chaotic to ordered oscillation, and thus a more efficient transport. The subsequent evolution of the flow, beyond the initial instability phase will be investigated in this study. The primary flow is assumed to consist of a sliding layer of negligible granular temperature placed above a high-shear basal layer of significant granular temperature. Each layer may grow or decrease in depth as the flow develops downstream and the two layers exchange mass across the interface. A flow instability behaving as a landslide with mixing confined to a thin layer near its base may evolve with time to become a fully-mixed debris flow, or vice-versa.Notation c _s shear wave speed in elastic medium - D nondimensional depth of steady-state sliding granular mass - d depth of slide mass - E _h depth-total kinetic energy of mean motion - E _m kinetic energy of the steady-state mean motion - E _T random motion energy or granular temperature - E _w ordered oscillation wave energy - e coefficient of restitution - f statistical quantity of granular material, reflecting dilute state - G shear modulus of elastic medium - g gravitational acceleration - g _o nondimensional gravitational acceleration - H nondimensional basal rapid granular flow depth - h depth of the basal rapid granular flow shear zone - J nondimensional granular particle diameter - K dynamic friction coefficient - P nondimensional pressure - P _ij uniform pressure tensor - p _o amplitude of perturbation pressure - p _ij total pressure tensor of granular flow - p _ij pressure tensor associated with instability wave - p _ij ^" random motion pressure tensor - S nondimensional wave number of shear wave - T granular temperature - T _ij uniform stress tensor - U uniform sliding velocity of a slide inx direction - U _B particle velocity in Bagnold's theory - U _b (y) uniform sliding velocity in basal rapid granular flow layer - u, v velocities inx direction andy direction, respectively - u, v ordered oscillation velocities of granular particles - u, v random motion velocities of granular particles - V nondimensional sliding velocity - W _D ,W _W energy terms - w ₀ mass transfer velocity at interface - heat dissipation - _w inelastic energy dissipation associated with instability waves - –K ² - wave number of perturbation wave, and _i are the real and imaginary parts of - ratio of shear velocity gradient over granular temperature - wave frequency of perturbation wave - solid fraction of granular material - _e bulk density of quasi-static granular material - _g bulk density of rapidly flowing granular material - p bulk density of granular particle - diameter of granular particle - _ij total stress in elastic medium - ' _ij perturbation stress in elastic medium - nondimensional parameter of particle - Rankine's earth pressure coefficient - * complex conjugate of the corresponding quantities - < > average over random motion period - Re {} the real part of complex quantity inside {}     

Elastic constants of two dental porcelains

The development of stress that affects the bonding in porcelain-fused-to-metal (PFM) systems can be influenced by the temperature dependence of the elastic constants of both systems. Instead of using the normal, static procedure, e.g. determining the slope of a stress-strain curve, and measuring the lateral and vertical strains, in this study the sonic resonance technique was used to determine the elastic moduli for two dental bodyporcelains. The sonic resonance technique involves the determination of both the flexural as well as the torsional resonance frequencies. From these values both Young's,Y, and shear moduli,G, are determined. Since two elastic constants are sufficient to describe completely the elastic response of isotropic materials, it was also possible to compute, by usingY andG, the bulk modulus,B, and the Poisson's ratio. Resonant frequency measurements taken at elevated temperatures resulted in correspondingly lower values for the elastic constants. Young's and shear moduli for two dental porcelains obtained in the range from 20° C (293 K) to 500° C (773 K) are presented in this study. These data may in the future be used for refined stress calculations in PFM systems.On study leave from University of Göttingen, D-3400 Göttingen, West Germany.Former NBS employee.