Ching S. Chang

Stress-strain relationship for granular materials based on the hypothesis of best fit

Abstract Stress strain relationships of a granular system are derived based on the premise that the mean field of displacement is the best fit of actual particle displacements. Based on the best fit hypothesis, two fundamental relationships are derived: (1) the average strain as a function of contact displacements, and (2) the mean field of contact force as a function of stress. These two relationships lead to a stress-strain law without the kinematic constraint of uniform strain. For an assembly with isotropic packing structure, closed-form stress-strain relationships are explicitly derived in which the modulus can be determined from the inter-particle stiffness. The difference between the derived stress-strain relationship and that derived from Voigt hypothesis is discussed. The effect of fabric variation is illustrated in the theory. Discrete element analyses are also performed and the results are used to compare with the estimated moduli from the proposed theory. In the comparison, the solution bounds for granular assembly system are illustrated.

Modeling Time-Dependent Behavior of Soft Sensitive Clay

The paper focuses on investigating the destructuration process during time-dependent stress-strain evolution. For this purpose, various oedometer tests and triaxial tests on intact and reconstituted samples of soft sensitive Vanttila clay were carried out. Based on experimental observations, a new elastic viscoplastic model, extended from the overstress theory of Perzyna, is developed. The proposed model accounts for inherent and induced anisotropy, interparticle bonds and bond degradation, and viscosity. The determination of model parameters is discussed, demonstrating how all model parameters can be determined in a straightforward way and no additional test is needed for the proposed model compared to the modified Cam clay model. The model is implemented into a finite-element code, which enables coupled consolidation analyses. The model is used to simulate various strain-rate and creep tests under one-dimensional and triaxial conditions on the intact samples of Vanttila clay. The comparisons between experimental results and simulations show that the model has good predictive ability on the time-dependent behavior of a soft sensitive clay.

Constitutive relation for a particulate medium with the effect of particle rotation

Abstract The constitutive behavior of granular assemblies is investigated taking into account the effect of particle rotation. Continuum fields are assumed by tensorial polynomial expansion for the discrete variables, namely, particle displacements and rotations. The strain measures for the packing are obtained from the particle displacement and rotation. From the principle of virtual work, the stress measures for the packing are expressed in terms of the contact forces, the contact moments and the geometric measures for the packing structure. The derived stress-strain relationship is then evaluated by an example of a randomly packed assembly of circular disks loaded in two different conditions. Deformation of the packing calculated from the constitutive equations is compared with the results obtained from the discrete method of computer simulation to show the applicability of the method.

Second-gradient constitutive theory for granular material with random packing structure

Abstract A second-gradient constitutive law for granular media is derived, in which stress is a function of the second order of strain gradient. The constitutive coefficients for granular materials with an isotropic packing structure are derived in explicit terms of inter-particle stiffness and particle size. In the present constitutive theory, the total number of elastic constants for the isotropic granular material is four: two of them are the usual Lame constants; the other two are associated with the second gradient of strain. The derived constitutive relationships illustrate the important role of micro-scale properties in the macro-scale behavior. The influence of inter-particle stiffness and particle size on the response of material due to the second gradient of strain is discussed. This paper also discusses the connection between the second-gradient theory and non-local theory. The second-gradient theory can be regarded as the first-order approximation of non-local theory.

Estimates of elastic moduli for granular material with anisotropic random packing structure

Abstract For randomly packed spheres, the packing structure can be characterized by a fabric tensor representing the orientation distribution of inter-particle contacts. The densities of inter-particle contacts are generally not the same for all inter-particle orientations, exhibiting the anisotropy of a packing structure. Utilizing the fabric tensor, elastic moduli of granular material with anisotropic packing structures are derived in explicit terms of inter-particle properties. The derivation is based on two different methods, namely the kinematic method and the static method. Results derived from kinematic and static methods provide, respectively, the upper and lower estimates of elastic moduli. Ranges of elastic moduli are discussed for granular materials with different inter-particle properties and fabric parameters. The close relationship between fabric tensor and material symmetry is demonstrated.

Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory

SummaryThis contribution is one in a series of two papers. In the current paper a constitutive law is developed that includes the micro-structural effects by particle displacement as well as particle rotation. Both degrees of freedom can be related to corresponding macroscopic kinematic continuum variables, where the resulting gradients of displacement are selected up to the fourth-order and the gradients of rotation up to the third order. The elastic micro-structural properties for an individual particle are used to derive the macro-level behavior for a fabric of equal-sized spherical particles, leading to a second-gradient micro-polar formulation. In this model, all coefficients are expressed in terms of particle stiffness and particle structure. It is shown that the second-gradient micro-polar model can be reduced to simpler forms, such as the classic linear elastic model, the second-gradient model and the Cosserat model. In the accompanying paper these reduced forms are treated in more detail by analyzing the corresponding dispersion relations for plane body wave propagation.

Effective elastic moduli of heterogeneous granular solids

Abstract A micro-mechanical model is employed to study the elastic stress-strain behavior of heterogeneous granular solids. The granular material is idealized as a collection of spherical particles interacting through inter-particle contacts. Based on this idealized model an equivalent continuum description of the granular solid is envisaged and the overall stiffness tensor of the granular solid is determined in terms of the stiffness of the inter-particle deformation. To facilitate the derivation of overall stiffness tensor, the granular solid is considered to be composed of continuum cells made of a single particle and the associated void space. A local stiffness tensor is defined for each cell. The local stiffness tensor is obtained in terms of the inter-particle stiffness, the number of contacts and the relative position of the neighboring particles. The local stiffness tensor is utilized to obtain the overall behavior of a representative volume of granular solid through the “self consistent” averaging technique. The overall stress and strain for the representative volume are determined as a volume average of the corresponding local quantities. To account for the heterogeneity of deformation in the granular medium, a “concentration” factor is defined for each cell. Based on the concept of volume averaging and the “concentration” tensor an overall stiffness tensor is derived for the granular solid. The applicability of the derived micro-mechanical model is evaluated by comparing its results with those obtained from the computer simulation method.

A micromechanical-based micropolar theory for deformation of granular solids

Abstract Granular material, perceived as a collection of particles, is modelled as a micropolar continuum taking account of the discreteness and microstructure of the material. The new feature of the proposed model is that the constitutive coefficients are derived in explicit terms of grain contact properties. In addition, the constitutive law, the equilibrium equations and the compatibility equations for granular material are derived to completely define a boundary value problem. Using the derived constitutive law and field equations, a procedure based on finite element analysis is described to obtain approximated solutions for boundary value problems. An example is shown for a granular packing under boundary pressure. Solutions of the example are compared with that obtained from the discrete element method to show the applicability of this method. Based on this model, the effects of particle rotation and couple stress on the deformation behavior of granular material are also studied.

Elastic material constants for isotropic granular solids with particle rotation

Abstract The granular material perceived as a collection of particles is modelled as a micropolar continua taking account of particle interaction and microstructure of the material. Explicit expressions of constitutive constants in terms of elastic inter-particle stiffness are derived for granular solids with Isotropie random fabric distribution. Based on the derived expressions for Isotropic material, six material constants are identified instead of two constants used in conventional elasticity. The derived constitutive constants in explicit terms of inter-particle contact properties provide a fundamental understanding of these constants and furnish useful inter-relations among these constants. The physical meaning of these material constants is discussed with emphasis on the role of particle spin. Finite element analysis incorporating these material constants is described and used to obtain solutions for boundary value problems. Examples are given for granular solids under boundary pressure to show the effects of inter-particle properties and material constants on the behavior of stress distribution and deformation in granular material.

Properties of granular packings under low amplitude cyclic loading

Abstract 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.