Mechanical properties of non-cohesive polygonal particle aggregates

We numerically investigate the effective material properties of aggregates consisting of soft convex polygonal particles, using the discrete element method. First, we construct two types of “sand piles” by two different procedures. Then we measure the averaged stress and strain, the latter via imposing a 10% reduction of gravity, as well as the fabric tensor. Furthermore, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution as well as the stress distribution. In the next step, elastic constants are determined, assuming Hooke’s law to be locally valid throughout the sand piles. We determine the relationship between invariants of the stress and strain tensor, observing that the behaviour is nonlinear. While linear elastic behaviour near the centre of the pile is compatible with our data, nonlinearity signals the transition to plastic behaviour near its surface. A similar behaviour was assumed by Cantelaube et al. (Static multiplicity of stress states in granular heaps. Proc R Soc Lond A 456:2569–2588, 2000). We find that the macroscopic stress and fabric tensors are not collinear in the sand pile and that the elastic behaviour is anisotropic in an essential way.     

Analysis of the influence of crushing on the behavior of granular materials under shear

The behavior of granular materials mainly depends on the mechanical and engineering properties of particles in its structural matrix. Crushing or breakage of granular materials under compression or shear occurs when the energy available is sufficient to overcome the resistance of the material. Relatively little systematic research has been conducted regarding how to evaluate or quantify particle crushing and how it effects the engineering properties of the granular materials. The aim of this study is to investigate the effect of crushing on the bulk behavior of granular materials by using manufactured granular materials (MGM) rather than using a naturally occurring cohesionless granular material. MGM allow changing only one particle parameter, namely the “crushing strength”. Four different categories of MGM (with different crushing strength) are used to study the effect on the bulk shear strength, stiffness modulus, friction and dilatancy angle “engineering properties”. A substantial influence on the stress–strain behavior and engineering properties of granular materials is observed. Higher confining stress causes some non-uniformity (strong variations/jumps) in volumetric strain and a constant volumetric strain is not always observed under large shear deformations due to crushing, i.e. there is no critical state with flow regime (with constant volumetric strain).     

A numerical photogrammetry technique for measuring microscale kinematics and fabric in Schneebeli materials

We present a numerical photogrammetry technique for obtaining semi–automated measurements concerning kinematic fields, i.e. translation and rotation of each “grain”, and fabric properties of a two-dimensional analogue granular material. An example is given in which the technique is applied in a biaxial compression test on a specimen consisting of 1,300 rods. The information that can be recorded by the technique is discussed along with its accuracy. This work is funded by the EU project Degradation and Instabilities in Geomaterials with Application to Hazard Mitigation (DIGA) in the framework of the Human Potential Program, Research Training Networks (HPRN-CT-2002-00220).     

Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method

Second-rank fabric tensors have been extensively used to describe structural anisotropy and to predict orthotropic elastic constants. However, there are many different definitions of, and approaches to, determining the fabric tensor. Most commonly used is a fabric tensor based on mean intercept length measurements, but star volume distribution and star length distribution are commonly used, particularly in studies of trabecular bone. Here, we investigate the effect of the fabric tensor definition on elastic constant predictions using both synthetic, idealized microstructures as well as a micrograph of a porous ceramic. We use an efficient implantation of a symmetric Galerkin boundary element method to model the mechanical response of the various microstructures, and also use a boundary element approach to calculate the necessary volume averages of stress and strain to obtain the effective properties of the media.     

On power-law type relationships and the Ludwigson explanation for the stress-strain behaviour of AISI 316 stainless steel

The influence of prior cold work on the stress strain behavior of a type 316 stainless steel is investigated at 300 K. The Ludwigson relation and the often cited explanation for the Ludwigson type stress strain behavior found in stainless steels to a changeover from planar slip to cross slip is reexamined. It appears that the Ludwigson type behavior is more a consequence of a structure sensitive “hardness state” of the material. This “hardness state” is expressed in terms of an equivalent plastic strain ɛ₀. The Swift equation that essentially incorporate such a correction term for strain and describes well, the stress strain curve and the work hardening in a type 316 stainless steel.     

On distortion of tensor and yield criteria of granular materials

The distortion of tensor defined by use of normals on the tensor ellipsoid and the mean distortion are introduced. It is explained that the distortion represents the unit shear strain for the finite strain tensor, and the shear-normal stress ratio for the stress tensor. The spatial mobilized plane which plays a very important role in the mechanics of granular materials is regarded as a plane on which the distortion of stress tensor coincides with its mean distortion, and the yield criteria of granular materials are considered applying the concept of the distortion of stress tensor.     

RVE computations with error control and adaptivity: the power of duality

The goal of computational homogenization is to obtain the macro-scale response, normally in terms of macro-scale stress for given macro-scale deformation, via RVE-computations. In this paper we investigate, in a systematic manner, the effects of Dirichlet and Neumann boundary conditions on the RVE. Adaptive computations are carried out with respect to, in particular, control of the error in the macro-scale stress tensor. This requires the corresponding dual solutions. As a new result, it is shown how the same dual solutions can be conveniently used in computing the algorithmic tangent stiffness tensor, thereby demonstrating the “power of duality”.     

Structural model of a microinhomogeneous cracked material

We develop a mathematical model of microscopically inhomogencous but macroscopically isotropic materials with statistically distributed components of tensors of stiffness and strength. In this model, the material is represented as the continuous set of “characteristic” (i.e., typical of a given material) disjoint microscopic domains (microvolumes). The microinhomogeneous material is identified with an “effectively homogeneous” material in such a way that, at the same points, the components of the displacement vector determined for these materials are equal. It is assumed that, for each “characteristic” microvolume the parameters of stiffness and strength of the material are constant and can be obtained as values of an arbitrary random variable distributed according to the Weibull law and averaged over a certain random interval of any length. The components of the tensor averaged as indicated above are also regarded as random variables distributed according to the normal law with the same probability of hitting any arbitrarily located “characteristic” microvolume. The model is based on the assumption that the material is isotropic both macroscopically and in any “characteristic” microvolume. The stress-strain state of the microinhomogeneous material is described by the “effective” (averaged over its volume) components of the stress tensor. The model takes into account cracks in the material if their length exceeds the size of the relevant “characteristic” volume. The model is justified for the case of an infinite microinhomogeneous cracked plane under uniaxial tension. It is shown that the parameters determining the stressed state of this plane are not independent in the vicinity of the crack tip. The relevant constraints are given by equations of the model. The choice of these parameters which ignores the indicated constraints leads to results contradicting well-known physical facts. By using the symmetry properties of the system under consideration and physical reasoning, we obtain equations for the determination of the size of “characteristic” domains and physically reasonable dependences of the maximal “effective” tensile stresses and their direction on the parameter of inhomogeneity of the material and average volume of defects. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Lviv. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 4, pp. 5–16, July–August, 1996.     

An Implicit integration scheme for generalized viscoplasticity with dynamic recovery

This paper is concerned with the development of a general implicit time-stepping integrator for the flow and evolution equations in a recent representative class of generalized viscoplastic models, involving both hardening and dynamic recovery mechanisms. To this end, the computational framework is developed on the basis of the unconditionally stable, backward Euler difference scheme. Its mathematical structure is of sufficient generality to allow a systematic treatment of several internal variables of the tensorial and scalar types. The matrix forms developed are directly applicable in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). The closed-form expressions for residual vectors and the algorithmic, (consistent) material tangent stiffness array are given explicitly, with the maximum matrix sizes “optimized” to depend only on the number of independent stress components, but not the number of internal state variables involved. Several numerical simulations are given to assess the performance of the developed schemes.     

A local constitutive model with anisotropy for ratcheting under 2D axial-symmetric isobaric deformation

A local constitutive model for anisotropic granular materials is introduced and applied to isobaric (homogeneous) axial-symmetric deformation. The simplified model (in the coordinate system of the bi-axial box) involves only scalar values for hydrostatic and shear stresses, for the volumetric and shear strains as well as for the new ingredient, the anisotropy modulus. The non-linear constitutive evolution equations that relate stress and anisotropy to strain are inspired by observations from discrete element method (DEM) simulations. For the sake of simplicity, parameters like the bulk and shear modulus are set to constants, while the shear stress ratio and the anisotropy evolve with different rates to their critical state limit values when shear deformations become large. When applied to isobaric deformation in the bi-axial geometry, the model shows ratcheting under cyclic loading. Fast and slow evolution of the anisotropy modulus with strain. Lead to dilatancy and contractancy, respectively. Furthermore, anisotropy acts such that it works “against” the strain/stress, e.g., a compressive strain builds up anisotropy that creates additional stress acting against further compression.     

Uncertainty of measurement results in various scales

The relationship between the concepts of “error” and “uncertainty” of measurement results is examined. The concepts of “standard uncertainty,” “combined standard uncertainty,” and “expanded uncertainty” are shown to be inapplicable in nonmetric scales of quantities and properties, in which the general concept of “uncertainty” in the broad sense is recommended. Translated from Izmeritel'naya Tekhnika, No. 5, pp. 29–30, May, 2000.     

Stress-induced anisotropy in sand under cyclic loading

The anisotropy of a granular material’s structure will influence its response to applied loads and deformations. Anisotropy can be either inherent (e.g. due to depositional process) or induced as a consequence of the applied stresses or strains. Discrete element simulations allow the interactions between individual particles to be explicitly simulated and the fabric can be quantified using a fabric tensor. The eigenvalues of this fabric tensor then give a measure of the anisotropy of the fabric. The coordination number is a particle scale scalar measure of the packing density of the material. The current study examines the evolution of the fabric of a granular material subject to cyclic loading, using two-dimensional discrete element method (DEM) simulations. Isotropic consolidation modifies and reduces the inherent anisotropy, but anisotropic consolidation can accentuate anisotropy. The ratio of the normal to shear spring stiffness at the particle contacts in the DEM model affects the evolution of anisotropy. Higher ratios reduce the degree of anisotropy induced by anisotropic consolidation. The anisotropy induced by cyclic loading depends on the amplitude of the loading cycles and the initial anisotropy.     

Macro deformation and micro structure of 3D granular assemblies subjected to rotation of principal stress axes

This paper presents a numerical investigation on the behavior of three dimensional granular materials during continuous rotation of principal stress axes using the discrete element method. A dense specimen has been prepared as a representative element using the deposition method and subjected to stress rotation at different deviatoric stress levels. Significant plastic deformation has been observed despite that the principal stresses are kept constant. This contradicts the classical plasticity theory, but is in agreement with previous laboratory observations on sand and glass beads. Typical deformation characteristics, including volume contraction, deformation non-coaxiality, have been successfully reproduced. After a larger number of rotational cycles, the sample approaches the ultimate state with constant void ratio and follows a periodic strain path. The internal structure anisotropy has been quantified in terms of the contact-based fabric tensor. Rotation of principal stress axes densifies the packing, and leads to the increase in coordination numbers. A cyclic rotation in material anisotropy has been observed. The larger the stress ratio, the structure becomes more anisotropic. A larger fabric trajectory suggests more significant structure re-organization when rotating and explains the occurrence of more significant strain rate. The trajectory of the contact-normal based fabric is not centered in the origin, due to the anisotropy in particle orientation generated during sample generation which is persistent throughout the shearing process. The sample sheared at a lower intermediate principal stress ratio \((b=0.0)\) has been observed to approach a smaller strain trajectory as compared to the case \(b=0.5\), consistent with a smaller fabric trajectory and less significant structural re-organisation. It also experiences less volume contraction with the out-of plane strain component being dilative.     

Nonequilibrium structural transitions as a mechanism of turbulence

A possible mechanism for turbulence is proposed and substantiated for the first time, whereby turbulence is considered as a nonequilibrium transition in ensembles of defects of the microscopic shear type, which are treated as real defects in the molecular structure of liquids. A statistical foundation is given for the evolution equations for the tensor order parameter, which characterizes an ensemble of such defects and has the meaning of nonequilibrium fluctuations of the strain rate. The types of macroscopic fluctuation modes of the strain rate are determined as self-similar solutions of the evolution equations for this tensor order parameter under conditions of nonequilibrium transitions, and qualitative correspondences are established between these solutions and real scenarios for the transition to turbulence. The Kolmogorov scaling laws (the natures of the “viscous” and “ inertial” intervals) are explained for fully developed turbulence. Pis’ma Zh. Tekh. Fiz. 23, 81–88 (July 12, 1997)     

Investigation of the influence of frictional viscosity regularization on quasi-static gas-solid flow predictions in a conical spouted bed with non-porous draft tube

The stress in the quasi-static particle flow is often modeled through the Mohr-Coulomb failure criterion. In the extension to complex three-dimensional flows, a granular viscosity is introduced through a tensorial rheology and the deviatoric frictional stress tensor is assumed aligned with the strain rate tensor. This granular viscosity is singular as the shear rate approaches zero, regardless of the local rheology. We discuss the influence of regularizing such a frictional viscosity on the particle circulation rate and other measured characteristics in a laboratory scale draft tube spouted bed. The friction between particles is modeled either with a constant Coulomb rheology or using a local particle pressure and strain-rate based friction known as $\mu \left(I\right)$-rheology. The predictions appear very dependent on the regularization parameter introduced by the method. The mean properties of the flow (e.g. circulation and pressure drop) monotonically converge towards the measurements when the regularization parameter tends to zero. In other respects, the two regularization models regarded in this study induced similar hydrodynamics within the spouted bed of interest. But the analysis of the conditional averages of the inertial number and the fraction of the solids in the quasi-static regime shows that the extent and staticity of the quasi-static region is sensitive to changes to the regularization parameter or regularization function.     

On finite damage: ductile fracture-damage evolution

The tensorial entities pertaining to the presence of a damage zone within a representative material sample are obtained from the micro-macro transition of kinematic quantities in finite deformation. It is shown that they reduce to a ‘small-damage’ tensor as the finite deformation measures reduce to their common infinitesimal counterpart.Based on an assumed local (microscopic) velocity field within a sample, the process of void growth from a spherical inclusion is then quantified by a tensorial damage evolution equation. This equation gives the rate of change of damage tensor as a function of the overall strain rate, actual damage state and some scale factor. The problem of oblate shape cavity growth in the direction of a higher (positive) overall strain component is examined.     

Statistical properties of a 2D granular material subjected to cyclic shear

This work focuses on the evolution of structure and stress for an experimental system of 2D photoelastic particles that is subjected to multiple cycles of pure shear. Throughout this process, we determine the contact network and the contact forces using particle tracking and photoelastic techniques. These data yield the fabric and stress tensors and the distributions of contact forces in the normal and tangential directions. We then find that there is, to a reasonable approximation, a functional relation between the system pressure, P, and the mean contact number, Z. This relationship applies to the shear stress τ, except for the strains in the immediate vicinity of the contact network reversal. By contrast, quantities such as P, τ and Z are strongly hysteretic functions of the strain, ε. We find that the distributions of normal and tangential forces, when expressed in terms of the appropriate means, are essentially independent of strain. We close by analyzing a subset of shear data in terms of strong and weak force networks.     

B-spline goal-oriented error estimators for geometrically nonlinear rods

We consider goal-oriented a posteriori error estimators for the evaluation of the errors on quantities of interest associated with the solution of geometrically nonlinear curved elastic rods. For the numerical solution of these nonlinear one-dimensional problems, we adopt a B-spline based Galerkin method, a particular case of the more general isogeometric analysis. We propose error estimators using higher order “enhanced” solutions, which are based on the concept of enrichment of the original B-spline basis by means of the “pure” k-refinement procedure typical of isogeometric analysis. We provide several numerical examples for linear and nonlinear output functionals, corresponding to the rotation, displacements and strain energy of the rod, and we compare the effectiveness of the proposed error estimators.     

A 3D fourth order fabric tensor approach of anisotropy in granular media

We propose a micromechanical approach for granular media, with a particular account of the texture-induced anisotropy and of the strain localization rule. The approach is mainly based on the consideration of a fourth order fabric tensor able to capture general anisotropy which can be induced by complex distribution of contacts. Incorporation of this fourth order fabric tensor in a suitable homogenization scheme allows to determine the corresponding macroscopic elastic properties of the granular material. For this purpose, in addition to the classical Voigt upper bound, a new kinematics-based localization rule is proposed. It generalizes the one formulated by Cambou et al. [B. Cambou, Ph. Dubujet, F. Emeriault, F. Sidoroff, Eur. J. Mech. A/Solids 14 (1995) 225–276] in the case of an isotropic contact distribution. The results of the complete model compare well to numerical simulations results when available [C.S. Chang, C.L. Liao, Appl. Mech. Rev. 47 (1 Part 2) (1994) 197–207] (case of isotropic distribution of contacts). Finally, the interest of the fourth order fabric tensor based approach combined with the proposed localization rule is shown for different distributions of contacts by comparing its predictions to those given by a second order fabric tensor approach.     

Inhomogeneous deformation of crystalline skeleton of syndiotactic polypropylene under uniaxial stretching

Structure changes of syndiotactic polypropylene (sPP) under uniaxial stretching are studied with the combination of micro-tensile tester and in situ wide angle X-ray diffraction (WAXD) measurement. Lamellae stacked “vertically” and “parallel” to the stretching direction (defined as “V” and “P” part) are separated on the basis of two-dimensional WAXD patterns. For all samples with different lamellar thickness, two critical points named as b ₁ and b ₂ were found in the stress–strain curves, while b ₁ and b ₂ points are the onsets of the rotation for the lamellae of “V” part and “P” part, respectively. The corresponding true stress and true strain for b ₂ point are bigger than that of b ₁ , demonstrating that for samples with initial isotropic lamellar orientation, inhomogeneous deformation of crystalline skeleton induced by uniaxial stretching is universal. And after b ₁ point, “stress-induced melting” always occurs simultaneously with lamellar slips. Furthermore, the relationship between lamellar thickness and the true stress for b ₁ and b ₂ point was also studied, illustrating a linear correlation between ln σ and 1/l (σ is the corresponding true stress, l is the lamellar thickness), consistent with Young’s model. However, the critical true strains for these two points did not change with the varying thickness.