Elastic Grains Research Articles

A rational multiscale nonlinear constitutive model for freeze–thaw rocks under triaxial compression

The stability and durability of rocks in cold regions are significantly impacted by the degradation of mechanical properties caused by freeze–thaw (F–T) environment. In this work, we shall propose a rational multiscale nonlinear constitutive model based on thermodynamics, micromechanics, and fractional calculus theory to describe the complete deformation and failure process of F–T rocks under triaxial compression. The F–T rocks at the mesoscale are regarded as consisting of porous matrix and cracks, while porous matrix is composed of the micropores and elastic solid grains at the microscale. According to experimental observations, we assume the F–T action mainly causes micropores growth and cracks opening, and mechanical damage is resulted from the initiation and propagation of cracks. In this context, the effects of F–T and mechanical damage on effective elastic properties of rocks can be quantitatively analyzed by using the two-step Mori–Tanaka (M-T) homogenization method. After subtly deriving the total free energy function of F–T rocks under compression, we systematically develop specific criteria for describing open cracks closure deformation, mechanical damage evolution and frictional sliding induced plastic distortion. Note that to correctly capture the plastic deformation characteristics, the non-orthogonal flow rule based on fractional differential calculations is employed. Following that, analytical analyses and numerical implementation of the proposed model are conducted. The performance of the model is evaluated by the simulations with experimental data on two kinds of F–T rocks, and discussions on parameters sensitivity and effects of fractional order are followed.

A combined SPH-DEM approach for extremely deformed granular packings: validation and compression tests

We present a novel smoothed particle hydrodynamics-discrete element method (SPH-DEM) approach for the simulation of deformable granular media. First, we show how the method converges to the analytical solution in a simple contact mechanics problem, namely the Hertz contact law for elastic spherical grains. Second, we analyze the evolution of a 2D packing of disks under uniaxial compression, displaying the evolution of key metrics of the packing such as the coordination number and the vertical stress. We show that the code produces data in quantitative agreement with what is known from literature. Finally, we demonstrate that our SPH-DEM coupling can be used to study packings of deformable grains from the onset of jamming to extremely compacted states, reaching packing fractions of ϕ≃0.995\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi \\simeq 0.995$$\\end{document}.

Friction-damage coupled models and macroscopic strength criteria for ice-saturated frozen silt with crack asperity variation by a micromechanical approach

This paper aims to present physical friction-damage coupled models and derive macroscopic nonlinear strength criteria for ice-saturated frozen silt. Two material scales are considered here. The material at the mesoscale is composed of a frozen silt matrix and embedded mesocracks. At the microscale, the frozen silt matrix is composed of elastic mineral grains and elastic ice crystals. Considering this microstructure, for the ice-saturated frozen silt, the plastic deformation is related to frictional sliding along the mesocracks, and damage is due to the propagation of mesocracks. We obtain the effective elastic properties with a two-step linear Mori-Tanaka homogenization procedure. For the second homogenization step, we deduce the macroscopic stress-strain relations and local thermodynamic forces associated with damage and plasticity. Then, an energy release-based damage criterion is developed to capture the evolution of damage, and a friction sliding criterion with an associated/nonassociated local plastic flow rule is introduced to describe the rate of plasticity. The friction sliding criterion considers the variation of surface asperities of mesocracks as a result of pressure melting and damage evolution. In this context, associated/nonassociated multiscale friction-damage models are established. Furthermore, associated/nonassociated macroscopic nonlinear strength criteria as inherent parts of the corresponding multiscale models are derived under a large range of compressive stresses. For application, a semi-implicit local numerical algorithm of the multiscale models is developed. The accuracy of associated/nonassociated multiscale models is assessed by comparing their numerical simulations with conventional triaxial compression (CTC) tests. Although the associated multiscale model correctly predicts the nonlinear strength behavior and axial deformation of the Lanzhou frozen silt under CTC tests with different temperatures, it fails to quantitatively reproduce volumetric deformation. Nonassociated multiscale model predictions with experimental data of volumetric deformation are significantly improved compared to the associated multiscale model.

Coarse–grained modeling with hierarchical deformable and rigid assemblages (HiDRA)

One of the major challenges in computational mechanics of materials remains to bridge the length-scale and time-scale gaps between the computational and experimental methods to study the microstructure of materials. Traditional molecular dynamics is a powerful tool at the nanoscale, but it does not allow to simulate such mechanical experiments as AFM microscopy dealing with sub-micrometric lengths and microsecond times. Coarse-grained (CG) modeling is one of the possible solutions to fill these gaps. CG modeling does not consider all atoms in a system, but the “pseudo-atoms” or “grains”, approximating groups of these atoms. This allows a significant reduction in the number of considered interactions and degrees of freedom and, consequently, saves the computational resources.In this work, the unique method for CG modeling named “Hierarchical Deformable and Rigid Assemblages” (HiDRA) is proposed. The grains are chosen as assemblages of the atoms in the original crystal lattice. The method allows incorporating the grains’ elasticity so that the overall strain uniformly distributes in the lattice to avoid the unexpected stress concentrations in the bonds between the grains. Grains interact via forces acting to the surface atoms. Equations of motion for the grains are derived taking the uniform strain of the grain as an independent variable. In-house code is developed to perform the simulations with elastic grains. As an application, the HiDRA method is used to study the longitudinal vibrations and full 3D dynamics of chains of one-dimensional grains. Phonon spectra show to what extent large grains can correctly describe wave propagation in molecular chains.

On the Poroelastic Biot Coefficient for a Granitic Rock

The Biot coefficient is a parameter that is encountered in the theory of classical poroelasticity, dealing with the mechanics of a fluid-saturated porous medium with elastic grains and an elastic skeletal structure. In particular, the coefficient plays an important role in the partitioning of externally applied stresses between the pore fluid and the porous skeleton. The conventional approach for estimating the Biot coefficient relies on the mechanical testing of the poroelastic solid, in both a completely dry and a fully saturated state. The former type of tests to determine the skeletal compressibility of the rock can be performed quite conveniently. The latter tests, which determine the compressibility of the solid material constituting the porous skeleton, involve the mechanical testing of the fully saturated rock. These tests are challenging when the rock has a low permeability, since any unsaturated regions of the rock can influence the interpretation of the compressibility of the solid phase composing the porous rock. An alternative approach to the estimation of the solid grain compressibility considers the application of the multi-phasic theories for the elasticity of composite materials, to estimate the solid grain compressibility. This approach requires the accurate determination of the mineralogical composition of the rock using XRD, and the estimation of the elasticity characteristics of the minerals by appealing to published literature. This procedure is used to estimate the Biot coefficient for the Lac du Bonnet granite obtained from the western region of the Canadian Shield.

A single grain boundary parameter to characterize normal stress fluctuations in materials with elastic cubic grains

A finite element analysis of intergranular normal stresses is performed in order to identify a possible statistical correlation between the intergranular normal stresses and the corresponding grain boundary type within a polycrystalline aggregate. Elastic continuum grains of cubic lattice symmetry are assumed in the analysis. Meaningful results are obtained by analyzing the first two statistical moments of grain boundary normal stresses obtained on several grain boundary types.Among the five macroscopic parameters (5D) describing a grain boundary, the orientation of the grain boundary plane relative to the two adjacent crystal lattices (4D) is found to be the most important property influencing the normal stresses. To account for it, a single new (1D) parameter E12 is introduced, which combines the geometrical aspect of grain boundary with its material properties and measures the average stiffness of grain boundary neighborhood along the grain boundary normal direction. It is demonstrated that E12, in combination with Zener elastic anisotropy index A, is able to accurately predict normal stress fluctuations on any grain boundary type in a material with cubic lattice symmetry. It is argued that largest normal stresses most likely form on grain boundaries whose normals are oriented along the stiffest direction in both adjacent grains (〈111〉 for crystals with A>1 or 〈001〉 for crystals with A<1). Moreover, it is shown that classification of grain boundaries according to their propensity to exhibit large normal stresses can be trivially reduced to the (analytical) calculation of the corresponding effective stiffness parameter E12.A few practical implications are discussed relevant to intergranular stress-corrosion cracking of Coincidence Site Lattice grain boundaries. For example, it is highlighted that in face-centered cubic materials the coherent Σ3 twin grain boundaries, which are known for their very high cracking resistance, nevertheless exhibit largest intergranular normal stresses, indicating that cracking resistance is associated with high grain boundary strength.

Silo discharge of mixtures of soft and rigid grains.

We study the outflow dynamics and clogging phenomena of mixtures of soft, elastic low-friction spherical grains and hard frictional spheres of similar size in a quasi-two-dimensional (2D) silo with narrow orifice at the bottom. Previous work has demonstrated the crucial influence of elasticity and friction on silo discharge. We show that the addition of small amounts, even as low as 5%, of hard grains to an ensemble of soft, low-friction grains already has significant consequences. The mixtures allow a direct comparison of the probabilities of the different types of particles to clog the orifice. We analyze these probabilities for the hard, frictional and the soft, slippery grains on the basis of their participation in the blocking arches, and compare outflow velocities and durations of non-permanent clogs for different compositions of the mixtures. Experimental results are compared with numerical simulations. The latter strongly suggest a significant influence of the inter-species particle friction.

The tensile strength of dust aggregates consisting of small elastic grains: constraints on the size of condensates in protoplanetary discs

ABSTRACT A consensus view on the formation of planetesimals is now exposed to a threat, since recent numerical studies on the mechanical properties of dust aggregates tend to dispute the conceptual picture that submicrometer-sized grains conglomerate into planetesimals in protoplanetary discs. With the advent of precise laboratory experiments and extensive computer simulations on the interaction between elastic spheres comprising dust aggregates, we revisit a model for the tensile strength of dust aggregates consisting of small elastic grains. In the framework of contact mechanics and fracture mechanics, we examine outcomes of computer simulations and laboratory experiments on the tensile strength of dust aggregates. We provide a novel analytical formula that explicitly incorporates the volume effect on the tensile strength, namely, the dependence of tensile strength on the volume of dust aggregates. We find that our model for the tensile strength of dust aggregates well reproduces results of computer simulations and laboratory experiments, if appropriate values are adopted for the elastic parameters used in the model. Moreover, the model with dust aggregates of submicrometer-sized grains is in good harmony with the tensile strength of cometary dust and meteoroids derived from astronomical observations. Therefore, we reaffirm the commonly believed idea that the formation of planetesimals begins with conglomeration of submicrometer-sized grains condensed in protoplanetary discs.

Accounting for Small-Scale Heterogeneity and Variability of Clay Rock in Homogenised Numerical Micromechanical Response and Microcracking

Clayey rocks have a complex microstructure with multiple characteristic lengths. Deformation under mechanical loading generally induces damage by microcracking, which essentially concerns the scale of mineral inclusions embedded in the clay matrix. The modelling of these materials is considered within the framework of a double scale approach, by numerical homogenisation, of the squared finite element method type. This allows a heterogeneous microstructure of the material to be taken into account and a distribution of morphological properties to be introduced. Emphasis is placed on the generation of microstructures satisfying experimental observations, and keeping a certain simplicity to fit into the framework of double scale modelling. The material characteristics and behaviour are defined at the grain scale: the mineralogical properties include the mineral phase proportions and the grain morphology, while the material constituents are represented by elastic grains separated by damageable cohesive crack models. Then, the overall microscale behaviour of the material under solicitation is derived from equilibrated elementary area (EA) configuration and computational homogenisation. The variability of the material response is studied with regard to small-scale aspects as microstructure variability, microstructure size, grain angularity, and properties of grain contacts. Deformation analyses at grain contacts emphasise a dominant shear deformation mode and the development of decohesion between grains. The latter induces microfaulting processes across the entire EA and strain softening of the overall response. Moreover, the improvement of microscale behaviour modelling opens new possibilities for more realistic multi-scale modelling and upscaled behaviour of heterogeneous rocks.

On the applicability of linear elastic fracture mechanics scaling relations in the analysis of intergranular fracture of brittle polycrystals

Crack propagation in polycrystalline specimens is studied by means of a generalized finite element method with linear elastic isotropic grains and cohesive grain boundaries. The corresponding mode-I intergranular cracks are characterized using a grain boundary brittleness criterion that depends on cohesive law parameters and average grain boundary length. It is shown that load–displacement curves for specimens with the same microstructure and for various cohesive law parameters can be obtained from a master load–displacement curve by means of simple linear elastic fracture mechanics scaling relations. This property is a consequence of the independence of intergranular crack paths from cohesive law parameters. Perfect scaling is obtained for cases characterized by the same grain boundary brittleness number, irrespective of its value, whereas scaling is approximated for cases with different but relatively large values of the grain boundary brittleness number. The former case corresponds to grain boundary traction profiles that are identical apart from a scale factor; in the latter case, a large grain boundary brittleness number implies similar, apart from a scale factor, traction profiles. By exploiting this property, it is demonstrated that computationally expensive simulations can be avoided above a certain grain boundary brittleness threshold value.

A comprehensive parametric study of grain-based models for rock failure process simulation

Grain and contact are the two key components in 3DEC-GBMs (grain-based models) because they control the micro-mechanical behavior and consequently the macro-mechanical behavior of rock. Three types of grains – rigid, elastic, and breakable grains – are considered in this study to explore the influence of grain properties on the mechanical behavior of rock. Equivalent 2D plane strain problems are solved using 3DEC to reduce computation time. A non-uniform distribution of grain size is used to generate numerical models using Neper, a software package for polycrystal generation and meshing. A comprehensive parametric study of properties of contact and grain of the three types of grains is conducted and calibration procedures are suggested for numerical modeling. The modeling results indicate that both the contact and the grain properties affect the macroscopic mechanical behavior of synthetic rock specimens. All the three types of grains produce reasonable results for pre-peak deformation (E and υ) and macroscopic strength (UCS, σt, C and ϕ) parameters. Each grain type has advantages and disadvantages in numerical modeling. Rigid grains cannot produce volumetric strain properly and a near-zero residual friction angle of contact is needed to capture post-peak deformation behavior due to the grain interlocking effect. Elastic grains also need a very low residual friction angle of contact and brittle rock deformation behavior cannot be captured. The complex calibration procedure and long computation time are the main issues of breakable grains, although it simulates well the volumetric strain, does not require a very low contact residual friction angle, and captures brittle rock behaviors better than other two types of grains. The findings from this study are useful for rock failure process simulation using the 3DEC-GBM modeling approach.

On the effect of grains interface parameters on the macroscopic properties of polycrystalline materials

In this paper, the influence of microscopic parameters on the macroscopic behaviour of polycrystalline materials under different loading configuration is investigated. Linear elastic grains with zero-thickness cohesive interfaces are considered at the microscale with in depth introduction of effective parameters. A multiscale method based on homogenisation technique is employed to bridge the scales. In order to minimize the homogenisation error, a representative volume element (RVE) of the microscopic structure is statistically determined to be used in the numerical analysis. For each loading condition of the RVE, several numerical examinations are conducted to illustrate the relationship between the microscopic parameters. Finally, the effects of microscopic critical fracture energies, maximum tensile and shear strengths of grain interfaces on the mechanical properties, i.e. stress-strain curve and yield surface at the macroscale are discussed in details. It is shown that macroscopic yield surface and stress-strain curves can be used to characterise the microscopic properties.

Micro-mechanical modelling of mechanical and electrical properties in homogeneous piezoelectric ceramic by using boundary integral formulations

Recent experiments on polycrystalline materials show that microcrystalline materials have a strong dependency ona grain size. In this study, mechanical and electrical properties of polycrystalline materials in micro level were studied by using averaging theorems. To completely understand the size-dependency of polycrystalline materials, an integral non-local approach that can predict the stress-strain relations for these materials was presented. In microcrystalline materials, crystalline and grain-boundary were considered as two separate phases. Mechanical properties of the crystalline phase were modelled using crystalline brittle material and is composed of randomly distributed and orientated single crystal anisotropic elastic grains. For microcrystalline materials, the surface-to-volume ratio of the grain boundaries is low enough to ignore its contribution to the elastic deformation. Therefore, the grain boundary phase was not considered in microcrystalline materials and mechanical properties of the crystalline phase were modelled using appropriate integral non-local approach. Finally, the constitutive equations for polycrystalline materials were implemented into a boundary integral equation and the results and some examples are provided for piezoelectric ceramic.

A cascade continuum micromechanics model for the effective elastic properties of porous materials

The elastic properties of porous materials with a disordered pore structure are estimated using the mean-field Eshelby homogenization scheme together with the principle of recurrence to generate a cascade of effective microstructures as a function of the porosity and the cascade level n. Starting with the Hashin–Shtrikman upper bound for porous materials, the proposed cascade micromechanics model generates a hierarchy of micro-structures which evolve from an initial configuration of a porous material with spherical pores embedded within an elastic solid phase consistent with the Mori–Tanaka matrix inclusion morphology to a porous material characterized by a hierarchic distribution of spherical elastic grains. The model is explicit and allows for an easy computational implementation. It predicts physically consistent threshold porosities, characteristic for the specific morphology of the porous material under consideration, beyond which the material loses its stiffness. The validity of the cascade micromechanics model is evaluated against experimental data for various materials ranging from foam to ceramics with different pore structures.

Pre-failure damage analysis for brittle rocks under triaxial compression

In this study pre-failure damage characteristics are investigated by conducting a series of conventional triaxial compression tests and corresponding numerical simulations for granite from Kirchberg (Saxony, Germany). First, lab test results are analysed. For evaluation and prediction of damage in the pre-failure range, damage indices are proposed considering increase of dissipation energy density and decrease of secant modulus. It can be concluded that the damage increases slowly before the reversal of volumetric strain and accelerates quickly afterwards. Then, a new failure criterion is deduced based on a correlation of maximum elastic strain energy density with uniaxial compressive strength and confining pressure. Finally, a micro-mechanical grain-based discrete element model using Voronoi blocks to represent minerals is set-up. It considers elastic grains and elasto-plastic contact deformations as well as inter- and intra-granular fracturing. The triaxial compression tests are simulated and the damage process including evolution of damage indices are investigated in detail. The proposed approaches can be used to predict and analyse limit and damage state of brittle rocks.

Heat Transfer in Composites Subjected to Temperature Variations

The heat transfer problem in the 2-phase composite material containing metallic and elastic phases, subjected to quick temperature variations, is the aim of theoretical analysis. The full description of the composite behaviour starts from the formulation of governing equations at 2-scale levels: micro-and macro-, passes through specification of the internal structure of the composite and finishes by numerical solution of the heat transfer problem through the considered material sample. The most important in the analysis are thermo-mechanical properties of the composite components creating the material. The big difference of the both phases properties (mismatch) can create additional difficulty in the exact thermal description of the composite. It is necessary also to specify by scanning electron microscopy (SEM) observations a real material internal structure, which includes: grain shapes and matrix, to create of the proper size of the Representative Volume Element (RVE) for numerical calculations.In the numerical example we analyse cermet, i.e. the composite build up of metallic matrix (cobalt) and tungsten carbide elastic grains, which exhibits high brittleness. Heat transfer across this very complex material causes heat flux concentrations in the metallic phase and further stress concentrations. These concentrations act as sources of damage initiators at the binder/carbide grains interfaces. The obtained results lead to the conclusion that the spatial distribution and content of the metallic phase first of all influence the heat transfer across the 2-phase composites.

Statistical strength criterion for materials with hexagonal close-packed crystal lattice

Dependences of concentration of the normal mi-crostresses arising from interaction of elastic anisotropic grains, on a type of a combined stress-strain state are in-vestigated. With use of the received dependences, for poly-crystals which grains have no cleavage planes, the statisti-cal strength criterion of the oriented fracture is suggested. The experimental data analysis for titan has shown that the criterion suggested provides more reliable strength prognosis for rigid stress states in comparison with known phenomenological criteria. Two parameters of a suggested criterion for hexagonal close-packed crystal lattice materi-als are determined for polycrystal model using the finite element analysis. Three others are suggested to be defined experimentally by means of a tension test performed on cylindrical specimens with notches of a various tip sharp-ness. DOI: http://dx.doi.org/10.5755/j01.mech.20.3.4964

Porous-grain-upper-boundary model and its application to Tarim Basin carbonates

Most of the carbonates in the Tarim Basin in northwest China are low-porosity and low-permeability rocks. Owing to the complexity of porosity in carbonates, conventional rockphysics models do not describe the relation between velocity and porosity for the Tarim Basin carbonates well. We propose the porous-grain-upper-boundary (PGU) model for estimating the relation between velocity and porosity for low-porosity carbonates. In this model, the carbonate sediments are treated as packed media of porous elastic grains, and the carbonate pores are divided into isolated and connected pores The PGU model is modified from the porous-grain-stiff-sand (PGST) model by replacing the critical porosity with the more practical isolated porosity. In the implementation, the effective elastic constants of the porous grains are calculated by using the differential effective medium (DEM) model. Then, the elastic constants of connected porous grains in dry rocks are calculated by using the modified upper Hashin-Shtrikman bound. The application to the Tarim carbonates shows that relative to other conventional effective medium models the PGU model matches the well log data well.

An analysis of competing toughening mechanisms in layered and particulate solids

The relative potency of common toughening mechanisms is explored for layered solids and particulate solids, with an emphasis on crack multiplication and plasticity. First, the enhancement in toughness due to a parallel array of cracks in an elastic solid is explored, and the stability of co-operative cracking is quantified. Second, the degree of synergistic toughening is determined for combined crack penetration and crack kinking at the tip of a macroscopic, mode I crack; specifically, the asymptotic problem of self-similar crack advance (penetration mode) versus $$90^{\circ }$$ symmetric kinking is considered for an isotropic, homogeneous solid with weak interfaces. Each interface is treated as a cohesive zone of finite strength and toughness. Third, the degree of toughening associated with crack multiplication is assessed for a particulate solid comprising isotropic elastic grains of hexagonal shape, bonded by cohesive zones of finite strength and toughness. The study concludes with the prediction of R-curves for a mode I crack in a multi-layer stack of elastic and elastic–plastic solids. A detailed comparison of the potency of the above mechanisms and their practical application are given. In broad terms, crack tip kinking can be highly potent, whereas multiple cracking is difficult to activate under quasi-static conditions. Plastic dissipation can give a significant toughening in multi-layers especially at the nanoscale.

Heat transfer and stress concentrations in a two‐phase polycrystalline composite structure. Part I: Theoretical modelling of heat transfer

AbstractThe aim of the paper is a numerical analysis of heat transfer in polycrystalline composites, containing metallic or elastic interfaces, i. e. metal‐ceramic composites (MCCs) or ceramic matrix composites (CMCs). An example of MCCs is a two‐phase material composed of brittle grains WC joined by the plastic binder Co. The work focuses also on the description of the heat concentrations in composites due to temperature changes, taking into account the real material internal structure, which includes grain shapes and interfaces. Under thermal loading the heat transfer in MMCs is mainly through metallic interfaces, where the heat flux concentrations occur due to different thermal properties of both phases. They can cause also stress concentrations and act as sources of damage initiators at the binder/carbide grains interfaces. The different contents of the interphase between elastic grains were taken into account in the analysis, starting from the interfaces of the small thickness: 2–4 μm and volume content approximately 10% (e. g. Co), to the higher values up to 30%. The major conclusion is the following: the volume content of the metallic phase (Co binder) and its architecture (spatial distribution) inside the material strongly influences the heat transfer through the composites.