arXiv subjects

Matthew R. Kuhn

Publications and source records attributed to Matthew R. Kuhn.

At least 19 recordsLinked to original sources

arXiv2026

Thermomechanics of dense granular materials: a particle-scale perspective

The paper presents a broad thermomechanic framework for the isothermal rate-independent constitutive behavior of dense granular materials. The essential quantities in this framework are directly measurable in discrete element (DEM) simulations: free energy, dissipation, stress, and strain. The paper proposes that energy and dissipation are governed by two sets of internal variables: fabric variables that control the reversible stiffness and structure variables associated with internal sliding. The relevant fabric variables are identified and measured with simulations. Two hypotheses are considered for the structure variables: the macro-scale irreversible strain and an aggregate measure of the micro-scale frictional forces among sliding contacts. Both hypotheses are tested with simulations, which allow direct calculation of the internal variables. The paper then demonstrates the manner in which the measured variables are applied in incremental constitutive models. Among other findings are the following. (1) Dissipation from contact sliding is pervasive and occurs in all directions of incremental loading. (2) Contact motions are not reversed by a reversal of the strain direction, and contacts continue to slide when loading is reversed. (3) The free energy can not be assumed smoothly differentiable; instead, Gâteaux derivatives must be used with irreversible effects. (4) Basic assumptions of elastoplasticity are contravened: no region of purely reversible strain exists, no uniform yield direction exists, no uniform flow direction exists, and irreversible strain is not proportional to the projected total strain. A three-mechanism elastoplasticity model, however, closely fit the DEM results, and methods are demonstrated for quantifying the model. The results emphasize that advanced constitutive models are needed for capturing the general incremental behavior of granular materials.

cond-mat.soft↗

arXiv2026

A thermomechanical framework for strongly nonlocal continua

The paper presents a consistent thermomechanical framework for strongly nonlocal continua, a type of generalized media in which a material's response at a point depends on deformation and temperature gradients within its neighborhood. Such dependence is responsible for localization phenomena having an intrinsic size, and nonlocal modeling also serves as a practical regularization tool to prevent mesh dependence. Nonlocality is of the integral type, using a kernel function that gives the relative influence of neighboring points on a central point. The paper develops three sources of nonlocality: (1) nonlocal momentum balance equations; (2) nonlocal conservation laws (including the first and second laws of thermodynamics); and (3) a material's nonlocal constitutive structure. Balance equations are derived using the principle of virtual power, permitting non-smooth force and displacement fields and non-smooth boundary surfaces with indistinct normal directions. The balance equations differ from those of classical local continua. Conservation laws also differ from those of local continua, with the mechanical power and heating at a point being averaged over its neighborhood. These laws are developed for processes that are sufficiently slow to minimize additional meso-scale kinetic energy due to internal turbulence. Stress, entropy, and dissipative forces are obtained as averaged derivatives of the free energy function with respect to strain, temperature, and internal variables. An example is presented of a stretched elastoplastic bar with a small defect, and nonlocality is shown to impart a characteristic size to the deformation pattern.

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arXiv2026

On average stress within sub-regions of granular media

The paper concerns computation of average stress within small sub-regions of a larger static granular assembly, where the sub-region's boundary is allowed to pass through the assembly's particles. An exact average is computed for certain stress components and for certain categories of sub-regions. The paper also identifies those choices of sub-regions for which exact stress components may be exactly computed, but when these conditions are not met, provides reasonable bounds on the error.

cond-mat.other↗

arXiv2024

Deformation Measures for Granular Materials

The paper presents a micromechanical representation of deformation in 2D granular materials. The representation is a generalization of K. Bagi's work and is based upon the void-cell approach of M. Satake. The general representation applies to a material region partitioned into polygonal subregions. This representation possesses a certain consistency that allows for a unique assignment of the contribution that each contact displacement makes to the average deformation of an assembly. The paper addresses construction of the particle graph and appropriate data structures for use with the Discrete Element Method. The approach is applied in a numerical simulation of a two-dimensional assembly of disks. The author presents results of the distributions of deformation and particle-group rotation, with a resolution of about a single particle diameter. Deformation was very nonuniform, even at low strains. Micro-bands, thin linear zones of intense rotation, were also observed.

cond-mat.soft↗

arXiv2020

Linear-frictional contact model for 3D discrete element simulations of granular systems

The linear-frictional contact model is the most commonly used contact mechanism for discrete element (DEM) simulations of granular materials. Linear springs with a frictional slider are used for modeling interactions in directions normal and tangential to the contact surface. Although the model is simple in two dimensions, its implementation in 3D faces certain subtle challenges, and the particle interactions that occur within a single time-step require careful modeling with a robust algorithm. The paper details a 3D algorithm that accounts for the changing direction of the tangential force within a time-step, the transition from elastic to slip behavior within a time-step, possible contact sliding during only part of a time-step, and twirling and rotation of the tangential force during a time-step. Without three of these adjustments, errors are introduced in the incremental stiffness of an assembly. Without the fourth adjustment, the resulting stress tensor is not only incorrect, it is no longer a tensor. The algorithm also computes the work increments during a time-step, both elastic and dissipative.

cs.CE↗

arXiv2019

Heterogeneity and patterning in the quasi-static behavior of granular materials

Heterogeneity is classified in five categories---topologic, geometric, kinematic, static, and constitutive---and the first four categories are investigated in a numerical DEM simulation of biaxial compression. The simulation experiments show that the topology and geometric fabric become more variable during loading. The measured fluctuations in inter-particle movements are large, they increase with loading, and they extend to distances of at least eight particle diameters. Deformation and rotation heterogeneity are large and are expressed in spatial patterning. Stress heterogeneity is moderate throughout loading.

cond-mat.soft↗

arXiv2019

Contact rolling and deformation in granular media

The paper considers rotations at different scales in granular materials: the rotations of individual particles, the rolling and rigid-rotation of particle pairs, the rotational interactions of a particle within its cluster of neighbors, and the rotation of material regions. Numerical, Discrete Element Method (DEM) simulations on two- and three-dimensional (2D and 3D) assemblies show that particle rotations are diverse, that they increase with strain until the material begins to soften, and that they are expressed in spatial patterns, even at small strains. The interactions of a pair of particles are a combination of three modes: a contact deformation mode, a contact rolling mode, and a mode of rigid pair motions. Definitions are presented for each mode, including four different definitions of contact rolling. A rolling curl is also defined, which describes the cumulative rolling of neighboring particles around a central particle or sub-region. At a larger scale, material deformation and rotation are measured within small sub-regions of material, and the material deformation can be attributed to separate contributions of contact rolling, contact deformation, and the rigid-rotation of particle pairs. The diversity and extend of contact rolling were measured in 2D and 3D simulations. A dominant rolling pattern was observed, which resembles the interactions of rolling gears. This pattern can extend to distances of at least six particle diameters from a central particle.

cond-mat.soft↗

arXiv2019

Stability, bifurcation, and softening in discrete systems: A conceptual approach for granular materials

Matrix stiffness expressions are derived for the particle movements in an assembly of rigid granules having compliant contacts. The derivations include stiffness terms that arise from the particle shapes at their contacts. These geometric stiffness terms may become significant during granular failure. The geometric stiffness must be added to the mechanical stiffnesses of the contacts to produce the complete stiffness. With frictional contacts, this stiffness expression is incrementally nonlinear, having multiple loading branches. To aid the study of material behavior, a modified stiffness is derived for isolated granular clusters that are considered detached from the rest of a granular body. Criteria are presented for bifurcation, instability, and softening of such isolated and discrete granular sub-regions. Examples show that instability and softening can result entirely from the geometric terms in the matrix stiffness.

cond-mat.soft↗

arXiv2019

Are granular materials simple? An experimental study of strain gradient effects and localization

Experiments test the dependence of shearing stress on the first two gradients of shear strain. The tests were conducted by direct numerical simulation using the Discrete Element Method (DEM) on a large two-dimensional (2D) assembly of circular disks. The assembly was coerced into non-uniformly deformed shapes by applying body forces to the material region. The tests show that shearing stress is affected by both the first and second gradients of shear strain, and the measured responses to strain and its gradients are all incrementally non-linear. The dilation rate is unaffected by strain gradients. Particle rotations, although highly erratic, are, on average, consistent with the mean-field rotation and unaffected by strain gradients. In independent unconstrained tests, the material was sheared without body forces so that localization could freely occur. Three localization patterns were observed: microbands at very small strains; non-persistent shear bands at moderate strains; and persistent bands at large strains. The observed features of microbands and shear bands are consistent with the measured influences of shearing strain and its first two derivatives.

cond-mat.soft↗

arXiv2019

Micro-mechanics of fabric and failure in granular materials

The paper addresses the underlying source of two forms of induced anisotropy in granular materials: contact orientation anisotropy and contact force anisotropy. A rational, mathematical structure is reviewed for the manner in which fabric anisotropy emerges and evolves during loading. Fabric is expressed as an orientation density, and transport phenomena such as convection, contact generation, and diffusion control the rate of fabric evolution during loading. The paper proposes specific measurable forms for all terms, based upon the micro-mechanics of particle interactions. Discrete element (DEM) simulations are used to verify and quantify these terms, so that the theory can be applied to general loading conditions. The DEM simulations are of densely packed durable spheres, and the emphasis is on soil behavior at large strains, specifically on fabric and strength at the critical state. Once the theory has been developed and quantified, it is applied to predict the effect of the intermediate principal stress on strength.

cond-mat.soft↗

arXiv2018

Implementation of the Jager contact model for discrete element simulations

In three-dimensional discrete element method (DEM) simulations, the particle motions within a granular assembly can produce bewildering sequences of movements at the contacts between particle pairs. With frictional contacts, the relationship between contact movement and force is non-linear and path-dependent, requiring an efficient means of computing the forces and storing their histories. By cleverly applying the principles of Cattaneo, Mindlin, and Deresiewicz, Jurgen Jager developed an efficient approach for computing the full three-dimensional force between identical elastic spheres that have undergone difficult movement sequences (J. Jager, New Solutions in Contact Mechanics. WIT Press: Southampton, U.K.). This paper presents a complete Jager algorithm that can be incorporated into DEM codes and also describes three special provisions for DEM simulations: (1) a method for handling particle pairs that undergo complex tumbling and twirling motions in three-dimensions; (2) a compact data structure for storing the loading history of the many contacts in a large assembly; and (3) an approximation of the Jager algorithm that reduces memory demand. The algorithm addresses contact translations between elastic spheres having identical properties, but it does not resolve the tractions produced by twisting or rolling motions. A performance test demonstrates that the algorithm can be applied in a DEM code with modest increases in computation time but with more substantial increases in required storage.

cond-mat.soft↗

arXiv2018

Dense granular flow at the critical state: maximum entropy and topological disorder

After extensive quasi-static shearing, dense dry granular flows attain a steady-state condition of porosity and deviatoric stress, even as particles are continually rearranged. The paper considers two-dimensional flow and derives the probability distributions of two topological measures of particle arrangement---coordination number and void valence---that maximize topological entropy. By only considering topological dispersion, the method closely predicts the distribution of void valences, as measured in discrete element (DEM) simulations. Distributions of coordination number are also derived by considering packings that are geometrically and kinetically consistent with the particle sizes and friction coefficient. A cross-entropy principle results in a distribution of coordination numbers that closely fits DEM simulations.

cond-mat.soft↗

arXiv2018

Transient rolling friction model for discrete element simulations of sphere assemblies

The rolling resistance between a pair of contacting particles can be modeled with two mechanisms. The first mechanism, already widely addressed in the DEM literature, involves a contact moment between the particles. The second mechanism involves a reduction of the tangential contact force, but without a contact moment. This type of rotational resistance, termed creep-friction, is the subject of the paper. Within the creep-friction literature, the term "creep" does not mean a viscous mechanism, but rather connotes a slight slip that accompanies rolling. Two extremes of particle motions bound the range of creep-friction behaviors: a pure tangential translation is modeled as a Cattaneo-Mindlin interaction, whereas prolonged steady-state rolling corresponds to the traditional wheel-rail problem described by Carter, Poritsky, and others. DEM simulations, however, are dominated by the transient creep-friction rolling conditions that lie between these two extremes. A simplified model is proposed for the three-dimensional transient creep-friction rolling of two spheres. The model is an extension of the work of Dahlberg and Alfredsson, who studied the two-dimensional interactions of disks. The proposed model is applied to two different systems: a pair of spheres and a large dense assembly of spheres. Although creep-friction can reduce the tangential contact force that would otherwise be predicted with Cattaneo-Mindlin theory, a significant force reduction occurs only when the rate of rolling is much greater than the rate of translational sliding and only after a sustained period of rolling. When applied to the deviatoric loading of an assembly of spheres, the proposed creep-friction model has minimal effect on macroscopic strength or stiffness. At the micro-scale of individual contacts, creep-friction does have a modest influence on the incremental contact behavior.

cond-mat.soft↗

arXiv2018

Investigation of cyclic liquefaction with discrete element simulations

A discrete-element method (DEM) assembly of virtual particles is calibrated to approximate the behavior of a natural sand in undrained loading. The particles are octahedral, bumpy clusters of spheres that are compacted into assemblies of different densities. The contact model is a Jager generalization of the Hertz contact, which yields a small-strain shear modulus that is proportional to the square root of confining stress. Simulations made of triaxial extension and compression loading conditions and of simple shear produce behaviors that are similar to sand. Undrained cyclic shearing simulations are performed with nonuniform amplitudes of shearing pulses and with 24 irregular seismic shearing sequences. A methodology is proposed for quantifying the severities of such irregular shearing records, allowing the 24 sequences to be ranked in severity. The relative severities of the 24 seismic sequences show an anomalous dependence on sampling density. Four scalar measures are proposed for predicting the severity of a particular loading sequence. A stress-based scalar measure shows superior efficiency in predicting initial liquefaction and pore pressure rise.

cond-mat.soft↗

arXiv2018

Stress-induced anisotropy in granular materials: fabric, stiffness, and permeability

The loading of a granular material induces anisotropies of the particle arrangement (fabric) and of the material's strength, incremental stiffness, and permeability. Thirteen measures of fabric anisotropy are developed, which are arranged in four categories: as preferred orientations of the particle bodies, the particle surfaces, the contact normals, and the void space. Anisotropy of the voids is described through image analysis and with Minkowski tensors. The thirteen measures of anisotropy change during loading, as determined with three-dimensional discrete element simulations of biaxial plane strain compression with constant mean stress. Assemblies with four different particle shapes were simulated. The measures of contact orientation are the most responsive to loading, and they change greatly at small strains, whereas the other measures lag the loading process and continue to change beyond the state of peak stress and even after the deviatoric stress has nearly reached a steady state. The paper implements a methodology for characterizing the incremental stiffness of a granular assembly during biaxial loading, with orthotropic loading increments that preserve the principal axes of the fabric and stiffness tensors. The linear part of the hypoplastic tangential stiffness is monitored with oedometric loading increments. This stiffness increases in the direction of the initial compressive loading but decreases in the direction of extension. Anisotropy of this stiffness is closely correlated with a particular measure of the contact fabric. Permeabilities are measured in three directions with lattice Boltzmann methods at various stages of loading and for assemblies with four particle shapes. Effective permeability is negatively correlated with the directional mean free path and is positively correlated with pore width.

cond-mat.soft↗

arXiv2018

Maximum disorder model for dense steady-state flow of granular materials

A flow model is developed for dense shear-driven granular flow. As described in the geomechanics literature, a critical state condition is reached after sufficient shearing beyond an initial static packing. During further shearing at the critical state, the stress, fabric, and density remain nearly constant, even as particles are being continually rearranged. The paper proposes a predictive framework for critical state flow, viewing it as a condition of maximum disorder at the micro-scale. The flow model is constructed in a two-dimensional setting from the probability density of the motions, forces, and orientations of inter-particle contacts. Constraints are applied to this probability density: constant mean stress, constant volume, consistency of the contact dissipation rate with the stress work, and the fraction of sliding contacts. The differential form of Shannon entropy, a measure of disorder, is applied to the density, and the Jaynes formalism is used to find the density of maximum disorder in the underlying phase space. The resulting distributions of contact force, movement, and orientation are compared with two-dimensional DEM simulations of biaxial compression. The model favorably predicts anisotropies of the contact orientations, contact forces, contact movements, and the orientations of those contacts undergoing slip. The model also predicts the relationships between contact force magnitude and contact motion. The model is an alternative to affine-field descriptions of granular flow.

cond-mat.soft↗

arXiv2018

Contact transience during slow loading of dense granular materials

The irregularity of particle motions during quasi-static deformation is investigated using discrete element (DEM) simulations of sphere and sphere-cluster assemblies. A total of three types of interparticle movements are analyzed: relative motions of particle centers, relative motions of material points of two particles at their contact, and the traversal of contacts across the surfaces of particles. Motions are a complex combination of rolling, sliding, and elastic distortion at the contacts, and all motions are highly irregular and variant, qualities that increase with increasing strain. The relative motions of particle centers diverge greatly from those of an affine displacement of the particles. The motions of the nonconvex sphere-cluster particles were more regular that those of the spheres. The paper also investigates the effect of the distance between two remote particles and their pair-wise relative displacements. Even for particle pairs separated by more than six intermediate particles, the relative motions do not conform with the mean deformation (affine) field. Force chains are shown to be transient features, which survive only briefly across elapsed strains.

cond-mat.soft↗

arXiv2018

The critical state of granular media: Convergence, stationarity, and disorder

Discrete-element simulations are used to monitor several micro-scale characteristics within a granular material, demonstrating their convergence during loading toward the critical state, their stationarity at the critical state, and the evolution of their disorder toward the critical state. Convergence, stationarity and disorder are studied in the context of the Shannon entropy and two forms of Kullback-Leibler relative entropy. Probability distributions of 20 aspects of micro-scale configuration, force and movement are computed for three topological objects: particles, voids and contacts. The probability distributions of these aspects are determined at numerous stages during quasi-static biaxial compression and unloading. Not only do stress and density converge to the critical state, but convergence and stationarity are manifested in all of the micro-scale aspects. The statistical disorder (entropy) of micro-scale movements and strains generally increases during loading until the critical state is reached. When the loading direction is reversed, order is briefly restored, but continued loading induces greater disorder in movements and strains until the critical state is reached again.

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