Behavior of depleted elastic networks: Comparison of effective-medium and numerical calculations

A soft departure from jamming: the compaction of deformable granular matter under high pressures

The high-pressure compaction of three dimensional granular packings is simulated using a bonded particle model (BPM) to capture linear elastic deformation. In the model, grains are represented by a collection of point particles connected by bonds. A simple multibody interaction is introduced to control Poisson's ratio and the arrangement of particles on the surface of a grain is varied to model both high- and low-frictional grains. At low pressures, the growth in packing fraction and coordination number follow the expected behavior near jamming and exhibit friction dependence. As the pressure increases, deviations from the low-pressure power-law scaling emerge after the packing fraction grows by approximately 0.1 and results from simulations with different friction coefficients converge. These results are compared to predictions from traditional discrete element method simulations which, depending on the definition of packing fraction and coordination number, may only differ by a factor of two. As grains deform under compaction, the average volumetric strain and asphericity, a measure of the change in the shape of grains, are found to grow as power laws and depend heavily on the Poisson's ratio of the constituent solid. Larger Poisson's ratios are associated with less volumetric strain and more asphericity and the apparent power-law exponent of the asphericity may vary. The elastic properties of the packed grains are also calculated as a function of packing fraction. In particular, we find the Poisson's ratio near jamming is 1/2 but decreases to around 1/4 before rising again as systems densify.

Universal scaling for disordered viscoelastic matter near the onset of rigidity

The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.

Phase transitions, percolation, fracture of materials, and deep learning

Percolation and fracture propagation in disordered solids represent two important problems in science and engineering that are characterized by phase transitions: loss of macroscopic connectivity at the percolation threshold p_{c} and formation of a macroscopic fracture network at the incipient fracture point (IFP). Percolation also represents the fracture problem in the limit of very strong disorder. An important unsolved problem is accurate prediction of physical properties of systems undergoing such transitions, given limited data far from the transition point. There is currently no theoretical method that can use limited data for a region far from a transition point p_{c} or the IFP and predict the physical properties all the way to that point, including their location. We present a deep neural network (DNN) for predicting such properties of two- and three-dimensional systems and in particular their percolation probability, the threshold p_{c}, the elastic moduli, and the universal Poisson ratio at p_{c}. All the predictions are in excellent agreement with the data. In particular, the DNN predicts correctly p_{c}, even though the training data were for the state of the systems far from p_{c}. This opens up the possibility of using the DNN for predicting physical properties of many types of disordered materials that undergo phase transformation, for which limited data are available for only far from the transition point.

Multifunctional twisted kagome lattices: Tuning by pruning mechanical metamaterials

This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus B=0 and shear modulus G>0, with either a generalized untwisted kagome lattice with B>0 and G>0 or with a honeycomb lattice with B>0 and G=0. These lattices exhibit jamming-like critical endpoints at which B, G, or both B and G jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of continuous rigidity percolation transitions at which both B and G begin to grow continuously from zero. The Poisson ratio and G/B can be continuously tuned throughout their physical range via random dilution in a manner analogous to "tuning by pruning" in random jammed lattices. These lattices can be produced with modern techniques, such as three-dimensional printing, for constructing metamaterials.

Jamming as a Multicritical Point

The discontinuous jump in the bulk modulus B at the jamming transition is a consequence of the formation of a critical contact network of spheres that resists compression. We introduce lattice models with underlying undercoordinated compression-resistant spring lattices to which next-nearest-neighbor springs can be added. In these models, the jamming transition emerges as a kind of multicritical point terminating a line of rigidity-percolation transitions. Replacing the undercoordinated lattices with the critical network at jamming yields a faithful description of jamming and its relation to rigidity percolation.

Mechanics of anisotropic spring networks

We construct and analyze a model for a disordered linear spring network with anisotropy. The modeling is motivated by, for example, granular systems, nematic elastomers, and ultimately cytoskeletal networks exhibiting some underlying anisotropy. The model consists of a triangular lattice with two different bond occupation probabilities, p(x) and p(y), for the linear springs. We develop an effective medium theory (EMT) to describe the network elasticity as a function of p(x) and p(y). We find that the onset of rigidity in the EMT agrees with Maxwell constraint counting. We also find beyond linear behavior in the shear and bulk modulus as a function of occupation probability in the rigid phase for small strains, which differs from the isotropic case. We compare our EMT with numerical simulations to find rather good agreement. Finally, we discuss the implications of extending the reach of effective medium theory as well as draw connections with prior work on both anisotropic and isotropic spring networks.

Effective-medium theory of a filamentous triangular lattice

We present an effective-medium theory that includes bending as well as stretching forces, and we use it to calculate the mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability p. We find a rigidity threshold p(b) which has the same value for all positive bending rigidity and a crossover characterizing bending, stretching, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at p(CF)=/~2/3 of the lattice when fiber bending rigidity vanishes.

Actively stressed marginal networks

We study the effects of motor-generated stresses in disordered three-dimensional fiber networks using a combination of a mean-field theory, scaling analysis, and a computational model. We find that motor activity controls the elasticity in an anomalous fashion close to the point of marginal stability by coupling to critical network fluctuations. We also show that motor stresses can stabilize initially floppy networks, extending the range of critical behavior to a broad regime of network connectivities below the marginal point. Away from this regime, or at high stress, motors give rise to a linear increase in stiffness with stress. Finally, we demonstrate that our results are captured by a simple, constitutive scaling relation highlighting the important role of nonaffine strain fluctuations as a susceptibility to motor stress.

Redundancy and cooperativity in the mechanics of compositely crosslinked filamentous networks

The cytoskeleton of living cells contains many types of crosslinkers. Some crosslinkers allow energy-free rotations between filaments and others do not. The mechanical interplay between these different crosslinkers is an open issue in cytoskeletal mechanics. Therefore, we develop a theoretical framework based on rigidity percolation to study a generic filamentous system containing both stretching and bond-bending forces to address this issue. The framework involves both analytical calculations via effective medium theory and numerical simulations on a percolating triangular lattice with very good agreement between both. We find that the introduction of angle-constraining crosslinkers to a semiflexible filamentous network with freely rotating crosslinks can cooperatively lower the onset of rigidity to the connectivity percolation threshold-a result argued for years but never before obtained via effective medium theory. This allows the system to ultimately attain rigidity at the lowest concentration of material possible. We further demonstrate that introducing angle-constraining crosslinks results in mechanical behaviour similar to just freely rotating crosslinked semflexible filaments, indicating redundancy and universality. Our results also impact upon collagen and fibrin networks in biological and bio-engineered tissues.

Nonlinear effective-medium theory of disordered spring networks

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor sites are connected by central-force springs. A lattice of N sites has O(√N) zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant κ are added with probability P=Δz/4, where Δz=z-4 and z is the average coordination number. The effective medium static NNN spring constant κ(m) scales as P(2) for P≪κ and as P for P≫κ, yielding a frequency scale ω*~Δz and a length scale l*~(Δz)(-1). To a very good approximation at small nonzero frequency, κ(m)(P,ω)/κ(m)(P,0) is a scaling function of ω/ω*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order ω*.

Anisotropic linear elastic properties of fractal-like composites

In this work, the anisotropic linear elastic properties of two-phase composite materials, made up of square inclusions embedded in a matrix, are investigated. The inclusions present a fractal hierarchical distribution and are supposed to have the same Poisson's ratio as the matrix but a different Young's modulus. The effective elastic moduli of the medium are computed at each fractal iteration by coupling a position-space renormalization-group technique with a finite element analysis. The study allows to obtain and generalize some fundamental properties of fractal composite materials.

Tangential-force model for interactions between bonded colloidal particles

Recently, Pantina and Furst [Phys. Rev. Lett. 94, 138301 (2005)] experimentally demonstrated the existence of tangential forces between bonded colloidal particles and the capability of these bonds to supporting bending moments. We introduce a model to be used in computer simulations that describes these tangential interactions. We show how the model parameters can be determined from experimental data. Simulations using the model are in agreement to the measurement by Pantina and Furst. Application of the model to an aggregate with fractal structure leads to more realistic behavior than using classical approaches only.

Sound propagation in a constrained lattice of beads: high-frequency behavior and dispersion relation

We report on acoustic wave propagation in a regular array of nominally identical beads under isotropic static stress. The weak polydispersity of the beads makes the contact lattice random. Time-frequency analysis of the acoustic signal is performed and allows measurement of the full lattice dispersion relation. Comparison with the theoretical prediction for a perfect triangular lattice gives an indication of the level of randomness in the contact lattice. The results extend, in a consistent way, a previous study restricted to long wavelength propagation [B. Gilles and C. Coste, Phys. Rev. Lett. 90, 174302 (2003)]: The contact lattice is ordered by increasing the stress, and the smaller the wavelength, the higher the stress required to get regular lattice behavior. Measurements involving ballistic propagation of the coherent wave, whatever its frequency, evidence reversible lattice behavior under compression and/or decompression. Nevertheless, correlations of short wavelength incoherent waves are a sensitive probe of disorder, and allow us to exhibit a small irreversible evolution of the lattice.

Effective medium theory of semiflexible filamentous networks

We develop an effective medium approach to the mechanics of disordered, semiflexible polymer networks and study the response of such networks to uniform and nonuniform strain. We identify distinct elastic regimes in which the contributions of either filament bending or stretching to the macroscopic modulus vanish. We also show that our effective medium theory predicts a crossover between affine and nonaffine strain, consistent with both prior numerical studies and scaling theory.

General up to next-nearest-neighbour elasticity of triangular lattices in three dimensions

We establish the most general form of the discrete elasticity of a two-dimensional triangular lattice embedded in three dimensions, taking into account up to next-nearest-neighbour interactions. Besides crystalline system, this is relevant to biological physics (e.g. red blood cell cytoskeleton) and soft matter (e.g. percolating gels, etc.). In order to correctly impose the rotational invariance of the bulk terms, it turns out to be necessary to take into account explicitly the elasticity associated with the vertices located at the edges of the lattice. We find that some terms that were suspected in the literature to violate rotational symmetry are, in fact, admissible.

Mean-field description of jamming in noncohesive frictionless particulate systems

A theory for kinetic arrest in isotropic systems of repulsive, radially interacting particles is presented that predicts exponents for the scaling of various macroscopic quantities near the rigidity transition that are in agreement with simulations, including the nontrivial shear exponent. Both statics and dynamics are treated in a simplified, one-particle level description and coupled via the assumption that kinetic arrest occurs on the boundary between mechanically stable and unstable regions of the static parameter diagram. This suggests that the arrested states observed in simulations are at (or near) an elastic buckling transition. Some additional numerical evidence to confirm the scaling of microscopic quantities is also provided.

Newtonian fluid meets an elastic solid: coupling lattice Boltzmann and lattice-spring models

We integrate the lattice Boltzmann model (LBM) and lattice spring model (LSM) to capture the coupling between a compliant bounding surface and the hydrodynamic response of an enclosed fluid. We focus on an elastic, spherical shell filled with a Newtonian fluid where no-slip boundary conditions induce the interaction. We calculate the "breathing mode" oscillations for this system and find good agreement with analytical solutions. Furthermore, we simulate the impact of the fluid-filled, elastic shell on a hard wall and on an adhesive surface. Understanding the dynamics of fluid-filled shells, especially near adhesive surfaces, can be particularly important in the design of microcapsules for pharmaceutical and other technological applications. Our studies reveal that the binding of these capsules to specific surfaces can be sensitive to the physical properties of both the outer shell and the enclosed fluid. The integrated LBM-LSM methodology opens up the possibility of accurately and efficiently capturing the dynamic coupling between fluid flow and a compliant bounding surface in a broad variety of systems.

Pressure dependence of the sound velocity in a two-dimensional lattice of Hertz-Mindlin balls: mean-field description

We study the dependence on the external pressure P of the velocities v(L,T) of long wavelength sound waves in a confined two-dimensional hexagonal close-packed lattice of 3D elastic frictional balls interacting via one-sided Hertz-Mindlin contact forces, whose diameters exhibit mild dispersion. The presence of an underlying long range order enables us to build an effective medium description, which incorporates the radial fluctuations of the contact forces acting on a single site. Due to the nonlinearity of Hertz elasticity, self-consistency results in a highly nonlinear differential equation for the "equation of state" linking the effective stiffness of the array with the applied pressure, from which sound velocities are then obtained. The results are in excellent agreement with existing experimental results and simulations in the high- and intermediate-pressure regimes. It emerges from the analysis that the departure of v(L)(P) from the ideal P(1/6) Hertz behavior must be attributed primarily to the fluctuations of the stress field, rather than to the pressure dependence of the number of contacts.

Fractures in heterogeneous two-dimensional systems

A two-dimensional triangular lattice with bond disorder is used as a testing ground for fracture behavior in heterogeneous materials in strain-controlled conditions. Simulations are performed with two interaction potentials (harmonic and Lennard-Jones types) and different breaking thresholds. We study the strain range where the fracture progressively develops from the first to the last breakdown. Scaling properties with the lattice size are investigated: no qualitative difference is found between the two interaction potentials. Clustering properties of the broken bonds are also studied by grouping them into disjoint sets of connected bonds. Finally, the role of kinetic energy is analyzed by comparing overdamped with dissipationless dynamics.

Nonlinear elasticity and pressure-dependent wave speeds in granular media

Following is an analysis of the small-strain nonlinear elasticity of granular media near states of zero stress, as it relates to the pressure-dependent incremental linear elasticity and wave speeds. The main object is elucidation of thep½dependence of incremental elastic moduli on pressurep, a dependence observed in numerous experiments but found to be at odds with thep½scaling predicted by various micromechanical models based on hertzian contact. After presenting a power-law continuum model for small-strain nonlinear elasticity, the present work develops micromechanical models based on two alternative mechanisms for the anomalous pressure scaling, namely: (1) departures at the single-contact level from the hertzian contact, due to point-like or conical asphericity; (2) variation in the number density of hertzian contacts, due to buckling of particle chains. Both mechanisms result inp½pressure scaling at low pressure and both exhibit a high-pressure transition top½scaling at a characteristic transition pressurep*. For assemblages of nearly equal spheres, a non-hertzian contact model for mechanism (1) and percolation-type model for (2) yield estimates ofp* of the formp* =cμˆ∝3. Herecis a non-dimensional coefficient depending only on granular-contact geometry, whileα≪ 1 is a small parameter representing spherical imperfections andμˆis an appropriate elastic modulus of the particles. Then, withRrepresenting particle radius andha characteristic spherical tolerance or asperity height, it is found thatα= (h/R)½for mechanism (1) as opposed toα=h/Rfor (2). Limited data from the classic experiments of Duffy & Mindlin on sphere assemblages tend to support mechanism (1), but more exhaustive experiments are called for. In addition to the above analysis of reversible elastic effects, a percolation model of inelastic ‘shake-down’ or consolidation is given. It serves to describe how prolonged mechanical vibration, leading to the replacement of point-like or inactive contacts by stiffer Hertz contacts may change the pressure-scaling behaviour of particulate media. The present analysis suggests that pressure-dependence of elasticity may provide a useful means of characterizing the state of consolidation and stability of dense particulate media.

New results in the theory of elasticity for two-dimensional composites

We bring together and discuss a number of exact relationships in two-dimensional (or plane) elasticity, that are useful in studying the effective elastic constants and stress fields in two-dimensional composite materials. The first of these dates back to Michell (1899) and states that the stresses, induced by applied tractions, are independent of the elastic constants in a two-dimensional material containing holes. The second involves the use of Dundurs constants which, for a composite consisting of two isotropic elastic phases, reduce the dependence of stresses on the elastic constants from three independent dimensionless parameters to two. It is shown that these two results are closely related to a recently proven theorem by Cherkaev, Lurie and Milton, which we use to show that the effective Young’s modulus of a sheet containing holes is independent of the Poisson’s ratio of the matrix material. We also show that the elastic moduli of a composite can be found exactly if the shear moduli of the components are all equal; a previously known result. We illustrate these results with computer simulations, where appropriate. Finally we conjecture on generalizations to multicomponent composite materials and to situations where the bonding between the phases is not perfect.

The membrane skeleton of erythrocytes. A percolation model

The spectrin network on the cytoplasmic surface of the erythrocyte membrane is modeled as a triangular lattice of spectrin tetramers. This network obstructs lateral diffusion of proteins and provides mechanical reinforcement to the membrane. These effects are treated in a systematic and unified manner in terms of a percolation model. The diffusion coefficient is obtained as a function of the fraction of normal spectrin tetramers for both static and fluctuating barriers. The elasticity of the network is calculated as a function of the fraction of normal spectrin and the ratio of bending to stretching energies. For static barriers, elasticity and lateral diffusion are incompatible: if a network is connected enough to be elastic, it is connected enough to block long-range lateral diffusion. The elasticity and the force required for mechanical breakdown go to zero at the percolation threshold; experimental evidence suggests the existence of a stability threshold at or near the percolation threshold. The model is qualitatively applicable to other cells with membrane skeletons, such as epithelial cells, in which localization of membrane proteins is essential to differentiation.