Session R16: Friction, Deformation, and Fracture

Effects of Particle Shape on the Deformation of Granular Pillars

Disordered systems are characterized by bulk properties such as yield strength and strain localization that are heavily influenced by local interactions. This general phenomenon can be observed in systems spanning a wide range of length scales, from nanoparticle assemblies to colloidal and granular packings, and can arise from several factors including friction and cohesion. In this study, we probe the effects of particle shape on the global and local behavior of a two-dimensional granular pillar undergoing uniaxial compression. This geometry allows for direct measurement of global material response, as well as tracking of all individual particles. The pillar is comprised of either discrete grains (monomers), pairs of grains bonded together (dimers), or groups of three bonded in a triangle (trimers). We find that pillars comprised of dimers are the strongest, and this effect is influenced by orientational ordering. In addition, while the three particle shapes cause the pillar to dilate at distinct rates, we find that local amorphous structure remains robust through the definition of a metric that quantifies local over-/under-packing. Finally, we highlight how particle shape and local structure impact the likelihood, severity, and spatial extent of local deformation.   

Scale and Stiffness Dependence of Static Friction on Amorphous Surfaces

Understanding the role that surface structure plays in determining friction at the interface of elastic solids is critical to the development of more efficient small scale mechanical devices and for extracting results from scanning probe microscopy experiments that rely on continuum predictions. Disagreement with continuum contact mechanics is expected at small length scales where atomic discreteness becomes important. We employ molecular simulation techniques to investigate the nature of contact at the atomic scale and to study the origin of frictional effects emerging from interatomic interactions. We focus on amorphous asperities to model realistic probing instruments with contact regions up to a micrometer across and tip radii orders of magnitude larger. The disordered composition of the interface leads to vanishing friction in the large contact limit for stiff surfaces because each atom experiences local lateral forces with random phase. The static friction coefficient scales with the number of contacting atoms $N_c$ as $1/\sqrt{N_c}$. Sufficiently weakening the lateral stiffness of the crystal compared to the characteristic stresses in the contact generates a transition to a regime with saturated average shear stress that is independent of contact size.   

Impact of solid and liquids balls on a solid surface: a unified description

We study experimentally the impact of ultra soft spherical gel balls of millimetric size $d_0$on a rigid substrate covered by a thin layer of liquid nitrogen to avoid viscous dissipation .The balls largely deform like a pancake at high impact velocities. We measure the maximally deformed size $d_{max}$ and the the time needed to reach this maximal size after impact $\tau_{max}$, versus the impact velocity $u_i$ for various elastic moduli . We do the same type of experiments with liquid droplets of various surface tensions . The experiments reveal a universal scaling behavior of the maximum deformation $\d_{max}/d_0} $ of both solid balls and liquid drops provided that both bulk and surface elasticity are properly taken into account. Moreover, we show that , in absence of viscous dissipation, the dynamics of the system can be understood as a conventional spring-mass system with a stiffness given by a combination of surface tension and bulk elasticity and a mass given by that of the ball (or drop) ; the deformation of the small ball (drop) during the impact linearly depends on the impact velocity, and the contact time scales as the period of this spring-mass system.   

A breakage mechanism for inhomogeneous biocolloids

Biocolloids, such as biofilms and free-floating microbial aggregates, constitute a large class of colloids with important applications in diverse areas of industrial, natural, and health sciences. In many of these applications it is often desirable to predict the distribution in space and in time of aggregate size, which depends in part on aggregate breakage. In the case of biocolloids, the internal structure may be inhomogeneous, and we may therefore wish to identify locations of likely breakage sites based upon this structure. We present a method to do so in the case of biocolloids in shear flow, by integrating the force density over the surface of the aggregate with respect to a specified breakage plane. We then apply this method in a simulation to generate a post-fragmentation density function for microbial aggregate fragmentation.   

Role of Structural Hierarchy in 2D Athermal Network Mechanics

Hierarchical materials are common in nature \footnote{Piechocka, I. et al., ``Structural Hierarchy Governs Fibrin Gel Mechanics'', \emph{Biophysical Journal} 98, Issue 10, 2281-2289}, and are of interest for various technical applications \footnote{Zheng, X. et al., ``Multiscale metallic metamaterials'', \emph{Nature Materials} 15: 1100-1106}. We assess the applicability of frame stiffness criteria pioneered by Maxwell and refined by Calladine when multiple, disparate length scales exist. We consider in particular a case in which an individual large-scale bond has a finer network structure with the same vertex arrangement as the large scale, but not necessarily equal connectivity. We present an experimental and computational study of the effect of connectivity at two scales on tensile stiffness for two-dimensional, dilute, hierarchical triangular lattices, and discuss observed energy storage at failure. For full connectivity on the small scale, behavior as large bonds are removed resembles theoretical results for a single-scale triangular filamentous lattice \footnote{Mao, X., et al., ``Elasticity of a filamentous kagome lattice'', \emph{Phys. Rev. E}, 87:042604}. Small-scale dilution causes far more abrupt softening, and an accompanying collapse of energy storage at failure.   

Polygonal crack patterns by drying thin films under quasi-two-dimensional confinement

Cracks patterns such as T/Y junction cracks in dried mud are ubiquitous in nature. Although the conditions for cracking in solids is well-known, cracks in colloidal and granular systems are more complex. Here we report the formations of polygonal cracks by drying thin films of corn starch ($\sim$ 10 $\mu$m in diameter) under quasi-2D confinement. We find there are two drying stages before the films are completely dried. Initially, a compaction front invades throughout the film. Then, a second drying stage "percolates" throughout the film with a characteristic branching pattern, leading to a dense packing of particles connected by liquid capillary bridges. Finally, polygonal cracks appear as the remaining liquid dries. The same drying kinetics occur for films with different thickness, $h$, except that fractal-like fracture patterns form in thin films, where the thickness is comparable to the particle size, while polygons form in thick films with many layers of particles. We also find that the average area of the polygons, $A$, in fully dried films scales with the thickness, $A\propto h^{\beta}$, where $\beta\approx 1.5$, and the prefactor depends on the initial packing fraction of the suspension. This form is consistent with a simple energy balance criterion for crack formation.   

Exactly solvable model for crack propagation in viscoelastic sheets: How to control a velocity jump leading to catastrophic failure

In viscoelastic solids, discontinuous transitions in the velocity of crack propagation have experimentally been observed in a narrow range of the energy release rate by using elastomers filled with carbon black particles. Although various theoretical studies have been performed on the crack propagation in viscoelastic solids, the physical mechanism of the transition has yet to be clarified. In this study, we propose a simple model for the crack propagation in viscoelastic solids for which an exact analytic solution exhibiting the velocity transition is available for the energy release rate as a function of the crack propagation velocity. On the basis of the exact expression, we provide an existence condition of the velocity transition and simple relationships useful as guiding principles to develop tough polymer materials, elucidating the physical mechanism of the transition. Our result implies that the discontinuous transition in the crack propagation velocity is a universal phenomenon that should be observed in a broad class of viscoelastic solids. An analogy between the velocity transition of our model and conventional discontinuous phase transitions are also discussed.   

Fiber networks below the isostatic point: fracture without stress concentration

The fracturing of elastic material in the over-damped condition is usually controlled by the stress concentration at crack tips. In this talk, we discuss our recent studies on the fracturing of fiber networks, which are below the central-force isostatic point and their deformations are dominated by bending of the fibers. We find that in these fiber networks there is a unique phase of fracturing that is characterized by fractal diffusive cracks, small avalanches, and a steady state of force chain generation and breaking. Most strikingly, there is no stress concentration at any scale in these networks. Interestingly, the entire phase is critical so no fine tuning is required to observe this phase. Finally I will discuss the implications on experiments.   

The interplay of crack hopping, delamination and interface failure in drying nanoparticle films

Films formed through the drying of nanoparticle suspensions release the build-up of strain through a variety of different mechanisms including shear banding, crack formation and delamination. In this talk I will show that important connections exist between these different phenomena: delamination depends on the dynamics of crack hopping, which in turn is influenced by the presence of shear bands. We also show that delamination does not occur uniformly across the film. As cracks hop they locally initiate the delamination of the film which warps with a timescale much longer than that associated with the hopping of cracks. The motion of a small region of the delamination front, where the shear component of interfacial crack propagation is believed to be enhanced, results in the deposition of a complex zigzag pattern on the supporting substrate.   

Collective nonaffine rearrangements in binary glasses during large-amplitude oscillatory shear

Using molecular dynamics simulations, we study the transient response of a three-dimensional binary Lennard-Jones glass subjected to periodic shear deformation. The cyclic loading is applied to slowly annealed, quiescent samples, which induces irreversible particle rearrangements at large strain amplitudes, leading to stress-strain hysteresis and a drift of the potential energy towards higher values. We find that near the critical strain amplitude, the amplitude of shear stress oscillations decreases after the first several cycles, which is accompanied with the formation of shear bands. In contrast, the initial response to cyclic shear involves disconnected clusters of atoms with large nonaffine displacements.   

Friction on a granular-continuum interface: Effects of granular media

We consider the frictional interactions of two soft plates with interposed granular material subject to normal and shear forces. The plates are soft photo-elastic material, have length 50 cm, and are separated by a gap of variable width from 0 to 20 granular particle diameters. The granular materials are two-dimensional rods that are bi-dispersed in size to prevent crystallization. Different rod materials with frictional coefficients between $0.04 < \mu < 0.5$ are used to explore the effects of inter-granular friction on the effective friction of a granular medium. The gap is varied to test the dependence of the friction coefficient on the thickness of the granular layer. Because the soft plates absorb most of the displacement associated with the compressional normal force, the granular packing fractions are close to a jamming threshold, probably a shear jamming criterion. The overall shear and normal forces are measured using force sensors and the local strain tensor over a central portion of the gap is obtained using relative displacements of fiducial markers on the soft elastic material. These measurements provide a good characterization of the global and local forces giving rise to an effective friction coefficient.   

Injection initiated fracture in soft solids.

Damage accumulation in soft materials under hydrostatic loading conditions is a primary injury mechanism in blast and blunt force trauma. A recently explored technique known as cavitation rheology (CR) provides a promising avenue for quickly and inexpensively approximating hydrostatic conditions via the reverse loading scenario, void pressurization. Past CR measurements of synthetic, polymeric materials at length scales from mm's to $\mu $m's have been found to correlate with both elastic modulus and fracture energy. The technique is performed via pressurization of fluid within a needle that is embedded within a material. This experimental setup allows crack evolution to be monitored, similar to traditional pre-notched failure samples. We observe a systematic evolution of crack morphology as a function of cross-link density in a soft elastomer. Crack shape is quantified using micro-computed tomography and shown to transition from being roughly penny-shaped to multi-lobed (predominately three) to spherical with decreasing crosslinking. Moduli are on the order of kPa. We describe this morphology evolution using a balance between the energetic costs of the strain energy in deforming the surrounding material and the intrinsic fracture energy necessary to form a new surface.   

Cracking Sheets into Shapes: Linear actuators from non-linear crack behavior

In recent years, the mechanics of highly deformable and soft complex structures have gained significant attention across interdisciplinary fields. A theoretical mechanics treatment of such systems, so as to include geometric non-linearities from large deflections, remains a challenging and timely task in order to unleash their full potentials for functionalization. With a focus on the design of thin plates patterned with cracks (cuts), we shall discuss how these structures reveal new effectively non-linear and anisotropic responses to external forces and strains, thus opening the path to practical problems in linear actuation. Sheets patterned with cuts, also known as Kirigami, that are thin enough to relieve localized stress through out-of-plane deformation, still lack fundamental models to describe their out-of-plane behavior. We propose to give a robust geometric and mechanical account of these structures that goes beyond past related attempts where inextensibility constraints were enforced. We will also demonstrate fundamentally new responses in our experiments, and explain how large scale geometric features are dominated by localization at the crack tip.   

Nonlinear response and crack propagation in Articular Cartilage modeled as a biopolymer double network

Articular cartilage (AC) is a soft tissue that covers the ends of bones to distribute mechanical load in joints. It is primarily composed of water, type II collagen, and large aggregating proteoglycans called aggrecan. Its fracture toughness is extremely high compared to synthetic materials, but the underlying physical mechanism is not well understood. Here we investigate how the toughness of AC depends on its microscale composition and structure by modeling it as a double network made of collagen and aggrecan embedded in a background gel, and by using rigidity percolation theory to characterize its mechanical response to shear and compressive (or tensile) strains. Our calculations of the mechanical moduli, as well as network-wide heat maps of local strains and energy show shear-stiffening and compression-softening with increasing applied strain, in good qualitative agreement with known experimental results. Notches are then introduced in the network to study crack propagation under shear and tensile strains for various applied loads. Preliminary results indicate a loading threshold above which the network will undergo catastrophic failure by fracturing. Our results may help to formulate a Griffith-like criterion for crack propagation and fracture in soft tissues.   

Effects of Elasticity and Inter-particle Friction in a Sheared Packing of Soft Particles

We build a rheometer to investigate experimentally centimeter-sized particles packed in 3D under either cyclic or steady shearing. Previously we have combined the cessation of shearing and internal imaging , to demonstrate the relaxation and residues of stress and the timescale of grain movements in the case of hydrogel particles. Here we extend our investigations to compare results using particles of different elastic moduli, and to identify how the inter-particle frictions changes the response. Mixture of slippery and frictional particles allows us to observe the transition from a fully lubricated packing, to its counterpart that reproduces the stick-slip behaviors as seen in sand flows dominated by frictions.