Modeling growth paths of interacting crack pairs in elastic media

The problem of predicting the growth of a system of cracks, each crack influencing the growth of the others, arises in multiple fields. We develop an analytical framework toward this aim, which we apply to the 'En-Passant' family of crack growth problems, in which a pair of initially parallel, offset cracks propagate nontrivially toward each other under far-field opening stress. We utilize boundary integral and perturbation methods of linear elasticity, linear elastic fracture mechanics, and common crack opening criteria to calculate the first analytical model for curved En-Passant crack paths. The integral system is reduced under a hierarchy of approximations, producing three methods of increasing simplicity for computing crack paths. The last such method is a major highlight of this work, using an asymptotic matching argument to predict crack paths based on superposition of simple, single-crack fields. Within the corresponding limits of the three methods, all three are shown to agree with each other. We provide comparisons to exact results and existing experimental data to verify certain approximation steps

Intrusion rheology in grains and other flowable materials

The interaction of intruding objects with deformable materials arises in many contexts, including locomotion in fluids and loose media, impact and penetration problems, and geospace applications. Despite the complex constitutive behaviour of granular media, forces on arbitrarily shaped granular intruders are observed to obey surprisingly simple, yet empirical 'resistive force hypotheses'. The physics of this macroscale reduction, and how it might play out in other media, has however remained elusive. Here, we show that all resistive force hypotheses in grains arise from local frictional yielding, revealing a novel invariance within a class of plasticity models. This mechanical foundation, supported by numerical and experimental validations, leads to a general analytical criterion to determine which rheologies can obey resistive force hypotheses. We use it to explain why viscous fluids are observed to perform worse than grains, and to predict a new family of resistive-force-obeying materials: cohesive media such as pastes, gels and muds.United States. Army Research Office (W911NF-14-1-0205)United States. Army Research Office (W911NF-15-1-0196

A finite element implementation of the nonlocal granular rheology

Inhomogeneous flows involving dense particulate media display clear size effects, in which the particle length scale has an important effect on flow fields. Hence, nonlocal constitutive relations must be used in order to predict these flows. Recently, a class of nonlocal fluidity models has been developed for emulsions and subsequently adapted to granular materials. These models have successfully provided a quantitative description of experimental flows in many different flow configurations. In this work, we present a finite element-based numerical approach for solving the nonlocal constitutive equations for granular materials, which involve an additional, non-standard nodal degree-of-freedom – the granular fluidity, which is a scalar state parameter describing the susceptibility of a granular element to flow. Our implementation is applied to three canonical inhomogeneous flow configurations: (1) linear shear with gravity, (2) annular shear flow without gravity, and (3) annular shear flow with gravity. We verify our implementation, demonstrate convergence, and show that our results are mesh independent.National Science Foundation (U.S.) (Grant NSF-CBET-1253228

Reference map technique for finite-strain elasticity and fluid-solid interaction

The reference map, defined as the inverse motion function, is utilized in an Eulerian-frame representation of continuum solid mechanics, leading to a simple, explicit finite-difference method for solids undergoing finite deformations. We investigate the accuracy and applicability of the technique for a range of finite-strain elasticity laws under various geometries and loadings. Capacity to model dynamic, static, and quasi-static conditions is shown. Specifications of the approach are demonstrated for handling irregularly shaped and/or moving boundaries, as well as shock solutions. The technique is also integrated within a fluid–solid framework using a level-set to discern phases and using a standard explicit fluid solver for the fluid phases. We employ a sharp-interface method to institute the interfacial conditions, and the resulting scheme is shown to efficiently capture fluid–solid interaction solutions in several examples.United States. Dept. of Energy. Office of Science (Computational and Technology Research, contract number DE-AC02-05CH11231)National Science Foundation (U.S.) (Grant DMS-0813648)National Science Foundation (U.S.) (Mathematical Sciences Postdoctoral Research Fellowship)Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Program)

Elastic sheets: Cracks by design

Different methods exist to control fracture in thin media in order to produce some desired shape or curved edge. Commonly, inhomogeneities are placed along a specific path to guide a fracture, such as scoring a material's surface or introducing a sequence of perforations. In some circumstances, the ability to guide fractures without altering the material is advantageous or even necessary, and could provide a key design tool in areas such as flexible electronics, thin films and monolayer materials. Writing in Nature Materials, Mitchell, Irvine and colleagues explore the possibility of guiding crack paths in thin, elastic sheets by draping them on surfaces with non-zero Gaussian curvature1. The out-of-plane elastic deformation imposed by the surface curvature causes an inhomogeneous stress distribution within the sheet. If a small crack is introduced, the pre-load in the membrane can cause the fracture to grow spontaneously. Depending on how the substrate geometry is chosen, the crack growth can be made to conform to a curved path and possibly arrest after a desired crack length has been reached. This opens up the possibility of a new methodology for incising two-dimensional shapes from sheets by fracturing them over a tailored bumpy substrate surface

Foreword on the special issue: from discrete particles to continuum models of granular mechanics

The mechanics of granular materials have remained a challenge to describe since the days of Coulomb, both theoretically and computationally. Granular media arise in a number of everyday circumstances (e.g., raw materials, soils, food/agriculture, pharmaceuticals, vehicle mobility, debris removal) and remain the second-most handled material by weight in global industry. Hence, the challenge of coming up with accurate and efficient ways to simulate a collection of grains is an issue of great import to multiple disciplines. There are two major perspectives. Ideally, one would like a continuum model able to describe the behavior of large collections of grains in a numerically efficient fashion. There is also the discrete perspective, where modeling arises at the individual particle level and the motion of every grain is tracked and evolved under mechanical laws

A hierarchy of granular continuum models: Why flowing grains are both simpleand complex

ranular materials have a strange propensity to behave as either a complex media or a simple media depending on the precise question being asked. This review paper offers a summary of granular flow rheologies for well-developed or steady-state motion, and seeks to explain this dichotomy through the vast range of complexity intrinsic to these models. A key observation is that to achieve accuracy in predicting flow fields in general geometries, one requires a model that accounts for a number of subtleties, most notably a nonlocal effect to account for cooperativity in the flow as induced by the finite size of grains. On the other hand, forces and tractions that develop on macro-scale, submerged boundaries appear to be minimally affected by grain size and, barring very rapid motions, are well represented by simple rate-independent frictional plasticity models. A major simplification observed in experiments of granular intrusion, which we refer to as the ‘resistive force hypothesis’ of granular Resistive Force Theory, can be shown to arise directly from rate-independent plasticity. Because such plasticity models have so few parameters, and the major rheological parameter is a dimensionless internal friction coefficient, some of these simplifications can be seen as consequences of scaling

Microscopic Description of the Granular Fluidity Field in Nonlocal Flow Modeling

A recent granular rheology based on an implicit "granular fluidity" field has been shown to quantitatively predict many nonlocal phenomena. However, the physical nature of the field has not been identified. Here, the granular fluidity is found to be a kinematic variable given by the velocity fluctuation and packing fraction. This is verified with many discrete element simulations, which show that the operational fluidity definition, solutions of the fluidity model, and the proposed microscopic formula all agree. Kinetic theoretical and Eyring-like explanations shed insight into the obtained form.Natioanal Science Foundation (U.S.) (Grant CBET-1253228

Nonlocal Constitutive Relation for Steady Granular Flow

Extending recent modeling efforts for emulsions, we propose a nonlocal fluidity relation for flowing granular materials, capturing several known finite-size effects observed in steady flow. We express the local Bagnold-type granular flow law in terms of a fluidity ratio and then extend it with a particular Laplacian term that is scaled by the grain size. The resulting model is calibrated against a sequence of existing discrete element method data sets for two-dimensional annular shear, where it is shown that the model correctly describes the divergence from a local rheology due to the grain size as well as the rate-independence phenomenon commonly observed in slowly flowing zones. The same law is then applied in two additional inhomogeneous flow geometries, and the predicted velocity profiles are compared against corresponding discrete element method simulations utilizing the same grain composition as before, yielding favorable agreement in each case

Soft catenaries

Using the classical catenary as a motivating example, we use slender-body theory to derive a general theory for thin filaments of arbitrary rheology undergoing large combined stretching and bending, which correctly accounts for the nonlinear geometry of deformation and uses integrated state variables to properly represent the complete deformation state. We test the theory for soft catenaries made of a Maxwell fluid and an elastic yield-stress fluid using a combination of asymptotic and numerical analyses to analyse the dynamics of transient sagging and arrest. We validate our results against three-dimensional finite element simulations of drooping catenaries, and show that our minimal models are easier and faster to solve, can capture all the salient behaviours of the full three-dimensional solution, and provide physical insights into the basic mechanisms involved.National Science Foundation (U.S.). Mathematical Sciences Postdoctoral Research Fellowship