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We probe flows of soft, viscous spheres near the jamming point, which acts as a critical point for static soft spheres. Starting from energy considerations, we find nontrivial scaling of velocity fluctuations with strain rate. Combining this scaling with insights from jamming, we arrive at an analytical model that predicts four distinct regimes of flow, each characterized by rational-valued scaling exponents. Both the number of regimes and values of the exponents depart from prior results. We validate predictions of the model with simulations.PACS numbers: 47.57. Bc, 83.50.Rp, 83.80.Iz The past few years have seen enormous progress towards understanding the static, "jammed" state that occurs when soft athermal particles are packed sufficiently densely that they attain a finite rigidity [1][2][3]. Such systems may flow when shear stresses are applied, and in seminal work, Olsson and Teitel addressed the relation between strain rate, shear stress and packing fraction in a simplified numerical model for the flow of soft viscous spheres [4]. When rescaled appropriately, the data for strain rateγ, shear stress σ and packing fraction φ were found to collapse to two curves, reminiscent of second order-like scaling functions, and a large length scale was found to emerge near jamming. Since then, qualitatively similar results have been obtained in simulations of a number of flowing systems [5][6][7][8][9], but there is little agreement on the actual value of scaling exponents, nor on the relation to jamming in static systems.Here we describe an analytical model that connects the scaling of static systems to the scaling of both the velocity fluctuations and the shear stress of flowing systems near jamming. The model is built around a "viscoplastic" effective strain γ eff = γ y + γ dyn , where γ dyn is a dynamic contribution set by the strain rate, and γ y stems from the (dynamical) yield stress and is controlled by the distance to jamming. We show that steady state power balance dictates nontrivial scaling of γ dyn with strain rate, and propose a nonlinear stress-strain relation that leads to a closed set of equations predicting a rich scaling scenario for flows near jamming. We verify central ingredients of the model and our predictions for the rheology numerically in Durian's bubble model for foams [11]. Our simple model captures and predicts the rheology and fluctuations starting from the microscopic interactions; it also indicates the need for, and provides, new ways to present and analyze rheological data near jamming.Numerical Model -The two-dimensional Durian bubble model stipulates overdamped dynamics in which the sums of elastic and dissipative forces on each bubble, represented by a disk, balance at all times [11]. Forces are pairwise and occur only between contacting bubbles. Elastic interaction forces are proportional to the disk overlap, f el ij = k(R i + R j − r ij ) α el , where r ij := r j − r i points from one bubble center to another and R i labels the radius of disk i. In the full model that we focus...

A numerical method for crack growth is described in which the crack is not regarded as a single discontinuity that propagates continuously. Instead, the crack is represented by a set of overlapping cohesive segments. These cohesive segments are inserted into finite elements as discontinuities in the displacement field by exploiting the partition-of-unity property of shape functions. The cohesive segments can be incorporated at arbitrary locations and orientations and are not tied to any particular mesh direction. The evolution of decohesion of the segments is governed by a cohesive law. The independent specification of bulk and cohesive constitutive relations leads to a characteristic length being introduced into the formulation. The formulation permits both crack nucleation and discontinuous crack growth to be modelled. The implementation is outlined and some numerical examples are presented.Keywords Crack growth, Fracture, Cohesive zones, Partitions of unity IntroductionIn conventional engineering fracture mechanics, crack growth is assumed to occur by the extension of a single dominant crack. However, there are a wide variety of circumstances where the fracture process involves the nucleation and growth of multiple crack-like flaws. For example, in heterogeneous materials, multiple cracks that initiate and grow in one phase may link up by nucleating cracks in another phase or by propagating across phase boundaries, e.g. [1]. Another example is the transition from subsonic to intersonic crack speeds via the nucleation of a micro-crack ahead of the main crack, [2]. Also, in quasi-brittle materials micro-cracking in front of the main crack tip plays a key role in setting the fracture toughness [3]. Hence, a need for analysing discontinuous crack growth arises in a wide variety of contexts.A cohesive surface methodology has emerged which permits the analysis of fracture processes in which there is no dominant flaw. In fact, an initial crack-like defect is not required since crack nucleation can occur naturally during the loading history. The basic assumption of the cohesive surface framework is that the separation process is confined to a set of discrete planes (or lines in a two-dimensional context). A constitutive relation is then specified for each cohesive surface that allows separation to occur.The cohesive approach to fracture was pioneered by Barenblatt [4], Dugdale [5] and Hillerborg et al. [6]. In these formulations, a dominant flaw was assumed present as in conventional engineering fracture mechanics, but a cohesive zone was introduced ahead of the existing crack tip. The relation between the work expended in this cohesive zone and that in the crack tip field is typically such that the stress singularity is cancelled and the near tip stresses are finite. The slip weakening model of Andrews [7] introduced in the geophysics literature did not require crack growth to be continuous. The cohesive framework was extended to finite deformations and to situations without an initial crack in [8]. Subsequ...

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