Reduced stress propagation leads to increased mechanical failure resistance in auxetic materials.

Materials with a negative Poisson ratio have the counterintuitive property of expanding laterally when they are stretched longitudinally. They are accordingly termed auxetic, from the Greek auxesis meaning to increase. Experimental studies have demonstrated auxetic materials to have superior material properties, compared with conventional ones. These include synclastic curvature, increased acoustic absorption, increased resilience to material fatigue, and increased resistance to mechanical failure. Until now, the latter observations have remained poorly understood theoretically. With this motivation, the contributions of this work are twofold. First, we elucidate analytically the way in which stress propagates spatially across a material following a localized plastic failure event, finding a significantly reduced stress propagation in auxetic materials compared with conventional ones. In this way, a plastic failure event occurring in one part of a material has a reduced tendency to trigger knock-on plastic events in neighboring regions. Second, via the numerical simulation of a lattice elastoplastic model, we demonstrate a key consequence of this reduced stress propagation to be an increased resistance to mechanical failure. This is seen not only via an increase in the externally measured yield strain, but also via a decreased tendency for plastic damage to percolate internally across a sample in catastrophic system-spanning clusters.

Slow Fatigue and Highly Delayed Yielding via Shear Banding in Oscillatory Shear.

We study theoretically the dynamical process of yielding in cyclically sheared amorphous materials, within a thermal elastoplastic model and the soft glassy rheology model. Within both models we find an initially slow accumulation, over many cycles after the inception of shear, of low levels of damage in the form strain heterogeneity across the sample. This slow fatigue then suddenly gives way to catastrophic yielding and material failure. Strong strain localization in the form of shear banding is key to the failure mechanism. We characterize in detail the dependence of the number of cycles N^{*} before failure on the amplitude of imposed strain, the working temperature, and the degree to which the sample is annealed prior to shear. We discuss our finding with reference to existing experiments and particle simulations, and suggest new ones to test our predictions.

Power law creep and delayed failure of gels and fibrous materials under stress.

Motivated by recent experiments studying the creep and breakup of a protein gel under stress, we introduce a simple mesoscopic model for the irreversible failure of gels and fibrous materials, and demonstrate it to capture much of the phenomenology seen experimentally. This includes a primary creep regime in which the shear rate decreases as a power law over several decades of time, a secondary crossover regime in which the shear rate attains a minimum, and a tertiary regime in which the shear rate increases dramatically up to a finite time singularity, signifying irreversible material failure. The model also captures a linear Monkman-Grant scaling of the failure time with the earlier time at which the shear rate attained its minimum, and a Basquin-like power law scaling of the failure time with imposed stress, as seen experimentally. The model furthermore predicts a slow accumulation of low levels of material damage during primary creep, followed by the growth of fractures leading to sudden material failure, as seen experimentally.

Ductile-to-brittle transition and yielding in soft amorphous materials: perspectives and open questions.

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

Model of Friction with Plastic Contact Nudging: Amontons-Coulomb Laws, Aging of Static Friction, and Nonmonotonic Stribeck Curves with Finite Quasistatic Limit.

We introduce a model of friction between two contacting (stationary or cosliding) rough surfaces, each comprising a random ensemble of polydisperse hemispherical bumps. In the simplest version of the model, the bumps experience on contact with each other only pairwise elastic repulsion and dissipative drag. These minimal ingredients are sufficient to capture a static state of jammed, interlocking contacting bumps, below a critical frictional force that is proportional to the normal load and independent of the apparent contact area, consistent with the Amontons-Coulomb laws of friction. However, they fail to capture two widespread observations: (i) that the dynamic friction coefficient (ratio of frictional to normal force in steady sliding) is a roughly constant or slightly weakening function of the sliding velocity U, at low U, with a nonzero quasistatic limit as U→0 and (ii) that the static friction coefficient (ratio of frictional to normal force needed to initiate sliding) increases ("ages") as a function of the time that surfaces are pressed together in stationary contact, before sliding commences. To remedy these shortcomings, we incorporate a single additional model ingredient: that contacting bumps plastically nudge one another slightly sideways, above a critical contact-contact load. With this additional insight, the model also captures observations (i) and (ii).

Shear-Driven Solidification and Nonlinear Elasticity in Epithelial Tissues.

Biological processes, from morphogenesis to tumor invasion, spontaneously generate shear stresses inside living tissue. The mechanisms that govern the transmission of mechanical forces in epithelia and the collective response of the tissue to bulk shear deformations remain, however, poorly understood. Using a minimal cell-based computational model, we investigate the constitutive relation of confluent tissues under simple shear deformation. We show that an initially undeformed fluidlike tissue acquires finite rigidity above a critical applied strain. This is akin to the shear-driven rigidity observed in other soft matter systems. Interestingly, shear-driven rigidity can be understood by a critical scaling analysis in the vicinity of the second order critical point that governs the liquid-solid transition of the undeformed system. We further show that a solidlike tissue responds linearly only to small strains and but then switches to a nonlinear response at larger stains, with substantial stiffening. Finally, we propose a mean-field formulation for cells under shear that offers a simple physical explanation of shear-driven rigidity and nonlinear response in a tissue.

Ductile and Brittle Yielding in Thermal and Athermal Amorphous Materials.

We study theoretically the yielding of sheared amorphous materials as a function of increasing levels of initial sample annealing prior to shear, in three widely used constitutive models and three widely studied annealing protocols. In thermal systems we find a gradual progression, with increasing annealing, from smoothly "ductile" yielding, in which the sample remains homogeneous, to abruptly "brittle" yielding, in which it becomes strongly shear banded. This progression arises from an increase with annealing in the size of an overshoot in the underlying stress-strain curve for homogeneous shear, which causes a shear banding instability that becomes more severe with increasing annealing. Ductile and brittle yielding thereby emerge as two limiting cases of a continuum of yielding transitions, from gradual to catastrophic. In contrast, athermal systems with a stress overshoot always show brittle yielding at low shear rates, however small the overshoot.

Edge-Induced Shear Banding in Entangled Polymeric Fluids.

Despite decades of research, the question of whether solutions and melts of highly entangled polymers exhibit shear banding as their steady state response to a steadily imposed shear flow remains controversial. From a theoretical viewpoint, an important unanswered question is whether the underlying constitutive curve of shear stress σ as a function of shear rate γ[over ˙] (for states of homogeneous shear) is monotonic, or has a region of negative slope, dσ/dγ[over ˙]<0, which would trigger banding. Attempts to settle the question experimentally via velocimetry of the flow field inside the fluid are often confounded by an instability of the free surface where the sample meets the outside air, known as "edge fracture." Here we show by numerical simulation that in fact even only very modest edge disturbances-which are the precursor of full edge fracture but might well, in themselves, go unnoticed experimentally-can cause strong secondary flows in the form of shear bands that invade deep into the fluid bulk. Crucially, this is true even when the underlying constitutive curve is monotonically increasing, precluding true bulk shear banding in the absence of edge effects.

Edge Fracture in Complex Fluids.

We study theoretically the edge fracture instability in sheared complex fluids, by means of linear stability analysis and direct nonlinear simulations. We derive an exact analytical expression for the onset of edge fracture in terms of the shear-rate derivative of the fluid's second normal stress difference, the shear-rate derivative of the shear stress, the jump in shear stress across the interface between the fluid and the outside medium (usually air), the surface tension of that interface, and the rheometer gap size. We provide a full mechanistic understanding of the edge fracture instability, carefully validated against our simulations. These findings, which are robust with respect to choice of rheological constitutive model, also suggest a possible route to mitigating edge fracture, potentially allowing experimentalists to achieve and accurately measure flows stronger than hitherto possible.

Understanding rheological hysteresis in soft glassy materials.

Motivated by recent experimental studies of rheological hysteresis in soft glassy materials, we study numerically strain rate sweeps in simple yield stress fluids and viscosity bifurcating yield stress fluids. Our simulations of downward followed by upward strain rate sweeps, performed within fluidity models and the soft glassy rheology model, successfully capture the experimentally observed monotonic decrease of the area of the rheological hysteresis loop with sweep time in simple yield stress fluids, and the bell shaped dependence of hysteresis loop area on sweep time in viscosity bifurcating fluids. We provide arguments explaining these two different functional forms in terms of differing tendencies of simple and viscosity bifurcating fluids to form shear bands during the sweeps, and show that the banding behaviour captured by our simulations indeed agrees with that reported experimentally. We also discuss the difference in hysteresis behaviour between inelastic and viscoelastic fluids. Our simulations qualitatively agree with the experimental data discussed here for four different soft glassy materials.

Shear Banding of Soft Glassy Materials in Large Amplitude Oscillatory Shear.

We study shear banding in soft glassy materials subject to a large amplitude oscillatory shear flow (LAOS). By numerical simulations of the widely used soft glassy rheology model, supplemented by more general physical arguments, we demonstrate strong banding over an extensive range of amplitudes and frequencies of the imposed shear rate γ[over ˙](t)=γ[over ˙]_{0}cos(ωt), even in materials that do not permit banding as their steady state response to a steadily imposed shear flow γ[over ˙]=γ[over ˙]_{0}=const. Highly counterintuitively, banding persists in LAOS even in the limit of zero frequency ω→0, where one might a priori have expected a homogeneous flow response in a material that does not display banding under conditions of steadily imposed shear. We explain this finding in terms of an alternating competition within each cycle between glassy aging and flow rejuvenation. Our predictions have far-reaching implications for the flow behavior of aging yield stress fluids, suggesting a generic expectation of shear banding in flows of even arbitrarily slow time variation.

Correlation lengths in hydrodynamic models of active nematics.

We examine the scaling with activity of the emergent length scales that control the nonequilibrium dynamics of an active nematic liquid crystal, using two popular hydrodynamic models that have been employed in previous studies. In both models we find that the chaotic spatio-temporal dynamics in the regime of fully developed active turbulence is controlled by a single active scale determined by the balance of active and elastic stresses, regardless of whether the active stress is extensile or contractile in nature. The observed scaling of the kinetic energy and enstrophy with activity is consistent with our single-length scale argument and simple dimensional analysis. Our results provide a unified understanding of apparent discrepancies in the previous literature and demonstrate that the essential physics is robust to the choice of model.

Age-dependent modes of extensional necking instability in soft glassy materials.

We study the instability to necking of an initially cylindrical filament of soft glassy material subject to extensional stretching. By numerical simulation of the soft glassy rheology model and a simplified fluidity model, and by analytical predictions within a highly generic toy description, we show that the mode of instability is set by the age of the sample relative to the inverse of the applied extensional strain rate. Young samples neck gradually via a liquidlike mode, the onset of which is determined by both the elastic loading and plastic relaxation terms in the stress constitutive equation. Older samples fail at smaller draw ratios via a more rapid mode, the onset of which is determined only by the solidlike elastic loading terms (though plastic effects arise later, once appreciable necking develops). We show this solidlike mode to be the counterpart, for elastoplastic materials, of the Considère mode of necking in strain-rate-independent solids.

Clustering and phase behaviour of attractive active particles with hydrodynamics.

We simulate clustering, phase separation and hexatic ordering in a monolayered suspension of active squirming disks subject to an attractive Lennard-Jones-like pairwise interaction potential, taking hydrodynamic interactions between the particles fully into account. By comparing the hydrodynamic case with counterpart simulations for passive and active Brownian particles, we elucidate the relative roles of self-propulsion, interparticle attraction, and hydrodynamic interactions in determining clustering and phase behaviour. Even in the presence of an attractive potential, we find that hydrodynamic interactions strongly suppress the motility induced phase separation that might a priori have been expected in a highly active suspension. Instead, we find only a weak tendency for the particles to form stringlike clusters in this regime. At lower activities we demonstrate phase behaviour that is broadly equivalent to that of the counterpart passive system at low temperatures, characterized by regimes of gas-liquid, gas-solid and liquid-solid phase coexistence. In this way, we suggest that a dimensionless quantity representing the level of activity relative to the strength of attraction plays the role of something like an effective non-equilibrium temperature, counterpart to the (dimensionless) true thermodynamic temperature in the passive system. However there are also some important differences from the equilibrium case, most notably with regards the degree of hexatic ordering, which we discuss carefully.

Discontinuous Shear Thickening in Biological Tissue Rheology.

During embryonic morphogenesis, tissues undergo dramatic deformations in order to form functional organs. Similarly, in adult animals, living cells and tissues are continually subjected to forces and deformations. Therefore, the success of embryonic development and the proper maintenance of physiological functions rely on the ability of cells to withstand mechanical stresses as well as their ability to flow in a collective manner. During these events, mechanical perturbations can originate from active processes at the single-cell level, competing with external stresses exerted by surrounding tissues and organs. However, the study of tissue mechanics has been somewhat limited to either the response to external forces or to intrinsic ones. In this work, we use an active vertex model of a 2D confluent tissue to study the interplay of external deformations that are applied globally to a tissue with internal active stresses that arise locally at the cellular level due to cell motility. We elucidate, in particular, the way in which this interplay between globally external and locally internal active driving determines the emergent mechanical properties of the tissue as a whole. For a tissue in the vicinity of a solid-fluid jamming or unjamming transition, we uncover a host of fascinating rheological phenomena, including yielding, shear thinning, continuous shear thickening, and discontinuous shear thickening. These model predictions provide a framework for understanding the recently observed nonlinear rheological behaviors in vivo.

Criteria for shear banding in time-dependent flows of complex fluids.

We study theoretically the onset of shear banding in the three most common time-dependent rheological protocols: step stress, finite strain ramp (a limit of which gives a step strain), and shear startup. By means of a linear stability analysis we provide a fluid-universal criterion for the onset of banding for each protocol, which depends only on the shape of the experimentally measured time-dependent rheological response function, independent of the constitutive law and internal state variables of the particular fluid in question. Our predictions thus have the same highly general status, in these time-dependent flows, as the widely known criterion for banding in steady state (of negatively sloping shear stress vs shear rate). We illustrate them with simulations of the Rolie-Poly model of polymer flows, and the soft glassy rheology model of disordered soft solids.

Constitutive model for the rheology of biological tissue.

The rheology of biological tissue is key to processes such as embryo development, wound healing, and cancer metastasis. Vertex models of confluent tissue monolayers have uncovered a spontaneous liquid-solid transition tuned by cell shape; and a shear-induced solidification transition of an initially liquidlike tissue. Alongside this jamming/unjamming behavior, biological tissue also displays an inherent viscoelasticity, with a slow time and rate-dependent mechanics. With this motivation, we combine simulations and continuum theory to examine the rheology of the vertex model in nonlinear shear across a full range of shear rates from quastistatic to fast, elucidating its nonlinear stress-strain curves after the inception of shear of finite rate, and its steady state flow curves of stress as a function of strain rate. We formulate a rheological constitutive model that couples cell shape to flow and captures both the tissue solid-liquid transition and its rich linear and nonlinear rheology.

Power fluctuations in sheared amorphous materials: A minimal model.

The importance of mesoscale fluctuations in flowing amorphous materials is widely accepted, without a clear understanding of their role. We propose a mean-field elastoplastic model that admits both stress and strain-rate fluctuations, and investigate the character of its power distribution under steady shear flow. The model predicts the suppression of negative power fluctuations near the liquid-solid transition; the existence of a fluctuation relation in limiting regimes but its replacement in general by stretched-exponential power-distribution tails; and a crossover between two distinct mechanisms for negative power fluctuations in the liquid and the yielding solid phases. We connect these predictions with recent results from particle-based, numerical microrheological experiments.

Criterion for extensional necking instability in polymeric fluids.

We study the linear instability with respect to necking of a filament of polymeric fluid undergoing uniaxial extension. Contrary to the widely discussed Considère criterion, we find the onset of instability to relate closely to the onset of downward curvature in the time (and so strain) evolution of the zz component of the molecular strain, for extension along the z axis. In establishing this result numerically across five of the most widely used models of polymer rheology, and by analytical calculation, we argue it to apply generically. Particularly emphasized is the importance of polymer chain stretching in partially mitigating necking. We comment finally on the relationship between necking and the shape of the underlying steady state constitutive curve for homogeneous extension.

Age-dependent transient shear banding in soft glasses.

We study numerically the formation of long-lived transient shear bands during shear startup within two models of soft glasses (a simple fluidity model and an adapted "soft glassy rheology" model). The degree and duration of banding depends strongly on the applied shear rate, and on sample age before shearing. In both models the ultimate steady flow state is homogeneous at all shear rates, consistent with the underlying constitutive curve being monotonic. However, particularly in the soft glassy rheology case, the transient bands can be extremely long lived. The banding instability is neither "purely viscous" nor "purely elastic" in origin, but is closely associated with stress overshoot in startup flow.