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.

Yield Stress Aging in Attractive Colloidal Suspensions.

We investigate the origin of yield stress aging in semidense, saline, and turbid suspensions in which structural evolution is rapidly arrested by the formation of thermally irreversible roll-resisting interparticle contacts. By performing optical tweezer three-point bending tests on particle rods, we show that these contacts yield by overcoming a rolling threshold, the critical bending moment of which grows logarithmically with time. We demonstrate that this time-dependent contact-scale rolling threshold controls the suspension yield stress and its aging kinetics. We identify a simple constitutive relation between the contact-scale flexural rigidity and rolling threshold, which transfers to macroscopic scales. This leads us to establishing a constitutive relation between macroscopic shear modulus and yield stress that is generic for an array of colloidal systems.

Stress hyperuniformity and transient oscillatory-exponential correlation decay as signatures of strength vs fragility in glasses.

We examine and compare the local stress autocorrelation in the inherent states of a fragile and a strong glass: the Kob-Andersen (KA) binary mixture and the Beest-Kramer-Santen model of silica. For both systems, local (domain-averaged) stress fluctuations asymptotically reach the normal inverse-volume decay in the large domain limit; accordingly, the real-space stress autocorrelation presents long-range power law tails. However, in the case of silica, local stress fluctuations display a high degree of hyperuniformity, i.e., their asymptotic (normal) decay is disproportionately smaller than their bond level amplitude. This property causes the asymptotic power law tails of the real-space stress autocorrelation to be swamped, up to very large distances (several nanometers), by an intermediate oscillatory-exponential decay regime. Similar contributions exist in the KA stress autocorrelation, but they never can be considered as dominating the power law decay and fully disappear when stress is coarse-grained beyond one interatomic distance. Our observations document that the relevance of power-law stress correlation may constitute a key discriminating feature between strong and fragile glasses. Meanwhile, they highlight that the notion of local stress in atomistic systems involves by necessity a choice of observation (coarse-graining) scale, the relevant value of which depends, in principle, on both the model and the phenomenon studied.

Anomalous elasticity and plastic screening in amorphous solids.

Amorphous solids appear to react elastically to small external strains, but in contrast to ideal elastic media, plastic responses abound immediately at any value of the strain. Such plastic responses are quasilocalized in nature, with the "cheapest" one being a quadrupolar source. The existence of such plastic responses results in screened elasticity in which strains and stresses can either quantitatively or qualitatively differ from the unscreened theory, depending on the specific screening mechanism. Here we offer a theory of such screening effects by plastic quadrupoles, dipoles, and monopoles, explain their natural appearance, and point out the analogy to electrostatic screening by electric charges and dipoles. For low density of quadrupoles the effect is to normalize the elastic moduli without a qualitative change compared to pure elasticity theory; for higher density of quadrupoles the screening effects result in qualitative changes. Predictions for the spatial dependence of displacement fields caused by local sources of strains are provided and compared to numerical simulations. We find that anomalous elasticity is richer than electrostatics in having a screening mode that does not appear in the electrostatic analog.

Frictional Granular Matter: Protocol Dependence of Mechanical Properties.

Theoretical treatments of frictional granular matter often assume that it is legitimate to invoke classical elastic theory to describe its coarse-grained mechanical properties. Here, we show, based on experiments and numerical simulations, that this is generically not the case since stress autocorrelation functions decay more slowly than what is expected from elasticity theory. It was theoretically shown that standard elastic decay demands pressure and torque density fluctuations to be normal, with possibly one of them being hyperuniform. However, generic compressed frictional assemblies exhibit abnormal pressure fluctuations, failing to conform with the central limit theorem. The physics of this failure is linked to correlations built in the material during compression from a dilute configuration prior to jamming. By changing the protocol of compression, one can observe different pressure fluctuations, and stress autocorrelations decay at large scales.

Key role of retardation and non-locality in sound propagation in amorphous solids as evidenced by a projection formalism.

We investigate acoustic propagation in amorphous solids by constructing a projection formalism based on separating atomic vibrations into two, "phonon" (P) and "non-phonon" (NP), subspaces corresponding to large and small wavelengths. For a pairwise interaction model, we show the existence of a "natural" separation lengthscale, determined by structural disorder, for which the isolated P subspace presents the acoustic properties of a nearly homogenous (Debye-like) elastic continuum, while the NP one encapsulates all small scale non-affinity effects. The NP eigenstates then play the role of dynamical scatterers for the phonons. However, at variance with a conjecture of defect theories, their spectra present a finite low frequency gap, which turns out to lie around the Boson peak frequency, and only a small fraction of them are highly localized. We then show that small scale disorder effects can be rigorously reduced to the existence, in the Navier-like wave equation of the continuum, of a generalized elasticity tensor, which is not only retarded, since scatterers are dynamical, but also non-local. The full neglect of both retardation and non-locality suffices to account for most of the corrections to Born macroscopic moduli. However, these two features are responsible for sound speed dispersion and have quite a significant effect on the magnitude of sound attenuation. Although it remains open how they impact the asymptotic, large wavelength scaling of sound damping, our findings rule out the possibility of representing an amorphous solid by an inhomogeneous elastic continuum with the standard (i.e., local and static) elastic moduli.

Origin of the Bauschinger Effect in Amorphous Solids.

We study the structural origin of the Bauschinger effect by accessing numerically the local plastic thresholds in the steady state flow of a two-dimensional model glass under athermal quasistatic deformation. More specifically, we compute the local residual strength, Δτ^{c}, for arbitrary loading orientations and find that plastic deformation generically induces material polarization, i.e., a forward-backward asymmetry in the Δτ^{c} distribution. In steady plastic flow, local packings are on average closer to forward (rather than backward) instabilities, due to the stress-induced bias of barriers. However, presumably due to mechanical noise, a significant fraction of zones lie close to reverse (backward) yielding, as the distribution of Δτ^{c} for reverse shearing extends quasilinearly down to zero local residual strength. By constructing an elementary model of the early plastic response, we then show that unloading causes reverse plasticity of a growing amplitude, i.e., reverse softening, while it shifts away forward-yielding barriers. This result in an inversion of polarization in the low-Δτ^{c} region and, consequently, in the Bauschinger effect. This scenario is quite generic, which explains the pervasiveness of the effect.

Rejuvenation and shear banding in model amorphous solids.

We measure the local yield stress, at the scale of small atomic regions, in a deeply quenched two-dimensional glass model undergoing shear banding in response to athermal quasistatic deformation. We find that the occurrence of essentially a single plastic event suffices to bring the local yield stress distribution to a well-defined value for all strain orientations, thus essentially erasing the memory of the initial structure. It follows that in a well-relaxed sample, plastic events cause the abrupt (nucleation-like) emergence of a local softness contrast and thus precipitate the formation of a band, which, in its early stages, is measurably softer than the steady-state flow. Moreover, this postevent yield stress ensemble presents a mean value comparable to that of the inherent states of a supercooled liquid around the mode-coupling temperature T_{MCT}. This, we argue, explains that the transition between brittle and ductile yielding in amorphous materials occurs around a comparable parent temperature. Our data also permit to capture quantitatively the contributions of pressure and density changes and demonstrate unambiguously that they are negligible compared with the changes of softness caused by structural rejuvenation.

Contact and macroscopic ageing in colloidal suspensions.

The ageing behaviour of dense suspensions or pastes at rest is almost exclusively attributed to structural dynamics. Here, we identify another ageing process, contact-controlled ageing, consisting of the progressive stiffening of solid-solid contacts of an arrested colloidal suspension. By combining rheometry, confocal microscopy and particle-scale mechanical tests using laser tweezers, we demonstrate that this process governs the shear-modulus ageing of dense aqueous silica and polymer latex suspensions at moderate ionic strengths. We further show that contact-controlled ageing becomes relevant as soon as Coulombic interactions are sufficiently screened out that the formation of solid-solid contacts is not limited by activation barriers. Given that this condition only requires moderate ion concentrations, contact-controlled ageing should be generic in a wide class of materials, such as cements, soils or three-dimensional inks, thus questioning our understanding of ageing dynamics in these systems.

Fluctuating Elasticity Fails to Capture Anomalous Sound Scattering in Amorphous Solids.

The fluctuating elasticity (FE) model, introduced phenomenologically and developed by Schirmacher [J. Non-Cryst. Solids 357, 518 (2011)JNCSBJ0022-309310.1016/j.jnoncrysol.2010.07.052], is today the only theoretical framework available to analyze low-temperature elastic acoustic scattering in glasses. Its existing formulations, which neglect the tensorial nature of elasticity and exclude long-range disorder correlations, predict that the acoustic damping coefficients obey the standard Rayleigh scaling law: Γ∼k^{d+1}, with k the acoustic wave vector, in dimension d. However, recent numerical data, supported by the analysis of existing experimental results, show that Γ does not obey this scaling law but Γ∼-k^{d+1}lnk. Here we analyze in detail how a fully tensorial FE model can be constructed as a long wavelength approximation of the elastic response of the discrete, atomistic, problem. We show that, although it incorporates all long-range correlations, it fails to capture the observed damping in two respects: (i) it misses the anomalous scaling, and predicts the standard Rayleigh law; (ii) it grossly underestimates the amplitude of scattering by about 2 orders of magnitude. This brings clear evidence that the small scale nonaffine displacement fields, although not simply reducible to local defects, play a crucial role in acoustic wave scattering and hence cannot be ignored.

Stress correlations in glasses.

We rigorously establish that, in disordered three-dimensional isotropic solids, the stress autocorrelation function presents anisotropic terms that decay as 1/r3 at long-range, with r being the distance, as soon as local stress fluctuations are normal, by which we mean that the fluctuations of stress, as averaged over spherical domains, decay as the inverse domain volume. Since this property is required for macroscopic stress to be self-averaging, it is expected to hold generically in all glasses and we thus conclude that the presence of 1/r3 stress correlation tails is the rule in these systems. Our proof follows from the observation that, in an infinite medium, when both material isotropy and mechanical balance hold, (i) the stress autocorrelation matrix is completely fixed by just two radial functions: the pressure autocorrelation and the trace of the autocorrelation of stress deviators; furthermore, these two functions (ii) fix the decay of the fluctuations of sphere-averaged pressure and deviatoric stresses with the increasing sphere volume. Our conclusion is reached because, in view of (ii), the normal decay of stress fluctuations is only compatible with both the pressure autocorrelation and the trace of the autocorrelation of stress deviators being integrable; in turn, due to the precise analytic relation (i) fixed by isotropy and mechanical balance, this condition demands the spatially anisotropic stress correlation terms to decay as 1/r3 at long-range.

Glass-like stress relaxation of a permanent gelatin network as a signature of pretransitional helix-coil fluctuations.

The stress response of permanently crosslinked gelatin gels was recently observed to display glass-like features, namely, a stretched-exponential behavior terminated by an exponential decay, the characteristic time scales of which increase dramatically with decreasing temperature. This phenomenon is studied here using a model of flexible polymer gel network where relaxation proceeds via elementary monomer exchanges between helix and coil segments. The relaxation dynamics of a full network simulation is found to be nearly identical to that of a model of independent strands, which shows that for flexible polymer gels in the range of elastic moduli of interest, both strand contour length disorder and elastic couplings are irrelevant. We thus focus on the independent strand model and find it not only to explain the observed functional form of the stress relaxation curves but also to yield predictions that match very satisfactorily the experimental measurements of final relaxation time and total stress drop. The system under study thus constitutes a rare case where the origin of glass-like behavior can be unambiguously identified, namely, as the signature of the enhancement of helix content fluctuations when approaching from above the mean-field helix-coil transition of strands.

Effects of disorder and chain stiffening on the elasticity of flexible polymer networks.

We examine how the distribution of contour lengths and the high-stretch stiffening of individual chain segments affect the macroscopic shear modulus of flexible polymer gels, using a two-dimensional numerical model in which polymer segments form a triangular network and disorder is introduced by varying their contour lengths. We show that, in the relevant parameter range: (i) the nonaffine contribution to the shear modulus is negligible, i.e., the Born approximation is satisfactory, and (ii) the shear modulus is dominated by the contribution originating from equilibrium chain tensions. Moreover, mechanical equilibration at the nodes induces specific correlations between the end-to-end distances and contour lengths of chain segments, which must be properly accounted for to construct reasonable estimates of chain pressure and shear modulus.

Inherent stress correlations in a quiescent two-dimensional liquid: Static analysis including finite-size effects.

After constructing a formalism to analyze spatial stress correlations in two-dimensional equilibrated liquids, we show that the sole conjunction of mechanical balance and material isotropy demands all anisotropic components of the inherent state (IS) stress autocorrelation matrix to decay at long range as 1/r^{2} in the large system size limit. Furthermore, analyzing numerical simulation data for an equilibrated supercooled liquid, we bring evidence that, in finite-sized periodic systems, the autocorrelations of pressure and shear stresses present uniform backgrounds of amplitudes proportional to the inverse cell area. These backgrounds bring relevant contributions to macroscopic IS stress fluctuations, with the consequence that the latter scale as inverse area, yet in an anomalous way, inconsistent with viewing an IS as equivalent, in the thermodynamic limit, to an ensemble of independent finite-sized subsystems. In that sense, ISs are not spatially ergodic.

Anomalous phonon scattering and elastic correlations in amorphous solids.

A major issue in materials science is why glasses present low-temperature thermal and vibrational properties that sharply differ from those of crystals. In particular, long-wavelength phonons are considerably more damped in glasses, yet it remains unclear how structural disorder at atomic scales affects such a macroscopic phenomenon. A plausible explanation is that phonons are scattered by local elastic heterogeneities that are essentially uncorrelated in space, a scenario known as Rayleigh scattering, which predicts that the damping of acoustic phonons scales with wavenumber k as kd+1 (in dimension d). Here we demonstrate that phonon damping scales instead as - kd+1 ln k, with this logarithmic enhancement originating from long-range spatial correlations of elastic disorder caused by similar stress correlations. Our work suggests that the presence of long-range spatial correlations of local stress and elasticity may well be the crucial feature that distinguishes amorphous solids from crystals.

Tensorial analysis of Eshelby stresses in 3D supercooled liquids.

It was recently proposed that the local rearrangements governing relaxation in supercooled liquids impress on the liquid medium long-ranged (Eshelby) stress fluctuations that accumulate over time. From this viewpoint, events must be characterized by elastic dipoles, which are second order tensors, and Eshelby fields are expected to show up in stress and stress increment correlations, which are fourth order tensor fields. We construct here an analytical framework that permits analyzing such tensorial correlations in isotropic media in view of accessing Eshelby fields. Two spherical bases are introduced, which correspond to Cartesian and spherical coordinates for tensors. We show how they can be used to decompose stress correlations and thus test such properties as isotropy and power-law scalings. Eshelby fields and the predicted stress correlations in an infinite medium are shown to belong to an algebra that can conveniently be described using the spherical tensor bases. Using this formalism, we demonstrate that the inherent stress field of 3D supercooled liquids is power law correlated and carries the signature of Eshelby fields, thus supporting the idea that relaxation events give rise to Eshelby stresses that accumulate over time.

Structural relaxation is a scale-free process.

We show that in deeply supercooled liquids, structural relaxation proceeds via the accumulation of Eshelby events, i.e. local rearrangements that create long-ranged and anisotropic stresses in the surrounding medium. Such events must be characterized using tensorial observables and we construct an analytical framework to probe their correlations using local stress data. By analyzing numerical simulations, we then demonstrate that events are power-law correlated in space, with a time-dependent amplitude which peaks at the alpha relaxation time τ(α). This effect, which becomes stronger near the glass transition, results from the increasingly important role of local stress fluctuations in facilitating relaxation events. Our finding precludes the existence of any length scale beyond which the relaxation process decorrelates.

Elastic signature of flow events in supercooled liquids under shear.

Using numerical simulation of a 2D Lennard-Jones system, we study the crossover from shear thinning to Newtonian flow. We find that the short-time elastic response of our system essentially does not change through this crossover, and show that, in the Newtonian regime, thermal activation triggers shear transformations, i.e., local irreversible shear events that produce Eshelby (long-ranged, anisotropic) deformation fields as previously seen in low-T glasses. Quite surprisingly, these Eshelby fields are found to persist much beyond the α-relaxation time, and shear thinning to coincide with the emergence of correlations between shear relaxation centers.

Ultrafast spherulitic crystal growth as a stress-induced phenomenon specific of fragile glass-formers.

We propose a model for the abrupt emergence, below temperatures close to the glass transition, of the ultrafast (glass-to-crystal) steady mode of spherulitic crystal growth in deeply undercooled liquids. We interpret this phenomenon as controlled by the interplay between the generation of stresses by crystallization and their partial release by flow in the surrounding amorphous visco-elastic matrix. Our model is consistent with both the observed ratios (∼10(4)) of fast-to-slow velocities and the fact that fast growth emerges close to the glass transition. It leads us to conclude that the existence of a fast growth regime requires both (i) a high fragility of the glassformer; (ii) the fine sub-structure specific of spherulites. It finally predicts that the transition is hysteretic, thus allowing for an independent experimental test.

Robustness of avalanche dynamics in sheared amorphous solids as probed by transverse diffusion.

Using numerical simulations, we perform an extensive finite-size analysis of the transverse diffusion coefficient in a sheared 2D amorphous solid over a broad range of strain rates at temperatures up to the supercooled liquid regime. We thus obtain direct qualitative evidence for the persistence of correlations between elementary plastic events up to the vicinity of the glass transition temperature T(g). A quantitative analysis of the data, combined with a previous study of the T and γ dependence of the macroscopic stress [Phys. Rev. Lett. 105, 266001 (2010)], leads us to conclude that the average avalanche size remains essentially unaffected by temperature up to T ~ 0.75T(g).