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Nonlinear stiffening is a ubiquitous property of major types of biopolymers that make up the extracellular matrices (ECM) including collagen, fibrin, and basement membrane. Within the ECM, many types of cells such as fibroblasts and cancer cells have a spindle-like shape that acts like two equal and opposite force monopoles, which anisotropically stretch their surroundings and locally stiffen the matrix. Here, we first use optical tweezers to study the nonlinear force-displacement response to localized monopole forces. We then propose an effective-probe scaling argument that a local point force application can induce a stiffened region in the matrix, which can be characterized by a nonlinear length scale R* that increases with the increasing force magnitude; the local nonlinear force-displacement response is a result of the nonlinear growth of this effective probe that linearly deforms an increasing portion of the surrounding matrix. Furthermore, we show that this emerging nonlinear length scale R* can be observed around living cells and can be perturbed by varying matrix concentration or inhibiting cell contractility.

Continuum mechanics describes compressive failure as a standard bifurcation in the response of a material to an increasing load: damage, which initially grows uniformly in the material, localizes within a thin band at failure. Yet, experiments recording the acoustic activity preceding localization evidence power-law distributed failure precursors of increasing size, suggesting that compressive failure is a critical phenomenon. We examine here this apparent contradiction by probing the spatial organization of the damage activity and its evolution until localization during compression experiments of 2D cellular solids. The intermittent damage evolution measured in our experiments is adequately described by a non-stationary depinning equation derived from damage mechanics and reminiscent of critical phenomena. In this description, precursors are damage cascades emerging from the interplay between the material's disorder and the long-range stress redistributions following individual damage events. Yet, the divergence of their characteristic size close to failure, which we observe in our experiments, is not the signature of a transition towards criticality. Instead, the system remains at a fixed distance to the critical point at all stages of the damage evolution. The divergence results from the progressive loss of stability of the material as it is driven towards localization. Thus, our study shows that compressive failure is a standard bifurcation for which the material disorder plays a marginal role. It also shows that precursory activity constitute by-products of the evolution towards localization and can serve to build a predictive method to assess the residual lifetime of structures., 29 pages, 10 figures including the supplementary information

The failure of brittle solids involves, before macroscopic rupture, power-law distributed avalanches of local rupture events whereby microcracks nucleate and grow, which are also observed in for an elastic interface evolving in a non-homogeneous medium. For this reason, it is tempting to relate failure to the depinning of an elastic interface. Here we compute the elastic kernel of the interface representing the damage field of a brittle solid. In the case of a damage model of rupture under compression, which implements the Mohr-Coulomb criterion at the local scale, we show that the elastic kernel is unstable, and hence is very different from the kernels of usual interfaces. We show that the unstable modes are responsible for the localization of damage along a macroscopic fault observed in numerical simulations. At low disorder, the most unstable mode gives the orientation of the macroscopic fault that we measure in numerical simulations. The orientation of the fault changes when the level of disorder is increased, suggesting a complex interplay of the unstable modes and the disorder.

We propose a novel approach inspired from non-local damage continuum mechanics to describe damage evolution and quasi-brittle failure of disordered solids. Heterogeneities are introduced at a mesoscopic continuous scale through spatial variations of the material resistance to damage. The central role played by the load redistribution during damage growth is analyzed by varying the interaction function used in the non-local model formulation. The spatio-temporal evolution of the damage field is obtained from energy conservation arguments, so that the formulation is thermodynamically consistent. We analytically determine the onsets of localization and failure that appear controlled by the redistribution function. Damage spreading is characterized through a complete statistical analysis of the spatio-temporal organization of the precursors to failure. The power law increase of the rate of energy dissipated by damage and an extracted correlation length close to failure supports the interpretation of quasi-brittle failure as a critical phenomena. Indeed, we establish a connection between our damage model and the evolution law of an elastic interface driven in a disordered medium. It allows to identify the order and control parameters of the critical transition, and capture the scale-free statistical properties of the precursors within the mean field limit. Finally, we experimentally investigate the coaction of localized dissipative events and elastic redistributions in disordered media via compression experiments of two-dimensional arrays of hollow soft cylinders. Our experimental observations show a quantitative agreement with the predictions derived following our approach.; Nous proposons une nouvelle approche inspirée des modèles d'endommagement non-locaux pour décrire la ruine des matériaux quasi-fragiles désordonnés. Les hétérogénéités matériaux sont introduites à une échelle continue mésoscopique via des variations spatiales de la résistance à l'endommagement alors que le mécanisme de redistribution des contraintes est décrit à travers une fonction d'interaction que l'on peut faire varier. L'évolution spatio-temporelle de l'endommagement est déterminée à partir du principe de conservation d'énergie et caractérisée via une étude statistique des précurseurs à la rupture. Cette approche nous permet de prédire la valeur des seuils de localisation et de rupture en fonction de la nature des redistributions. A l'approche de la rupture, nous mettons également en évidence une augmentation en loi de puissance du taux d'énergie dissipée ainsi qu'une longueur de corrélation, supportant l'interprétation de la rupture quasi-fragile comme un phénomène critique. En effet, nous démontrons que notre model d'endommagement s'apparente à la loi d'évolution d'une interface élastique évoluant dans un milieu désordonné. Cette analogie nous permet d'identifier les paramètres d'ordre et de contrôle de cette transition critique et d'expliquer les invariances d'échelle des fluctuations dans la limite champ moyen. Enfin, nous appliquons ces concepts théoriques à travers l'étude expérimentale de la compression d'un empilement bidimensionnel de cylindres élastiques. Notre approche permet de décrire de façon quantitative la réponse mécanique non-linéaire du matériau, et en particulier la statistique des précurseurs ainsi que la localisation des déformations.