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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

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 are known to mechanically stretch their surroundings that locally stiffens the matrix. Although the bulk nonlinear elastic behaviors of these biopolymer networks are well studied, their local mechanical responses remain poorly characterized. Here, to understand how a living cell feels the nonlinear mechanical resistance from the ECM, we mimic the cell-applied local force using optical tweezers; we report that the local stiffening responses in highly nonlinear ECM are significantly weaker than responses found in bulk rheology, across two orders of magnitude of the locally applied force since the onset of stiffening. With a minimal model, we show that a local point force application can induce a stiffened region in the matrix, which expands with increasing magnitude of the point force. Furthermore, we show that this stiffened region behaves as an effective probe upon local loading. The local nonlinear elastic response can be attributed to the nonlinear growth of this effective probe that linearly deforms an increasing portion of the matrix.

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.