Biopolymer networks, such as those constituting the cytoskeleton of a cell or biological tissue, exhibit a nonlinear strain-stiffening behavior when subjected to large deformations. Interestingly, rheological experiments on various in vitro biopolymer networks have shown similar strain-stiffening trends regardless of the differences in their microstructure or constituents, suggesting a universal stiffening mechanism. In this article, we use computer simulations of a random network comprised of cross-linked biopolymer-like fibers to substantiate the notion that this universality lies in the existence of two fundamental stiffening mechanisms. After showing that the large strain response is accompanied by the development of a stress path, i.e., a percolating path of axially stressed fibers and cross-links, we demonstrate that the strain stiffening can be caused by two distinctly different mechanisms: 1) the pulling out of stress-path undulations; and 2) reorientation of the stress path. The former mechanism is bending-dominated and can be recognized by a power-law dependence with exponent 3/2 of the shear modulus on stress, whereas the latter mechanism is stretching-dominated and characterized by a power-law exponent 1/2. We demonstrate how material properties of the constituents, as well as the network microstructure, can affect the transition between the two stiffening mechanisms and, as such, control the dominant power-law scaling behavior.

We show that chevron-notched samples offer an attractive approach to the measurement of fracture toughness in micron-scale samples of brittle materials and use the method to characterize quartz and nanocrystalline alumina. Focused ion beam milling is used to carve bend bars of rectangular cross-section a few micrometres wide and containing a notch with a triangular ligament. Load-controlled testing is conducted using a nanoindentation apparatus. If the notch is appropriately machined, cracks nucleate and propagate in a stable fashion before becoming unstable. Sample dimensions are measured using a scanning electron microscope, and are used as input in finite element simulations of the bars' elastic deformation for various crack lengths. The calculated compliance calibration curve and the measured peak load then give the local fracture toughness of the material. Advantages of the method include a low sensitivity to environmental subcritical crack growth, and the fact that it measures toughness at the tip of a sharp crack situated in material unaffected by ion-milling. The approach is demonstrated on two materials, namely, monolithic fused quartz and nanocrystalline alumina Nextele 610 fibres; results for the latter give the intrinsic grain boundary toughness of alumina, free of grain bridging effects.

Actin filaments assemble into network-like structures and play an important role in various cellular mechanical processes. It is known that the response of actin networks cross-linked by stiff proteins is characterized by two distinct regimes: (i) a linear stress–strain response for small deformations and (ii) a power law relation between elastic shear modulus and stress with an exponent 3/2 for large deformations. If the response is interpreted by the entropic single filament model, this hardening is attributed to the reduction of the thermal filament fluctuations as the filaments undergo affine stretching. By contrast, here we numerically study the elastic properties of a discrete, fully three-dimensional model for an isotropic filamentous networks, where athermal filaments of finite length are interconnected by rigid cross-links. By analyzing the network microstructure, the network connectivity is quantified in terms of the ratio of the mean filament length and the mean cross-linking distance along the filament. We derive a scaling relation for the initial network shear modulusi.e., regime ias a function of the network connectivity. When sheared to large strains, the simulated networks exhibit nonlinear strain hardening characterized by a 3/2 power lawregime ii. The origin of the strain hardening is shown to be associated now with the stretching of percolating and nearly fully extended stress paths in the network. In addition, we show that the strain at the onset of the strain hardening regime only depends on the network connectivity.

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