Elastic Instability behind Brittle Fracture

We argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a nonlinear elastic instability, which cannot be captured without accounting for geometrically precise description of finite elastic deformation. As a prototypical problem we consider a homogeneous elastic body subjected to tension and assume that it is weakened by the presence of a free surface which then serves as a location of cracks nucleation. We show that in this maximally simplified setting, brittle fracture emerges from a symmetry breaking elastic instability activated by softening and involving large elastic rotations. The implied bifurcation of the homogeneous elastic equilibrium is highly unconventional for nonlinear elasticity as it exhibits strong sensitivity to geometry, reminiscent of the transition to turbulence in fluids. We trace the postbifurcational development of this instability beyond the limits of applicability of scale-free continuum elasticity and use a phase-field approach to capture the scale dependent subcontinuum strain localization, signaling the formation of actual cracks.

Active elasticity drives the formation of periodic beading in damaged axons

In several pathological conditions, such as coronavirus infections, multiple sclerosis, Alzheimer's and Parkinson's diseases, the physiological shape of axons is altered and a periodic sequence of bulges appears. Experimental evidences suggest that such morphological changes are caused by the disruption of the microtubules composing the cytoskeleton of the axon. In this paper, we develop a mathematical model of damaged axons based on the theory of continuum mechanics and nonlinear elasticity. The axon is described as a cylinder composed of an inner passive part, called axoplasm, and an outer active cortex, composed mainly of F-actin and able to contract thanks to myosin-II motors. Through a linear stability analysis we show that, as the shear modulus of the axoplasm diminishes due to the disruption of the cytoskeleton, the active contraction of the cortex makes the cylindrical configuration unstable to axisymmetric perturbations, leading to a beading pattern. Finally, the nonlinear evolution of the bifurcated branches is investigated through finite element simulations.

More views of a one-sided surface: mechanical models and stereo vision techniques for Möbius strips

Möbius strips are prototypical examples of ribbon-like structures. Inspecting their shapes and features provides useful insights into the rich mechanics of elastic ribbons. Despite their ubiquity and ease of construction, quantitative experimental measurements of the three-dimensional shapes of Möbius strips are surprisingly non-existent in the literature. We propose two novel stereo vision-based techniques to this end—a marker-based technique that determines a Lagrangian description for the construction of a Möbius strip, and a structured light illumination technique that furnishes an Eulerian description of its shape. Our measurements enable a critical evaluation of the predictive capabilities of mechanical theories proposed to model Möbius strips. We experimentally validate, seemingly for the first time, the developable strip and the Cosserat plate theories for predicting shapes of Möbius strips. Equally significantly, we confirm unambiguous deficiencies in modelling Möbius strips as Kirchhoff rods with slender cross-sections. The experimental techniques proposed and the Cosserat plate model promise to be useful tools for investigating a general class of problems in ribbon mechanics.