Integrated Calendar

Cracks, the major vehicle for material failure, tend to accelerate to high velocities in brittle materials. In three-dimensions, cracks generically undergo a micro-branching instability at about 40% of their sonic limiting velocity. Recent experiments showed that in sufficiently thin systems cracks unprecedentedly accelerate to nearly their limiting velocity without micro-branching, until they undergo an oscillatory instability. Despite their fundamental importance and apparent similarities to other instabilities in condensed-mater physics and materials science, these dynamic fracture instabilities remain poorly understood. They are not described by the classical theory of cracks, which assumes that linear elasticity is valid inside a stressed material and invokes an extraneous local symmetry criterion to predict crack paths. Here we develop a theory of two-dimensional dynamic brittle fracture capable of predicting arbitrary paths of ultra-high-speed cracks in the presence of elastic nonlinearity without extraneous criteria. We show that cracks undergo a dynamic oscillatory instability controlled by small-scale elastic nonlinearity near the crack tp. This instability occurs above an ultra-high critical velocity and features an intrinsic wavelength that increases proportionally to the ratio of the fracture energy to an elastic modulus, in quantitative agreement with experiments. This ratio emerges as a fundamental scaling length assumed to play no role in the classical theory of cracks, but shown here to strongly influence crack dynamics. The degree of universality of the instability is also demonstrated. Those results pave the way for resolving other long-standing puzzles in the failure of materials.

To bridge the microscopic molecular motion in complex structures with the macroscopic diffusion given by the Magnetic Resonance (MR) signal, we propose a general stochastic model for molecular motion in a magnetic field. In this model, the Fokker-Planck equation governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad hoc selection of a propagator. The boundary conditions that implicitly describe the microstructure of the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal.
In the second part of the talk, we will utilize our knowledge about the Bloch Torrey equation to quantify the effect of self-diffusion on multi-parametric sequences, such as those used for Magnetic Resonance Fingerprinting (MRF). We propose a signal simulation approach, generate dictionaries based parameter estimation that replaces the Bloch equation with the Bloch-Torrey equation, and accounts for protocol and scan dependent parameters. We apply this framework to a Multi Spin Echo (MSE) protocol and quantify the diffusion encoding introduced by the spoiler gradients in this sequence. We further show that increasing the spoiler strength would allow detecting the diffusion by including the diffusion effect in the dictionary.