Intergranular fracture in polycrystals is often simulated by finite elements coupled to a cohesive zone model for the interfaces, requiring cohesive laws for grain boundaries as a function of their geometry. We discuss three challenges in understanding intergranular fracture in polycrystals. First, 3D grain boundary geometries comprise a five-dimensional space. Second, the energy and peak stres...

By a combination of geometrical and configurational analysis we study the properties of absolute minimal and equilibrium states of general Mumford-Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results concern the mathematical relations between physical quantities as energy release rate and energy concentration for 3D cracks with complex shap...

Thin-film fracture can be used as a nano-fabrication technique but, generally, it is a stochastic process that results in non-uniform patterns. Crack spacings depend on the interaction between intrinsic flaw populations and the fracture mechanics of crack channeling. Geometrical features can be used to trigger cracks at specific locations to generate controlled crack patterns. However, while th...

The receptor-rich postsynaptic membrane of the elasmobranch electric organ was fixed by quick-freezing and then viewed by freeze-fracture, deep-etching, and rotary-replication . Traditional freeze-fracture revealed a distinct, geometrical pattern of shallow 8.5-nm bumps on the E fracture-face, similar to the lattice which has been seen before in chemically fixed material, but seen less clearly ...

A new computational model for brittle fracture and fragmentation has been developed based on finite element analysis of non-linear elasticity equations. The proposed model propagates the cracks by splitting the mesh nodes alongside the most over-strained edges based on the principal direction of strain tensor. To prevent elements from overlapping and folding under large deformations, robust geo...

The receptor-rich postsynaptic membrane of the elasmobranch electric organ was fixed by quick-freezing and then viewed by freeze-fracture, deep-etching and rotary-replication. Traditional freeze-fracture revealed a distinct, geometrical pattern of shallow 8.5-nm bumps on the E fracture-face, similar to the lattice which has been seen before in chemically fixed material, but seen less clearly th...

With the growth in interest in geometrical ideas it is important to be clear about the nature of geometrical reasoning and how it develops. This paper provides an overview of three theoretical frameworks for the learning of geometrical reasoning: the van Hiele model of thinking in geometry, Fischbein’s theory of figural concepts, and Duval’s cognitive model of geometrical reasoning. Each of the...