Quasistatic crack growth based on viscous approximation: a model with branching and kinking.

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking

A comparison of delamination models: Modeling, properties, and applications

This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed

Adaptive isogeometric analysis for phase‐field modeling of anisotropic brittle fracture

The surface energy a phase‐field approach to brittle fracture in anisotropic materials is also anisotropic and gives rise to second‐order gradients in the phase field entering the energy functional. This necessitates C 1 continuity of the basis functions which are used to interpolate the phase field. The basis functions which are employed in isogeometric analysis (IGA), such as nonuniform rational B‐splines and T‐splines naturally possess a higher order continuity and are therefore ideally suited for phase‐field models which are equipped with an anisotropic surface energy. Moreover, the high accuracy of spline discretizations, also relative to their computational demand, significantly reduces the fineness of the required discretization. This holds a fortiori if adaptivity is included. Herein, we present two adaptive refinement schemes in IGA, namely, adaptive local refinement and adaptive hierarchical refinement, for phase‐field simulations of anisotropic brittle fracture. The refinement is carried out using a subdivision operator and exploits the Bézier extraction operator. Illustrative examples are included, which show that the method can simulate highly complex crack patterns such as zigzag crack propagation. An excellent agreement is obtained between the solutions from global refinement and adaptive refinement, with a reasonable reduction of the computational effort when using adaptivity

Bounds on the effective behaviour of a square conducting lattice

A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behaviour of a square two-dimensional lattice of such resistors when both types coexist in fixed proportions in the lattice. The macroscopic behaviour is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds, following in the footsteps of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focuses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is 1/2. We conjecture that the derived bounds are optimal for all proportions