Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major concern in structural designs. Computational modeling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals, and biological soft tissues. This chapter provides an extensive overview of the literature on the so-called phase-field fracture/damage models (PFMs), particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFMs are the regularized versions of the variational approach to fracture which generalizes Griffith's theory for brittle fracture. They can handle topologically complex fractures such as initiation, intersecting, and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFMs. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFMs are presented. Unresolved issues for further investigations are also documented.

Phase-field modeling of fracture

Localisation studies have been carried out for an unconstrained, elasto-plastic, strain-softening Cosserat continuum. Because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement. A finite, constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions. Because of the existence of rotational degrees-of-freedom in a Cosserat continuum additional wave types arise and wave propagation becomes dispersive. This has been investigated analytically and numerically for an elastic Cosserat continuum and an excellent agreement has been found between both solutions.

Localisation in a Cosserat continuum under static and dynamic loading conditions

Cracking elements: A self-propagating Strong Discontinuity embedded Approach for quasi-brittle fracture

Summary
Nonlocal integral and/or gradient enhancements are widely used to resolve the mesh dependency issue with standard continuum damage models. However, it is reported that whereas the structural response is mesh independent, a spurious damage growth is observed. Accordingly, a class of modified nonlocal enhancements is developed in literature, where the interaction domain increases with damage. In this contribution, we adopt a contrary view that the interaction domain decreases with damage. This is motivated by the fact that the fracture of quasi-brittle materials typically starts as a diffuse network of microcracks, before localizing into a macroscopic crack. To ensure thermodynamics consistency, the micromorphic theory is adopted in the model development. The ensuing microforce balance resembles closely the Helmholtz expression in a conventional gradient damage model. The superior performance of the localizing gradient damage model is demonstrated through a one-dimensional problem, as well as mode I and II failure in plane deformation. For all three cases, a localized deformation band at material failure is obtained. This article is protected by copyright. All rights reserved.

Localizing gradient damage model with decreasing interactions

A geometrically regularized gradient-damage model with energetic equivalence

https://www.tpbin.com/Uploads/Subjects/cc23e097-122e-40a0-af62-4d9e74e2358a.pdf

Stress-based nonlocal damage model

From diffuse damage to strain localization from an Eikonal Non-Local (ENL) Continuum Damage model with evolving internal length

Identification of an equivalent viscous damping function depending on engineering demand parameters

The study of a reinforced concrete beam tested under four-point bending is proposed in this article. An analysis of the behaviour of this beam from the global response down to local information such as cracking is performed. In order to describe the progressive degradation of the beam, a damage model is used, associated to a stress-based non-local regularisation method. A post-treatment of the finite element analysis is then performed to characterise the cracking pattern (crack spacing and crack opening). In this article, two different methods of post-treatment are compared: the topological search and continuous/discontinuous crack opening and the global/local analysis. Results show the capability of both methods to give a good estimation of cracking. Finally, the main advantages and drawbacks of both methods are underlined.

/pdf/cracking-analysis-of-reinforced-concrete-structures-38pa3mpuj6.pdf

Cédric Giry

Papers

Stress-based nonlocal damage model

From diffuse damage to strain localization from an Eikonal Non-Local (ENL) Continuum Damage model with evolving internal length

Identification of an equivalent viscous damping function depending on engineering demand parameters

Cracking analysis of reinforced concrete structures

Topological search of the crack pattern from a continuum mechanical computation