Lecturers:
Dr. Ing. Marco PAGGI, Politecnico di Torino short curriculum
Dr. Ing. Simone PUZZI, Politecnico di Torino short curriculum

The course intends to provide an overview of advanced engineering applications of nonlinear crack models widely used in Fracture Mechanics, namely the cohesive crack model and the bridged crack model.

The first two lectures are dedicated to the cohesive crack model, which is today one of the most used numerical models for the analysis of nonlinear crack propagation problems. After a brief account of the main features and hypotheses of the cohesive crack model for Mode I and Mixed Mode crack propagation, the mathematical details regarding its implementation in the Finite Element Method are carefully discussed.

Several engineering applications will be presented:
(i) analysis of Mixed Mode crack propagation in concrete beams;
(ii) evaluation of the anchorage bearing capacity in concrete and mortar using pull-out tests;
(iii) study of crack propagation in gravity dams;
(iv) analysis of stability of delamination in layered beams.

The last two lectures are devoted to the bridged crack model, which is broadly used in the modelling of fiber-reinforced materials and has also been applied to flexural elements reinforced by means of FRP.

After an introductory part dealing with the mathematical details of the model in the case with multiple reinforcements, the behaviour of a reinforced beam is described in detail, evidencing the rising of snap-back and snap-through instabilities.

Advancements of the bridged crack model, including reinforcements at multiple scales (e.g. aggregates/fibers and steel reinforcements in RC) and behaviour under cyclic loading, are also included in the lectures.

The course is subdivided in 2 topics and 4 lectures, as follows:

Topic 1 - lecture 1: Modelling Mode I cohesive crack propagation using the Finite Element Method (Marco PAGGI, 2 h) The main features and hypotheses of the cohesive crack model for Mode I crack propagation are briefly introduced. Then, the details regarding its implementation in the Finite Element Method are presented. Engineering applications illustrating the ductile-to-brittle size-scale transition in the mechanical behaviour of concrete beams under three-point bending complete the lecture.

Topic 1 - lecture 2: Advanced applications of the cohesive crack model in Mode Mixity (Marco PAGGI, 2 h) The mathematical theory for the Mixed Mode formulation of the cohesive crack model and its implementation in the Finite Element Method are briefly introduced. Then, the following engineering applications are presented, in comparison with experimental results: Mixed Mode crack propagation in concrete beams; evaluation of the anchorage bearing capacity in concrete and mortar using pullout tests; study of crack propagation in gravity dams; analysis of stability of delamination in layered beams.

Topic 2 - lecture 1: Bridged crack model: multiple reinforcements (Simone PUZZI, 2 h) The mathematical theory of the bridged crack model is revisited in the case of flexural elements with multiple reinforcements. The main features and hypotheses are briefly recalled. Then, the flexural behaviour of reinforced elements is analysed and described in detail. The ductile-to-brittle size-scale transition is evidenced and interpreted.

Topic 2 - lecture 2: Advanced applications of the bridged crack model (Simone PUZZI, 2 h) In the first part of the lecture, the model is extended to the case of flexural elements with reinforcements at multiple scales. Then, the behaviour of composite beams subjected to cyclic loading is studied. The combined effects of crack length, brittleness number, and fiber number on the cyclic behavior of the flexural element are described and the dissipation capacity of the reinforced beam is evaluated.

This course is an outcome of the ILTOF - Innovative Learning and Training on Fracture project. The course is maintained free of charge by the TCN Consortium for higher education. The ILTOF project has been funded with support from the European Commission. This publication reflects the views only of the Author(s), and the Commission cannot be held responsible for any use which may be made of the information contained therein.