The present course aims at introducing the fundamental concepts of Linear Elastic Fracture Mechanics (LEFM) and their applications in Civil Engineering. The solution to elastic problems of cracked structural elements is presented, as well as both the stress-field and the energy-based fundamental approaches. The singular stress field in the crack tip vicinity is determined and the instability conditions for brittle crack propagation are defined. The size-scale effects and the related ductile-to-brittle transition are described using nonlinear crack models, and then applied to the assessment of the brittle behaviour in reinforced concrete structures. After introducing the basic concepts of Fractal Geometry, the problem of fatigue resistance and fatigue crack growth is introduced. Some applications to metallic structures are eventually proposed.

The course has the scope of providing the students with all the conceptual elements and tools, which are necessary to perform the brittle behaviour assessment of next-generation reinforced concrete structures (Hybrid RC, FRP-bar RC, and High-strength PC). In the case of metallic structures subjected to cyclic loading, the fatigue assessment is illustrated for both resistance and subcritical crack growth.

Fundamental notions are required from the basic mathematical courses (Mathematical Analysis 1 and 2, Geometry, Rational Mechanics, Numerical Methods: Solution to ordinary and partial differential equations, eigenvalue/eigenvector problems; Statics, kinematics, and dynamics of rigid bodies) and engineering courses (Theory of Elasticity, Reinforced and prestressed concrete).

The program of the course is subdivided into the following topics: (1) Introduction to Fracture Mechanics: Fracture energy; Griffith energy criterion (2) Plane stress and plane strain conditions: Airy stress function in cartesian and polar coordinates; Thick-walled cylinder; Circular hole in a plate subjected to tension (3) Stress-singularity at the crack tip: Westergaard method (complex potentials); Mixed-mode conditions; Stress-intensity factors (4) Stress-singularity at the vertex of a re-entrant corner: Williams method (series expansions); Irwin fundamental theorem (5) Extension of the plastic zone at the crack tip; Ductile-to-brittle size-scale transition; Brittleness number (6) Bridged crack model and the problem of minimum reinforcement; Applications to fibre-reinforced materials (7) Cohesive crack model and snap-back instability; Applications to plain concrete (8) Overlapping crack model and the problem of maximum reinforcement; Applications to steel-bar reinforced and to prestressed concrete (9) Applications of Fractal Geometry to strength of materials and fracture mechanics: Lacunar versus invasive fractality (10) Scale effects on the mechanical properties and fractal cohesive crack model: Renormalization group theory; Multi-fractal scaling laws (11) Fatigue resistance and Woehler curve; Fatigue crack growth and Paris law; Fatigue limit and fatigue threshold (12) Fracto-emissions as seismic precursors

The theoretical classes are followed by practical exercises carried out in the Computational Laboratories. In such practical classes, numerical methods are applied for the solution of the problems already treated from an analytical point of view during the theoretical classes. Commercial finite element codes as well as software developed ad-hoc within the research activity are provided by the teacher.

All the slides shown during the lectures are available on the course website for the students regularly registered. Reference text books: A. Carpinteri, “Advanced Structural Mechanics”, Chaps. 11 and 12, CRC Press, New York, 2017. A. Carpinteri, “Structural Mechanics: A Unified Approach”, Chaps. 19 and 20, Chapman & Hall, London, 1997. A. Carpinteri, “Fracture and Complexity”, Springer Nature, Berlin, 2021.

Modalità di esame: Prova orale obbligatoria;

Exam: Compulsory oral exam;

... The exam consists in an oral interview, during which the student replies to scientific and technical questions related to the course topics. In this way, the level of preparation of the student is carefully assessed on both theoretical and practical aspects.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.