


Politecnico di Torino  
Academic Year 2017/18  
01NZIMX Plasticity and fracture 

Master of sciencelevel of the Bologna process in Civil Engineering  Torino 





Subject fundamentals
The present course aims at introducing the fundamental concepts of Theory of Plasticity and Fracture Mechanics. Once ductile and brittle structural collapses are defined, the methodologies for the analysis structures in the of elastoplastic field and for the solution of elastic problems of cracked structural elements are described. In such a context, the most important static and energetic parameters for the evaluation of the stress field around a crack tip as well as the instability condition for the crack propagation are defined. The sizescale effects affecting the phenomena of cracking and crushing are emphasized. Some practical applications to concrete and reinforced concrete structures are proposed.

Expected learning outcomes
 Acquisition of the fundamental concepts of the subject, that will permit students to analyze, at least from a qualitative point of view, complex phenomena related to the mechanical response of structural elements (transitions between different collapse mechanisms, sizescale effects, ...)
 Learning of the analytical steps at the base of the fundamental models of plastic analysis of structures and fracture mechanics. Application of such models to specific case study analyzed during the course.  Capacity to analyze and solve, with autonomy, real problems and case study different from those presented during the course. At this purpose exercises are assigned to students, to be solved independently and presented at the exam.  Acquisition of skills in linear and nonlinear finite element modelling. At this purpose, several practical exercises are carried out in the computer laboratory. Furthermore, an exercise is assigned to each student, that has to be solved independently and presented at the exam.  Learning of the experimental techniques and methodologies for material characterization and for structural monitoring. At this purpose, visits to the experimental laboratory are foreseen. 
Prerequisites / Assumed knowledge
Fundamental notions on the basic mathematical courses (Analysis 1 and 2, Geometry, Analytical Mechanics, Numerical Methods: solution of ordinary and partial differential equations, eigenvalue/eigenvector problems ; statics, kinematics and dynamics of rigid bodies,..) and engineering courses (Structural Mechanics, Structural Engineering: study of isostatic and statically indeterminate structures; static, kinematic and constitutive equations for beams and plates) are required.

Contents
The program of the course is subdivided in the following five topics:
(1) PLASTICITY (18 h): elastoplastic flexure; incremental plastic analysis of beam systems; theorems of plastic limit analysis; proportionally loaded beam systems; nonproportionally loaded beam systems; cyclic loads and shakedown; deflected rectangular and circular plates; strip method. (2) PLANE STRESS AND PLANE STRAIN CONDITIONS(18 h): plane stress and plane strain conditions in cartesian coordinates, deep beam; plane stress and plane strain conditions in cylindrical coordinates, thickwalled cylinder; circular hole in a plate subjected to tension; concentrated force acting on the edge of an elastic halfplane; Analytical functions; KolosoffMuskhelishvili method; elliptical hole in a plate subjected to tension. (3) LINEAR ELASTIC FRACTURE MECHANICS (18 h): Griffith’s energy criterion; Westergaard’s method; Williams’ method; fundamental relation between the energy and the stress approaches, crack branching criteria in Mixed Mode condition; plastic zone at the crack tip; size effects and ductiletobrittle transition. (4) NONLINEAR FRACTURE MECHANICS (18 h): cohesive crack model, snapback phenomenon; advanced applications of the cohesive crack model: Mixed Mode crack propagation; overlapping crack model for concrete in compression; bridged crack model, minimum reinforcement; advanced applications of the bridged crack model: repeated loading. (5) ADVANCED APPLICATIONS (12 h): sizescale effects on the plastic rotational capacity of reinforced concrete beams in bending; application of Fractal Geometry to damaging and cracking phenomena; Multifractal Scaling Law (MFSL), size effects in the international standard codes; composite and hierarchical materials; fatigue crack growth and residual life of structures subjected to repeated loads, Paris’ law; energy emissions from brittle fracture; structural monitoring by the acoustic emission technique: seismic precursors. 
Delivery modes
The theoretical classes are supported by practical exercises carried out in the computer laboratories. In such experiences, numerical methods are applied for the solution of the problems treated from an analytical point of view during the theoretical classes. Commercial finite element codes as well as software developed adhoc within the research activity are provided by the teacher. In details, the proposed exercises are: (1) incremental plastic analysis of a statically indeterminate beam, elastoplastic analysis of a 2D frame; (2) numerical solution by means of a finite element approach of the following problems: deepbeam, thickwalled cylinder, concentrated force acting on the edge of an elastic halfplane, circular hole in a plate subjected to tension; (3) numerical modelling of the stress singularities, numerical evaluation of the stress intensity factor for the threepoint bending scheme, examples of crack propagation; (4) numerical examples of cohesive crack propagation in both Mode I and Mixed Mode, applications of the bridged crack model.
Besides, the students are asked to attend some experimental tests carried out in the Materials and Structures Laboratory of the Politecnico di Torino. In particular, the proposed tests are: tension test on steel rebars, compression test on concrete and rock specimens controlled by the circumferential strain, threepointbending test for the evaluation of the fracture energy for concretelike materials. All the aforementioned tests will be monitored by means of the acoustic emission technique in order to localize the cracking pattern within the specimens. In this context, some methodologies largely used in seismology (such as the GutenbergRichter law) will be applied to analyse the acoustic emission wave amplitude distribution, in order to characterize the damage growth and to identify the material critical conditions leading to the structural collapse. 
Texts, readings, handouts and other learning resources
Handouts:
 all the slides shown during the lectures are available for regularly registered students on the course website.  A. Carpinteri, "Development of realistic concrete models including scaling effects", Final Report to the Commission of the European Communities, Reactor Safety Programme 198587, Ispra, Italy, 1989.  A. Carpinteri, "Fractal nature of material microstructure and size effects on apparent mechanical properties", Internal Report, Laboratory of Fracture Mechanics, Politecnico di Torino, N. 1/92, 1992. Book:  A. Carpinteri, "Structural Mechanics: A Unified Approach", Chapters 18, 19 e 20, Chapman & Hall, 1997. 
Assessment and grading criteria
The exam consists in an oral examination during which the student presents and discusses the individual exercises assigned during the course and replies to theoretical questions. In this way, the level of preparation of the student is carefully verified on both the theoretical and the practical aspects.

