Nanotechnology Engineering

BIAGIO CARBONI

The course provides a methodological preparation for the setting up and solving of continuum mechanics problems and provides the tools for the solution of technical problems concerning the mechanics of solids and structures. The students' learning is ascertained through an exam consisting of a written and an oral test in order to verify that the student is able to solve elementary problems of solid mechanics and slender elastic structures.
In particular, the topics cover (1) the mechanics of deformable bodies (displacement and deformation: compatibility equations) forces and efforts: postulate, lemma and Cauchy theorem, balance equations, constitutive equations: linear and isotropic and fibro-reinforced elastic materials. of resistance), (2) the elastic theory of the solid of Saint-Venant (The elastic problem of the continuous beam-shaped: exact solution and solution to Saint-Venant (extension, flexure, uniform torsion and non-uniform bending), (3) elements of beams mechanics and structural mechanics (Displacements and strain measurements, Forces and torques: balance equations, Relations between the monodimensional beam model and the 3D body, The kinematic and equilibrium problems of the constrained planar beam, The elastic problem: displacement method. The thermo-elastic response of beams in extension-bending. The technical beam theories: Euler-Bernoulli beam and Timoshenko beam).

The main objective of the course is to provide students with the fundamental knowledge of continuum mechanics through the study of the theories and methods used to formulate and solve problems of deformation and equilibrium in continuous systems, such as deformable bodies and complex structures.

The first part of the course focuses on introducing the concepts of deformation, stress, and their mathematical formulations, starting from the description of infinitesimal deformation and transformations of volume and area. Students are expected to acquire the ability to understand and apply the definitions of stretch, shear angle, and strain tensor in order to describe the behavior of deformable bodies under the action of internal and external forces.

The second part of the course addresses the balance laws of linear and angular momentum, studying stress tensors (such as the Cauchy and Piola–Kirchhoff tensors) and introducing the balance equations in the material (Lagrangian) configuration. Students will learn how to formulate and solve problems related to stress and strain in rigid and deformable bodies by applying the theory of boundary value problems, and they will develop the necessary skills to analyze and interpret equilibrium conditions and the mechanical response of complex systems.

The third part of the course focuses on the analysis of the mechanical properties of materials, with particular attention to linear elastic and hyperelastic behavior, as well as the introduction of constitutive relations governing the response of a material to mechanical and thermal loads. Students will learn the theory of isotropic and hyperelastic materials and will develop the ability to formulate the elastic problem for three-dimensional continuous systems.

Finally, the fourth part of the course addresses the problems related to the planar linear beam, formulating the elastic problem within a displacement-based approach. Students are expected to develop both theoretical and practical skills to calculate and interpret the mechanical characteristics of constrained beam systems subjected to external loads. These notions are fundamental for understanding aspects related to beam dynamics.

To approach the course in Continuum Mechanics, a solid understanding of the topics covered in the Mathematics I and General Physics courses is required, with particular reference to concepts of linear algebra, differential and integral calculus, and vector and matrix theory. Specifically, students must have a strong grasp of: operations with vectors and matrices, solving systems of linear equations, knowledge of coordinate transformations, and computational techniques to solve problems in geometry and linear algebra. Furthermore, a solid foundation in differential and integral calculus is essential for understanding differential equations and conservation laws, which will be frequently used throughout the course.
At the beginning of the course, theoretical reviews will be provided on the mathematical and physical tools necessary for understanding the topics covered, to help students address any specific or general gaps in knowledge. Students must become familiar with the concept of tensors, differentiation, and integration of scalar and vector functions, in order to successfully tackle topics such as deformation, stress, and the conservation of energy and momentum, which are fundamental for the analysis of materials and structures within the context of continuum mechanics.

ENGLISH
1. Introduction to Continuum Mechanics
1.1Discrete dynamical systems: deformation, energy dissipation, harmonic oscillator
2. Theory of Deformation
2.1 Deformation gradient: properties
2.2 Measures of deformation: stretch and shear angle
2-3 The Cauchy–Green deformation tensor
2.4 Principal stretches and principal stretch directions
2.5 Polar decomposition of the deformation gradient: pure stretch and rigid rotation
2.6 Volume and area transformations: volumetric and areal stretch
Green–Lagrange strain tensor and linearization
Displacement gradient, infinitesimal strain tensor
3. Theory of Stress
3.1 Body forces and surface forces from reference to current configuration, and mass conservation
3.2 Cauchy stress
3.3Cauchy’s lemma and the fundamental theorem of stress (Cauchy–Poisson theorem)
3.4 The Cauchy stress tensor
3.5 Balance of linear momentum and angular momentum
3.6 The nominal stress tensor (First Piola–Kirchhoff stress tensor)
3.7 The Second Piola–Kirchhoff stress tensor and linearization of the balance equations
4. Constitutive Relations
4.1 Hyperelasticity
4.2 Constitutive symmetry restrictions: monoclinic and orthotropic materials
4.3 Elastic isotropy
4.4 Lamé constants and engineering constants
4.5 Vectorial and tensorial forms of the isotropic hyperelastic law
5. Theory of the Planar Linear Beam
5.1 Mechanical and kinematic properties of constraints
5.2 Isostatic/isokinematic, hypostatic/hypokinematic, hyperstatic/hyperkinematic systems
5.3 Compatibility equations for Euler–Bernoulli and Timoshenko beams
5.4 Indefinite equilibrium equations
5.5 Generalized linear elastic constitutive law
5.6 The elastic problem in the displacement-based approach (elastic curve)
5.7 Beam dynamics: natural frequencies, vibration modes

The teacxher will provide handouts on the topics covered at the end of each lesson.

Teaching will take place exclusively in person, except in the case of unavailability of classrooms due to graduation sessions. Google Meet and Zoom platforms may be used for office hours. Lectures will primarily rely on the use of the blackboard by the instructor to explain the topics. The theoretical discussion will be followed by a practical section in which exercises will be carried out. Both theoretical explanations and, especially, the exercise sessions will be characterized by strong collective interaction. Students will also be invited to voluntarily perform exercises prepared beforehand at home.

Attendance to the course is not mandatory. However, the highest possible participation is strongly encouraged.

The exam consists of a practical and a theoretical part. The practical part will be written, while the theoretical part may be taken either in written or oral form, depending on the number of attendees.
For regular attendees, the exam can be divided into two partial exams (exoneri):
First partial exam: Around the beginning of November, it will cover the first half of the course (up to the analysis of deformation). The partial exam consists of a practical test (exercises) and an oral exam. Both tests must be passed with a minimum score of 18/30 to be exempt from the topics covered in this first partial exam.
Second partial exam: At the end of the course, it will cover the remaining part of the program. This partial exam will also include a practical test and an oral exam.
On the date of the second partial exam, students will also have the opportunity to retake the first partial exam if they have not previously taken it, or to take it for the first time. Afterward, they can proceed with the second part of the exam during the January and February exam sessions.
After the January and February exam sessions, it will no longer be possible to take the exam in separate partial exams. Students will have to take the entire exam, which will consist of:
A written practical exam: covering the entire program.
A theoretical exam, which will be conducted either in written or oral form, depending on the number of attendees. If the number is large, the theoretical exam will be written; otherwise, it will be possible to take it orally.
The practical exam content will consist of exercises that will be extensively covered during the course, for which numerous examples will be provided on Classroom. The theoretical exam will consist of questions and proofs related to the topics covered in the lectures.

The evaluation methods described above may undergo modifications, which will be communicated and agreed upon in class.

The questions for the learning assessment will consist of demonstrations carried out by the instructor on the various topics, as well as questions aimed at evaluating the understanding of the methodologies presented from both a theoretical and practical/applicative perspective. The types of exercises will be extensively explained by the instructor in class, and numerous examples will be provided on the course's Classroom site.