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Mathematical Physics
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
The discipline is located on the border between applied mathematical sciences and experimental sciences and is precisely the union of the mathematical and physical mentality; this allows you to transform a physical problem into a mathematical one and, after solving it, to physically interpret the result, systematically and rigorously translating a mechanical or thermomechanical system into equations, solving it and discussing the results. Thus, at the end of the course, the student will be able to face and solve numerous problems related to the motion and equilibrium of continuous systems; furthermore he will have acquired general knowledge on partial differential equations and some methodologies for their resolution.
Course Prerequisites
Compulsory preliminary teaching:
Mathematical Analysis, Geometry, Physics
Teaching Methods
The course is carried out entirely in the classroom including exercises which will be interactive.
Assessment Methods
The exam consists of a written and an oral test. The written test lasts 150 minutes, with the binding outcome for the next oral test: it is based on 4 closed-ended questions and 4 open-ended questions, and focuses on solving one or more practical problems related to motion and equilibrium of material point systems and rigid material bodies in inertial and non-inertial reference systems. The oral test focuses instead on a discussion of the theoretical foundations necessary to solve the same problems.
The outcome of the exam will result from the sum of partial scores reported in the written and oral test.
The written test aims to ascertain the student's ability to apply the knowledge acquired during the course; the oral exam aims to verify the level of knowledge and understanding of the course contents and to evaluate the autonomy of judgment, learning ability and communication ability.
The final grade will be awarded according to the following evaluation criteria:
30 - 30 cum laude: complete, in-depth and critical knowledge of the topics, excellent language skills, complete and original interpretative skills, full ability to independently apply knowledge to solve the proposed problems;
26 - 29: complete and in-depth knowledge of the topics, excellent language properties, complete and effective interpretative skills, able to independently apply knowledge to solve the proposed problems;
24 - 25: knowledge of the topics with a good degree of command, good language skills, correct and sure interpretative skills, good ability to correctly apply most of the knowledge to solve the proposed problems;
21 - 23: adequate knowledge of the topics but limited mastery of them, satisfactory language skills, correct interpretative ability, more than sufficient ability to independently apply the knowledge to solve the proposed problems;
18 - 20: basic knowledge of the main topics, basic knowledge of technical language, sufficient interpretative ability, sufficient ability to apply the basic knowledge acquired;
Insufficient: does not have an acceptable knowledge of the topics covered during the course.
Texts
1. T. Ruggeri: Introduzione alla Termomeccanica dei Continui, 2^ edizione, Monduzzi editoriale, Milano, 2013
2. F. Andreussi: Problemi di Calcolo Vettoriale e Tensoriale, Editrice Tecnico Scientifica, Pisa, 1974
3. M. Ciarletta & D. Iesan: Elementi di Meccanica dei Continui con Applicazioni, Pitagora editrice, Bologna, 1997
Contents
1. Elements of vector and tensor calculus (2 credits)
Reference systems and general information on free vectors - Dyad - Operations on vectors - Applied vectors - Resultant and resultant polar moment - Continuous systems - Characteristic vectors and scalar invariant - Plane and parallel applied vector systems - Matrix operators and Cartesian components - Identity operator - Kronecker and Levi-Civita symbols: properties and relations - Product of a scalar for a matrix operator - Sum of two operators - Product of two operators - Transposed operator - Trace of an operator - Determinant of an operator: expression of the determinant in the case of n = 3 - Inverse operator - Complementary operator - Some notable identities of matrix operators: some notable identities in the case n = 3 - Scalar product between operators - Symmetric and antisymmetric operators: dual vector associated with an antisymmetric operator, symmetric and antisymmetric parts of an operator - Deviatoric and isotropic part of an operator - Rotation operator - Orthogonal similarity transformations: basis change matrices, principal invariants of an operator - Eigenvalues and eigenvectors of an operator: eigenvalues and invariants of the powers of an operator, eigenvalues and eigenvectors for symmetric operators, diagonalization of an operator, Hamilton-Cayley theorem, relations between invariants and derivatives of the main invariants in the case n = 3 - Tensor product: semi-Cartesian representation of an operator, eigenvalues and eigenvectors of a tensor product in the case n = 3 - Sign definite operators: square root operator of a positive definite operator - Polar Theorem
2. Deformation and kinematics of a continuous body; acting forces (1.2 credits)
Configuration of a continuum - Deformation gradient operator - Deformation operators - Inverse deformation operator - Linear expansion coefficient - Slips - Surface expansion coefficient - Volume expansion coefficient - Incompressible bodies - Homogeneous deformation - Small deformations - Velocity and acceleration - Velocity gradient operator - Notes on rigid kinematics - Forces in a continuum - Stress tensor and Cauchy theorem - Examples of Cauchy stress tensor: pressure, simple tension, simple shear
3. Balance laws and general constitutive principles in continuum mechanics (1.3 credits)
Law of conservation of mass: Lagrangian formulation, Eulerian formulation - Cardinal equations: boundary conditions - Principle of virtual works - General balance laws: transport theorem, energy balance law, balance laws of thermomechanics in Eulerian form, invariance Galilean (optional), Lagrangian formulation of balance laws, momentum balance law in Lagrangian form and first Piola-Kirchhoff tensor, boundary conditions in Lagrangian variables, energy balance law in Lagrangian variables – Physical interpretation of Piola-Kirchhoff tensor, second Piola-Kirchhoff tensor, power of internal forces in terms of Piola-Kirchhoff tensors - General principles for constitutive laws: the principle of material indifference, the entropy principle
4. Continuous elastic, thermoelastic and fluid media (1.5 credits)
Elastic bodies: consequences of the principle of material indifference in the elastic case - Thermoelastic bodies: principles of material indifference in thermoelasticity, field equations of thermoelasticity, consequences of the principle of entropy in thermoelasticity - Linear elasticity: equations of isotropic linear elasticity - Ideal fluids and Euler equations: boundary conditions in the case of ideal fluids, work of internal forces in an ideal fluid - Fourier-Navier-Stokes dissipative fluids - Entropy principle for a fluid - Some special cases of fluids: Fourier-Navier fluids- Incompressible Stokes, compressible Euler fluids and linearized equations - Fluid equations in the Lagrangian formulation