The main objective of the course is to acquire knowledge of mathematical and physical modeling of physical systems and to develop critical skills for their application  in solving engineering problems.

The goal of part [1] of the Course is to provide the student with an overall understanding of the basics of advanced Continuum Mechanics and Continuum Thermodynamics. The student is expected to become knowledgable about all the main concepts and ideas, so that he might be ready to use them in more application-oriented contexts, which are part of the program in Mathematical Engineering, in particular for the "scientific computation track".

The goal of part [2] of the course is to make the student understand, both in a theoretical and an application oriented setting, the construction of perturbative approaches with a rigorous application of asymptotic series.

Particular focus is given to the understanding of the boundary layer theory, the construction of  the multi-scale methods, and the analysis of bifurcation problems and wave propagation phenomena in physical system models.

The student is expected:

- to master the mathematical develoment of all the mathematical and physical methods exposed during the course;

- to use dimensional analysis and variational principles to model the governing equations at the physical problem at hand;

- to develop the critical skills for selecting the most adapted method and for providing a rigorous derivation of the required mathematical solution;

- to use widespreadly adopted symbolic programming languages for object and modeling management (e.g. Mathematica);

- to develop a good capability of handling and working with tensorial quantities, of writing balance equations and boundary conditions for simple problems in elasticity theory;

- to understand the structure and the construction of basic theories of Continuum Mechanics;

- to become articulate and fluent in expressing and communicating all the key ideas and proofs of the basic theorems and concepts.

PART 1

Cauchy stress tensor and mechanical balance equations (reminder).

Theorem of kinetic energy and power of internal stresses. Piola transformation for vector and tensor fields. Piola stress tensor: definition and properties.

Elastic solids. Constitutive relation. Local balance equations in a material form. Material frame indifference. Symmetry group. Isotropicity. Representation formula. Strain energy in finite elasticity. Strain energy for isotropic materials. Examples: Blatz-Ko, Mooney-Rivlin, Ogden, Neo-Hookean. Internal constraints and reactive stresses. Examples: incompressibility, inextensibility in a given direction.

Linearization of constitutive relation for elasticity. Infinitesimal strain tensor. Elasticity tensor. Linear constitutive relation. Symmetry group. Isotropic linear elasticity. Lame' constants. Navier equations. Wave propagation in linear isotropic elasticity: transverse and longitudinal wave popagation. Simple shear, uniform extensions and uniform tractions. Poisson's ratio and Young's modulus.

Energy balance for continuum thermodynamics. Entropy inequality. Axioms of Continuum Thermodynamics. Constitutive relations for thermoelasticity. Free energy and entropy inequality. Coleman and Noll method and consequences. Linear thermoelasticity. Field equations for linear thermoelasticity. Fourier's law. Heat equation. Symmetry group for elastic fluids.

Constitutive relations for materials with stress tensor depending on F and its time derivative. Viscoelastic materials. Fluids. Viscous fluids. Newtonian and non-Newtonian fluids.

Tensor algebra and general coordinates in continuum mechanics. Natural basis and dual basis. The metric tensor. Covariant and controvariant components. Covariant derivative and Christoffel symbols.

PART 2

Asymptotic Analysis. Classification of singular points in ordinary differential equations. Irregular singular points: dominant balance method. Introduction to regular and singular perturbation theory.

Asymptotic regimes. Dimensionless Analysis: Buckingham theorem. Examples of dimensionless number in physical system models. Boundary layer theory. Matching conditions: Van Dyke rule and intermediate variable method. Quasilinear problems: applications of Erdelyi theorem. WKB approximation.

Bifurcations and instabilities. Fixed points and their linear stability analysis. Geometrical theory of bifurcation points. Multiple scale expansion and its application to nonlinear oscillators. Introduction to hydrodycamic instability problems. Basic notions of variation calculus. Elastic functionals and Euler-Lagrange equations.

Weakly nonlinear analysis. Applications of weakly nonlinear analysis in linear elastic problems of one-dimensional objects. Application of multiple-scale methods for wave propagation. The theorem of stationary phase and the effect of dispersion. Group velocity. The evolution of a wave envelope. Effect of nonlinearities: solitonic solutions in the Korteweg-deVries equation.

Basic ideas of calculus. Numerical series. Theory of ordinary differential equations and basic notions about partial differential equations. Elementary knowledege of classical Rational Mechanics and Continuum Mechanics (forces, balance equations, Cauchy stress tensor, kinematics of continua).

The exams wil be organized in agreement with the Academic Calendar and the Rules of the School of Engineering.

The verification of knowledge and capabilities acquired by the student is based, for each part, on a written test of about 2h, followed by an oral part.

The two written tests can be taken together during the same exam session, or separately in different exam sessions.

The final grade is based on an overall evaluation of the two parts of the exam.

The written test for part [1] consists in an exercise on Finite Elasticity theory, which requires the student to be able to work with tensorial constitutive relations and writing and solving simple equilibrium problems. This will provide a basis on which to evaluate the capability of the student to apply correctly knowledge and comprehension acquired during the course.

In the oral part of the exam the student will show his knowledge and his communication skills through appropriate exposition of all the main concepts and correct presentation of the proofs of all the main results and theorems.

The written test for part [2] consists in two written exercises based on physical problems, in which the student is expected to provide answers to few itemized questions.

In the oral part the student is expected to answer theoretical questions on course topics, and to verify student's skills at making connections between course topics and previous courses.

The teachers might place lecture notes in the Beep area of the Course.