The aim of the course is to teach the mathematical foundations to handle continuous material: gases, inviscid, viscous and viscoelastic liquids, elastic and viscoelastic solids. This course is considered a pre-requisite for all the courses dealing with the mechanics of fluids and solids the student will encounter in the following terms.

The student will be able to understand and describe the behaviour of continuum systems. The student will be able to translate in mathematical terms the problems dealing with continuum materials, deducing suitable mathematical models. Conversely, examining the model, the student will be able to foresee the mechanical properties of the related material and the properties of the solution of the mathematical problem.

Fundamentals of differential and integral calculus and of partial differential equations.

Kinematics of continua
Lagrangean and Eulerian coordinates
Deformation gradient
Polar decomposition
Cauchy--Green strain tensors.
Rate of strain tensors.
Singular surfaces and the generalization of Gauss and Stokes' theorems.

Balance equations
Reynolds' theorem, Rankine'Hugoniot's conditions.
Balance equations in Lagrangean and Eulerian coordinates
Mass, momentum and energy balance.
Entropy and Clausius-Duhem inequality.
constraints: incompressibility and inestensibility
Principle of objectivity

Constitutive equations
Elastic and iperelastic solids.
Linear elasticity.
Inviscid fluids. Euler's equation. Bernoulli's theorem. Kelvin's theorem. Acoustic waves.
Viscous fluids. Navier-Stokes equation. Helmholtz'Hodge decomposition.
Non-Newtonian fluids: shear thinning. n-grade fluids
Fading memory. Linear viscoelasticity. Spring-dashpot models and integral models.

The course is not suited for simple exercises to be developed in a limited time. However, several deeper problems will be proposed to be developed by the students at home. These problems will represent part of the final exam.

Notes available on the web
N. Romano and R. Lancellotta, Continuum Mechanics Using Mathematica: Fundamentals, Applications, and Scientific Computing, Birkhauser 2005.
I.S. Liu, Continuum Mechanics, Springer Verlag, 2002.

The exam will consist of an oral discussion and of the solution of 5 exercises and 1 "problem" distributed over the different chapters of the program. At least one of the exercises should include some numerical visualization script. Each exercise is worth 4 or 5 points and the problem is worth 6 or 7 points depending on their difficulty.

The exam has the aim of accertaining the kowledge of the subjects listed in the official programme and the skill in applying the theory and his methods to the solution of exercices. For each student the exam is consituted by

1- a writted exam consisting of
1a- two exercises on the first two chapters of the lecture notes (deformations and kinematics od continua)
1b- a theoretical question on chapters 3-5 of the lecture notes (balance equations, and their applications to continuum mechanics and constitutive equation)
During the written exams, that last two hours, students can not bring any book or notes and scientific calculators are useless

2- a set of exercises to be solved individually at home (every student has his own set of exercises different from each other). The set consists of
2a- three problems on the behaviour of some continua (elastic solids, fluids, viscoelastic media) described in chapters 6-8 of the lecture notes
2b- practical identification of a constitutive equation from experimental data
2c- deduction of a classical continuum mechanics equation in curvilinear coordinates, following the contents of the appendix of the lecture notes

Every exercise contributes to the final result witha score that ranges between 3 and 4 according to its difficulty, apart from the theoretical question at item 1b that is worth 5 points and the lenghtiers curvilinear equation question at item 2c that is worth 7 or 8 punti, again as a function of its difficulty.