Elements of Mathematical Physics

Module MODULO II

The aim of the course is to give the main aspects of classical continuum mechanics. 

In particular, the course aims to allow the student to acquire the following skills:

-knowledge and understanding:  knowledge of results and fundamental methods  in continuum mechanics. Skill of understanding problems and to extract the major features. Skill of reading, understanding and analyzing a subject in the related literature and present it in a clear and accurate way. 

-applying knowledge and understanding: skill of elaborating new example or solving novel theoretical exsercise,  looking for the most appopriate methods and applying them in an appropriate way. 

-making judgements: To be able of devise proposals suited to correctly interprete complex problems in  fluid dynamics and elasticity. To be able to formulate autonomously  adequate judgements on the applicablity of studied models to theoretical or real situations. 

-communication skills: skills of presenting arguments, problems, ideas and solutions in mathematical terms with clarity and accuracy and with procedures  suited for the audience, both in an oral and a written form. Skill of clearly motivating the choice of the strategy, method and contents, along with the employed computational tools.

learning skills: reading and analyzing a subject in the literature involving continuum material systems. To tackle in an autonomuous way the systematic study of arguments not previously treated.  To acquire a degree of autonomy such that the student can be able to start with an autonomuos reserach activity.

Frontal lectures.

If restrictions will be introduced because the COVID pandemic, le lectures will be given in a mixed way or only online and some changes could be introduced to assure the accomplishiments foreseen for the course. 

Learning assessment may also be carried out on line, should the conditions require it.

IMPORTANT: in order to guarantee equal opportunities  to the students with handicap and/or any form of disability, such students may ask to talk to the teacher to program suitable actions. The interested students may also contact  prof. Patrizia Daniele o, the delegate in the Department of Mathematics and Computer Science for students with handicap and/or any form of disability.

2. Elements of tensorial calculus. Polar decomposition theorem. Tensorial analysis. Continuous media. Kinematics of a continua: deformations, deformation gradient, Reynolds transport theorem and its applications. Equations of balance of mass, linear momentum, angolar momentum and energy in global and local form, classification of forces acting on a continuum. Cauchy stress tensor. Clausius-Duhem inequality. Constitutive equations for fluids and elastic media. Material frame indifference. Special motions of fluids and their stability. Balance equation for elastic media in Lagrangian form and Piola-Kirchhoff stress tensor. Linear elasticity.  

   Euler equations for gas dynamics. Isentropic gas dynamics. Shock-waves. Euler equations and Navier-Stoker-Fourier equations as hydrodynamic limit of the transport Boltzmann equation by applying the Chapman-Enskog expansion. 

    
   

1. M. Gurtin, An introduction to continuum mechanics, Academic Press

2. I. Müller, Thermodynamics, Pitman

2. C. Truesdell, W. Noll, The non-linear field theories of mechanics, Springer

If necessary the exam will be  online.