


Politecnico di Torino  
Academic Year 2016/17  
03BOWNG Continuum Mechanics 

Master of sciencelevel of the Bologna process in Mathematical Engineering  Torino 





Subject fundamentals
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 prerequisite for all the courses dealing with the mechanics of fluids and solids the student will encounter in the following terms.

Expected learning outcomes
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.

Prerequisites / Assumed knowledge
Fundamentals of differential and integral calculus and of partial differential equations.

Contents
Kinematics of continua
Lagrangean and Eulerian coordinates Deformation gradient Polar decomposition CauchyGreen 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 ClausiusDuhem 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. NavierStokes equation. Helmholtz'Hodge decomposition. NonNewtonian fluids: shear thinning. ngrade fluids Fading memory. Linear viscoelasticity. Springdashpot models and integral models. 
Delivery modes
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.

Texts, readings, handouts and other learning resources
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. 
Assessment and grading criteria
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 35 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 68 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. 
