82201 - Physics of Solids and Fluids

Learning outcomes

During this course, the student will learn the main physical laws governing the mechanical and thermal behavior of continuum media. In particular, he will be able to tackle problems concerning the equilibrium and the dynamics of solid elastic media, of viscid with applications in Physics of the Earth.

Course contents

The mechanics of continuum media provides several applications to the study of natural phenomena occurring in solid and fluid materials, throughout the Universe. Our ability to describe these phenomena requires the introduction of tensors (e.g. strain and stress) and constitutive relations between them, which provide a “complete” system of equations.

The scope of the course is:

• to introduce the main concepts and theorems of tensor algebra for the study of stress and strain in solid elastic materials, in viscous and inviscid fluids (with mention to plastic and viscoelastic materials);
• to obtain the equations governing the equilibrium and the dynamics of solids and fluids;
• to describe the mechanisms of heat transmission by conduction and convection;

to provide a wide range of applications to geophysical phenomena which take place in the in the Earth system.

MOLECULAR FUNDAMENTALS: Continuum model of the matter: Fluid materials: Temperature and thermal agitation, Pressure in a fluid at the equilibrium state: Archimede's and communicating vessel principle. Microphysical models of viscosity, thermal conduction, osmotic diffusion, surface tension. Heat propagation equation. Solid materials: microphysical models of compressibility, specific heat and thermal expansion.

FLUIDS IN EQUILIBRIUM: Equation of state and specific heat for a generic substance; equilibrium of a compressible medium; adiabatic gradient; latent heat and phase transitions. Applications (adiabatic temperature gradient in the atmosphere, in the ocean, in the Earth mantle; gravitational stability; potential temperature and density. Role of phase transitions).

THERMAL CONDUCTION: the conduction equation (Fourier law), heat flow, Lagrangian and Eulerian description: the material time derivative; the heat transmission equation, radioactive heat production. Applications (geotherms in the continental and in the oceanic lithosphere, seafloor isostatic topography. Solidification of lakes: the Stefan problem).

MECHANICS OF CONTINUUM MEDIA: “zero dimensional” conceptual models of elastic, viscous and viscoelastic materials. Definition of “continuum” medium: the minimum elementary volume. Definition of a tensor of rank k, the Kronecker delta and the permutation symbol, the e-delta identity: The deformation tensor: geometric interpretations of its components. Strain eigenvalues and eigenvectors; isotropic and deviatoric strain components. Body forces and surface tractions; the stress tensor, the Cauchy relation. Conservation laws for a continuum medium: conservation of mass, equations of motion and angular momentum. Symmetry of the stress tensor: principal stresses and stress axes. Normal stresses and shear stresses, isotropic and deviatoric stress components, mean pressure. The energy equation.

ELASTIC SOLIDS: Constitutive relationships for elastic materials. Isotropic materials: bulk modulus and rigidity, Lamé constants. The inverse constitutive relation, Young and Poisson moduli, their thermodynamic bounds. Elasto-dynamics: The Cauchy-Navier equation, rotational and irrotational elastic waves. Applications (simple stress configurations in the Earth’s crust: lithostatic pressure, uniaxial stress and uniaxial strain profiles. Classification of tectonic regimes. Friction and shear failure).

FLUIDS: Constitutive relationship for a Newtonian fluid: dynamic viscosity. Energy and Entropy equations. The Navier-Stokes equation, the Euler equation (inviscid fluids). The Boussinesq approximation. Examples of stationary and transient laminar flows. Poiseuille flow in a cylindrical conduit: the Reynold number and transition to turbulence. The Bernoulli equation: stationary flows and irrotational transient flows.

Lecture notes of Proff. Maurizio Bonafede and Maria Elina Belardinelli, available online (Insegnamenti online, IOL) at the end of the lessons regarding each chapter.

The following textbooks cover most of the arguments presented in the lecture notes:

F. Reif - Fisica statistica (in La fisica di Berkeley, Vol. 5), McGraw Hill, Newton, Massachusetts, 1967.

D. Turcotte e G. Schubert, Geodynamics, Cambridge University Press, 2014.

P. K. Kundu, Fluid mechanics, Academic Press, San Diego, California, 1990.

Y. C. Fung, Foundations of solid mechanics, Prentice Hall, Englewood Cliffs, New Jersey, 1965.

Teaching methods

Classroom lectures

Assessment methods

Oral Examination: typically 3 questions are proposed to assess the student's theorical knowledge of the main equations of Physics applied to the study of dynamic and thermal behaviour of fluids and solids. One question may require solving a problem similar to those proposed by the teacher during the exercise lectures. The mark is obtained as the mean among the evaluation of each of the three answers. The grading scale is the following

18-20 Knowledge of a limited number of faced subjects, which turns out with the teacher help only;

21-23 Knowledge of a limited number of faced subjects;

24-26 Knowledge of the majority of faced subjects and fair command of the basis instruments;

27-29 Knowledge of the faced subjects and command of the basis instruments;

30-30L Knowledge of the faced subjects, ability to argue and connect in complex problems, command of the basis instruments.

Office hours

See the website of Maria Elina Belardinelli

See the website of Eleonora Rivalta

See the website of Massimo Nespoli