82201 - FISICA DEI SOLIDI E DEI FLUIDI

Academic Year 2017/2018

  • Moduli: Maurizio Bonafede (Modulo 2) Maria Elina Belardinelli (Modulo 1)
  • Teaching Mode: Traditional lectures (Modulo 2) Traditional lectures (Modulo 1)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Physics (cod. 8007)

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 and inviscid fluids, 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 widerange of applications to geophysical phenomena which take place in the in the Earth interior, in the oceans and in the atmosphere.

Module 1 (prof. Maurizio Bonafede)

FLUIDS IN EQUILIBRIUM: Equation of state and specific heat for a generic substance; 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.

FLUIDS: Constitutive relationship for a Newtonian fluid: dynamic viscosity. Energy and Entropy equations. The Navier-Stokes equation, the Euler equation (inviscid fluids). Acoustic waves. Examples of stationary and transient laminar flows. Poiseuille flow in a cylindrical conduit: the Reynold number and transition to turbulence. Stoke’s formula and its applications in Geophysics. The Bernoulli equation: stationary flows and irrotational transient flows. The Boussinesq approximation. Gravity waves at the surface of a homogeneous fluid: dispersion relation, “deep water” and “shallow water” approximations. Dynamic pressure in a gravity wave. The trajectories of fluid particles. The role of surface tension: capillary waves. Stationary waves in closed basins (“seiches”). Gravity waves along the interface of immiscible fluids with different density: barotropic and baroclinic waves.

Raleigh convection: the Raleigh number and the Prandtl number. Transition to instability. Geometrical structure of convective cells. Theory of turbulent flows. Applications (the mean motion equation, Reynold’s stresses, eddy viscosity, the f-plane approximation, geostrophic flow, thermal wind, Ekman spiral at the ocean surface and at the base of the atmosphere).

 

Module 2 (Prof. Maria Elina Belardinelli)

ELASTIC SOLIDS: Strain energy in isothermal and adiabatic processes. Constitutive relationships for thermo-elastic materials. Isotropic materials: bulk modulus and rigidity, Lamé constants. The inverse constitutive relation, Young and Poisson moduli, their thermodynamic bounds. Relationship between thermal expansion and thermo-elastic coefficients. Isothermal and adiabatic elastic constants. Applications (simple stress configurations in the Earth’s crust: lithostatic pressure, uniaxial stress and uniaxial strain profiles. Stress profiles in sedimentary basins: the role of heat flow. Stress profiles in erosional environments. Classification of tectonic regimes. Plane deformation models: the role of friction and Anderson’s theory of faulting).

Elasto-dynamics: The Cauchy-Navier equation, rotational and irrotational elastic waves, the polarization of a plane wave. Applications (reflection and refraction of a plane elastic wave, free surface and welded boundary conditions).

Readings/Bibliography

Lecture notes of Prof. Maurizio Bonafede, available online (AMS Campus) 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.

Office hours

See the website of Maria Elina Belardinelli

See the website of Maurizio Bonafede