Propagation of Mechanical Waves in Deformable Solids

Proponent: Prof. Carlo Giovanni Lai

Course learning outcomes/abstract: the course begins with a review of a few mathematical concepts including Fourier and Laplace transforms, Bessel and Hankel functions, spherical harmonics.
Next, the two main classes of wave motion represented by hyperbolic and dispersive waves are introduced as they constitute the theoretical framework for the remaining of the course. Afterwards wave propagation in elastic waveguides will be thoroughly discussed. Problems examined include vibrations of strings, longitudinal and flexural waves in beams and in thin membranes. A selected number of both free and forced vibration problems will be analyzed with reference to infinite and finite 1D and 2D structural members. The second part of the course focuses on problems of wave propagation in unbounded continua and half-spaces. Topics include wave motions with polar and axial symmetry, propagation of waves in non-homogeneous media, surface Love and Rayleigh waves, the solution of the Lamb problem including a discussion of the differences between 2D versus 3D radiation (Huyghens’ principle). The dispersive properties of surface waves will be illustrated with examples from geophysics and seismology in the solution of relevant forward and inverse problems. The following subject to be studied is wave propagation in linear dissipative continua which includes Boltzmann equations, the elastic-viscoelastic correspondence principle for time-invariant boundary conditions and a discussion on the implications of the principle of physical causality. In connection to the theory of viscoelasticity ideas from fractional calculus will also be introduced. The course ends with the analysis of moving loads applied at the free-surface of an elastic half-space under subcritical and supercritical regimes. The subject is closely related to the study of the vibrational impact induced by fast and super-fast trains. Time permitting and if of interest to the participants the course will also provide a brief introduction to wave propagation in saturated poroelastic media (Biot’s theory).

Goals: Scope of the course is to provide advanced theoretical knowledge on the phenomena associated with the propagation of mechanical waves in elastic, viscoelastic and poroelastic continua. The course is addressed to scholars working in different, yet interacting research fields including geophysics, seismology, geotechnical engineering and structural mechanics. Thus it is a cross-domain course aiming at bridging the gaps among the above disciplines when studying the propagation of mechanical disturbances in natural and artificial materials. Although the emphasis of the course is on mathematical and constitutive modeling, an effort is made by the instructor to illustrate applications of the theories to the solution of real-world problems. At the end of the course the participants should have mastered the tools of analysis in order to independently investigate their research topics.

Number of hours and planning: 32 hours (4 hours of class per week for a total of 8 weeks)

Period: Time period: January 19th January – 10th March 2023

PhD courses involved: 1. Design, Modeling and Simulation in Engineering (DICAr); 2. Earth and Environmental Sciences (DSTA); 3. Understanding and Managing Extremes (IUSS Pavia)

Delivery mode and location ( in presence, on line, ecc): Standard lectures in presence

Language: english

Evaluation criterial: Assignments will be handed over and graded during the course

Credits (CFU): 4