Unit MATHEMATICAL MODELS FOR APPLICATIONS
- Course
- Mathematics
- Study-unit Code
- 55A00066
- Curriculum
- Matematica per le applicazioni industriali e biomediche
- Teacher
- Luigi Vergori
- Teachers
-
- Luigi Vergori
- Hours
- 42 ore - Luigi Vergori
- CFU
- 6
- Course Regulation
- Coorte 2021
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione modellistico-applicativa
- Academic discipline
- MAT/07
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Introduction to continuum mechanics. Kinematics and dynamics of deformable bodies. Nonlinear Elasticity and derivation of the linearized theory from the nonlinear one. Elements of fluid dynamics: inviscid and viscous fluids, Navier-Stokes equations, exact solutions.
- Reference texts
- S. Forte, L. Preziosi, M. Vianello: Meccanica dei Continui. Springer-Verlag Italia, 2019.
- Educational objectives
- The course aims at introducing the basics of continuum mechanics, the theory of elasticity and fluid dynamics. At the end of the course, the student should be able to:
1) specialize the equations of balance to the various types of fluids and solids according to the most appropriate constitutive classes for the mechanical characterization of the material response;
2) solve equilibrium problems for solids and fluids and study the stability of the equilibrium configurations;
3) derive the basic equations of fluid dynamics, especially the Navier-Stokes equations, and find exact solutions to initial and boundary value problems. - Prerequisites
- Calculus. Linear algebra. Elements of classical mechanics.
- Teaching methods
- Chalk and board lectures
- Other information
- Although the attendance is not compulsory, it is recommended.
- Learning verification modality
- The final exam consists in a written and an oral parts. The written part of the exam will last two hours and consist in the solution of some exercies similar to the ones solved in class. This part of the exam aims at checking whether or not the student is able to make the calculations necessary to the study of the material response of real bodies. The oral part will consist in some questions regarding the theory developed in class. The oral part of the exam aims at ensuring that the student has acquired the basics of continuum mechanics.
For information on the services for students with special needs visit the page http://www.unipg.it/disabilita-e-dsa - Extended program
- Tensor algebra: second-order tensors, symmetric, skew-symmetric, orthogonal and positive definite tensors, rotations. Real and tensor isotropic functions with real or tensor variables. Representation theorems for isotropic functions. Kinematics of deformable bodies. Balance equations of mass, linear and angular momenta, and energy, Cauchy theorem. Constitutive theory: constitutive classes, material symmetry, frame indifference, incompressibility. Isotropic elastic solids: Cauchy stress tensor, equilibrium configurations and their stability. Derivation of the linear theory of elasticity from the nonlinear theory: infinitesimal shear modulus, Young's and bulk moduli, Poisson's ratio. Dymamics of inviscid and viscous fluids: Euler and Navier-Stokes equations, laminar flows.