MATHEMATICAL PHYSICS
Academic year and teacher
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- Versione italiana
- Academic year
- 2017/2018
- Teacher
- ARIANNA PASSERINI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/07
Training objectives
- Aim of the course is introducing topics in Continuum Solid Mechanics, specifically in Elasticity and Elasto-plasticity, with a focus on the mathematical analysis of the models and on the existence results.
At the end of the course, the student will have acquired, on one hand, notions and tecniques in the modeling of solid continua, and on the other hand, he will have gained experience in the application of abstract methods in Functional Analysis and Calculus of Variations to mechanically motivated (especially non-linear) problems. Prerequisites
- In order to fruitfully attend the course, the student should have notions in the following areas (in addition to the first-two-years undergraduate teachings):
Continuum Mechanics (only basic/general notions are needed)
Functional Analysis (Banach-, Hilbert-, Sobolev spaces; however more or less extensive references will be made during the course when necessary) Course programme
- Review of Continuum Mechanics: balance laws, virtual power principle, action principle, weak formulation of the boundary value problem; isotropic materials; hyperelastic solids (8 h).
Linear Elasticity: existence and uniqueness for quasi-static solutions (4 h).
Nonlinear Hyperelasticity: non-convexity, quasi-/poly-convexity; Ogden materials; weak topologies, lower semicontinuity, direct method; J. Ball approach to existence of solutions of the displacement-traction problem (10 h).
Plasticity: phenomenology, internal variables, yield surfaces, plastic flow rule, dissipation function, rate-independent evolutions (10 h).
Existence and unicity for small-strain elastoplasticity (4 h)
Finite-strain plasticity: modeling, existence results (6 h) Didactic methods
- The course consists of class lectures on the mentioned subjects
Learning assessment procedures
- The final exam consists in an oral discussion on the subjects of the course
Reference texts
- Lecture notes;
P.G. Ciarlet. Mathematical elasticity. Vol. I: Three-dimensional elasticity. North-Holland Publishing Co., 1988.
M. E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua. Cambridge University Press, 2009.
W. Han, B.D. Reddy, Plasticity, Mathematical Theory and Numerical Analysis. Springer 2013.
