Continuum Mechanics

Enrollment in year 3.

Passed problem-solving examination is a prerequisite for the theoretical part of the examination.

Theory of elasticity. Kinematics of deformation, strain tensor. Stress tensor: contact forces, Cauchy equations. Hooke law: symmetry of isotropic body, harmonic approximation, set of elastic constants, Navier equations, elastic theory of crystals. Elastic theory of plates: Monge parametrization, equilibrium, elastic instability due to in-plane stress. Elastic theory of rods: torsion and bending, general description of deformed rods, Kirchhoff theory. Elastic waves.

Hydrodynamics. Ideal fluids: continuity equation, Euler equation, hydrostatics, convection; flow lines, Bernoulli equation; circulation, Kelvin theorem; potential flow, compressible and incompressible fluids, Potential flow of incompressible fluid. Flow past obstacle, drag in potential flow, d'Alembert paradox. Viscous fluids: viscous stress tensor, Navier-Stokes equation, energy dissipation. Hydrodynamic similarity, Reynolds number, Stokes formula. Boundary layer: Prandtl theory, Blasius equation, backflow. Hydrodynamic instabilities. Turbulence.

L. D. Landau in E. M. Lifshitz, Theory of Elasticity (Butterworth Heinemann, Oxford, 1986).
L. D. Landau in E. M. Lifshitz, Fluid Mechanics (Butterworth Heinemann, Oxford, 1987).
S. Timoshenko in J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).
M. H. Sadd, Elasticity: Theory, Applications, and Numerics (Elsevier Butterworth Heinemann, Amsterdam, 2005).
A. E. Green in W. Zerna, Theoretical Elasticity (Dover, New York, 1992).
R. W. Soutas-Little, Elasticity (Dover, Mineola, 1999).
P. K. Kundu, Fluid Mechanics (Academic Press, San Diego, 1990).
E. Guyon, J.-P. Hulin, L. Petit in C. D. Mitescu, Physical Hydrodynamics (Oxford University Press, New York, 2001).
J. H. Spurk, Fluid Mechanics: Problems and Solutions (Springer, Berlin, 1997).
H. Ockendon in J. R. Ockendon, Viscous flow (Cambridge University Press, Cambridge, 1995).
D. J. Tritton, Physical Fluid Dynamics (Van Nostrand Reinhold, New York, 1977).
U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995).

Systematic introduction into the elasticity theory based on strain and stress tensors. Phenomenological theory of laws of motion and of deformation energy for continuous media, phenomenological theory of viscous media. Overview of selected examples in elastomechanics and hydrodynamics.

Knowledge and understanding: Knowledge of laws of motion for solids and fluids, knowledge of the connection between strain and stress tensors as well as the basics of hydrodynamics. Understanding of instabilities and the role of non-linear terms in equations of motion.
Application: Students will learn to recognize, define, and solve problems in elastomechanics and hydrodynamics in condensed-matter physics, physics of materials, biophysics, geophysics, and meteorology.
Reflection: The course relies on a systematic use of the phenomenological description of the state of continuous media, helping the students learn to recognize the relevant degrees of freedom and conservation laws and apply them.
Transferable skills: Tensorial description of physical systems. Einstein convention. Consolidation of vector calculus and skills needed to solve partial differential equations.

Lectures, tutorials, homeworks and consultations.

2 written tests (mid-term and end-term) applied towards the problem-solving examination, problem-solving examination
Theoretical examination
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

[1] A. Šiber in P. Ziherl, Many-body contact repulsion of deformable disks, Phys. Rev. Lett. 110, 214301 (2013).
[2] T. Dotera, T. Oshiro in P. Ziherl, Mosaic two-lengthscale quasicrystals, Nature 506, 208 (2014).
[3] M. Krajnc in P. Ziherl, Theory of epithelial elasticity, Phys. Rev. E 92, 052713 (2015).
[4] L. Athanasopoulou in P. Ziherl, Phase diagram of elastic spheres, Soft Matter 13, 1463 (2017).
[5] J. Rozman, M. Krajnc in P. Ziherl, Collective cell mechanics of epithelial shells with organoid-like morphologies, Nat. Commun. 11, 3805 (2020).