Continuum Mechanics - NMMO401
Title in English: Mechanika kontinua
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017 to 2018
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English
Teaching methods: full-time
Guarantor: Mgr. Vít Průša, Ph.D.
prof. RNDr. Jan Kratochvíl, DrSc.
Class: M Mgr. MA
M Mgr. MA > Povinně volitelné
M Mgr. MOD
M Mgr. MOD > Povinné
M Mgr. NVM
M Mgr. NVM > Volitelné
Classification: Mathematics > Mathematical Modeling in Physics
Incompatibility : NMOD012
Interchangeability : NMOD012
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download syllabus-sis.pdf Syllabus. Mgr. Vít Průša, Ph.D.
Annotation -
Last update: T_MUUK (14.05.2013)
The course presents concept of continuum media, notion of its deformation and stress, conservation laws, constitutive equations, elastic solids and simple fluids.
Course completion requirements -
Last update: Mgr. Vít Průša, Ph.D. (09.10.2017)

Pass the exam and get credits for the tutorials.

Conditions for getting the credit for the tutorials:

1) Attendance.

2) Solving homework problems.

You can enroll for the exam only if you already got credits for the tutorials.

Literature - Czech
Last update: Mgr. Vít Průša, Ph.D. (05.04.2016)

Gurtin, M. E., E. Fried, and L. Anand (2010). The mechanics and thermodynamics of continua. Cambridge: Cambridge University Press.

Ogden, R. W. (1984). Nonlinear elastic deformations. Ellis Horwood Series: Mathematics and its Applications. Chichester: Ellis Horwood Ltd.

Truesdell, C. and K. R. Rajagopal (2000). An introduction to the mechanics of fluids. Modeling and Simulation in Science, Engineering and Technology. Boston, MA: Birkhauser Boston Inc.

Brdička, M., Sopko, B., Samek, L. (2011). Mechanika kontinua, Praha: Academia.

Maršík, F. (1999): Termodynamika kontinua, Praha: Academia.

Requirements to the exam -
Last update: Mgr. Vít Průša, Ph.D. (09.10.2017)

You can enroll for the exam only if you already got credits for the tutorials.

The exam is an oral exam, and it consists of three parts:

1) Proof of a simple theorem. The theorem will be specified at the end of the semester. Typically, you will be asked to prove a theorem that has been formulated during the lecture without a proof. (The proof can be usually found in standard textbooks.) In presenting the proof, you can use your own notes.

2) Presentation of a solution to a problem discussed in a scientific paper. The objective is to show that you know what is the paper about, and what are the used methods and conclusions. The paper will be specified at the end of semester. Typically there will be a list of papers from which you can choose the paper that is most readable/conveninent/interesting for you. In discussing the problem, you can use your own notes!

3) During our conversation we will definitely encounter some notions from the field of continuum mechanics. You will be asked to explain some of the notions. Detailed list of the definitions/theorems and concepts you are expected to know will be specified at the end of the semester.

More information is available on the lecture's webpage.

Syllabus -
Last update: Mgr. Vít Průša, Ph.D. (05.04.2016)

Please see the attached PDF file for a nicely typeset syllabus.


  • Linear algebra.

∗ Scalar product, vector product, mixed product, tensor product. Transposed matrix.

∗ Tensors. Inertia tensor as an example of a tensorial quantity in mechanics.

∗ Cofactor matrix cof A and determinant det A. Geometrical interpretation.

∗ Cayley-Hamilton theorem, characteristic polynomial, eigenvectors, eigenvalues.

∗ Trace of a matrix.

∗ Invariants of a matrix and their relation to the eigenvalues and the mixed product.

∗ Properties of proper orthogonal matrices, angular velocity.

∗ Polar decomposition. Geometrical interpretation.

∗ Spectral decomposition.

  • Elementary calculus.

∗ Matrix functions. Exponential of a matrix. Skew-symmetric matrices as infinitesimal generators of the group of rotations.

∗ Representation theorem for scalar valued isotropic tensorial functions and tensor valued isotropic tensorial functions.

∗ Gateaux derivative, Fr echet derivative. Derivatives of the invariants of a matrix.

∗ Operators ∇, div and rot for scalar and vector fields. Operators div and rot for tensor fields. Abstract definitions and formulae in Cartesian coordinate system. Identities in tensor calculus.

  • Line, surface and volume integrals.

∗ Line integral of a scalar valued function, line integral of a vector valued function.

∗ Surface integral of a scalar valued function, surface integral of a vector valued function, surface Jacobian.

∗ Volume integral, Jacobian matrix.

  • Stokes theorem and its consequences.

∗ Green identities.

∗ Potential vector field, path independent integrals, curl free vector fields. Characterisation of potential vector fields.

∗ Helmholtz decomposition, v = −∇φ + rot A.

∗ Korn equality.

  • Elementary concepts in classical physics.

∗ Newton laws.

∗ Galilei principle of relativity, non-inertial reference frame.

∗ Fictitious forces (Euler, centrifugal, Coriolis).


  • Basic concepts.

∗ Notion of continuous body. Abstract body, placer, configuration.

∗ Reference and current configuration. Lagrangian and Eulerian description.

∗ Deformation/motion χ.

∗ Local and global invertibility of the motion/deformation.

∗ Deformation gradient F and its geometrical interpretation. Polar decomposition of the deformation gradient and its geometrical interpretation.

∗ Deformation gradient for simple shear, pure bending and inflation of a hollow cylinder.

∗ Displacement U.

∗ Deformation of infinitesimal line, surface a volume elements. Concept of isochoric motion.

∗ Lagrangian velocity field V , Eulerian velocity field v. Material time derivative of Eulerian quantities.

∗ Streamlines and pathlines (trajectories).

∗ Relative deformation/motion.

∗ Spatial velocity gradient L, its symmetric D and skew-symmetric part W. Relation of D and W to the time derivatives of the relative deformation/motion.

  • Strain measures.

∗ Left and right Cauchy-Green tensor, B and C.

∗ Green-Saint-Venant strain tensor E, Euler-Almansi strain tensor e. Geometrical interpretation.

∗ Linearised strain ε.

  • Compatibility conditions for linearised strain ε.

  • Rate quantities.

∗ Rate of change of Green-Saint-Venant strain, rate of change of Euler-Almansi strain and their relation to D.

∗ Rate of change of infinitesimal line, surface and volume elements. Divergence of Eulerian velocity field and its relation to the change of volume.

  • Kinematics of moving surfaces.

∗ Lagrange criterion for material surfaces.

∗ Formula for the projection of the velocity of the surface to the direction parallel to the normal to the surface.

  • Reynolds transport theorem. Reynolds transport theorem in the presence of discontinuities.


  • Mechanics.

∗ Balance laws for continuous medium as counterparts of the classical laws of Newtonian physics of point


∗ Concept of contact/surface forces. Existence of the Cauchy stress tensor T (tetrahedron argument).

∗ Pure tension, pure compression, tensile stress, shear stress.

∗ Balance of mass, linear momentum and angular momentum in Eulerian description.

∗ Balance of angular momentum and its implications with respect to the symmetry of the Cauchy stress tensor.

∗ Balance of mass, linear momentum and angular momentum in Lagrangian description.

∗ First Piola-Kirchhoff tensor T R . Piola transformation.

∗ Formulation of boundary value problems in Eulerian and Lagrangian descripition, transformation of

traction boundary conditions from the current to the reference configuration.

  • Elementary concepts in thermodynamics of continuous medium.

∗ Internal energy, heat flux.

∗ Balance of total energy in Eulerian and Lagrangian description.

∗ Balance of internal energy in Eulerian and Lagrangian description.

∗ Referential heat flux.

  • Boundary conditions.

  • Geometrical linearisation. Incompressibility condition in the linearised setting. Specification of the boundary

conditions in the linearised setting.

  • Balance laws in the presence of discontinuities.


  • Pressure and thermodynamic pressure. Derivation of compressible and incompressible Navier-Stokes fluid

model via the representation theorem for tensor valued isotropic tensorial functions.

  • Cauchy elastic material. Derivation via the representation theorem for tensor valued isotropic tensorial func-


  • Physical units, dimensionless quantities, Reynolds number.


  • Archimedes law.

  • Deformation of a cylinder (linearised elasticity). Hooke law.

  • Inflation of a hollow cylinder made of an incompressible isotropic elastic solid. (Comparison of linearised theory

and fully nonlinear theory.)

  • Isothermal atmosphere versus swimming pool. (Difference between the notion of the pressure in the case of

compressible and incompressible Navier-Stokes fluid model.)

  • Stokes formula via dimensional analysis.

  • Waves in compressible Navier-Stokes fluid.

  • Waves in linearised isotropic elastic solid.

  • Stability of the rest state of the incompressible Navier-Stokes fluid.

  • Pressure driven flow of incompressible Navier-Stokes fluid in a pipe of circular cross section.
Entry requirements -
Last update: Mgr. Vít Průša, Ph.D. (18.05.2018)

Fundamentals of linear algebra, multivariable calculus.