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Course, academic year 2023/2024
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Continuum Mechanics - NMMO401
Title: Mechanika kontinua
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: yes / unlimited
Key competences: 4EU+ Flagship 3
State of the course: taught
Language: English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: Mgr. Vít Průša, Ph.D.
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
Files Comments Added by
download syllabus-continuum-mechanics-2020.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. (04.10.2021)

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. (26.09.2020)

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

Topics/notions printed in italics are not a part of the exam. All other topics/notions are expected to mastered at the level of a reflex action.

  • Preliminaries.

    • Linear algebra.

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

      • Tensors.

      • Cofactor matrix cof���� and determinant det����. 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.

    • Elementary calculus.

      • Matrix functions. Exponential of a matrix.

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

      • Gâteaux derivative, Fréchet 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 γφdX, line integral of a vector valued function γv • dX.

      • Surface integral of a scalar valued function SφdS, surface integral of a vector valued function Sv • dS, surface Jacobian.

      • Volume integral, Jacobian matrix.

    • Stokes theorem and its consequences.

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

      • Korn equality.

    • Elementary concepts in classical physics.

      • Newton laws.

      • Galilean invariance, principle of relativity, non-inertial reference frame.

      • Fictitious forces (Euler, centrifugal, Coriolis).

  • Kinematics of continuous medium.

    • 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, condition det���� > 0.

      • Deformation gradient ���� and its geometrical interpretation. Polar decomposition ���� = ℝ���� of the deformation gradient and its geometrical interpretation.

      • Relative deformation gradient.

      • Deformation gradient and polar decompostion for simple shear.

      • 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 ${\ensuremath{\frac{{\mathrm{d}}{}}{{\mathrm{d}}{t}}}}$ of Eulerian quantities.

      • Streamlines and pathlines (trajectories).

      • Spatial velocity gradient ����, its symmetric part ���� and skew-symmetric part ����.

    • Strain measures.

      • Left and right Cauchy–Green tensor, ���� and . Hencky strain.

      • Green–Saint-Venant strain tensor ����, Euler–Almansi strain tensor ����. Geometrical interpretation.

      • Linearised strain $\linstrain$.

    • Compatibility conditions for linearised strain $\linstrain$ in ${\ensuremath{{\mathbb R}}}^2$. Compatibility conditions for linearised strain $\linstrain$ in ${\ensuremath{{\mathbb R}}}^3$.

    • Rate quantities.

      • Rate of change of Green–Saint-Venant strain, rate of change of Euler–Almansi strain and their relation to the symmetric part of the velocity gradient ����.

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

      • Objective derivatives of tensorial quantities (Oldroyd derivative, Truesdell derivative).

    • Kinematics of moving surfaces.

      • Lagrange criterion for material surfaces.

    • Reynolds transport theorem.

      • Reynolds transport theorem for the volume moving with the medium.

      • Reynolds transport theorem in the presence of surface discontinuities.

  • Dynamics and thermodynamics of continuous medium.

    • Mechanics.

      • Balance laws for continuous medium as counterparts of the classical laws of Newtonian physics of point particles.

      • Concept of contact/surface forces. Existence of the Cauchy stress tensor ���� (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 regarding the symmetry of the Cauchy stress tensor. Proof of the symmetry of the Cauchy stress tensor.

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

      • First Piola–Kirchhoff stress tensor ����R and its relation to the Cauchy stress tensor ����. 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.

      • Specific internal energy e, energy/heat flux jq.

      • Balance of total energy in the Eulerian and Lagrangian description.

      • Balance of internal energy in the Eulerian and Lagrangian description.

      • Referential heat flux Jq.

      • Specific Helmholtz free energy, specific entropy.

    • 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.

  • Simple constitutive relations.

    • Pressure and thermodynamic pressure, engineering equation of state. Derivation of compressible and incompressible Navier–Stokes fluid model via the representation theorem for tensor valued isotropic tensorial functions. Complete thermodynamical description of a compressible viscous heat conducting fluid – Navier–Stokes–Fourier equations.

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

    • Green elastic material. (Hyperelastic solid.) Relation between the specific Helmholtz free energy and the Cauchy stress tensor for an elastic solid. Rate type formulation of constiutive relations for a hyperelastic solid.

    • Physical units, dimensionless quantities, Reynolds number.

  • Simple problems in the mechanics of continuous medium.

    • Archimedes law.

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

    • Inflation of a hollow cylinder made of an incompressible isotropic elastic solid. (Comparison of the linearised elasticity theory and fully nonlinear theory.)

    • Waves in compressible Navier–Stokes fluid.

    • Waves in the linearised isotropic elastic solid.

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

    • Drag acting on a rigid body moving with a uniform velocity in 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.

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