Syllabus
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 ∫γφ dX, line integral of a vector valued function ∫γv • dX.
Surface integral of a scalar valued function ∫Sφ dS, 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 ℂ. 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.)
Annotation
The course presents concept of continuum media, notion of its deformation and stress, conservation laws, constitutive equations, elastic solids and simple fluids.