All lecture notes have been updated (25/06). Minor changes have been made.

The next examination date is scheduled for February 3, 2020.

The term "vector" is well established - the precise representation of vectors, however, requires not only the introduction of related coefficients but also of the related basis system. The vector itself can be represented with respect to different basis systems. In other words, the vector itself is independent of the particular basis system chosen which is fundamental for applications of vector (and tensor) calculus to continuum physics.

The course starts with basic vector properties, operations and transformations, whereby representations are typically referred to a Cartesian basis system. Based on this basic vector algebra, related properties and operators are introduced for second order tensors. This includes additive, multiplicative and spectral decompositions as well as the Cayley-Hamilton theorem and properties of special second order tensors. Moreover, isotropic tensor functions are discussed. By analogy with second order tensors, fourth order tensors are introduced thereafter, including Kelvin and Voigt representations.

The course continues with basic vector and tensor analysis, including directional derivatives, gradient-, divergence-, and rotation-operations. Based on this, integral theorems are discussed, which are most relevant for, e.g., transformations of balance relations in continuum physics.

Finally, general basis systems are introduced and applied to vector and tensor algebra and analysis. This includes basis systems which are neither normalised nor orthogonal and which may be either fixed in space or change in space.

In particular, the course includes the following chapters

1. Introduction

2. Vector Algebra

3. Tensor Algebra of Second Order Tensors

4. Special Second Order Tensors and Operators

5. Eigenvalue Problems of Second Order Tensors

6. Permutation Tensor

7. Cayley-Hamilton Theorem

8. Tensor Algebra of Fourth Order Tensors

9. Special Fourth Order Tensors and Operators

10. Vector and Tensor Analysis

11. Tensor Functions

12. Integral Theorems

13. General Basis Systems

The exercises of the course include, amongst others, computational aspects and programming of selected topics of tensor calculus.

Semester |
Lecturer |
Date |
Location |
---|---|---|---|

SS 2019 |
Wednesday, 12:00-13:30 |
MB-Building, |

Documents |
Contents |
---|---|

Chapter 0 | Contents |

Chapter 1 | Introduction |

Chapter 2 | Vector Algebra |

Chapter 3 | Tensor Algebra |

Chapter 4 | Special Second Order Tensors and Operators |

Chapter 5 | Eigenvalue Problems of Second Order Tensors |

Chapter 6 | Permutation Tensor |

Chapter 7 | Cayley-Hamilton Theorem |

Chapter 8 | Tensor Algebra of Fourth Order Tensors |

Chapter 9 | Special Fourth Order Tensors and Operators |

Chapter 10 | Vector and Tensor Analysis (updated) |

Tensor Functions | |

Chapter 12 | Integral Theorems |

Chapter 13 | General Basis Systems |

Semester | Date | Location |
---|---|---|

SS 2019 |
Tuesday, 13:15-14:45 and |
MB-Building, |

The documents accompanying the tutorials are to be used in preparation of each exercise. All participants are requested to become acquainted with the exercises dealing with the subject of the respective tutorial. This will ensure that all participants are on the same level of knowledge for each individual lesson.

Tutorial No. |
Date |
Description |
Files |
---|---|---|---|

1 | 09.04.19 | Basis systems and transformations (Whiteboard-based tutorial in the lecture room) |
Handout |

2 | 16.04.19 + 23.04.19 |
Programming based on index notation | Handout Framework |

3 | 23.04.19 | Finite rotations | Handout Framework |

4 | 29.04.19 | Additive tensor decompositions (Whiteboard-based tutorial in the lecture room) |
Handout |

5 | 07.04.19 | Finite rotation with Euler angels | Handout Framework |

6 | 14.05.19 | Eigenvalue problems and Cayley-Hamilton theorem (Whiteboard-based tutorial in the lecture room) |
Handout |

7 | 15.05.19 | Eigen-projection with a perturbation algorithm | Handout Framework |

8 | 21.05.19 | Polar decomposition | Handout Framework |

9 | 28.05.19 + 29.05.19 |
Micromechanics | Handout Framework |

10 | 04.06.19 + 11.06.19 |
Tensor derivatives (updated) (Whiteboard-based tutorial in the lecture room) |
Handout |

11 | 18.06.19 + 25.06.19 |
Material model based on eigenvalues (final) | Handout Framework |

12 | 26.06.19 + 10.07.19 |
Generalisation of orthonormal basis systems (Whitboard-based tutorial in the lecture room) |
Handout |

*** An examination will be provided on February 3, 2020 at 10.00 o'clock. Room: To be announced.**

In order to register for the exam send an email with your name and matriculation number to isabelle.noll@tu-dortmund.de.

Deadline for the registration is 20.01.2020.

*** The final examination is a written test (60minutes).**

*** No lecture notes or text books are allowed. **

*** You should bring your pocket calculator along.**** **

- J. Betten, Tensorrechnung für Ingenieure, Teubner, 1987
- R. De Boer, Vektor- und Tensorrechnung für Ingenieure, Springer, 1982
- E. Klingbeil, Tensorrechnung für Ingenieure, BI Wissenschaftsverlag, 1989
- M. Itskov, Tensor Algebra and Tensor Analysis for Engineers, Springer, 2015
- R. M. Bowen and C.-C. Wang, Introduction to Vectors and Tensors, Dover Publications, 2008
- C. Eringen, Continuum Physics, Academic Press, 1971

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