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Tensor Calculus (19/6)

Announcements

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

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

Contents

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.

Lectures

Semester

Lecturer

Date

Location

SS 2019

Prof. Dr.-Ing. Andreas Menzel

Wednesday, 12:00-13:30
and Tuesday, 16:15-17:45

MB-Building,
Room 165

Accompanying documents for the lectures

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)

Chapter 11

Tensor Functions
Chapter 12 Integral Theorems
Chapter 13  General Basis Systems

Tutorials

Semester Date Location
SS 2019

Tuesday, 13:15-14:45 and
Wednesday, 12:00-13:30

MB-Building,
Room 163
(CIP-Pool)

Accompanying documents for the tutorials

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

Examinations

* 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.

References

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