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Tensor Calculus (19/6)

Announcements

A question time regarding Matlab / programming problems will be offered in the CIP pool every Tuesday between 15.00h and 16.00h, starting on May 14.

Contents

The term "vector" is well known, but for the precise definition it is not sufficient to introduce a triple of three (real) numbers. As shown in this lecture, a definition of the underlying base coordinate system is required first. For this reason, the lecture puts an emphasis on index notation, although symbolic notation is also introduced.

The most commonly used vector products are introduced in Cartesian coordinates. Co- and contra-variant bases are then introduced by making use of curvilinear coordinates and their relation to the Cartesian system. The transformation between bases and vector coordinates for the Cartesian bases, co- and contra-variant bases as well as their physical counterparts are elaborated in detail.

With the help of transformations of Cartesian coordinate systems – e.g., rotation and reflection matrices – we introduce the term "tensors". Moreover, we show that vectors can, in certain cases, be considered as first order tensors. Examples for so-called pseudo-vectors that do not comply with tensorial transformation rules are also presented.

In the course of dealing with tensor algebra, the relations obtained for vectors are generalised to higher-order tensors. New operations like the dyadic product or the (multiple) contraction of tensors in co-, contra- and mixed-variant representations are introduced. Regarding symmetric second-order tensors, we show unique decompositions of these tensors, eigenvalue problems, tensor powers, as well as selected tensor functions of interest.

The tensor calculus introduced thereafter covers differential operators applied to tensor fields. Based on the nabla operator, the corresponding operations such as the gradient, divergence, and rotation are derived. Moreover, derivatives of tensor functions with respect to tensor coordinates as well as generalised differentiation relations – i.e. the Fréchet and Gâteaux derivatives – are covered. The second fundamental topic of tensor calculus is the integration of tensor fields and the application of different integration rules.

Finally, the interconnections between the above named relations are elaborated by means of variational 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
(will be continually updated)
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

Chapter 11

Tensor Functions
Chapter 12 Integral Theorems
Chapter 13  General Basis Systems (updated)

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
(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

* The examination will be on August 19, 2019 at 10.00 o'clock.

*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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