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2015 | Buch

Kapitel lesen Erstes Kapitel lesen

Tensor Algebra and Tensor Analysis for Engineers

With Applications to Continuum Mechanics

verfasst von: Mikhail Itskov

Verlag: Springer International Publishing

Buchreihe : Mathematical Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This is the fourth and revised edition of a well-received book that aims at bridging the gap between the engineering course of tensor algebra on the one side and the mathematical course of classical linear algebra on the other side. In accordance with the contemporary way of scientific publications, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this monograph addresses graduate students as well as scientists working in this field. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Vectors and Tensors in a Finite-Dimensional Space

Abstract

We start with the definition of the vector space over the field of real numbers.

Mikhail Itskov

Chapter 2. Vector and Tensor Analysis in Euclidean Space

Abstract

In the following we consider a vector-valued function \(\varvec{x}\left( t\right) \) and a tensor-valued function \(\mathbf {A}\left( t\right) \) of a real variable \(t\).

Mikhail Itskov

Chapter 3. Curves and Surfaces in Three-Dimensional Euclidean Space

Abstract

A curve in three-dimensional space is defined by a vector function

Mikhail Itskov

Chapter 4. Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors

Abstract

So far we have considered solely real vectors and real vector spaces.

Mikhail Itskov

Chapter 5. Fourth-Order Tensors

Abstract

Fourth-order tensors play an important role in continuum mechanics where they appear as elasticity and compliance tensors.

Mikhail Itskov

Chapter 6. Analysis of Tensor Functions

Abstract

Let us consider a real scalar-valued function \(f\left( {{\mathbf {A}}}_1,{{\mathbf {A}}}_2,\ldots ,{{\mathbf {A}}}_l\right) \) of second-order tensors \({{\mathbf {A}}}_k\in {\mathbf {L}}{\text {in}}^n \, \left( k=1,2,\ldots ,l\right) \).

Mikhail Itskov

Chapter 7. Analytic Tensor Functions

Abstract

In the previous chapter we discussed isotropic and anisotropic tensor functions and their general representations.

Mikhail Itskov

Chapter 8. Applications to Continuum Mechanics

Abstract

Let us consider an infinitesimal vector \(\mathrm{{d}}\varvec{ X }\) in the reference configuration of a material body and its counterpart \(\mathrm{{d}}\varvec{ x }\) in the current configuration.

Mikhail Itskov

Chapter 9. Solutions

Abstract

In this chapter, solutions of the exercises are presented.

Mikhail Itskov

Backmatter

Titel: Tensor Algebra and Tensor Analysis for Engineers
verfasst von: Mikhail Itskov
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-16342-0
Print ISBN: 978-3-319-16341-3
DOI: https://doi.org/10.1007/978-3-319-16342-0

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