Tensores

Modern Birkhäuser classics, 2017

The purpose of the present chapter is to provide a systematic introduction of the basic concepts and definitions of the tensors of strains and stress. The presentation begins with a section, in which we briefly present the elements of tensor analysis. Tensor analysis enables one to present in a simple and elegant form the fundamantals of continuum mechanics, and it is used systematically subsequently throughout the book. We then introduce the definition of the tensors of strains and stress, which characterize a continuum, in a reference frame and in an actual frame without any assumptions on the smallness of strains.

K24389 Illustrating the important aspects of tensor calculus, and highlighting its most practical features, Physical Components of Tensors presents an authoritative and complete explanation of tensor calculus that is based on transformations of bases of vector spaces rather than on transformations of coordinates. Written with graduate students, professors, and researchers in the areas of elasticity and shell theories in mind, this text focuses on the physical and nonholonomic components of tensors and applies them to the theories. It establishes a theory of physical and anholonomic components of tensors and applies the theory of dimensional analysis to tensors and (anholonomic) connections. This theory shows the relationship and compatibility among several existing definitions of physical components of tensors when referred to nonorthogonal coordinates. The book assumes a basic knowledge of linear algebra and elementary calculus, but revisits these subjects and introduces the mathematical backgrounds for the theory in the first three chapters. In addition, all field equations are also given in physical components as well. Comprised of five chapters, this noteworthy text: • Deals with the basic concepts of linear algebra, introducing the vector spaces and the further structures imposed on them by the notions of inner products, norms, and metrics • Focuses on the main algebraic operations for vectors and tensors and also on the notions of duality, tensor products, and component representation of tensors • Presents the classical tensor calculus that functions as the advanced prerequisite for the development of subsequent chapters • Provides the theory of physical and anholonomic components of tensors by associating them to the spaces of linear transformations and of tensor products and advances two applications of this theory Physical Components of Tensors contains a comprehensive account of tensor calculus , and is an essential reference for graduate students or engineers concerned with solid and structural mechanics.

2010

Tensor Analysis, 2018

In Sect. 2.2.4 the Cauchy stress tensor T was defined. The stress tensor is the "original" tensor as the word tensor means stress. We shall use the definition of the stress tensor as an introduction to the general concept of tensors. We consider a body of continuous material and a material surface A in the body. At a place r a positive side of the surface is defined by a unit vector n as a normal pointing out from the surface. In a Cartesian coordinate system Ox with base vectors e k the normal vector n has the components: n k ; i.e. n ¼ n k e k : The contact force on the positive side of the surface is represented by the stress vector t with Cartesian components: t i ; i.e. t ¼ t i e i : The contact forces on positive coordinate surfaces through the place r are the stress vectors t k with Cartesian components T ik ; i.e. t k ¼ T ik e i : The components T ik are called the coordinate stresses. The Cauchy stress theorem by Eq. (2.2.27

Theoretical and Applied Mechanics, 2008

An objective of this paper is to reconcile the "symmetry" approach with the "symmetry groups" approach as these two different points of view presently coexist in the literature. Here we will be concerned exclusively with linearly elastic materials. The starting point for an analysis of the inherent symmetry of elastic materials is the notion of a symmetry transformation. Particularly, we paid attention to the compliance tensor for cubic and hexagonal crystals.

HAL (Le Centre pour la Communication Scientifique Directe), 2019

Tensor calculus is introduced to Physics and Mechanical engineering students in 2D and 3D and applied to anisotropic elasticity such as in condensed matter physics approaching the subject from the practical tool aspect point of view. It provides powerful mathematical techniques to tackle many aspects of Vector Calculus, Continuum mechanics, Solid State Physics, Electromagnetism...

Mathematical Methods in the Applied Sciences, 2012

With the words continuum mechanics, we indicate a classical field theoretic representation (long wavelength approximation) of the macroscopic mechanical behavior of bodies extended in space, under the action of external agencies. In this setting, the description of changes in placements including strain effects plays a prominent role.

Transactions of the VŠB - Technical University of Ostrava, Mechanical Series

This article is devoted to an analysis of scalar, vector and tensor fields, which occur in the loaded and deformed bodies. The aim of this article is to clarify and simplify the creation of an understandable idea of some elementary concepts and quantities in field theories, such as, for example equiscalar levels, scalar field gradient, Hamilton operator, divergence, rotation and gradient of vector or tensor and others. Applications of those mathematical terms are shown in simple elasticity and plasticity tasks. We hope that content of our article might help technicians to make their studies of necessary mathematical chapters of vector and tensor analysis and field theories easier.

This booklet contains an explanation about tensor calculus for students of physics and engineering with a basic knowledge of linear algebra. The focus lies mainly on acquiring an understanding of the principles and ideas underlying the concept of 'tensor'. We have not pursued mathematical strictness and pureness, but instead emphasise practical use (for a more mathematically pure resumé, please see the bibliography). Although tensors are applied in a very broad range of physics and mathematics, this booklet focuses on the application in special and general relativity.

2017

This book was born with the vocation of being a tool for the training of engineers in continuum mechanics. In fact, it is the fruit of the experience in teaching this discipline during many years at the Civil Engineering School of the Technical University of Catalonia (UPC/BarcelonaTech), both in undergraduate degrees (Civil Engineering and Geological Engineering) and postgraduate degrees (Master and PhD courses). Unlike other introductory texts to the mechanics of continuous media, the work presented here is specifically aimed at engineering students. We try to maintain a proper balance between the rigor of the mathematical formulation used and the clarity of the physical principles addressed, although always putting the former at the service of the latter. In this sense, the essential vector and tensor operations use simultaneously the indicial notation (more useful for rigorous mathematical proof) and the compact notation (which allows for a better understanding of the physics of...

2009

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