Physics
Tensors
Tensors are mathematical objects that generalize scalars, vectors, and matrices. In physics, tensors are used to represent physical quantities that have both magnitude and direction, and they play a crucial role in describing the behavior of physical systems in various coordinate systems. They are essential for formulating the laws of physics in a coordinate-independent manner.
Written by Perlego with AI-assistance
Related key terms
1 of 5
12 Key excerpts on "Tensors"
eBook - PDFTensor Calculus and Applications
Simplified Tools and Techniques
- Bhaben Chandra Kalita(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
Generally, from a mathematical point of view, these physical quan-tities conveniently need to describe by means of referring them to some coordinate system. Eventually, Tensors which are independent of any par-ticular coordinate system are the appropriate tools to adopt in this con-sideration. Hence, physical laws of continuum mechanics are expressed in terms of tensor equations. Usually, tensor transformations are linear and homogeneous, and if they (laws of continuum mechanics) are expressed in the form of tensor equations in one coordinate system, they remain valid in any other coordinate system. This invariance of tensor equations under coordinate transformations is one of the principal reasons (similar to gen-eral theory of relativity) for the utility of Tensors in continuum mechan-ics. Of course, Cartesian Tensors are sufficient to deal with continuum mechanics, and hence, it can be developed with reference to Cartesian Tensors. In Euclidean space of three dimensions, the number of components of a tensor of order n is 3 n . A vector is a tensor of order one with 3 1 components. 124 Tensor Calculus and Applications The stress and strain are the second-order Tensors having 3 2 = 9 components in general. The physical quantities (mentioned earlier) involved in the study of continuum mechanics are the stresses and strains, which are invariably related to the deformation of media and bodies. Second-order Tensors are also known as “dyadics,” and the quantities in continuum mechanics are represented by dyadics [11]. Definition Stress: The forces occurring in a bulk of the material medium proportional to the mass of the substance (e.g., gravity, magnetic force, centrifugal force) are known as body forces, which are measured per unit volume. The forces acting over the volume of bounding surface of a body and mea-sured in units of force per unit area are called surface forces.
eBook - PDF- J. N. Reddy(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Like physical vectors, Tensors are more general objects that possess a magnitude and multiple direction(s) and satisfy rules of tensor addition and scalar multiplication. In fact, physical vectors are often termed the first-order Tensors. As will be shown shortly, the specification of a stress component (i.e., force per unit area) requires a magnitude and two directions – one normal to the plane on which the stress component is measured and the other is its direction – to specify it uniquely. This chapter is dedicated to the study of the elements of algebra and calculus of vectors and Tensors. Useful elements of the matrix theory and eigenvalue prob-lems associated with second-order Tensors are discussed. Index and summation notations, which are extensively used throughout the book, are also introduced. Those who are familiar with the material covered in any of the sections may skip them and go to the next section or to Chapter 3. i 10 VECTORS AND Tensors 2.2 Vector Algebra In this section, we present a review of the formal definition of a geometric (or physical) vector, discuss various products of vectors and physically interpret them, introduce index notation to simplify representations of vectors in terms of their components as well as vector operations, and develop transformation equations among the components of a vector expressed in two different coordi-nate systems. Many of these concepts, with the exception of the index notation, may be familiar to most students of engineering, physics, and mathematics and may be skipped. 2.2.1 Definition of a Vector The quantities encountered in analytical descriptions of physical phenomena may be classified into two groups according to the information needed to specify them completely: scalars and nonscalars. The scalars are given by a single number. Nonscalars have not only a magnitude specified, but also additional information, such as direction.
eBook - PDF- J. N. Reddy(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
Like physical vectors, Tensors are more general objects that are endowed with a magni- tude and multiple direction(s) and satisfy rules of tensor addition and scalar mul- tiplication. In fact, physical vectors are often termed the first-order Tensors. As will be shown shortly, the specification of a stress component (i.e., force per unit area) requires a magnitude and two directions – one normal to the plane on which the stress component is measured and the other is its direction – to specify it uniquely. 8 2.2 Vector Algebra 9 This chapter is dedicated to a review of algebra and calculus of physical vectors and Tensors. Those who are familiar with the material covered in any of the sections may skip them and go to the next section or Chapter 3. 2.2 Vector Algebra In this section, we present a review of the formal definition of a geometric (or phys- ical) vector, discuss various products of vectors and physically interpret them, in- troduce index notation to simplify representations of vectors in terms of their com- ponents as well as vector operations, and develop transformation equations among the components of a vector expressed in two different coordinate systems. Many of these concepts, with the exception of the index notation, may be familiar to most students of engineering, physics, and mathematics and may be skipped. 2.2.1 Definition of a Vector The quantities encountered in analytical description of physical phenomena may be classified into two groups according to the information needed to specify them completely: scalars and nonscalars. The scalars are given by a single number. Non- scalars have not only a magnitude specified but also additional information, such as direction. Nonscalars that obey certain rules (such as the parallelogram law of addition) are called vectors. Not all nonscalar quantities are vectors (e.g., a finite rotation is not a vector).
- Adrian Burd(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
11 Tensors We have already seen that physical quantities can be described mathematically by scalars or vectors. A scalar describes the situation where there is a single number at each point of space—a temperature, or a concentration of a chemical, for example. Vectors require both a magnitude and a direction in order to specify them—a velocity, a force, etc. There are still other physical objects that cannot be described by these two types of objects. As an example, let us consider how a body deforms under both stress and pressure. This is a question that is important in the study of geophysical fluid dynamics, where one is interested in the motions of packets of air or water, and also in geology and geophysics, where one wants to understand how rocks will deform under the forces acting on them. Let us look at a cubical volume in a fluid and think about the forces acting on a point in the upper face (Figure 11.1). There can be a force parallel to the z axis that gives rise to a normal stress σ zz , and there are also tangential forces that deform the surface in the x or y directions τ zx and τ zy respectively. So, each face of the cube can have three forces acting on it. If we shrink the cube down to an infinitesimal point, then we can write the forces acting at that point as a matrix, σ ij = ⎛ ⎝ σ xx τ xy τ xz τ yx σ yy τ yz τ zx τ zy σ zz ⎞ ⎠ , (11.1) called the stress tensor. This matrix has nine entries, so we need something other than a vector (which in three dimensions has only three components) to represent it—we need a tensor. 11.1 Covariant and Contravariant Vectors To better understand Tensors we need to go back and take another look at vectors. This is because we have not told the whole story concerning vectors. In particular, we want to look again at how vectors transform under coordinate changes. Recall that although the components of a vector may change when we change coordinates, the magnitude of the vector does not.
eBook - PDFApplied Elasticity
Matrix and Tensor Analysis of Elastic Continua
- J D Renton(Author)
- 2002(Publication Date)
- Woodhead Publishing(Publisher)
2 Cartesian Tensors One of the advantages of vector equations is that they can be written without reference to any coordinate system. A physical entity such as a force can be represented by a single mathematical entity - a vector. Likewise, the total state of stress (or strain) at a point can be thought of as a single entity and represented by a matrix. Relationships between stress and strain matrices can be written and manipulated in a coordinate-free form, as seen in section 1.7 for example. This does not mean that a force is a vector and a state of stress is a matrix; these are merely representations. Tensor notation is a very useful alternative form of representation. Tensor equations can also be thought of as coordinate-free in that they are not written in relation to specific coordinates but can be interpreted in any system. For this to be possible, all the terms in the equation must transform from one coordinate system to another in the same way. A tensor is defined by the way in which it transforms, so that in a tensor equation each term, which may be a product of Tensors, must be the same kind of tensor. Because the transformations are linear, sums and differences of such terms will transfonn in the same way. Tensor equivalents of the vectors and matrices in the last chapter can be found and other entities, which are beyond the scope of such descriptions, can be expressed by Tensors. The general theory is not limited to Cartesian coordinates, although the following chapter will be devoted exclusively to such Cartesian Tensors. 2.1 VECTOR AND MATRIX REPRESENTATION The notation used here is essentially the same as that of the subscripted elements in section 1.1. The numerical value of the subscript is associated with a particular coordinate axis. Thus the coordinates of a point can be represented by x,, where each ordinate has a particular value of/.
eBook - PDF- Harold Jeffreys, Bertha Jeffreys(Authors)
- 1999(Publication Date)
- Cambridge University Press(Publisher)
Chapter 3 Tensors We know that intellectual food is sometimes more easily digested, if not taken in the most condensed form. It will be asked, To what extent can specialized notations be adopted with profit ? To this question we reply, only experience can tell. p. C A J O R I , History of Mathematical Notations, p. 77 3*01. In this chapter we develop the theory of Tensors in a simple and restricted form. In many branches of physics the tensor notation in this form provides a compact mathematical expression, and familiarity with it is a preparation for the complete theory, involving the use of oblique axes, curvilinear coordinates and space of more than three dimensions; it is also an introduction to the ideas of matrix algebra. General tensor theory is indispensable as the mathematical apparatus of the theory of relativity, and matrix algebra in quantum mechanics and much of classical physics find their clearest expression in this notation. In the applications made in this chapter the physical ideas involved are simple, and practice in using the notation in this way is extremely valuable before proceeding to the applications of its complete form to theories where the physical ideas themselves are more difficult to grasp. 3*02. Transformation of coordinates. Contraction. We have defined a vector A by the transformation property which is equivalent to A i — lyA'^ (2) A vector can also be called a tensor of the first order. A scalar is a tensor of zero order. Now if we consider the set of nine products A i B k we notice that the scalar and vector products are particular linear combinations of these products. If we form a similar set for the components referred to new axes AW^lylvAtBto (3) AtBt-tyuAiBi. (4) In the same way as we use the transformation property to define a vector we now use these relations to define a tensor of the second order.
- Tomas B. Co(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Based on this convention, we denote Tensors by a letter with double underlines. We also limit our discussion to spaces with a maximum of three dimensions. Using the three Cartesian unit vectors, we have T = T xx (δ x δ x ) + T xy δ x δ y + T xz ( δ x δ z ) + · · · + T zz ( δ z δ z ) (4.15) where the scalar T ij is called the (i, j )−component of tensor T . A special tensor, called the unit tensor, is defined by δ = δ x δ x + δ y δ y + δ z δ z (4.16) When this tensor operates on any vector, the result is the same vector. EXAMPLE 4.4. Stress Tensors. Consider a material contained in a 3D domain. Stress is defined as the amount of force F , applied at a point p , per unit area of a fixed plane S that includes the point p . In general, the force F may be on an oblique angle θ with the plane S. We can identify the plane S by a unit vector n (S) that is perpendicular to S, known as its unit normal vector. Then the stress τ ( p , S) ( i.e., at p with respect to plane S ) can be decomposed into two additive vectors, τ ( p , S) = τ n ( p , S) + τ s ( p , S) where τ n ( p , S), called the normal stress, is pointed along the direction of the unit normal vector, and τ s ( p , S), called the shear stress, is pointed along a direction perpendicular to the unit normal vector (see Figure 4.6). If the material is non- uniform in terms of properties and applied forces, then the stress vector needs to be localized by using infinitesimal planes dS instead of S. The vector τ is also known as the traction vector. 8 A vector v can be considered a tensor only when it acts as an operator on another vector, say, a (which in our case is via dot products), v a = v · a Otherwise, a vector is mostly just an object having a magnitude and direction. Also, the n th -order tensor described in Definition 4.3 is still limited to the space of physical vectors, and the functional operations are limited to dot products.
eBook - PDF- Pijush K. Kundu, Ira M. Cohen(Authors)
- 2001(Publication Date)
- Academic Press(Publisher)
Cartesian Tensors Chapter 2 1. Scalars and Vectors 24 2. Rotation of Axes: Formal Definition of a Vector 25 3. Multiplication ofMatrices 28 4. Second-Order Tensor 29 5. Contraction and Multiplication 31 6. Force on a Surface 32 Example 2.1 34 7. Kronecker Delta and Alternating Tensor 35 8. Dot Product 36 9. Cross Product 36 10. Operator V: Gradient, Divergence, and Curl 37 11. Symmetric and Antisymmetric Tensors 38 12. Eigenvalues and Eigenvectors of a Symmetric Tensor 40 Example 2.2 40 13. Gauss' Theorem 42 Example 2.3 43 14. Stokes' Theorem 45 Example 2.4 46 15. Comma Notation 46 16. Boldface vs Indicial Notation 47 Exercises 47 Literature Cited 49 Supplemental Reading 49 /. Scalars and Vectors In fluid mechanics we need to deal with quantities of various complexities. Some of these are defined by only one component and are called scalars, some others are defined by three components and are called vectors, and certain other variables called Tensors need as many as nine components for a complete description. We shall assume that the reader is familiar with a certain amount of algebra and calculus of vectors. The concept and manipulation of Tensors is the subject of this chapter. A scalar is any quantity that is completely specified by a magnitude only, along with its unit. It is independent of the coordinate system. Examples of scalars are temperature and density of the fluid. A vector is any quantity that has a magnitude and a direction, and can be completely described by its components along three specified coordinate directions. A vector is usually denoted by a boldface symbol, for example, x for position and u for velocity. We can take a Cartesian coordinate system x,X2,X3, with unit vectors a 1 , a 2 , and a 3 in the three mutually perpendicular directions (Figure 2.1). (In texts on vector analysis, the unit vectors are usually denoted 24
eBook - PDF- Pijush K. Kundu, Ira M. Cohen(Authors)
- 2010(Publication Date)
- Academic Press(Publisher)
Chapter 2 Cartesian Tensors 1. Scalars and Vectors . . . . . . . . . . . . . . 25 2. Rotation of Axes: Formal Definition of a Vector . . . . . . . . . . . . . . . . . . . . . . . 26 3. Multiplication of Matrices . . . . . . . . 29 4. Second-Order Tensor . . . . . . . . . . . . 30 5. Contraction and Multiplication . . 32 6. Force on a Surface . . . . . . . . . . . . . . . 33 Example 2.1 . . . . . . . . . . . . . . . . . . . . 35 7. Kronecker Delta and Alternating Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 8. Dot Product . . . . . . . . . . . . . . . . . . . . . 37 9. Cross Product . . . . . . . . . . . . . . . . . . . 38 10. Operator ∇ : Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . . . . . . . . . 38 11. Symmetric and Antisymmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 40 12. Eigenvalues and Eigenvectors of a Symmetric Tensor . . . . . . . . . . . . . . . . 41 Example 2.2 . . . . . . . . . . . . . . . . . . . . 42 13. Gauss’ Theorem . . . . . . . . . . . . . . . . . 44 Example 2.3 . . . . . . . . . . . . . . . . . . . . 45 14. Stokes’ Theorem . . . . . . . . . . . . . . . . . 47 Example 2.4 . . . . . . . . . . . . . . . . . . . . 48 15. Comma Notation . . . . . . . . . . . . . . . . 49 16. Boldface vs Indicial Notation . . . . . 49 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 50 Literature Cited . . . . . . . . . . . . . . . . . 51 Supplemental Reading . . . . . . . . . . . 51 1. Scalars and Vectors In fluid mechanics we need to deal with quantities of various complexities. Some of these are defined by only one component and are called scalars, some others are defined by three components and are called vectors, and certain other variables called Tensors need as many as nine components for a complete description. We shall assume that the reader is familiar with a certain amount of algebra and calculus of vectors. The concept and manipulation of Tensors is the subject of this chapter.
eBook - PDF- Heinz Schade, Klaus Neemann, Andrea Dziubek, Edmond Rusjan, Andrea Dziubek, Edmond Rusjan(Authors)
- 2018(Publication Date)
- De Gruyter(Publisher)
5 Representation of Tensor Functions 5.1 The Basic Idea of the Representation of Tensor Functions 1. Modeling a physical process often means to find a function relating two Tensors. From the context we can assume that such a function exists, but it is not known and it can often be determined only from experiments. Typical examples of such functions are the stress–displacement relation T = f ( D ) for an elastic solid, where the (sym-metric) stress tensor T depends on an (also symmetric) displacement tensor D ; or the yield-stress condition σ = f ( T ) in plasticity theory, which relates the experimentally determined yield-stress σ to the three-dimensional stress state described by the stress tensor T . The functions f or f in the examples are called, according to the order of the ten-sors, scalar-valued or tensor-valued tensor functions. Contrary to what one may ex-pect, these functions cannot have an arbitrary form, because the transformation prop-erties of a tensor impose certain restrictions. For example, if in a given Cartesian co-ordinate system σ = f ( T ij ) and T ij = f ij ( D kl ), then, under a change of the coordinate system, substituting the transformed coordinates ̃ T ij into the function f must result in the same scalar σ = f ( ̃ T ij ) , and similarly, substituting the transformed coordinates ̃ D kl into the function f must give the transformed coordinates ̃ T ij = f ij ( ̃ D kl ) . If we as-sume that the Tensors are polar, then, according to the transformation equations (2.17), a scalar-valued function f must satisfy f ( T mn ) = f ( α im α jn T ij ) , (a) and a tensor-valued function f is restricted by the condition α mi α nj f mn ( D kl ) = f ij ( α pk α ql D pq ) . (b) 2. How to evaluate conditions of the type (a) or (b) and how to use them to construct functions relating Tensors is the subject of a theory called the theory of the representa-tion of tensor functions.
- Anthony E. Armenakas(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
Chapter 2 Strain and Stress Tensors 2.1 The Continuum Model Bodies are composed of a large number of discrete particles (atoms, molecules) in constant motion. Solids differ from liquids and liquids from gases in the spacing of these particles and in the amplitude of their motion. In studying the behavior of bodies the assumption is usually made that the material is distributed in the space which it occupies without leaving gaps or empty spaces. In other words, it is assumed that at every instant of time, there is a particle at every point of the space occupied by the body at that time . This model is referred to in the literature as continuum and it is used in all engineering disciplines because it is mathematically convenient. It permits integration and differentiation of the quantities describing the behavior of a body which are functions of the space coordinates. An infinitesimal portion of a continuum is called a particle . In this text we study the behavior of deformable solid bodies subjected to external loads on the basis of the continuum model. 2.2 External Loads Consider a body initially in a reference undeformed and unstressed state of 0 mechanical and thermal equilibrium at the uniform temperature T . In this state the † †† body is not subjected to external loads and heat does not flow in or out of it because its temperature is uniform. Subsequently, the body is subjected to one or more of the following external loads: 1 . Body forces 2 . Surface forces 1 2 3 3 . A temperature field T ( x , x , x ) ††† 4. Specified components of displacements of some particles of the surface of the body † When a body is in a state of mechanical equilibrium, its particles do not accelerate; that is, the sum of the forces acting on any portion of the body and the sum of their moments about any point vanish. †† When a body is in a state of thermal equilibrium, heat does not flow in or out of it; that is, the temperature of all its particles is the same.
eBook - PDFFinite Elasticity And Viscoelasticity: A Course In The Nonlinear Mechanics Of Solids
A Course in the Nonlinear Mechanics of Solids
- Aleksey Drozdov(Author)
- 1996(Publication Date)
- World Scientific(Publisher)
Chapter 1 Tensor calculus In this Chapter we introduce basic concepts and establish the main as-sertions in the tensor theory which are employed in the mechanics of continua. Our exposition of the tensor algebra and analysis is based on the so-called direct tensor notation which allows arbitrary curvilinear coordinate frames to be used. Section 1 is concerned with Eulerian and Lagrangian coordinate frames. In Sec-tion 2, basic concepts of the tensor algebra are provided. Section 3 deals with the tensor calculus. We introduce the nabla-operator and covariant derivatives of Tensors and prove the Stokes theorem. In Section 4, we discuss corotational derivatives, i.e. time derivatives of objective Tensors which are indifferent with respect to superimposed rigid motions. Finally, Section 5 is concerned with scalar and tensor functions of Tensors. 1. Geometry of Motion This Section is concerned with the geometrical description of motion in the mechanics of continua. We introduce Eulerian and Lagrangian coordinates and establish a correspondence between setting a motion with respect to these coordinate frames. We derive expressions for tangent vectors of the main and dual bases of Lagrangian coordinate frames, develop formulas for the volume element, and establish several properties of covariant and contravariant objects. 1.1. Description of Motion The main problem in the mechanics of continua is to describe motion of a medium under the action of external loads which are assumed to be given. Our objective is to find what kind of motion the medium performs (i.e. to derive a law of motion and distributions of velocities and accelerations at different points), as well as to determine the stress distribution. To answer these ques-tions various coordinate frames are introduced, where the motion is described. 1 Two different kinds of coordinate frames are employed. The first is Eulerian (spatial) coordinate frame, which is fixed and immobile in space.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
