Closely associated with tensor calculus is the indicial or index notation. A tensor of order zero zeroth order tensor is a scalar simple number. A tensor is a linear mapping of a vector onto another vector. Mathematica has sufficient functions to correctly compute the curl of a vector or tensor if the definitions given in the attached file are followed. First we note that a basis for a tensor of rank m, n is provided by the tensor product of m vectors and n oneforms. Whats special about vectors is that they can be resolved into components to see whats really happening in each pl.
A gentle introduction to tensors washington university in. From this example, we see that when you multiply a vector by a tensor, the result is another vector. Yet not any 3n scalars qualify to define a tensor a scalar, a vector, and a 2nd order tensor have 1, 3, and 9 components, respectively. A second order tensor is one that has two basis vectors standing next to each other, and they satisfy the same rules as those of a vector hence, mathematically, tensors are also called vectors. Tensor series expansion of a spherical function for the use in constitutive theory of materials containing orientable particles. The tht establishes a bijection between the real space.
A tensor may consist of a single number, in which case it is referred to as a tensor of order zero, or simply a scalar. Transposedivdottx,y,z, levicivitatensor3, x, y, z is all you need after defining the tensor t, for example. The glyph includes visual encoding for physically signi. Second, tensor theory, at the most elementary level, requires only. Let us consider the transformation of the coordinate frame by an arbitrary rigid rotation. The components of a vector ai, a first order tensor, are said to be. Each component represents a magnitude for that particular plane and direction. The stress tensor and strain tensor are both secondorder tensor fields, and are related in a general linear elastic material by a fourthorder elasticity tensor field. This equation is the transformation law of a second rank tensor. Derivative of inverse of second order tensor with respect to tensor using indices 1 tensor scalar field laplacian, rot and multiplying by vector in cartesian and spherical coordinates. The derivatives of scalars, vectors, and secondorder tensors with respect to secondorder tensors are of considerable use in continuum mechanics. A semiintuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and similar higherorder vector products.
Doitpoms tlp library tensors in materials science what. An introduction to tensors for students of physics and engineering joseph c. Lagrange scalar densities which are concomitants of a pseudoriemannian metrictensor, a scalar field and their derivatives of arbitrary order are considered. Two examples, together with the vectors they operate on, are. Consider a continuous medium of material, such as an elastic medium e. A secondorder tensor is one that has two basis vectors standing next to each other, and they satisfy the same rules as those of a vector hence, mathematically, tensors are also called vectors. Assuming that there is no internal torque, we obtain that p ij p ji. An introduction to tensors for students of physics and. A tensor hankel transform tht is defined for vector fields, such as displacement, and secondorder tensor fields, such as stress or strain.
The off diagonal terms of the permeability tensor can be calculated from the definition of a second order cartesian tensor. Where lambda is a fourthorder tensor, called the mobility tensor which is essentially the inverse of the relaxation time of the polymer fluids, rho is the fluid density, q is anisotropic viscosity matrix that is related to viscous dissipation, l is coupling parameter between the velocity gradient field and the structural tensor field, c is. A fourthorder tensor relates two secondorder tensors. Many of the concepts will be familiar from linear algebra and matrices. A formal definition of the transpose which does not rely on any particular coordinate system is as follows. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. The trace of a second order tensor a is defined as the dot product of a. A basic operations of tensor algebra the tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems. It may be regarded as a definition, or as resulting from the fact that the tensor is a vectorvalued function of a tensor. It is this chapter that provides the foundations for tensor applications in physics. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler, which characterizes simplified approaches to a problem as the poor mans way to do x. Pdf the hankel transform of first and secondorder tensor. Tensors for beginners albert tarantola september 15, 2004 1 tensor notations the velocity of the wind at the top of eiffels tower, at a given moment, can be represented by a vector v with components, in some local, given, basis, vi i 1,2,3. The present text is supposed to be the first part of a series of documents about.
In detail, the tensor quantifying stress in a 3dimensional solid object has components that can be conveniently represented as a 3. Rn is the giving of a k,ltensor in every point ofu, varying smoothly with the point. Tensor simple english wikipedia, the free encyclopedia. Transposes,leigenvaluesandinvariants ofthirdordertensors. Let us generalize these concepts by assigning nsquared numbers to a single point or ncubed numbers to a single.
To introduce the engineer to the very important discipline in applied mathematicstensor methods as well as to show the fundamental unity of the different fields in continuum mechanicswith the unifying material formed by the matrixtensor theory and to present to the engineer modern engineering problems. This is a general property of all second order tensors. The definition of a tensor as a linear operator is prevalent in physics. Tensors provide a mathematical framework for solving physics problems in areas such as elasticity, fluid mechanics and general relativity. Doing so, the 81 components of a fourth order tensor are stored in a 9. Take the x 3 as a rotation axis, the forces that can produce torques are p 12 and p 21. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. What is the geometrical interpretation of a second order tensor.
Matrix notation of such relations is only possible, when the 9 components of the secondorder tensor are stored in columns. Mirrorimaging of vectors as a second order tensor mapping example of a tensor the combination u linearly transforms a vector into another vector and is thus a secondorder tensor3. A secondorder stress tensor takes a vector unit normal to a stress vector. Thus, if e is a fourthorder tensor, a typical element is denoted by eijk. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. The derivatives of scalars, vectors, and second order tensors with respect to second order tensors are of considerable use in continuum mechanics.
A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. In general, there are two possibilities for the representation of the. Definition of a tensor 6 valid tensor relationship, each term, whether on the left or right side of the equation, must have the same free indices each in the same position. Definition a secondorder tensor t is defined as a bilinear function from two copies of a vector space v into the space of real numbers. A system with two indices is called a second order system. Scalars, vectors and tensors a scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. A vector is a bookkeeping tool to keep track of two pieces of information typically magnitude and direction for a physical quantity. The nine components of a secondorder tensor are generally stored in a threebythree matrix. The firstorder tensor is symbolized with a boldface letter and by an arrow at the top part of the vector, i. The hankel transform of first and second order tensor fields. What is the geometrical interpretation of a second order.
Udayanandan associate professor department of physics nehru arts and science college, kanhangad 1. Four of the nine components of the stress tensor acting on a small. The tensor transformation law may be used to obtain the values of the property as the direction of the cause driving force changes. This statement serves as tensors definition an nth order tensor has thus 3n components. Quantities that have magnitude and two directions, e. The stress tensor and strain tensor are both second order tensor fields, and are related in a general linear elastic material by a fourth order elasticity tensor field. For the sake of comprehension we consider again the zeroth, first, and second order tensors. Tensors article about tensors by the free dictionary. A tensor is a generalization of a scalar a pure number representing the value of some physical quantity and a vector a geometrical arrow in space, and a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. Introduction to tensor calculus a scalar eld describes a onetoone correspondence between a single scalar number and a point. It would probably be more helpful to him if your answer just focused on tensors perhaps with an example which includes the special case of a tensor of order 3, while making it clear that it generalizes to higher orders and not on misconceptions that he.
Mirrorimaging of vectors as a second order tensor mapping example of a tensor the combination u linearly transforms a vector into another vector and is thus a second order tensor3. Whenever we want to represent some physical quantity mathematically, we need to see how much information is needed to specify the value of that quantity. For example, consider a force f applied to a spanner at a distance r from the centre of the nut, fig. As a first example, we define the covariant coordinate differential, dxi, to be. Syllabus tensor analysisintroductionde nitionde nition of di erent rank tensorscontraction and direct productquotient rulepseudo tensors. Tensors of the order zero and one are per definition symmetric traceless and have no skewsymmetric part. Typically, when tensor mathematics is applied, the meaning of each index has been defined beforehand. A tensor of order zero zerothorder tensor is a scalar simple number. Decomposition and visualization of fourthorder elastic.
The nine components of a second order tensor are generally stored in a threebythree matrix. Secondorder scalartensor field equations in a four. Transposedivdottx,y,z, levicivitatensor3, x, y, z is all you need after defining the tensor t. The rank or order of a tensor is defined by the number of directions and hence the dimensionality of the array required to describe it. The determinant of a second order tensor a is defined to. For the sake of comprehension we consider again the zeroth, first, and secondorder tensors. Dec 19, 2016 well i presume that you must be having a bit idea about the 1st order tensors. For reasons which will become apparent, a scalar may be thought of as an array of dimension zero same as the order of the tensor. Derivative of inverse of secondorder tensor with respect to tensor using indices 1 tensor scalar field laplacian, rot and multiplying by vector in cartesian and spherical coordinates. The hankel transform of first and secondorder tensor fields.
The secondorder and higherorder tensors are symbolized with a boldface letter. I hope i will be able to nalize and publicize the document for the next level in the near future. An ndimensional vector eld is described by a onetoone correspondence between nnumbers and a point. Dec 09, 2017 less of a focus on physics in this one.
Chapter 2 of aris, appendix a of bsl the algebra of vectors and tensors will be described here with cartesian. At this early stage in these notes, we generically define tensor as an. A a, transforming any second order tensor a into itself. The order of a tensor is the same as the number of subscripts needed to write a typical element. The most general secondorder eulerlagrange tensors derivable from such a lagrangian in a fourdimensional space are constructed, and it is shown that these eulerlagrange tensors may be obtained from a lagrangian which is at most of.
You must already be familiar with the concept of scalars and tensors. Take a covariant derivative of the second bianchi identity, and sum over ciclic permutations of four indices abcd. Second order tensors examples of second order tensors scalar multiplication and addition contraction and multiplication the vector of an antisymmetric tensor canonical form of a symmetric tensor reading assignment. We notice that a scalar is a tensor of rank zero, a vector is a first rank tensor, the 3by3 array just defined is a second rank tensor, etc. A second order identity for the riemann tensor 3 proof. Introduction to tensor calculus for general relativity mit. In more particular models, the stress tensor is symmetric. Tensor article about tensor by the free dictionary. An example of a scalar would be the mass of a particle or object. If a certain free index is covariant contravariant in one term, it must be covariant contravariant in all terms. A second order stress tensor takes a vector unit normal to a stress vector.
The word tensor comes from the latin word tendere meaning to stretch. In the following, we will restrict to secondorder tensor. We propose a glyph to visualize asymmetric secondorder twodimensional tensors. Isotropic tensors a tensor which has the special property that its components take the same value in all cartesian coordinate systems is called an isotropic tensor. For example, consider a force f applied to a spanner at a distance r. In general, there are two possibilities for the representation of the tensors and the tensorial equations. Matrix notation of such relations is only possible, when the 9 components of the second order tensor are stored in columns. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations the directional derivative provides a systematic way of finding these derivatives. Note that double contraction is used here, while single contraction was used to define the identity second order tensor transforming any vector into itself. Four of the nine components of the stress tensor acting on a small cubic. Any set of 9 numbers that transform like this under rotations form the components of a tensor. A fourth order tensor relates two second order tensors. Take the x 3 as a rotation axis, the forces that can produce torques are p 12.
1157 1533 1562 88 1129 797 253 1549 825 467 1469 1521 1217 1227 767 584 605 362 1340 332 1344 788 1421 84 706 197 1464 237 1440