Oct 16, 2014 lectures for transport phenomena course at olin college this lecture describes what the stress tensor is. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors. Chapter 3 the stress tensor for a fluid and the navier. You will be able to use the notion of a tensor to describe what happens.
The stress strain relation in a crystal is a more complex situation that can also be described in terms of tensors. Thus, the word tensor without any qualifier means second order tensor. The second invariant of the viscous stress tensor is iit. Doitpoms tlp library tensors in materials science what is. In the case of a tensor of rank 2 there are three quantities that are invariant to coordinate transformations. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. For more comprehensive overviews on tensor calculus we recommend 58, 99, 126, 197, 205, 319, 343. On tensors and equations of the electromagnetic field. In continuum mechanics, the cauchy stress tensor, true stress tensor, or simply called the stress tensor is a second order tensor named after augustinlouis cauchy.
Syllabus tensor analysisintroductionde nitionde nition of di erent rank tensors contraction and direct productquotient rulepseudo tensors general tensors metric tensors. In detail, the tensor quantifying stress in a 3dimensional solid object has components that can be conveniently represented as a 3. 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. 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. The stress tensor pij is defined as the jth component of the forces within the body acting on an. Representing a force in three dimensions requires three numbers, each referenced to a. The order of a tensor is called its rank and is defined by its law of transformation under a change of coordinates. To get all of the elements of the stress tensor in the new coordinate system. In continuum mechanics, stress tensors are second rank contravariant ten sors see chap. From this tensor, in the form of its divergences, the emf equations follow. Several properties of the stress tensor remain unchanged by a change in coordinates. Like rank 2 tensors, rank 3 tensors may be called triads.
Second rank tensor an overview sciencedirect topics. This tensor is referred to as the stress tensor and denoted as t whilst its components in a cartesian coordinate. The stress tensor california institute of technology. Understanding the role of the metric in linking the various forms of tensors1 and, more importantly, in di.
The calculus of matrices is presented in 44, 114, 350, for example. 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. These invariants are closely related to important quantities. In a general model of continuous media, where it is not assumed that the stress tensor is symmetric, this means that we need 9 scalar quantities to characterize the state of. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. Note that in chapter ii it was shown that the last two. For a general vector x x 1,x 2,x 3 we shall refer to x i, the ith component of x. Tensor mathematics cfd direct architects of openfoam. Latter relationship still holds in the case of inhomogeneous stress or body forces, see textbooks. A second rank tensor has in general 9 independent elements. A fourth rank tensor is a fourdimensional array of numbers.
You must already be familiar with the concept of scalars and tensors. It is this chapter that provides the foundations for tensor applications in physics. In three dimensions, the bingham model can be generalized by introducing the second invariants of the stress and rateofstrain tensors. If the dependence of w on the green deformation tensor, c, is known, the cauchy true stress tensor, t, can be readily obtained by means of the relation 10 24 where b is the left cauchygreen tensor also called the finger tensor and. A vectorwith one indexis a tensor of the first rank, and a scalarwith no indexis a tensor of zero rank. Similar labels, which are much less common in use, may be attached to higher rank tensors. A second rank tensor is defined here as a linear vector function, i. Tensor math allows us to solve problems that involve tensors. Strain is a dimensionless measure of local deformation. The 4d, 4th rank, antisymmetric tensor and the 4d equivalent.
Stress nine quantities are required to define the state of stress at a point. This disambiguation page lists articles associated with the title stress tensor. The state variables pertaining to the generic material element are f, f p, and the hardening parameter. Geoscientists have undertaken mapping of the earths crustal strain or stress fields using a great variety of field data. Statistics and geometry of the eigenspectra of three.
Four of the nine components of the stress tensor acting on a small cubic. Our focus is on relating deformation to quantities that can be measured in the. A gentle introduction to tensors washington university in. An introduction to tensors for students of physics and. Jun 16, 2018 in this video, i continue my introduction to tensors by talking about the transformation property that defines tensors, that tensors are invariant under a change of coordinate system. The index i may take any of the values 1, 2 or 3, and we refer to the. They are used to describe the mechanical properties of rigid or deformable bodies in terms of inertia, stress or strain. For a given euclidean manifold, a tensor field is uniquely determined when we specify its kind i. They typically relate a vector to another vector, or another second rank tensor to a scalar. Stress tensor stress is a second rank tensor quantity. Tensor algebra operations for making new tensors from. A third rank tensor would look like a threedimensional matrix. The stress tensor the general state of stress in any homogeneous continuum, whether.
Each component represents a magnitude for that particular plane and direction. More generic names for higher rank tensors, such as polyad, are also in use. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. Physical significance of the stress tensor youtube. The 4d, 4th rank, antisymmetric tensor and the 4d equivalent to the cross product or more fun with tensors richard r. On covariances of eigenvalues and eigenvectors of secondrank. An introduction to tensors for students of physics and engineering joseph c. Piezoelectricity is described by a third rank tensor.
A second rank tensor may be thought of as a vectorvalued function of a vector. Introduction in mechanics, dynamics, statistics, etc. Thus, a complete description of the emf is an asymmetric tensor of the second rank. Furthermore, the stress tensor would have to be diagonal in any coordinate frame because, clearly, the fluid doesnt know which frame we choose to use to describe the stress tensor. When a stress is applied, the crystal will distort slightly and this relation of strain to stress is, for small stress, a linear one. Tensors of higher rank are required to fully describe properties that relate two second rank tensors e.
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