Deformation gradient tensor an overview sciencedirect. This is an introductory discussion of the stress tensor. Here, the stress is defined more formally, cauchys law cauchys law states that there exists a cauchy stress tensor. Ive spent a few weeks scouring the internet for a an explanation of tensors in the context of engineering mechanics. Physical significance of the stress tensor youtube. The stress tensor is a representation of stress on three mutually perpendicular planes in a coordination system. 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. Stress measures this disambiguation page lists articles associated with the title stress tensor. In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields tensors that may vary over a manifold, e. The traction vector and stress tensor special thanks to d.
Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are. Writing the tensor in terms of indexed components, the rank tells you the number of indices required. Thus, it follows that, irrespective of the orientation of the principal axes, the trace of the stress tensor at a given point is always equal to the sum of the principal stresses. What are the best books to learn tensors from for the first. By expressing the deviatoric shear stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above cauchy equations will lead to the navierstokes equations below. The transform applies to any stress tensor, or strain tensor for that matter. Cauchygreen strain tensor an overview sciencedirect. Cauchy s assumption consists in introducing the stress vector, denoted by tm, t, as the elementary force df acting on an internal facet of the cm, isolated by a virtual cut. For this reason, an isotropic state of stress is also referred to as a hydrostatic state of stress. However, several other measures of stress can be defined. State of stress stress tensor complete notes on stress.
Derivation of cauchy stress pge 334 reservoir geomechanics. I 1 is the trace of the cauchy stress tensor and is very often replaced by the mean stress. Tensor is the mathematical tool used to express these physical quantities. The stress tensor and the relativistic stressenergy tensor. 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.
Coordinate transformations of 2nd rank tensors were discussed on this coordinate transform page. The stress tensor is a way of describing the stress throughout the object precisely. The cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations where the differences in stress distribution in most cases can be neglected. You know, the ones every engineering student know and love stress, strain, etc. Cauchy, who knew full well the di erence between a balance principle and a constitutive relation, stated the result clearly and proudly. This is primarily aimed at the graduate student and postgraduate researcher with a subset appropriate for part iiiiv engineering undergraduate students.
It holds for any continuum no matter how the stress tensor t is connected with the rate of strain. Any physical property that can be quanti ed is called a physical quantity. Introduction this page introduces hydrostatic and deviatoric stresses. Cauchy s theorem of the existence of the stress tensor, published in 1823. It is important to note that this stress tensor is always defined in the deformed state of the material, and is known as the cauchy stress tensor. Piolakirchhoff stress tensor, in continuum mechanics.
In fact, this lack of symmetry and lack of a clear physical meaning makes it uncommon for the pk1 stress to be used in the modeling of materials. A graphical representation of this transformation law is the mohrs circle for stress. So a scalar is a zero rank tensor, a list of numbers aka vector is a rank 1 tensor, a 2 dimensional grid aka matrix is a rank 2 tensor, and higher rank things are just called tensors. It is mathematicalsorrybut it gives the barebones definitions that are needed to do differential geometry. Thats sometimes called the stress vector but is better called the traction vector, the word stress being reserved for the tensor. This page covers principal stresses and stress invariants. In case the deformations are small and the bady is elastic, then relations e1 are linear. Both mathematical and engineering mi stakes are easily made if this crucial difference is not recognized and understood. So the stress matrix, for this arrangement of the axes, is given by. The state of stress at a given point of the earths crust, at a given moment, is represented by a second order tensor. A first course in rational continuum mechanics 1st edition. The magnitudes of the components of the stress tensor depend on how we have defined the orthogonal x 1, x 2 and x 3.
The stress tensor the general state of stress in any homogeneous continuum, whether. Aug 29, 20 state of stress at a point stress tensor consider a three dimensional body in equilibrium under the action of a set of external forces p1,p2,p3, the stresses developed in different planes vary. The first index denotes the direction of the normal, or perpendicular, stresses to the plane across which the contact force acts, and the second index denotes the direction of the component of force see text. This is cauchy s equation of motion and a is the acceleration. For large deformations, also called finite deformations, other measures of stress, such as the first and second piolakirchhoff stress tensors, the biot stress tensor, and the kirchhoff stress tensor, are required. Pollard who pioneered this exercise in 1991 introduction stress concentrations in the earth lead to the development of structures such as faults, folds, and joints. Stress balance principles 04 properties of the stress tensor. Continuum mechanicsstress tensor wikibooks, open books for. The symmetry of the stress tensor a polar fluid is one that is capable of transmitting stress couples and being subject to. It is an attribute of matter, radiation, and nongravitational force fields. Lecture 3 the concept of stress, generalized stresses and.
This stress tensor is often decomposed into two parts. Tensors are referred to by their rank which is a description of the tensor s dimension. Both tensor and vector quantities are denoted by boldface letters. A zero rank tensor is a scalar, a first rank tensor is a vector. Hence the pk1 stress tensor is not symmetric, and this restricts its use as an alternative stress measure to the cauchy stress measure. Because the stress tensor is uniform, the force is the same, but now the normal component. I really, really love manifolds, tensors, and forms.
One example of isotropic stress is the stress arising in fluid at rest, which cannot support shear stress, in which case. The stress tensor the secondorder tensor which we will be examining has. Our focus is on relating deformation to quantities that can be measured in the. The hydrostatic stress is related to volume change, while the deviatoric stress is related to shape change. Indices of the stress tensor may be raised or lowered with abandon by balanced application of the metric tensor. The general stress tensor has six independent components and could require us to do a lot of calculations. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation.
The mechanical action exerted by the segment cut on this facet, with outward unit normal nm, t, is therefore a possible modeling of the cohesive forces. In our daily life we see large number of physical quantities. If we think of a tensor as a linear operator taking vectors into vectors, we dont need a set of basis vectors for the denition of a tensor. The system of equations is completed by a constitutive equation, linking the stress tensor to a measure of. This problem set is designed to be a tutorial for stress analysis at a point in a rock mass or. Reactions upon containers and submerged obstacles 3.
If the material is elastic, the object will return to its initial shape and size when these forces are removed. Stress is always simply \force area\, but some complexity does arrise because the relative orientation of the force vector to the surface normal dictates the type of stress. This is purely a notational convenience to allow us to write the 4th order elasticity tensor on a. On the dual variable of the cauchy stress tensor in. The plastic behaviour of materials is often independent of a hydrostatic stress and this feature necessitates the study of the deviatoric stress. But i cannot find any explanations of tensors without running into abstract formalisms like homomorphisms and inner product spaces. Physics stack exchange is a question and answer site for active researchers, academics and students of physics.
For the cauchy stress tensor at poin 1,1,1 of a body, expressed with respect to the orthonormal system fei, e2, es, with components units are mpa 1055 5 5 10 a find principal stresses and maximum shear stresses by solving the characteristic equation of the stress b find the planes on which the stress traction vector has no normal component, and also the planes on which 5051. We shall now turn to the useful strain measure in viscoelasticity, namely the relative cauchy green strain tensor, which is obtained from 1. For infinitesimal deformations of a continuum body, in which the displacement vector and the displacement gradient 2nd order tensor are small compared to unity, i. If it helps, i found a whole appendix the classical cauchy stress tensor and equations of motion in the book the geometry of physics. To make things easier it can be rotated into the principal stress tensor by a suitable change of axes. The two are subsets of any given stress tensor, which, when added together, give the original stress tensor back. Cauchy s stress theorem states that in a stress tensor field there is a traction vector. The viscous stress tensor of a fluid vanishes under hydrostatic conditions. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. The stress tensor and the relativistic stress energy tensor classical stress tensor. If the constitutive law linking the second piolakirchhoff stress tensor and the right cauchy green strain tensor derives from a potential, then the cauchy stress tensor and the logarithm of the left cauchy green strain tensor are linked by a related potential.
The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. 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. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all. Also discussed are vector components and the properties of determinants. If an internal link led you here, you may wish to change the link to point directly to the intended article. To define this, let us assign the position coordinates of the particle at time. Uniaxial tension this is the situation for a simple tensile test, figure 5. This definition will become important when we deal with small versus large deformation mechanics. In detail, the tensor quantifying stress in a 3dimensional solid object has components that can be conveniently represented as a 3. 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. The historical choice is to caste vectors as column vectors, which are implicitly contravariant vectors.
The matrix of this linear operator or components of the tensor are given by 1. The last section provided the first step toward finding the stress on an arbitrary plane in continuous media. Recall that this transformation can be written in terms of the pullback of. All nonrelativistic balance equations, such as the navierstokes equations, can be derived by beginning with the cauchy equations and specifying the stress tensor through a constitutive relation. Truesdell stress rate of the cauchy stress the relation between the cauchy stress and the 2nd pk stress is called the piola transformation.
Chapter 1 17 pages has some interesting examples which demonstrate how tensors arise naturally, namely the symmetric stress tensor in elasticity, the antisymmetric inertia tensor for rigid bodies, and crossproduct vectors which arise in electromagnetism. The most commonly used measure of stress is the cauchy stress tensor, often called simply the stress tensor or true stress. We can then think of a tensor as a linear operator, which takes an arbitrary vector a into a new vector ta. Doitpoms tlp library stress analysis and mohrs circle. Tensors 9 of 45 stress in tensor in 2dimensions duration. For example, one side of the eraser might be pulled on tensile stress while the under is squeazed compressive stress.
The trace of the stress tensor, is a scalar, and, therefore, independent of the orientation of the coordinate axes. What are the best books to learn tensors from for the. Tensor, dyadic, and engineering approaches by pei chi chou, vector and tensor analysis with applications by. There are quite a few wellwritten books on these topics like those by jaunzemis 1, malvern 2, fung 3, sokolnikoff 4 etc. It follows from this definition that the stress in a cauchyelastic material does not depend on the path of deformation or the history of deformation, or on the time taken to achieve that deformation or the rate at which the state of deformation is reached. Everything here applies regardless of the type of stress tensor. A second rank tensor looks like a typical square matrix. Tensors, differential forms, and variational principles. The concept of stress, generalized stresses and equilibrium problem 31.
When the force vector is normal to the surface, as shown to the right, the stress is called normal stress and represented. The important property of a physical quantity is that it can be measured and expressed in terms of a mathematical quantity like number. Solid objects will deform when adequate forces are applied to them. The cauchy stress is symmetric, but the deformation gradient is not. A number of sets of stress tensor invariants are compared in ref. A handbook on tensors in geology on free shipping on qualified orders. Hence, the relative strain tensor history is a function of t and the time lapse and thus, the extra stress tensor in a simple fluid is a function of t only. Continuum mechanicsobjective stress rates wikiversity. Further simplification based on experiments observations. An introduction for mathematicians and physicists by paul renteln. The 2nd order cauchy stress tensor may be written in matrix form as. Developed by gregorio riccicurbastro and his student tullio levicivita, it was used by albert einstein to develop his theory of general relativity. The stressenergy tensor, sometimes called the stressenergymomentum tensor or the energymomentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of newtonian physics.
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