A cauchyelastic material is also called a simple elastic material it follows from this definition that the stress in a cauchyelastic material does not depend on the path of deformation or the history of. He used the very first tensor when he was working in what is now called the cauchy stress theorem, a result that describes the stress caused by a tension in materials. Stress balance principles 03 the cauchy stress tensor. Use of cartesian tensors occurs in physics and engineering, such as with the cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the cauchy momentum equation will lead to the navierstokes equations. May 30, 2014 a lecture from lectures on continuum physics. The components in a second coordinate system with base vectors e. Contemporary plasticity theories and their related material models for finite deformations are either based on additive decomposition of a strainrate tensor or on multiplicative decomposition of a deformation gradient tensor into an elastic part and a plastic part. Grabiner is more technically challenging than many books on the history of mathematics.
Cauchy, who knew full well the di erence between a balance principle and a constitutive relation, stated the result clearly and proudly. A more sophisticated example is the cauchy stress tensor t, which takes a direction v as input and produces the stress tv on the surface normal to this vector for output, thus expressing a relationship between these two vectors. Stress balance principles 04 properties of the stress tensor. The forces encountered in this book are constrained to surface and body forces. The tensor s components, in a threedimensional cartesian coordinate system, form the matrix whose columns are the stresses forces per unit area acting on the e 1, e 2, and e 3 faces of the cube. For a given strain, it is seen that if in a given basis one of the eigenvectors of the stress tensor has a zero component, say the. A zero rank tensor is a scalar, a first rank tensor is a vector. Everything here applies regardless of the type of stress tensor. The history from the reference configuration is then ux,t. Continuum mechanicsobjective stress rates wikiversity. In elasticity, he originated the theory of stress, and his results are nearly as valuable as those of simeon poisson. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials.
A history of the theory of elasticity and of the strength of mate. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement. Stress components taking cauchy s law to be true it is proved below, the components of the stress tensor with respect to a cartesian coordinate system are, from 1. Cauchygreen strain tensor an overview sciencedirect. Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are. Since the cauchy and first piolakirchhoff stress tensors correspond to. This derivation assumes that there are no body moments. Why is the cauchy stress tensor symmetrical at equilibrium. From the standpoint of the nonlinear continuum mechanics, the former theories, which are used to model hypoelasticplastic. Relation of the cauchy stress tensor to applied forces. Augustinlouis cauchy project gutenberg selfpublishing. Cauchys stress tensor is defined in the deformed configuration and is thus not practical to use for large deformation analysis or experimental measures.
An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar. If the constitutive law linking the second piolakirchhoff stress tensor and the right cauchygreen strain tensor derives from a potential, then the cauchy stress tensor and the logarithm of the left cauchygreen strain tensor are linked by a related potential. Cauchy stress tensor an overview sciencedirect topics. Coordinate transformations of 2nd rank tensors were discussed on this coordinate transform page. Therefore, we need to develop alternative stress tensors. The meaning of the components of the cauchy stress tensor becomes.
The origins of cauchys rigorous calculus by judith v. Subtracting the mean normal stress from the stress tensor produces the deviatoric stress. Nov 28, 2016 the symmetry of the cauchy stress tensor is obtained from the balance of angular momentum. The origins of cauchys rigorous calculus dover books on. In mathematics, ricci applied tensors to differential geometry during the 1880s and 1890s. 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. If there are body moments, the only way that angular momentum can be conserved is if the stress tensor is asymmetric. Cauchygreen tensor is the pullback of the riemannian metric on space by the deformation. Ive been trying to understand the derivation for the cauchy momentum equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation im guessing they assume the reader knows it already. The 180 year old stress theory by cauchy is found to be insufficient to serve as a basis for a modern understanding of material behaviour.
Cauchys stress theorem states that in a stress tensor field there is a traction vector. There are some di erent methods and processes to prove the existence of stress tensor and presentation of the cauchy. Isotropic and deviatoric stress the stress tensor can be divided into two parts. However, several other measures of stress can be defined. The kirchhoff stress is just the cauchy stress scaled by the volume change. Augustinlouis cauchy, in full augustinlouis, baron cauchy, born august 21, 1789, paris, francedied may 23, 1857, sceaux, french mathematician who pioneered in analysis and the theory of substitution groups groups whose elements are ordered sequences of a set of things. The transform applies to any stress tensor, or strain tensor for that matter. If you pick three perpendicular surfaces of these cubes you can def. Introduction to stress and equations of motion comsol multiphysics. Lecture 3 the concept of stress, generalized stresses and. Sometimes you may also encounter the kirchhoff stress. Alexandre laurent cauchy 17921857, who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, and eugene francois cauchy 18021877, a publicist who also wrote several mathematical works.
A natural question is how the cauchy stress tensor relates to applied forces. Tensor, dyadic, and engineering approaches by pei chi chou, vector and tensor analysis with applications by. A cauchy elastic material is also called a simple elastic material. I imagine you have already looked into the wikipedia stuff so ill just explain how i understand it personally. Explode the collection of balls into free bodies as. We note that t o has the same direction as the cauchy stress vector t the first piolakirchhoff stress tensor also known as the lagrangian stress tensor is a. The symmetry of the cauchy stress tensor is obtained from the balance of angular momentum. A graphical representation of this transformation law is the mohrs circle for stress. Cauchy tetrahedron argument and the proofs of the existence of. 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. 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. He was one of the greatest of modern mathematicians. Tensors are referred to by their rank which is a description of the tensor s dimension. The history of the calculus and its conceptual development, carl b.
The 2nd order cauchy stress tensor may be written in matrix form as. He was one of the first to state and rigorously prove theorems of calculus, rejecting the. Augustinlouis cauchy french mathematician britannica. Pdf cauchys stress theory in a modern light researchgate. Examples are given of several special stress fields possessing particular structures, and the stress power is introduced. Neither textbooks nor personal discussions ever raised, or even permitted, the question. Introduction to continuum mechanics fourth edition, 2010. Cauchy had two brothers, alexandre laurent cauchy 17921857, who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849. The first piolakirchhoff is an unsymmetric tensor, and is for that reason less attractive to work with. For large deformations, also called finite deformations, other measures of stress, such as the first and second piolakirchhoff stress tensors, the biot stress. In general, these tractions are both parallel and perpendicular to the normal vectors of the faces.
Stress components taking cauchys law to be true it is proved below, the components of the stress tensor with respect to a cartesian coordinate system are, from 1. On the dual variable of the cauchy stress tensor in. Cauchygreen strain tensor an overview sciencedirect topics. A year or two of calculus is a prerequisite for full appreciation of grabiners work. Think of dividing some continuous material into super small cubes. The stress is defined as the force in the current configuration acting on the original area. Linear elasticity in linear elasticity theory for homogeneous bodies, the stress tensor. 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 corresponding referential equations of motion are obtained in terms of the piolakirchhoff stress tensor. Cauchy was the son of louis francois cauchy 17601848 and mariemadeleine desestre. 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. A more complex example is the cauchy stress tensor t, which takes a directional unit vector v as input and maps it to the stress vector t v, which is the force per unit area exerted by material on the negative side of the plane orthogonal to v against the material on the. The most commonly used measure of stress is the cauchy stress tensor, often called simply the stress tensor or true stress. Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all. Cauchys development of the stress tensor is discussed along with cauchys spatial equations of motion. The cauchy stress tensor for a material subject to an. To answer this question, we can turn to an analysis first done by.
For large deformations, also called finite deformations, other measures of stress, such as the first and second piola kirchhoff stress tensors, the biot stress tensor, and the kirchhoff stress tensor, are required. In physics, a cauchyelastic material is one in which the stress at each point is determined only by the current state of deformation with respect to an arbitrary reference configuration. It follows from this definition that the stress in a cauchy elastic 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. Sometimes general curvilinear coordinates are convenient, as in highdeformation continuum mechanics, or even necessary, as in general relativity. At a certain orientation of the faces, however, the tractions are only parallel to the. The cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A second rank tensor looks like a typical square matrix. To answer this question, we can turn to an analysis first done by a. Pdf the 180 year old stress theory by cauchy is found to be insufficient to serve as a basis for. An alternative framework for developing material models. On the dual variable of the cauchy stress tensor in isotropic. For example, j means the abstract cauchy stress tensor, while. The stress tensor is a matrix that specifies the tractions on three mutually perpendicular faces of an infinitesimal cube.
The stress vector t o, defined by the preceding equation, is a pseudostress vector in that, being based on the undeformed area, it does not describe the actual intensity of the force df, which acts on the deformed area da dan. Continuum mechanics conservation laws what if the stress tensor is not symmetric. Chapter 3 the stress tensor for a fluid and the navier stokes. This page covers principal stresses and stress invariants. By assuming inviscid flow, the navierstokes equations can further simplify to the euler equations. The concept of stress, generalized stresses and equilibrium problem 31. Voigt used tensors for a description of stress and strain on crystals in 1898 14, and the term tensor first appeared with its modern meaning in his work.
In terms of a coordinate basis or fixed frame of reference, a tensor can be represented as an organized. If an internal link led you here, you may wish to change the link to point directly to the intended article. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. In particular, the trace of the cauchygreen strain tensor, tr g, accounts for local changes in curve lengths while the determinant det g.
Stress measures this disambiguation page lists articles associated with the title stress tensor. This is noted in a wonderful, if somewhat old fashioned book, well worth some study, arnold. If the stress tensor were not symmetric, there will. The divergence of the stress tensor can be written as. Cauchy s theorem of the existence of the stress tensor, published in 1823. Symmetry of stress tensor which implies that only six stress components are needed to describe the state of stress at a point in a continuum. Cauchy s stress tensor is defined in the deformed configuration and is thus not practical to use for large deformation analysis or experimental measures.
1445 1335 1341 135 220 489 1119 703 932 944 473 680 949 1427 331 87 501 830 807 224 1213 955 930 754 1266 1148 1317 281