Cauchy continuum

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Cauchy continuum

The tensor relates a unit-length direction vector n to the traction vector T n across an imaginary surface perpendicular to n :.

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The unit vector is dimensionless. The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress.

The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations : It is a central concept in the linear theory of elasticity.

For large deformations, also called finite deformationsother measures of stress are required, such as the Piola—Kirchhoff stress tensorthe Biot stress tensorand the Kirchhoff stress tensor. According to the principle of conservation of linear momentumif the continuum body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations Cauchy's equations of motion for zero acceleration.

At the same time, according to the principle of conservation of angular momentumequilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetricthus having only six independent stress components, instead of the original nine. However, in the presence of couple-stresses, i. There are certain invariants associated with the stress tensor, whose values do not depend upon the coordinate system chosen, or the area element upon which the stress tensor operates.

These are the three eigenvalues of the stress tensor, which are called the principal stresses. Only surface forces will be discussed in this article as they are relevant to the Cauchy stress tensor.

In particular, the contact force is given by. Cauchy's stress principle asserts [6] : p. In specific fields of continuum mechanics the couple stress is assumed not to vanish; however, classical branches of continuum mechanics address non- polar materials which do not consider couple stresses and body moments.

This equation means that the stress vector depends on its location in the body and the orientation of the plane on which it is acting. Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, i. Cauchy's fundamental lemma is equivalent to Newton's third law of motion of action and reaction, and is expressed as. The state of stress at a point in the body is then defined by all the stress vectors T n associated with all planes infinite in number that pass through that point.

This equation implies that the stress vector T n at any point P in a continuum associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.

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To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area d A oriented in an arbitrary direction specified by a normal unit vector n Figure 2. The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane with unit normal n.Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.

The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atomsand so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate.

Fundamental physical laws such as the conservation of massthe conservation of momentumand the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the material under investigation is added through constitutive relations.

Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed.

These physical properties are then represented by tensorswhich are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience. Materials, such as solids, liquids and gases, are composed of molecules separated by space. On a microscopic scale, materials have cracks and discontinuities.

However, certain physical phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies.

Continuum mechanics

A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material. The validity of the continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exists.

When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the representative volume element RVE size, one employs a statistical volume element SVEwhich, in turn, leads to random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements SFE.

The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous. Specifically for fluidsthe Knudsen number is used to assess to what extent the approximation of continuity can be made.

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Consider car traffic on a highway, with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a partial differential equation PDE for the density of cars.

cauchy continuum

The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. Cars do not appear and disappear. But via the Leibniz integral rule. Consequently, conservation derives the first order nonlinear conservation PDE.The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. Note that only we use column vectors in cartesian coordinate system above for clarity, but the equation is written using physical components which are neither covariants "column" nor contravariants "row".

After an appropriate change of variables, it can also be written in conservation form :. Let us start with the generalized momentum conservation principle which can be written as follows: "The change in system momentum is proportional to the resulting force acting on this system".

It is expressed by the formula: [6]. Surface forces act on walls of the cubic fluid element. For each wall, the X component of these forces was marked in the figure with a cubic element in the form of a product of stress and surface area e. It requires some explanation why stress applied to the walls covering the coordinate axes takes a minus sign e.

The second element requiring explanation is the approximation of the values of stress acting on the walls opposite the walls covering the axes. This approximation results from the application of the Taylor's formula for function approximate i. In this way we obtained the sought after approximation of tension on the opposite wall.

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There are mass forces acting on the inside of the control volume. Applying Newton's second law i th component to a control volume in the continuum being modeled gives:. Then, based on the Reynolds transport theorem and using material derivative notation, one can write. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows.

The main step not done above in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes F i. Here j and s have same number of dimensions N as the flow speed and the body acceleration, while Fbeing a tensorhas N 2. In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations.

A significant feature of the Navier—Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.

Regardless of what kind of continuum is being dealt with, convective acceleration is a nonlinear effect. Convective acceleration is present in most flows exceptions include one-dimensional incompressible flowbut its dynamic effect is disregarded in creeping flow also called Stokes flow.

Both interpretations give the same result. This representation can be contrasted to the one in terms of the tensor derivative. The vector calculus identity of the cross product of a curl holds:. Lamb in his famous classical book Hydrodynamics[8] still in print, used this identity to change the convective term of the flow velocity in rotational form, i. The Cauchy momentum equation becomes:.While you can initiate many of these measures yourself, an addiction is hard to beat on your own, especially when temptation is always within easy reach.

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Cauchy stress tensor

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cauchy continuum

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Cauchy momentum equation

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My company (Tortuga Backpacks) compiles quotes from 3rd party reviews on a Reviews page with links back to the original article. We also use these quotes on our product page as social proof. Definitely a great idea Fred. A client (plumbing company) already has in place a next day follow-up phone call to make sure that everything has been done well and the client is happy.

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cauchy continuum

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Richard's prediction: Ah, Arsenal. They win, they lose, they win, they lose. They are so inconsistent and I am trying to work out what they are due - a loss, I think - sorry Gunners fans.

This early spell is when Sam Allardyce usually has a big effect on results, but I think Liverpool might alter that statistic on this occasion. Media playback is not supported on this deviceBecause of their Europa League tie on Thursday, Allardyce will not have had much time to train with his squad, Everton ship a lot of goals and boy do Liverpool like scoring them at the moment.

The Reds record in front of goal is just staggering, as is the amount of chances they are creating at the moment too,It will not be one-way traffic - it is a derby, and I know what kind of games they are - but even if it is a lot tighter, I cannot see Liverpool making lots of chances without taking a couple of them. Richard's prediction: I think this will be a cracking game - they always seem to be - and I think there will be a few goals.

I am backing Liverpool to win it, because they are at home. I just think United have hit a run of form, and their win at Arsenal last week was very comfortable and impressive.

If Kompany is missing, that will give United more opportunities from set-pieces and, although City will create plenty of chances themselves, David de Gea is obviously in such fine form in goal for Jose Mourinho's side. The key will be what Mourinho tries tactically.

You know he will have a plan, and he will have had a team in mind for the past three or four matches too.

So Pogba's suspension is a blow for him as far as that is concerned, but I just have a feeling United will go and beat City, in not a particularly good game. Richard's prediction: City look absolutely unstoppable - what a pleasure it has been watching them this season.

If anyone can stop them, it's Jose Mourinho, but I don't think that he can. He beat comedian Nish Kumar who got six correct results, but with no perfect scores, giving him 60 points.

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