Exploring Fluid Mechanics: Understanding a Second Order Tensor Quantity

[Cyrus]

Hello,

Please help me out here as I self study fluid mechanics. I ran into what they are calling a second order tensor quantity, which seems to be fancy words for a 3x3 matrix of sigmas and rhos, for shear and normal stress. They have a picture of a cube, with all the positive stresses indicated on it. Now they have a convention like sigma_xx or sigma_xy. I get that. I don't get the picture of the cube, though. Is it merely an illustration? See they start with the premise that you are looking locally at point C in space. And you make a plane of area deltaA, perpendicular to the x axis, then you find the shear and normal stress to the unit normal. Then they repeat for the other two orthogonal planes. Thats fine. Then the say there are an infinite number of planes that can pass through point C and have different shear values, FINE! Then they say that any stress can be found for any plane provided that you now know these three mutually perpendicular stress planes...ok i kinda see what they mean, but I am not tooo sure...I could use some explaining on that point. Also, this damn box!? Is point C somewhere inside this box of dimensions dx,dy,dz?? Why is this thing now a box, I thought we just cared about flat planar areas that passed through point c? Is point C now surrounded by this box? Or is the box there just to explain to my stupid self the purpose of the x,y subsripts, because I have a feeling its going to be used as a local control surface for some reason.

Thanks for your help,

(This stuffs still great fun so far! I am just annoyed at the unclarity)

 

[Astronuc]

Well, stress (like pressure) is force/unit area, and one has three dimensions.

With three dimensions mutually orthogonal, each direction (dimension) has a surface associated with it formed by the other two dimensions. Thus the y,z-plane is perpendicular to x-direction, the x,z-plane to the y-direction, and the x,y-plane to the z-direction (in Cartesian cooridates). So with three dimensions there is an incremental volume (cube) with 6 faces (surface areas, planes) associated with the volume.

There are normal stress $\sigma_{xx}, \sigma_{yy}, and \sigma_{zz}$ which act normal to a plane (surface) in tension or compression, and there are shear stress, e.g. $\sigma_{xy} = \tau_{xy}$ which acts parallel to a plane (surface). The $\sigma_{xx}$ stress acts in the x-direction on the y,z plane (surface).

 

[Cyrus]

I realize that astronuc, but my question was why the cube? I already know what the stresses and planes represnt, but I am asking WHY are there 6 of them constructed the way they are?

 

[PerennialII]

To satisfy the balance and equilibrium equations continuum mechanics is build upon in a 3D spatial description. That accounts for the 6 planes, the cube is a differential geometrical element.

 

[Cyrus]

Ok, and it is dx,dy,dz in dimension, but is it always true that dx,dy,dz are equal in magnitude? I don't see why they have to be, I have yet to run into a definition stating that they are.

 

[Astronuc]

I realize that astronuc, but my question was why the cube? I already know what the stresses and planes represnt, but I am asking WHY are there 6 of them constructed the way they are?

As PerennialII pointed out, a cube is an element in differential geometry, and it is the simplest volume element in Cartesian coordinates. What other volume element geometry would one propose?

If one used cylindrical or spherical coordinates (geometry) then one could use an annular or spherical volume segment.

The cube has 6 faces in 3D. In a static system, equal an opposing forces are applied on opposite faces of a volume element.

 

[PerennialII]

Following a typical differential geometric treatment their size is really not that much of an issue, the differential element under question being the smallest element in the continuum treatment's "radar", "resolution" etc. But if you think about the sizing of the different axes, nothing is gained by making them different sizes since it does not increase the "information content" of the solution of the problem (or its formulation)(ok, I'd say you could do it, but can't really see for what reason unless you'd be doing something really exotic, anisotropic, mixing different equilibrium concepts, material models etc., and even in such cases the basic formulation can be retained & formulated in a "simple" differential cube).

 

[farimani]

Vector base continuum non-cauchian theorem

Symmetry of stress tensor ( ) is the start point of this survey. As we know in non-magnetic fields this symmetry is always valid and everybody has been accepting that for 200 years after Cauchy.
While in quantum or discrete mechanics (statistical mechanics) it’s not evident. Also there are some continua situations, in them, that symmetry isn’t valid. So what is the reason of this difference? What is the problem?

Why a differential tetrahedron element is used? Nobody cares. But it’s the point because this element just differentiate the position not the orientation!
And this result is not valid for any arbitrary volume, we will show the Cauchy principle is just a approximate solution of sigma. However we don’t know the exact one but it doesn’t matter.

In next we will try to find the most general shape for the relation. It’s just a claim to say vector form of equations is a valid shape for Cauchy–Riemann rules and we can’t prove it yet.
We will generate the conservative rule equations without assumption of linear mapping. And we show the computational cost advantages of vector base continuum in compare whit tensor.
Some constitutive equations will be written in vector base form and we regenerate some well-known deformation such as navier-stockse