Is Pressure a Scalar Quantity or a Vector?

[Ahsan Khan]

second thing i want to know after i answered for first is that, once i know the force in Newton in each side then how we calculate different stresses on one of.the surface.of definite direction of cube.thanks

 

[Studiot]

In order to pursue this form of analysis you need to realize that cubes have six faces not three.

You need to consider the forces on all six faces, plus anybody forces acting and any accelerations imposed on the source object of the cube.

This is the most general situation and leads to the equations of fluid and solid continuum mechanics.

Several simplifying assumptions can often be made, but we haven't defined whether our cube is part of the water in a stagnant pond or an element of the skin of a high velocity spacecraft .

 

[Ahsan Khan]

In order to pursue this form of analysis you need to realize that cubes have six faces not three.

You need to consider the forces on all six faces, plus anybody forces acting and any accelerations imposed on the source object of the cube.

I too already realized the six faces of the cube.Why i repeatedly talks three faces, since the opposite faces need not mentioning.whenever we apply any force on any plane surface we just see the force and and one of the area.

 

[Ahsan Khan]

further whatever be the sides there must be some way to find stresses on the surfaces of the cube so as to obtain stress tensor. i am looking for the meyhod to do so.and i am still waiting for my two queires of earlier posts.I like to mention to studiot that my system is in equilibrium under no acceleration.

 

[Studiot]

In order to answer either of your questions about stress states in this cube you need to consider a 'free body'

For the cube itself this means including the opposite faces.

For some selected point within the cube this means considering a plane cutting through the cube so that it includes the selected point. Three of the cube faces plus the cutting plane then form the free body.

Do you really need to do this in three dimensions to gain an understanding?

 

[Ahsan Khan]

Studiot in your last post i don't think i get something new,what i studied and understand about stresses is that they are different at different points in a system.to find a stress tensor at a point you need to imagine an small cube at the very point now to find tensor at the point of interest we need to find stresses(three on each surfaces) on the three surfaces.rest i mentioned in earlier posts.

 

[Ahsan Khan]

so i will like to say chose any point you like and with respect to the cube so formed answer my two questions.they will answer quench all my delima with stress tensor and thus to understand pressure.Regards

 

[mechprog]

If you don't mind intrusions at such a later stage...
Pressure has a great role to play not only in mechanics but also in thermodynamics (and probably greater) and thermodynamicists do not have much concern for vectors and tensors- you know. Take pressure as scalar or not mechanics has a better substitute- stress, but that will be inefficient in thermodynamics.
If I were to define pressure I would do so by
$$\bold{\sigma_n} = p \bold{\hat{n}}$$
just like we do so
$$\vec{v}=v\hat{\epsilon_s}$$
in case of speed and velocity.
And you know stresses are better dealt with tensors (usually cartesian), so if need insight into that there are many books that deal with them together (like Borg's Matrix tensor methods in continuum mechanics or freely available http://www.math.odu.edu/~jhh/counter2.html" )

 

[Studiot]

imagine an small cube at the very point now

How can you have a cube at a point?
A cube is a three dimensional object.

You can have your point of interest in the centre of the cube, at one of the vertices (this is conventional) or somewhere else.

I already asked you to consider this but you did not answer.

I really don't know where you are coming from on this, since your original question arose from a discussion with 12th graders, but you are studying (what?) tensors at university.

I know of four approaches to derive the formula you apparently seek.

The simplest is known as the Engineers' method and involves direct calculation with forces and moments and some geometry/trigonometry (direction cosines). It uses significant simplifications. The free body diagram is used (cube and cutting plane as previously described)

The next method is the simple continuum mechanics method. This involves simple manipulation of partial differentials and Cauchy's method, but does not need tensors. It can be extended to allow for body forces. Again this uses the cube.

The full continuum mechanics method does not work on the cube, but works directly with points and displacements. It allows body forces and accelerations to be included, the simpler formula appear as special cases by setting these equal to zero. Volume and Surface integrals are used, along with Greens theorem.

Finally there are energy methods involving Gauss' theorem and more intricate partial differential manipulation.

 

[Dale]

The full continuum mechanics method does not work on the cube, but works directly with points and displacements. It allows body forces and accelerations to be included, the simpler formula appear as special cases by setting these equal to zero. Volume and Surface integrals are used, along with Greens theorem.

I know the engineer's method, but would be very interested in this one.

 

[Ahsan Khan]

Ok now i consider the point at the centre of the cube.And I will prefer engineer's method.And I hope thise methods are somewhat lengthy but will give me what i need and I think the tradational engineer's method can be found somewhere in the net.Thank you all guys.I hope i can understand how to make stress tensor matrix after studying engineers method and also hope you people will continue your support in case i ask for further help!

 

[Studiot]

It will take quite a bit of writing out. I will try to post something over the weekend.

 

[Dale]

No worries if you cannot.