Solid/Fluid Mechanics

yaronben

Hi,

In my research i am trying to develop a new model for plastic deformation, and i suspect there is a strong similarity between

plasticity and turbulent flow !.

My question is: if there is workd done trying to apply Reynolds decomposition (Reynolds Stress) to the governing equations and solved plasticity problem in Solids?

thanks,

Yaron B.S.

boneh3ad

On the contrary, plastic deformation is much more akin to laminar flow than turbulent flow. Unless of course you have a strange, exotic material of some kind, but even then I can't imagine as solid having properties that would support being modeled as turbulent.

yaronben

from the "fluid dynamics" point of view you probably right but:
continuum mechanics states that "neighbours material cells/atom/molecolus" remain "neighbours" my claim is that plastic deformation take place only when there is relative motion between neighbours meaning continuum does not hold in plasticity,
which resembles to turbulent flow in fluid dynamics where small fluctoation are added:
u=u+u' etc.

Studiot

Whilst I appreciate you may wish to preserve confidentiality in your work, you need to provide more context for proper comment to be offered.

This, however, is an area of knowledge that is wide open for much fruitful research.

Here are some thoughts.

Two parameters of importance in flow mechanics are distance and time.

In gas dynamics and hydraulics distances are large and timescales are short in relation to the size of the fluid element or parcel. Fluids usually travel quite a long way in a relatively short time.

In plastic flow it is the other way round. Fluid elements do not move very far from their start point and often take a very long time to do this even to geological timescales.
I can only think of one exception which is in the field of viscous damping.



I have supplied some context of my own in the attachment, but please let us know if you are talking about contact mechanics, rheological rock mechanics, asphalt machanics, viscous damping or what.

Feel free to use the PM system.

Attached Thumbnails

 

boneh3ad

I think you have some misconceptions about what turbulence is, then. A laminar flow can (and often is) represented with the u=U+u' notation. The important distinction is the nature of u'. In a turbulent flow, u' is large, broadband and involves a great deal of vorticity and mixing. That doesn't seem to me to be possible. In a laminar flow, u' is smaller, typically represents only a few characteristic frequencies (unless you impose some kind of forcing) and does not involve the mixing.

Indeed, looking at the idea of the Reynolds number provides some further insight. The Reynolds number represents the ratio of inertial forces to viscous forces in the fluid. For turbulence, you generally need a large Reynolds number. A viscoplastic solid would have such a high effective viscosity than I can't imagine you could get turbulent plastic deformation of a solid.

One only need look at the example of a boundary layer to see that neighboring fluid particles in a laminar flow need not remain neighbors.

All that said, Reynolds stresses can (and sometimes are) used in laminar flows. For example, if you do some manipulation of the equations, you find that they are one of the primary energy generation mechanisms in a viscous laminar flow for Tollmien-Schlichting waves.

Studiot

The implications of this are that for most purposes it does not matter whether the flow is laminar or turbulent. Quite likely we will not be around long enough to see the difference anyway.

So I would not place too much emphasis on this issue.

yaronben

first i would like to thank you for the comments its encouraging.
my interest is in plastic deformation metals for example (aluminum, iron).
it seems to me that both:solid and fluid mechanics relies on the same principal, the principal of continuum mechanics, and yet the ideas and work done along the way in those fields are very diverse i wonder why? my attempt is to borrow some ideas from one field(fluid mechanics) to the other(solid mechanics). while in solid mechanics the distinction between elastic deformation and plastic deformation relies on the material model(consitutive model), in fluid mechanics the distinction between laminar and turbulent flow is the Reynolds decomposition-->u=U+u' + the closure problem. so i am trying to find whether an atempt to implement this idea was made?

Studiot

Where to start?

Continuum mechanics has a much wider application than just solids and 'fluids'

The stress paths in foundation engineering developed by Boussinesq and Coulomb follow similar trajectories to electrostatic potential mapping and jukowsi aerofoil theory and hydraulic flow nets in dams etc. All are approachable by conformal mapping techniques.

There is another branch of (flow) mechanics - that of granular materials - flour, sugar, grain cement etc and soil mechanics. Contrast this with stress analysis within rocks formations.

100 years ago continuum mechanics did not really exist. Mechanics was really divided into statics and dynamics. Continuum mechanics were developed in the next 50 years, at the same time as the mathematical apparatus for it (Vector and Tensor formulations etc)

Between 50 and 30 years ago there was a link in both Physics (eg Mechanical properties of Solids and Fluids by R C Stanley) and Engineering (eg Applied Mechanics by Low). However continuum mechanics was not emphasised.

This was because our use of solids is essentially a statics issue, for which the equations of equilibrium are available, whereas the problems of fluids is essentially a dynamics one.

There is also the issue of scale - a ship or an aircraft or even a dam is tiny compared to an ocean or lake. But solid objects - even large bridges - are within compass of the system.

Around 50 - 30 years ago several developments changed matters.

Firstly the serious development of fracture mechanics and dislocation theory offered new ways to approach solids.

Secondly the advent of computers allowed the development of finite element methods. This is a natural use or extension of continuum mechanics so the technique blossomed in this form.

But all this was via elastic methods.

Plasticity was considered more difficult. You will find small treatment in Continuum Mechanics by Spencer.

In the engineering world, however plastic methods of structural analysis were introduced into codes with so called limit state design. These acknowledge that in a real structure not all parts of the members first reach plasticity at the same time affording considerable reserves of strength. these are now routinely available, even for reinforced concrete.

As regards modern sources

Mechanique des Materiaux Solides by Lemaitre and Chaboche
Or english translation by Cambridge University Press

Has an in depth treatment of plasticity, developed from continuum mechanics.

Elasticity, Fracture and Flow by Jaeger has some good mathematical work

Elasticity, Plasticity and the Structure of Matter by Houwink and Decker has a materials approach.

Journal Sources include

The International Journal of Solids and Structures, formerly published by Pergammon.

Finally remember also that Reynolds Number division into laminar and turbulent is strictly only valid for Newtonian fluids type 2 on my sketch.

Come back if you have any further questions.

boneh3ad

This is just not true. That is NOT the difference between laminar and turbulent flow in fluid mechanics. Read my previous post. Using a model like u=U+u' is perfectly valid.

When you are referring to the Reynolds decomposition, that does not actually imply anything about the flow other than it may have a fluctuation component (as many flows do). It is also used in developing the Reynolds-averaged Navier-Stokes equations (RANS). Of course the RANS equations are non-physical approximations of the real governing equations and rely wholly on empirical data or a turbulence model to feed in a few constants to get an approximate, time-averaged solution.

The equation is equally valid in laminar and turbulent flow but is only really used in turbulent flows because of the difficulty in solving them. The standard Navier-Stokes equations can be "exactly" solved numerically in an economically feasible amount of time for most if not all laminar flow problems. RANS is just a way of coping with the fact that this isn't true for turbulent flow problems.

With all that said, you probably could find a way to apply a similar transformation to your governing equations for plastic deformation and create a Reynolds-averaged plastic deformation equation if you do desired and end up with an analogous quantity to the Reynolds stresses in a fluid. That said, it is up to you to determine if that is even desired. In fluids it is usually used for turbulemcr modeling for problems that are intractable without averaging. If you are looking for exact solutions to your problems, I would imagine looking elsewhere would be your best bet.

Studiot

Now I come to think of it the plasticflow of paste type materials through a nozzle is pretty laminar.

Cooks exploit this with cake icing and there used to be a toothpate that extruded as stripes as an advertising ploy.