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Solid/Fluid Mechacnics

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.

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Amit Pandey's picture

Plasticity flow rule ~ Darcy's law (fluid mechanics)

 

 Flow plasticity theory

In 1934, Egon Orowan, Michael Polanyi and Geoffrey Ingram Taylor, roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations. The more correct mathematical theory of plasticity, flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation.

 Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments[1] on the flow of water through beds of sand. It also forms the scientific basis of fluid permeability used in the earth sciences, particularly in hydrogeology.

 

Ref: wiki 

Mubeen's picture

Some months ago I attended a lecture in the Conference (83rd Annual Meeting of Association of Applied Mathematics and Mechanics),

From vortices to dislocations: How fluid mechanics can inspire solid mechanics
by: Markus Scholle

The link to the abstract is:

http://www3.mathematik.tu-darmstadt.de/GAMM/abstracts/Section-6_bta406@GAMM.pdf

I would strongly recommend you to check the details in the conference proceedings.

83rd Annual Scientific Conference of the International Association of Applied Mathematics and Mechanics

http://onlinelibrary.wiley.com/journal/10.1002/%28ISSN%291617-7061/

Perhaps the proceedings for 2012 will appear in December this year. 

If necessary you can find the contact details of the author by Googling his name and the university's name (Hochschule Heilbronn)

 --

Mubeen.

Amit Acharya's picture

There are big conceptual similarities between the averaging ideas that produce the Reynolds stress for turbulent flows and averaging dislocation mechanics to produce plasticity. In the presence of high inertia, one would of course get the Reynolds stress in linear momentum for solids too - however, even for quasi-static motions, similar ideas are useful - see Sec. 6 of

www.imechanica.org/node/9289

 

As for similarities with vorticity evolution (and differences - vortices do not move w.r.t the material but dislocations do), see eqns 5 - 13 of

http://www.imechanica.org/node/12936

Amit Pandey's picture

Dr Acharya

In some of his lectures Dr Desai (http://www.u.arizona.edu/~csdesai/)use to mention the similarities between the normality rule (plastic strain proportional to potential function) and Darcy's law.  I would like your comments or reading related to it.

 AP 

 

 

 

 

Amit Acharya's picture

Amit P,

I am not sure I understand the details of what you mean. With this limited understanding of your question, here is what I can say:

Darcy's law is a constitutive assumption in seepage which says that the vectorial flow-rate through a porous medium is proportional to the spatial gradient of the head. The normality rule in plasticity is another phenomenological statement that the tensorial direction of the macroscopic plastic strain rate is in the direction of the gradient, with respect to the stress, of the plastic potential - there is no reference to a spatial gradient of any sort. Moreover, beyond this assumption there is no commitment to the magnitude of the plastic strain rate through this law, at least in rate-independent plasticity.

Thus, e.g., if the head was uniform there would be no groundwater flow. In macroscopic plasticity even for spatially homogeneous motions there can be plastic flow.

- AA

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