# Continuum mechanics in solids

• Sivasakthi
I mean, what are the limitations? SteamKingYou can develop general equations for deformations, but there are usually some limitations associated with the type of deformation you are interested in. For example, you might be interested in elastic deformation, but if the material is in a state of equilibrium, then the general equation will be incorrect. You might also be interested in plastic deformation, but if the material is in a state of equilibrium, then the general equation will be incorrect. In general, the equations will be correct at the level of detail you are interested in, but they might not be correct at the level of detail where the physical processes are happening. ChetIn summaryf

#### Sivasakthi

Hi all..
I am currently doing some works on the continuum mechanics. And trying to study the macroscopic behavior of solids ( for simplicity, taken homogeneous materials) upon the action of external force ( which is the stress; pressure).
How is it possible to account for the changes that can be seen in macroscopic scale be related to that of the internal dimensions?
I have started with meso scale. But the errors/ uncertainties me made to go to much smaller dimensions ; involving quantum mechanics.
I am totally confused in which way I should proceed further.
Writing down hamitonian and eqn of motion of such a system, will it be possible to involve all the possible lengths ( in micro and nano scales) all together and study the errors caused by them?
Kindly help...

Your post sounds like a word salad of technical terms mixed together. I couldn't make any sense out of it.

If you are trying to figure out how homogeneous materials deform under the application of loads, try studying a text on the theory of elasticity. I know continuum mechanics is the term in vogue now, but these elasticity guys knew their stuff (all done before computers, you know), as far as deflections and deformations go.

Here's a reference to get you started:
https://archive.org/details/TheoryOfElasticity

Hi all..
I am currently doing some works on the continuum mechanics. And trying to study the macroscopic behavior of solids ( for simplicity, taken homogeneous materials) upon the action of external force ( which is the stress; pressure).
How is it possible to account for the changes that can be seen in macroscopic scale be related to that of the internal dimensions?
I have started with meso scale. But the errors/ uncertainties me made to go to much smaller dimensions ; involving quantum mechanics.
I am totally confused in which way I should proceed further.
Writing down hamitonian and eqn of motion of such a system, will it be possible to involve all the possible lengths ( in micro and nano scales) all together and study the errors caused by them?
Kindly help...
I think your question is "how do we quantitatively relate things that are happening on very small length and time scales to things that are happening macroscopically over much larger length and time scales?" This is currently a very active area of research, particularly with regard to viscoelastic materials that also crystallize when deformed. The approach that is being developed is to use a hierarchy of models, starting at the smallest scales (say with molecular dynamics), and building to larger length scales like mesoscale, and finally macroscale. Of course, in passing from one scale to the next higher scale, there is going to typically be some parameterization involved, but based on the results of modeling calculations at the smaller scale. There was a study headed up by the math department at Clemson U. to coordinate work in this area about 10 years ago. One guy who I know of who was involved in this was Antony Beris of the ChE Dept. at the Univ. of Delaware. Another guy was Tony McHugh at Univ. of Illinois. I also seem to remember a study funded by the NSF; maybe it was the same study.

Chet

Thanks Chet..
But my confusion is how can we include all those length scales all together . I do think their degree of impact upon macro scales are different. So is it possible to take a general length scale and form an equation for the dynamics? Or do we have to consider them separately?

SteamKing, it is not just to know how do these materials deform; but yeah that definitely is the thing that I need. Above all it is that I want to know how to approach that method of dynamics.
Anyways thanks a lot for that reference you made :)

Thanks Chet..
But my confusion is how can we include all those length scales all together . I do think their degree of impact upon macro scales are different. So is it possible to take a general length scale and form an equation for the dynamics? Or do we have to consider them separately?
Are you asking about how to develop rheological constitutive equations for materials? Must it start at the microscale, or can it be developed from some general principles (e.g., principle of material objectivity) together with experimental data? Are you looking for equations that apply to any and all deformations, or are there some constraints on the class of deformations for some physical system you are studying?

Chet

The macroscopic behavior depends on the type of material we take. So, I guess there should definitely be constraints for the different materials ( whether or not these constraints are of same degree needs to be checked).
Actually I am confused if we can have a general equation or approach for the deformations of any material.
For eg; if we take into account the micro scale, will the effect be same for any material? If not, then what can be other factors that we need to consider along with it.
I wish to know if the same can be for other scales too..
I don't know if my doubt is relevant. Still, I need this to be cleared for further progress .

The macroscopic behavior depends on the type of material we take. So, I guess there should definitely be constraints for the different materials ( whether or not these constraints are of same degree needs to be checked).
Actually I am confused if we can have a general equation or approach for the deformations of any material.
For eg; if we take into account the micro scale, will the effect be same for any material? If not, then what can be other factors that we need to consider along with it.
I wish to know if the same can be for other scales too..
I don't know if my doubt is relevant. Still, I need this to be cleared for further progress .
To do it macroscopically, you need to take advantage of some knowledge you have about the particular material you are dealing with. For example, consider the cases of viscous fluids and purely elastic solids. Obviously, the rheological behavior of these will be different macroscopically. The macroscopic behavior of Newtonian fluids is parameterized entirely in terms of the fluid viscosity, which is measured experimentally. The form of the rheological equation also makes use of what we know: the stress tensor in the fluid is determined by the rate of deformation tensor, and is linear in the rate of deformation tensor. On the other hand, for a linearly elastic solid, the macroscopic behavior is parameterized in terms of the Young's modulus and the Poisson ratio, both of which are measured experimentally. And the stress tensor in the solid is determined by the strain tensor, and is linear in the strain tensor. Now, if you wish to predict the viscosity of a Newtonian fluid (rather than measuring it experimentally), you need to go to smaller scale systems and consider what the molecules are doing. The same thing for Young's modulus and Poisson ratio.

Chet