# micromechanics of composite materials

This blog focuses on the micromechanics modeling of composite materials.

### why micromechanics cannot give the exact value?

Moduli (Young's modulus, shear modulus, bulk modulus, Poisson's ratio) of a linear elastic particulate composite material (with randomly distributed particles, and perfect bonding between the particles and the matrix) should have a fixed value once the properties of particles and the matrix, the particle volume fraction, are given.

My question is: why there is not a micromechanics model that can give the exact value of the moduli of composites?

Currently all the micromechanics models, like dilute solution, Mori-Tanaka method, self consistent method, generalized self consistent method, differential method, etc., can give only estimations.

### Re: why micromechanics cannot give the exact value?

Henry,

There are certain classes of random composites that have exact solutions in three-dimensions. You can find examples in Prof. Graeme Milton's book The Theory of Composites. You can find further exact relations (mostly for 2-D cases) at Graeme's webpage.

I talked to Graeme about this issue today and his feeling was that the exact nature of a "random" composite is not known. As a result there is a plethora of results for various types of random composites starting with the work of Kramer(?) on perfectly random composites where there is no point-to-point correlation at all.

However, such a perfectly random composite is hard to imagine in any physical situation. In fact the PDEs describing the defomation (or electromagnetism, etc.) of any material need a certain amount of smoothness to be well-posed. In other words, any "random" composite necessarily needs to contain statistical information about the correlation between various points.

The volume fraction just provides a first order correlation function. One has to consider correlation functions of all orders to get an exact solution. This requirement necessarily forces us to opt for approximations.

Biswajit

### Thanks a lot. This problem

Thanks a lot.

This problem has puzzled me for quite a long time, now I am relieved.

### size effects in energetic composite materials

Solid propellants and high explosives can be considered as composite materials with energetic particles in polymeric binder matrix.

These energetic materials display strong particle size effects. For example:
(1) Large particles debond earlier than small ones in high explosives.
(2) A mix of large and small particles gives much higher explosiveness than small particles only, at a fixed volume fraction of energetic particles.

Particle size strongly influences the behavior of high explosives, but the classical composite theories cannot predict the size effect since the theories involve no intrinsic material lengths.

### Asymmetric behaviors

The properties of a composite material may be different for the material subject to tension and compression.

### Computational Mesomechanics of Composites

For references:

New Book: Computational Mesomechanics of Composites
http://imechanica.org/node/1361

Leon Mishnaevsky Jr.
http://www.geocities.com/CapeCanaveral/8807/

### Re: why micromechanics cannot give the exact value?

Rutgers in 1962 has a review paper on the equations used to relate viscosities with volume concentration of disperse systems.  It is about 40 pages long, so even the volume fraction is not simple to account for.  But testing such materials is also rather difficult.

Another area concerns compacted/tabletted systems, where predictions of properties of two component tablets (+ porosity = three components) are still not straightforward.

### Time scale, size scale and imperfections in "real" materials

Hi Henry,

I agree well with the point micromecanics cannot give the exact value. I did a few works on molecular dynamics simulation, so I want to say some thoughts on this topic within my best knowledge. The main differences lied in micro modeling and macro study may be in three aspects: the time scale, size scale and the imperfections modeling in real materials. As we know most MD simulations are conducted under ps scale and high strain rate (10^6~10^12), it is formidale to simulate the macro experiments due to limited computation resource,especially for quasistatic problems. Of cource there is nothing wrong with MD method, but it is still hard to build general correlations between micro modeling and experiments under current conditions. The second one is the size scale, we know at the atomistic scale, the contiuum assumption is not suitable, and most materials behaves big differently at micro and sub-micro scales.  So giving a link between multi scales is sitll an open question, not even the exact value for macro problems. Onthe defect initially induced. As far as I know, since the real materials are always full of various defects, the micro modeling method cannot yet simulation these  imperfections.

So I think the micro modeling methods should not put more emphasis on the exact values. Perhaps the mechanicians would benifit most from these methods for some problems that contiuum theory hard to solve, such as the deformation underlying mechanism based on discrete things, like dislocation .etc, and predictions of mechanical behaviors at macro level.

Kejie

### Accuracy when volume fraction approaches 1

How accurate is the micromechanics estimation when the particle volume fraction approaches 1?

### Effective stiffness of composite

Dear mechanician,

Thank you for useful discussion. I would have one question that: Is it possible to calculate the Exact Value of Effective Stiffness of Fully Periodic medium by One Unit Cell ?

Thank you very much

Regards

### VAMUCH

Yes. You can calculate the exact value of effective stiffness of fully periodic medum by one unit cell. If you use VAMUCH , a general-purpose micromechanics code, you will always get the same effective properties no matter whether you use one unit cell or multiple unit cells.

### micro mechanics of composite

Dear Mechanician,

my area of research  prediction  the debonding in composites

how can  i found that which unit cell to be choosen for analysis of debonding and how to get traction on unit cell 