Hi Mohammed,

Rule of mixtures as mentioned by Aditya is a good keyword but needs more detailed explanation (see below). Bonding effects are important, while small length scale should not pose a problem. Are the nanotubes aligned or randomly oriented? In the case of alignment, the elastic properties of the composite are directionally dependent (its a tensor) and a single elastic constant might not be sufficient to describe the behaviour well.

Now back on the "rule of mixtures". It normally refers to a law according to which:

E(composite)=f(matrix)E(matrix)+f(reinforcement)E(reinforcement).

with E being Youngs modulus and f being volume fraction: f(m)+f(r)=1. However, this kind of rule should NOT be used. In fact, the modulus of the composite may assume values in a range depending on the topology of the system. The best one can do in general is to give lower and upper bounds of the modulus. See for instance Hashin-Shtrikman bounds. Should the lower and upper bounds turn out to be sufficiently close to one another, you might use the average (Hill's average) of the bounds. In the other case, i.e. the bounds are not sufficiently close, you might make reasonable assumptions (based on microstructural knowledge) to calculate so-called effective modulus or apply a self-consistent scheme. See for instance textbook [Beltzer, Acoustics of Solids, Springer,1988].

Best, Frank

Hi,

If you have a two or three dimensional model/image of the same; might I suggest the use of a free FEM software (http://www.oofem.org/en/oofem.html). Another approach would be to calculate it manually, using the rule of mixtures. 

Many articles have already used it for similar (not same) problems. My only concern being at such small scales, the effectiveness of this method might be limited. Also, it does not take into account any bonding effects between matrix and filler (assumes homogeneous particle adhesion).

I hope it helps. 

http://onlinelibrary.wiley.com/doi/10.1111/str.12088/full

Hi Mohammed,

Rule of mixtures as mentioned by Aditya is a good keyword but needs more detailed explanation (see below). Bonding effects are important, while small length scale should not pose a problem. Are the nanotubes aligned or randomly oriented? In the case of alignment, the elastic properties of the composite are directionally dependent (its a tensor) and a single elastic constant might not be sufficient to describe the behaviour well.

Now back on the "rule of mixtures". It normally refers to a law according to which:

E(composite)=f(matrix)E(matrix)+f(reinforcement)E(reinforcement).

with E being Youngs modulus and f being volume fraction: f(m)+f(r)=1. However, this kind of rule should NOT be used. In fact, the modulus of the composite may assume values in a range depending on the topology of the system. The best one can do in general is to give lower and upper bounds of the modulus. See for instance Hashin-Shtrikman bounds. Should the lower and upper bounds turn out to be sufficiently close to one another, you might use the average (Hill's average) of the bounds. In the other case, i.e. the bounds are not sufficiently close, you might make reasonable assumptions (based on microstructural knowledge) to calculate so-called effective modulus or apply a self-consistent scheme. See for instance textbook [Beltzer, Acoustics of Solids, Springer,1988].

Best, Frank

Perhaps make a sample and get it measured using dynamic mechanical thermal analysis (DMTA aka DMA)?

Your University may have this capability; you may be able to get some help from another University (e.g. Imperial College Eng Dept have some excellent alloy / composite capabilities); there are a number of contract labs that perform this as a service; instrument vendors may also do some free testing for you if you are likely to need this measurement regularly and will potentially buy one.

Please also be aware that elastic modulus is temperature & frequency dependent (in fact, there is often some superposition of the two). If you are quoting a single value (for design or other purposes) then I'd suggest a standard method - e.g. ATSM, BSi, DIN, other. And ensure you are collecting data in the linear viscoelastic range (LVR) where response is independent of strain - you can find this region via a quick strain sweep.

If you are new to the topic then you might want to get some extra help from a materials scientist.

Hi Mohammad,

In addition to Mr. Robson's suggestions, I just wanted to addthat if you need actual measurements you can also try RFDA system (resonance frequency damping). If you do not have access to this system we can do measurements for you. NO money or any other expectaions are involved. All you have to do is to mail your samples to us, and then we will return them back to you.

It depends on the degree of accuracy that you want to reach. Generally you should know the filler orientation, cylinder shaped filler aspect ratio, the interface properties, volume fraction, mechanical properties of the composite material components. In a FEM environment a more careful study on the homogenized mechanical properties of the composite should be done using periodic boundary conditions applied on accurate representative cubic volumes from statistical point of view.

The MODULUS OF ELASTICITY is a characteristic property of a supposedly bulk- homogeneous solid.   However, one can determine such property of various non-homogeneous materials,i.e., metallic alloys, composite materials like in your case, by simply applying a tensile test , using the universal tensile machine, to determine the stress & the corresponding strain of the tested solid piece.   Then by using Hook's law you can determine the MODULUS OF ELASTICITY , the direct proportionality constant of  the stress & the strain.   

ASTM E8, EN ISO 6892-1:2009, other according to Tensile Testing of alloys.

Hi, I meant theoretical young's modulus not  experimental ...thanks

Mohammed naeem Mohammed: sorry for missunderstandig.

As far as I understand, the question is about "calculation" of elastic properties of a "nano" composite , not the measurement of it.

You can do it simply by linear, geometric and harmonic mixture laws. I am not sure, how well this approach is applicable to a nanoparticle reinforced composites, it is widely used in other composites with microscale reinforcement.

If you want a more accurate solution considering the particle size, size distribution, shape of the reinforcement, geometrical alignment distribution; you have to use "Mean Field Homogenization" techniques.

Representative Volume Element finite element simulations may provide local effects which is not accesible from mean field approaches. However, their applicability to nanoscale reinforcement is questionable.

I think the best approach is to measure it before calculating it. For this, I can also suggest RFDA method. You can contact me if you need a few measurements to be performed freely. We are also able to do perform high temperature RFDA.

Hi Mohammed

The MODULUS OF ELASTICITY FOR NANOCOMPOSITE MATERIALS (Aluminium as a matrix and carbon nano tube as a reinforced ); Ec=VAl EAL+VC EC , where

Ec is the theoretical Modulus of elasticity of the composite.

VAl is the volume fraction of Al in the composite

EAL is the experimental Modulus of elasticity of Al.

VC is the volume fraction of C in the composite.

EC is the experimental Modulus of elasticity of C.

Dear All,

Elastic modulus should be estimated experimentally as the first step in critical assessment of theoretical/simulation approach. To obtain the absolute value of elastic moduli one should also consider available geometrical size of a nanocomposite specimen. Thus, one may select either resonant (exponentially damped sinusoidal signals) method and/or subresonant (phase lag between the stress and strain signals) and/or impulse technique.

The selection of the experimental techniue is determined by the shape of nanocomposite samples. The anisotropy effect comes later, as a natural constraint and indication how to design the optimal shape of a sample to estimate Young's modulus, E, and/or shear modulus, G. The estimation of elastic modulus from intrinsic values of two components, i.e. Al and CNT, may yield unreliable results.