I am determining the mathematical relationship to indicate the changes in Young's Modulus and Compression Strength of NiTi Shape Memory Alloys by the incorporation of porosity. I have prepared 4 samples with 40, 50, 60, 70 percent porosities and measured their Young's Modulus and Compression Strength. Can I develop a mathematical relationship between these three variables using these results?
Generally, there is. for example, the relation between Young’s modulus and Porosity can be expressed by Nielsen Eq.:
E = (E0 * (1 - P) ^ 2) / (1 + P / (f - 1))
where E is Young’s modulus, E0 is Young’s modulus for pore-less sample, P is volume fraction of porosities, and f is shape factor.
I think you can find the " f " value by fitting the best curve for your results based on mentioned equation.
How about linear regression? I think its simplest form of mathematical relationship.
Just feed the data to Excel or OpenOffice Calc, or to Scilab/Matlab and use appropriate function like REGLIN
Generally, there is. for example, the relation between Young’s modulus and Porosity can be expressed by Nielsen Eq.:
E = (E0 * (1 - P) ^ 2) / (1 + P / (f - 1))
where E is Young’s modulus, E0 is Young’s modulus for pore-less sample, P is volume fraction of porosities, and f is shape factor.
I think you can find the " f " value by fitting the best curve for your results based on mentioned equation.
Yes. Young;s modulus and comp. strength of engg. brittle materials reduce with increasing porosity. For various relationship you can refer an excellent book by Ashby & Gibson on Cellular Solids.
Thank you all for your valuable answers. Dear Mehdi Shahedi Asl, Can you provide me the reference of the equation you mentioned. I'll be thankful.
Obviously, the shape of the pores will have an effect on the effective properties. Analytical solutions for the effective elastic properties of solid with cavities of various shapes are given e.g. in the review article by Kachanov, Tsukrov and Shafiro, Applied Mechanics Reviews, Vol 47, No 1S, p. 151. The review article contains analytical solutions to basic cases and also to some rather exotic cases such as a mixture of randomly distributed hypotrocoids and cracks . The same authors have also published Handbook Elasticity Solutions in 2003. The Handbook might also contain information on this topic.
Generally, I am more familiar with Roy Rice's book Porosity of Ceramics, but the same generally holds for metals as well. The empirical formula for E (and G and H as well) of an exponential works well, using E0 as the Young's modulus at 0 porosity. It is of the form E = E0 * exp (-b*P), where P is the volume fraction porosity, and b is a dimensionless constant, typically between 2 and 6 for most materials. When b*P is small, the first two terms of Taylor expansion works, making the linear expression, E = E0 * (1 - b*P). These have held up well in my experience with semimetals and semiconductors up to nearly the green density of the powder compact, usually around P = 0.4. I don't know if this will work as well for your specimens with higher still porosity. At that point, the microstructure likely may have a much larger influence on the E.
Also, if your E is measured by compression, like your compression strength is, then the test results may be very different than E by other tests, such as tension or a dynamic test such as sonic resonance. I recall a test of Mg2Si where the published E from compression was about 30-40% less than all other published results I found, typically by dynamic tests. This isn't to say it is a bad test, just be careful in comparing.
I am surprised that brittle fracture does not take place in the shear planes. It could have a lowering of the compressive stress
Thank you very much R. D. Schmidt and Pedro Marcal, I will be in touch with you after my exams in this week.
Please peruse the attachment for various relationships between elastic moduli and porosities.
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