Young's modulus changes with the change of temperature as reported in the previous studies. But how to explain it from thermal expansion? Is there any theoretical prediction or explanation at atomic scale?
For a good discussion of the temperature variation of eleastic constants in general is given by H.M. Ledbetter and R.P. Reed J. Phys. Chem. Ref. data vol 2, 531 (1973). But these are based on thermodynamic considerations. Now a days the right approach would be to do ab initio DFT and phonon calculations to get elastic constants from which Young's modulus can be calculated. However for higher temperature one need to consider anharmonicity which is more difficult. Quasi-harmic approximation may help but not always.
Dear Cheng
Kindly refer to the following.Hope you find something useful in the following link.
https://books.google.co.in/books?id=MlrMA1qrOHAC&pg=PA281&lpg=PA281&dq=theoretical+relation+between+young%27s+modulus+and+temperature&source=bl&ots=2AUF7E8sBE&sig=SHrjzlUGjYspMmvcoIxuDYlyfiE&hl=en&sa=X&ved=0CFoQ6AEwCWoVChMIk8LDqc7hxgIVkG2OCh3vJgkm#v=onepage&q=theoretical%20relation%20between%20young's%20modulus%20and%20temperature&f=false
Dear Cheng
Kindly refer to the following.Hope you find something useful in the following link.
https://books.google.co.in/books?id=MlrMA1qrOHAC&pg=PA281&lpg=PA281&dq=theoretical+relation+between+young%27s+modulus+and+temperature&source=bl&ots=2AUF7E8sBE&sig=SHrjzlUGjYspMmvcoIxuDYlyfiE&hl=en&sa=X&ved=0CFoQ6AEwCWoVChMIk8LDqc7hxgIVkG2OCh3vJgkm#v=onepage&q=theoretical%20relation%20between%20young's%20modulus%20and%20temperature&f=false
If you do not like long ab-initio calculations then there is another way to tackle the problem. First of all we can easily guess how the elastic constants should vary with temperature. The temperature causes dilatation of the lattice and therefore weakening of bonds. So elastic constant should decrease with temperature. At zeo temperature the curve should have a contant slope (Nerst's theorem). It should then curve down and decrease with increasing temperature. At higher temperature the curve will become almost a straight line. But very close to melting the curve will definitely deviate from linear behaviour again. We also know that the single crystal elastic constants are related with the polycrystalline (isotropic) elastic moduli, Young's modulus E, Poission rato \nu, bulk modulus B and shear modulus G. The isotropic moduli are related to each other and only two of them like B and G or E and \nu are independent. So one can only consider two of these. There are some approximate analytical expressions for their dependence. For that read the following articles:
1. Q. Liu and Q. He, Acta Physica Polonica, 112, 69 (2007)
2. J. L. Tallon, J. Phys. Chem. Solids 41, 837 (1980)
For experimental data please look at:
3. H.M. Ledbetter, and E.R. Naimon, J. Phys. Chem. Data, vol. 3, No. 4 (1974).
Dear Guangming Cheng,
The first answer of Tapan Chatterji leads in a fundamental direction. A more practical approach lies in the fact that for the case of isotropic metals the intrinsic properties are interrelated via Grüneisen's constant (see Section 2.2.4 of the reference given in the link below). The outcome of the discussion based on Grüneisen's constant is that the coefficient of thermal expansion is inversely proportional to the value of the elastic constants of the solids studied for a rather large range of temperatures.
http://www.delftacademicpress.nl/m017.php
Materials Science in Design and Engineering
http://www.delftacademicpress.nl/m017.php
Dear Dr. Tapan Chatterji and Dr. P. Van Mourik,
Thank you very much for your explanation. We have done the experiments and try to find the relationship between the elastic modulus and temperature. And we want to elucidate the mechanism from atomic scale view. We mainly focused on single crystalline materials with special loading direction. There are some references showing nonlinear relationship between them but we would like to explain it in details. Now I am back to the study of this issue and I will carefully read the references you recommended. I will keep in touch with you if I have some questions.
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