Here is an easy one:
Assuming a constant frequency, non resonance test and recalling that:
f = frequency
t = time
delta = phase lag
stress = Sigma = Sigma0 sin(2 pi f t + delta)
strain = epsilon = epsilon0 sin(2 pi f t + delta)
storage modulus = E' = Sigma0/epsilon0 cos(delta)
loss modulus = E'' = Sigma0/epsilon0 sin(delta)
and further assuming that delta is a non periodic function of temperature T.
My understanding is that since there is a pi/2 phase lag between cos and sin and since the 'instantaneous' value of delta is the same for E' and E'', the inflection point of E' has to be located at the peak of E''. However, I've not been able to find this explicitly stated anywhere in the literature.
Is this reasoning right or am I missing something?
If this is right, the corollary to this question would be that : the glass transition temperature obtained at the peak of E'' would be the same as that obtained at the inflection point of E'.
Thanks,
Laurent
Hi Laurent,
there may be some small differences in the experimental raw data because they are not "clean" of noise. In theory this is supposed to be the case. Are you working on experimental data? There is a really good theoretical book by Wineman and Rajagopal on viscoelastic materials.
Hi Valeria,
Right now, I mainly had theoretical considerations in mind. I am working on the modelling of mechanical properties of polymers and their composites across relaxations phases and I was considering the many definitions and experimental approaches related to Tg, their mutual relationships and their relationship to my model.
In terms of experimental data, I have made some exploratory DMTA runs on epoxies, but I am mainly working with published results.
I'm going to give a try on finding the book you recommended.
Thanks you very much and have a nice day!
Laurent
You're welcome. I have seen both definitions in the literature, actually, but I know that the tests have noise, so the two numbers may not coincide. Good luck in finding the book, it is really very good, but definitely very theoretical. Real data is more complex. Have a nice day yourself!
Hi Laurent,
I would recommand reading the following papers to have a comprehensive look about the the viscoelastic phenomena around the glass transition temperature.
Regards
P.S. Please, remember that glass transition is a kinetics driven phenomenon: it is identified under specific temperature/pressure histories.
- GRASSIA L., D'AMORE A. (2006). Costitutive law describing the phenomenology of subyield mechanically stimulated glasses. PHYSICAL REVIEW E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS, vol. 74, p. 21504, ISSN: 1539-3755
- GRASSIA L., D'AMORE A. (2009). On the interplay between viscoelasticity and structural relaxation in glassy amorphous polymers. JOURNAL OF POLYMER SCIENCE. PART B, POLYMER PHYSICS, vol. 47, p. 724-739, ISSN: 0887-6266
- GRASSIA L., D'AMORE A. (2009). The Relative Placement of Linear Viscoelastic Functions in Amorphous Glassy Polymers.. JOURNAL OF RHEOLOGY, vol. 53, p. 339-356, ISSN: 0148-6055, doi: 10.1122/1.3056631
- GRASSIA L., D'AMORE A., SIMON S.L. (2010). On the Viscoelastic Poisson's ratio of Amorphous Polymers. JOURNAL OF RHEOLOGY, vol. 54, p. 1009-1022, ISSN: 0148-6055
Thank you Pr. D'Amore,
Those article are now part of my current reading list. It is still a bit too early to share much of the modelling approach I'm envisioning but the kinetic nature of the glass transition (and other relaxations) is a conceptual foundation of the approach.
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