I want to theoretically calculate the CTE of particle loaded glass reinforced vinyl ester composite? Which relation will give better results?
Let us suppose that the composite components (denoted by subscript i) are sepately heated from a reference temperature Tº to T, exceeding Tº by ΔT = T-Tº. Let us also accept that the volumetric thermal expansion coefficient, defined as αi = (1/ρi)·[dρi / d(ΔT)], can be taken as approximately constant (αi ≈ αiº). At the temperature T: ρi = ρiº·exp(-αiº·ΔT), where ρiº is the component density for the reference temperature.
The following correlation reasonably estimates the density (ρ) of a non porous composite for the temperature T: 1/ρ ≈ ∑fi·wi / ρi, where fi are empirical dimensionless fitting coefficients taken as temperature independent, wi are mass fractions, and ρiº are densities for each component (∑wi = ∑fi = 1).
We may now write: 1/ρ ≈ ∑fi·wi / [ρiº·exp(-αiº·ΔT)]. For the composite we may possibly analogously define its thermal expansion coefficient (α ≈ αº), so that: ρ = ρº·exp(-αº·ΔT), We now can write: 1/[ρº·exp(-αº·ΔT)] ≈ ∑fi·wi / [ρiº·exp(-αiº·ΔT)]. Here it can be noticed that 1/ρº ≈ ∑fi·wi / ρiº, and we can eliminate ρº from our previous expression: ∑fi·wi / ρiº ≈ [exp(-αº·ΔT)]·∑fi·wi / [ρiº·exp(-αiº·ΔT)]. Taking, for simplicity, gi ≡ fi·wi / ρiº, we shall have: exp(αº·ΔT) ≈ {∑ [gi·exp(αiº·ΔT)] / ∑gi}. After differentiation (to ΔT), ΔT dependency can be conveniently eliminated for small ΔT (by taking the limit for ΔT → 0); and we obtain: αº ≈ ∑ (αiº·gi) / ∑gi. In absence of composition-dependent experimental density data for the composite, we may reasonably take fi ≈ 1, what corresponds to accept that volume additivity approximately holds. We thus obtain a mixture rule to estimate the volumetric thermal expansion coefficient of a composite, after those of its components: αº ≈ ∑ (αiº·wi / ρiº) / ∑(wi / ρiº).
Note that the expansion was considered isotropic, so that the volumetric thermal expansion coefficient is three times the linear coefficient (β). So, in terms of linear expansion coefficients, we also can write: βº ≈ ∑ (βiº·wi / ρiº) / ∑(wi / ρiº). Since these coefficients can be taken as approximately constant (βi ≈ βiº; β ≈ βº):
β ≈ ∑ (βi·wi / ρiº) / ∑(wi / ρiº)
Alternatively, in terms of the composite density, given as 1/ρº ≈ ∑wi / ρiº, one would have:
β ≈ ρº·∑ (βi·wi / ρiº)
Please find below some relevant information:
http://aquila.usm.edu/cgi/viewcontent.cgi?article=1083&context=masters_theses
http://www.dtic.mil/dtic/tr/fulltext/u2/a109131.pdf
http://www.fbe.deu.edu.tr/ALL_FILES/Tez_Arsivi/2005/YL_t1855.pdf
I hope it helps
Thanks Carlos.
Can you please give me the reference for this approach, if published?
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