NONLINEAR COMPLEX MODULUS IN BITUMENS

Transcription

1 NONLINEAR COMPLEX MODULUS IN BITUMENS J. Stastna, K. Jorshari and L. Zanzotto Bituminous Materials Chair University of Calgary 2500 University Drive NW Calgary, Alberta T2N 1N4 Canada

2 Large Amplitude Oscillations - LAOS Input: shear strain - sinusoidal simple shear (γ 0 >> 1) γ () t = γ sin( t) 0 ω Output: shear stress - periodic (FFT or DFT) τ () t = τ ( ω, γ ) sin ( nω t + δ ( ω γ ) n = 1 odd n 0 n, 0

3 Higher harmonic moduli: G n = ( τ / γ ) cosδ,g = ( τ / γ ) sinδ n 0 n n n 0 n τ ( t) = γ0 [ G n ( ω, γ0 ) sin( nωt) + Gn ( ω, γ0 ) cos( nωt) ] n= 1 odd

4 Linear viscoelastic limit, γ 0! 0 : n = 1, and G'1 G'(γ ), G''1 G''(γ ) Nonlinear viscoelastic constitutive equations Many available - a few practical - none satisfactory in both shearing and elongational flows Wagner's modification of Lodge's rubberlike- liquid : memory function can be factorized memory function = linear viscoelastic memory x damping function

5 In LAOS the damping function, h, is a double periodic function of the current time and the elapsed time (material has memory). By developing, h, into a double Fourier series One obtains nonlinear harmonic moduli.

6 G 1 ( ω, γ ) 0 = h 0,0 G lin B * 2 cosβ 2 G 1 for n 3, odd ( ) * ω, γ 0 = h 0,0G lin B2 sin β2 G n ( ω, γ ) 0 = C * n 1 cos γ n 1 B * n+ 1 cosβ n+ 1 G n * * ( ω, γ 0 ) = Cn 1 sin γ n 1 Bn 1 sin βn 1 + +

7 Here, G lin and G lin are the linear dynamic moduli, and B * n * * ( iβ ), C = C exp( iγ ) * = Bn exp n n n are complex functions generated by the double Fourier expansion of the damping function, and h 0,0 is the averaged damping function over the domain( π ω π ω) x( π ω, π ω),.

8 Kazatchkov's hypothesis ( observed in some polymer melts) : G 1 = ( ω, γ ) = G ( ω) h( γ ) o lin 0 then it should be possible to obtain G'1 ( and G''1) from linear viscoelastic moduli and the damping function ( usually exp( αγ 0 ) ( b ) or 1/ 1+ aγ 0 ) by simple shifting.

9 "Spoiling" terms G 1 ( ω, γ ) 0 = h 0,0 G lin B * 2 cosβ 2 G 1 ( ) * ω, γ 0 = h 0,0G lin B2 sin β2

10 Fourier decomposition and reconstruction of the shear stress. Frequency 1Hz, strain amplitude 4.

11 Lissajous figure Frequency 1Hz, strain amplitude 4

12 Same as in previous Fig. except the rate of strain is used, and some points are interpolated.

13 Maximum shear stress. Base asphalt, T = 27C.

14 Maximum shear stress. Base asphalt with 4% SBS, T = 44C

15 Maximum shear stress. Base asphalt with 6% SBS, T = 50C.

16 Maximum shear stress. Base asphalt with 4% EVA, T = 34C.

17 First harmonic modulus G'1. Base asphalt, T = 27C.

18 First harmonic modulus G''1. Base asphalt, T = 27C

19 Third harmonic modulus G'3. Base asphalt, T = 27C.

20 Third harmonic modulus G''3. Base asphalt, T = 27C.

21 Fifth harmonic modulus G'5. Base asphalt, T = 27C

22 Fifth harmonic modulus G''5. Base asphalt, T = 27C.

23 First harmonic modulus G'1. Base asphalt with 6% SBS, T = 50C.

24 First harmonic modulus G'1. Base asphalt with 4% EVA, T = 34C.

25 First harmonic modulus G''1. Base asphalt with 4% EVA, T = 34C.

26 Exponential damping function - absolute term in double Fourier series.

27 Fractional damping function of Soskey and Winter - absolute term in double Fourier series.

28 Exponential damping subtracted from G'1. Base asphalt, T = 27C.

29 Exponential damping subtracted from G''1. Base asphalt, T = 27C.

30 Exponential damping subtracted from G'1. Base asphalt with 6% SBS, T = 50C.

31 Exponential damping subtracted from G''1. Base asphalt with 6% SBS, T = 50C.

32 Comparison of linear viscoelastic master curve with LAOS. First harmonic modulus G'1. Base asphalt, T = 27C.

33 Comparison of linear viscoelastic master curve with LAOS. First harmonic modulus G''1. Base asphalt, T = 27C

34 Comparison of linear viscoelastic master curve with LAOS. First harmonic modulus G'1. Base asphalt with 6% SBS, T = 50C.

35 Comparison of linear viscoelastic master curve with LAOS. First harmonic modulus G''1. Base asphalt with 6% SBS, T = 50C.