MSE 383, Unit 3-3. Joshua U. Otaigbe Iowa State University Materials Science & Engineering Dept.

Transcription

1 Dynamic Mechanical Behavior MSE 383, Unit 3-3 Joshua U. Otaigbe Iowa State University Materials Science & Engineering Dept. Scope Why DMA & TTS? DMA Dynamic Mechanical Behavior (DMA) Superposition Principles (TTS) DMA is extremely important in design of polymers >> e.g., in a tire, high temperatures contribute to rapid degradation and wear TTS enables prediction of long-term data from short-term data Recall steady stress or steady strain states DMA quantifies response of polymers to dynamic loads at moderate ω amp stress strain 3π/2 time π/2 π ω t Elastic Solids Strain is in-phase (or in-step) with applied stress e.g., f=f o sin ωt >> where ω = angular frequency (2π x frequency in Hz) Then, strain will vary similarly >> γ= γ o sin ωt (perfect elastic body) 3.3.1

2 Viscoelastic Solids Strain lags behind stress (as in creep) Strain is out-of-phase >> i.e., γ = γ o sin (ωt + δ) (viscoelastic body) amp stress strain 3π/2 δ/ω time δ π/2 π ω t If γ = γ o sin (ωt), then >>f = f o sin (ωt + δ) (d = phase angle or lag) f = f o sin (ωt) cos (δ) + f o cos (ωt) sin (δ) f may be considered to consist of 2 parts >> f o cos (δ) in-phase with the strain >> f o sin (δ) 90 o out-of-phase with strain f = γ G 1 sin (ωt) + γ G 2 cos (ωt) δ where: G 1 = f o /γ o cos δ (in-phase component) G 2 = f o /γ o sin δ (out-of-phase component) >> G 1 = REAL (or in-phase) component of G* >> G 2 = IMAGINARY (energy loss ) component of G* >> G* = f/γ (maximum values) Using complex number notation, we can write: γ = γ o exp iωt (3.21) f = f o exp i(ωt + δ) (3.22) Dividing (3.22) by (3.21) yields: f/γ = G* = f o /γ o exp iδ = f o /γ o (cos δ + i sinδ) = G 1 + ig

3 Definition of G & tan δ Summary of DMA G 1 = storage modulus, a measure of the energy stored in the sample due to the applied strain (10 3 Mpa) G 2 = loss modulus, a measure of the energy dissipated as heat (10 Mpa) G 2 /G 1 = tan δ = mechanical damping or loss tangent (0.01) G 2 <<G 1 in most cases G 1 = in-phase stress / strain = f o /γ o cos δ = G* cos (δ) G 2 = out-of-phase stress / strain = f o /γ o sin δ = G* sin (δ) G* = (G G 2 2 ); tan δ = G 2 / G 1 G* = G 1 + ig 2 (where I = (-1)) See class text for similar compliance relations, J* = J 1 ij 2 DMA predictable by MAXWELL model (cf. Creep and SR) It can be shown for this model that: G Gω τ Gωτ 1 = ; G = ; tanδ = ω τ 1 + ω τ ωτ If interested, see Ward, 1983; Ferry 1980 for details. ILLUSTRATE. Figure shows correct qualitative features in the case of G 1 and G 2 but not for tan δ 3.3.3

4 How Are Dynamic Mechanical Properties Measured? 2 broad groups determined by ω range of testing >> Resonance devices free oscillation (e.g., torsion pendulum) forced oscillation >> Non-resonance devices direct measurement of forced oscillation wave and pulse propagation Read torsion pendulum in class text, pp 116ff** Practical Applications of DMA Torsion Pendulum Selection of rubber compound for tire >> Low G2 to minimize energy dissipation and resultant heat build-up Selection of materials for engine applications & shock absorbers >> large G2 to dissipate vibrational energy rather than transmit it Studying various transitions in polymers >> ILLUSTRATE Most common device for measuring dynamic mechanical properties Measurements are made in shear and often used for soft and hard solids in a low frequency range of about Hz. Schematic of Torsion Pendulum & Decay Curve 3.3.4

5 A A 1 A 2 A The sample is given an initial torsional displacement and the frequency and amplitude decay of the oscillation are observed on release. G 1 is determined from the sample geometry, moment of inertia of the oscillating mechanism and the observed period of oscillation. For e.g., with a cylindrical specimen length L and radius R, G 8 LI = π R P where: I = moment of inertia of the oscillating mechanism P = observed period of oscillation = 2π/ω The damping or logarithmic decrement,, is calculated from the amplitude decay of the oscillations as follows: A1 A2 Ai 1 Ai = ln = ln = ln = ln A A A n A For small (i.e. < 1 for polymers) 2 3 i+ 1 i+ n = π tan δ tan δ = /π (and hence G 2 can be calculated) See Nielsen (1974); Ferry (1980) for additional material on this and other devices

6 Typical tan δ vs T Curves for Polymers (Otaigbe, Polymer Eng. Sci. 31(2), 104 (1991) Additional Applications of DMA Human tissues engineering (high damping reduces stresses in motion) (cf. PU s) PU elastomers in modern footwear applications >> high damping cushions shock waves Time-Temperature Superpositioning Short times = low T Long Time = high T σ speed of testing σ temperature T > Tg ε ε Explain silly putty, rubber in liquid N 2 TTS (or WLF equation) used to quantify equivalence of time & temp Applicable to many viscoelastic response tests >> e.g., creep, SR, DMA, etc. ILLUSTRATE & explain 3.3.6

7 Time-Temperature Superpositioning WLF equation relates horizontal shifts along log time scale to temperature a T is amount of horizontal shift Shifting a constant T curve along log time axis = dividing x-axis values by a T Storage compliance of poly-n-octyl methacrylate in the transition zone between glasslike and rubberlike consistency, plotted logarithmically against frequency at 24 temperatures as indicated Composite curve obtained by plotting the data of Fig Temperature dependence of the shift factor a T with reduced variables, representing the behavior over an used in plot, chosen empirically; curve, from extended frequency scale at temperature T o=100 C. equation

8 Shift Factor, a T WLF Equation a T = t T /t To (for the same response) t T = time taken to reach a specific response (e.g., modulus) at temp T t To = time taken to reach a same response (e.g., modulus) at reference temp T o Note: a T is independent of mechanical test for a specific polymer If T o = T g, then a T for most amorphous polymers is log 10 a T ( T Tg ) = ( T T ) g (T s in Kelvins) valid for T g < T < T g +100 o C Numerical values change if T o T g Relation Between a T, Viscosity and Relaxation Time log 10 a T = log 10 (τ/τ o ) = log 10 (η/η o ) ILLUSTRATE application of WLF equation Important Notes on TTS and WLF Equations WLF equation is empirical but based on free volume theory of transitions Correction factor of T g /T (at T > T g ) x E dervies from kinetic theory of rubber elasticity >> E r T absolute (ideal rubber elasticity) WLF equation not valid for crystalline polymers and below T g >> a vertical shift proposed for this case still open to question For crystalline polymers and below T g, time-temperature relation is: lna T H 1 1 = R T T o (based on Arrhenius eqn. for rate processes) TTS is subjective and requires experience 3.3.8

9 Important Notes on TTS and WLF Equations, Cont d Worked Example 1 The damping for PMMA at a frequency of 1 cycle s -1 is located at 130 o C. At what temperature would the peak be located if measurements were made at 1000 cycles s -1? For PMMA, Tg = 105 o C. Solution Recall that frequency is reciprocal time (i.e. ω = sec -1 ) ω Tg t ( T T ) T g log10 at = log = log = ω t ( T T ) T Tg Shifting the measurements at 1 cycle s -1 to Tg (i.e. finding the frequency at which the peak would be located at Tg) yields: g ω log ω ( ) = ( ) = 569. ω ω = ω 130 = 1 cycle s -1 (given) ω 105 = (1 cycle s -1 ) (2.03 x 10-6 ) = 2.03 x 10-6 cycles s

10 Worked Example 1, Cont d Worked Example 2 End of Lecture Now shifting from Tg to T, we have log ( 105) = = T ( T 105) T = 156 o C A plastic material has a viscosity of 10 3 Pa.s at 0 C. If it obeys the WLF equation, what is its viscosity at 25 C? Assume the viscosity is Pa.s at T g. Suggest a more exact solution to the problem. Solution Recall η T. ( T Tg) log at = log = η ( T Tg) Tg Let T g = x C; η Tg = Pa.s ( ) log 10 = x ( 273 x) x = 1752 K Now shift from Tg to 25 C (298 K) yields η. ( ) log = ( ) η 25 = 1.55 x Pa.s Note: A more exact value of η T can be obtained from the following relation: a T = η Τ /η Tg x (T g ϑ Tg )/(T ϑ T ) g = Tg or reference temperature Read Ch. 4, Class Text Additional optional reading >> Nielsen (1993) >> Ward (1983) >> Ferry (1980)