Equilibrium creep compliance

Property Equilibrium Creep Compliance J° = A.O//10 Longest Relaxation Time ko... 

As discussed in Chapters 10 and 11, rheology can be very sensitive to the microstructure of liquids. For example, the viscosity of entangled polymer melts depends on molecular weight to the 3.4 power, T]o Ml . Equilibrium creep compliance is very sensitive to molecular weight distribution. The yield stress and low frequency G are good indicators of the flocculation state of colloids. Extensional viscosity can be an important indicator of bread dough quality (Padmanabhan, 1993). 

Figure 4.16 Creep compliance (strain per unit imposed tensile stress) versus time for glassy polyvinylchloride after aging for various times after a quench from equilibrium at 90°C to a glassy state at 20°C. The master curve with many symbols is the superposition of all the curves and is obtained by a horizontal shift. The pluses were obtained by reheating to 90 C after 1000 days of aging, and then quenching again to 20°C, followed by one day of aging. This result shows that the aging process is thermoreversible. (From Struik 1976, with permission from the New York Academy of Sciences.)...

In the same way, the difficulty involved in obtaining the equilibrium shear compliance from the extrapolation of creep experiments to infinite time leads to the determination of by means of an expression analogous to Eq. (8.71) (42),... 

The / = 0 intercept of the long-time creep compliance is a measure of the stored elastic energy in flow, and is called the steady state compliance J q-The time-dependent strain of a viscoelastic solid in creep is sketched as the bottom curve in Fig. 7.24. The long-time creep compliance of any solid is simply a time-independent compliance /eq that is the reciprocal of its equilibrium modulus Geq. 

The equilibrium shear compliances Je indicated at long times are all about 1.0 X lO" " Pa except for that of the PB-1 sample, which is unexpectedly somewhat higher. The fact that the PB-1 softening dispersion is found at shorter times than that of the PB-2 elastomer has to be attributed to its higher cis to trans ratio of placements, 0.80 as opposed to 0.67, which reflects a lower Tg. The fact that the rate of creep of the Viton elastomer is approximately five orders of magnitude slower in the softening dispersion than that of the PB-2 is believed to be a reflection of a T, which is 70°C higher. What is most unexpected is that... 

The Epons 828,1001,1002,1004, and 1007 fully cured with stoichiometeric amounts of DDS are examples of well-characterized networks. Therefore, mechanical measurements on them offer insight into the viscoelastic properties of rubber networks. The shear creep compliance J t) of these Epons were measured above their glass temperatures [11, 12, 14]. From the statistical theory of rubber elasticity [1-5, 29-33] the equilibrium modulus Ge is proportional to the product Tp, where p is the density at temperature T, and hence the equilibrium compliance is proportional to (Tpy Thus J t) is expected to be proportional to and J(t)Tp is the quantity which should be compared at different temperatures. Actually the reduced creep compliance... 

With constant stress, G t) = Gy, where creep strain y t) is constant [y(t) = Gq/G] for a Hookean solid. It would be directly proportional to time for a Newtonian liquid [(y(0 = Go/r])t]. Here t is the initial time at which recovery of the viscoelastic material begins. For a viscoelastic fluid, when stress is applied, there is a period of creep that is followed by a period of recovery. For such liquids, strains return back toward zero and finally reach an equilibrium at some smaller total strain. For the viscoelastic liquid in the creep phase, the strain starts at some small value, but builds up rapidly at a decreasing rate until a steady state is reached. After that the strain simply increases linearly with time. During this linear range, the ratio of shear strain to shear stress is a function of time alone. This is shear creep compliance, J t) The equation of shear creep compliance can be written as follows ... 

Special specimen preparation as with tensile testing. However, the extraction of intrinsic mechanical parameters from creep indentation data is analytically complex [3, 4]. Confined compression or unconfined compression tests require preparation of cylindrical cored specimens of tissue and underlying bone. With unconfined compression, the free draining tissue edges and low aspect ratio, layered nature of the test specimen may introduce error. Compression of a laterally confined specimen by a porous plunger produces uniaxial deformation and fluid flow. Confined compression creep data has been analyzed to yield an aggregate equilibrium compressive modulus and permeability coefficient [5] and uniaxial creep compliance [6]. 

Poly(vlnyl acetate) PVAc 349 8.86 101.6 305 [67,68] aj s of the entire softening dispersion from dynamic mechanical J f) with 10equilibrium with ambient moisture and the water absorbed lower the sample s Tg. Its T-dependence Is considerably weaker than either of the shift factors obtained from creep compliance In dried samples given Immediately below. 

The slight time difference between the two shift functions arises from the time position of the reference equilibrium creep curve chosen for the determination of b (A,A ). This position is about 1.5 lO s for a creep compliance value of J(t)= 10 Pa" (Figure 4). Due to the long preconditioning and the long creep time, the exact position of this curve on the time scale may be somewhat uncertain. If the time position of 1.5 lO s is reduced to lO s, the two curves in Figure 9 will coincide. 

The Voigt model will exhibit a creep compliance which will start from zero and asymptotically approach a constant deformation known as the equilibrium strain. The Maxwell model, on the other hand, will eventually creep at a constant rate. 

The linear viscoelastic properties G(t)md J t) are closely related. Both the stress-relaxation modulus and the creep compliance are manifestations of the same dynamic processes at the molecular level in the liquid at equilibrium, and they are closely related. It is not the simple reciprocal relationship G t) = 1/J t) that applies to Newtonian liquids and Hookean solids. They are related through an integral equation obtained by means of the Boltzmann superposition principle [1], a link between such linear response functions. An example of such a relationship is given below. 

As with the simple models from Chapter 3, each different mechanical model can be described by a differential equation. The differential equation governing the response for any mechanical model may be obtained by considering the constitutive equations for each element as well as the overall equilibrium and kinematic constraints of the network. Once the differential equation is obtained, the response of the model to any desired loading can be examined by solving the differential equation for that particular loading. The solution for simple creep or relaxation loading will provide the creep compliance or the relaxation modulus for the given model. In this... 

Figure 18 shows a direct comparison between the measured hardness and other gel characteristics. As mentioned above, the hardness correlates well with the measured equilibrium modulus and the inverse of the creep compliance. The correlation with adhesion properties is more complex. There is a striking similarity between the dependence... 

In self-bodied emulsions the structures take a finite time to reach a state of equilibrium and thus there is this additional ageing factor related to the continuous phase, creep compliances falling and viscosities rising [196]. An idealized diagram of the relationship between rj and J and the mixed emulsifier concentration in oil-in-water emulsions is shown in Fig. 8.40. 

Figure 4.6 Sketch showing general features of creep compliance curves for a viscoelastic melt and a crosslinked elastomer.The strain in the melt approaches a straight line (Eg. 4.22) with a slope of 1/ 7o and an intercept of J°, while that in the elastomer approaches an equilibrium...

Figure 5.9 shows the creep and recoverable compliances of a metallocene, linear, low-density polyethylene at 150 °C [36]. This polymer has a polydispersity index (Af, /M ) of about two. The data shown start in the transition from the plateau to the terminal zones, and the last few points are in the terminal zone, which corresponds here to steady flow. Note that the experiment had to be continued for about 2.5 hours to reach steady state, where J t) = /° +tlr)g and /j (f) = /°. (/r is the recoverable compliance defined in Eq. 4.26.) It is of interest to compare the creep compliance of an entangled polymer melt with that of a cross-linked elastomer. The latter cannot flow, so at long times /(t) approaches a constant value, the equilibrium compliance, /°. 

An analogous procedure based on (18.25) can be used to extract the and Ji = 1/Gj in a generalized Voigt-Kelvin model from creep data, (t). It requires a knowledge of the equilibrium elastic compliance, (oo). Its development will be left as an end-of-chapter exerdse. 

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