Time dependence mechanical relaxation

Above Tk the time-dependent mechanical properties of a polymer are determined fundamentally by the distribution of relaxation or retardation... 

Temperature dependence proton relaxivity 188 relaxation rate 144-5 Temperature-sensitive contrast agents 218-19 Th -" 368 Ti "" 347 Tilt angle 242 Time constant 14 Time correlation functions 76 Time-dependent mechanism 14 Time-dependent perturbation methods 23 Time-dependent perturbation theory 45-8 Titanium(III) 115,134-5,161 TPEN 224 TPPS 219... 

The object of any degradation test, and also tests of such time-dependent mechanical properties as creep and stress relaxation, is directly or indirectly to predict service life. If time were not important, tests could be made under the expected service conditions and the test continued for the desired lifetime. With expected lifetimes sometimes measured in tens of years, this is clearly not a viable option in most cases, and some form of extrapolation, and usually acceleration, will be needed. 

Time-Dependent Mechanical Properties 4.1 Creep and Stress Relaxation... 

In chapter 5 the use of NMR spectroscopy in studying the various kinds of motion that can take place in solid polymers is described and in the present chapter the way that these motions contribute to the time-dependent mechanical properties of the polymers is considered. In chapter 9 the effects of motion on the dielectric properties and the use of measurements of dielectric relaxation in the study of relaxation mechanisms are considered. 

Viscoelastic behavior is a time-dependent mechanical response and usually is characterized with creep compliance, stress-relaxation, or dynamic mechanical measurements. Since time is an additional variable to deformation and force, to obtain unique characterizing functions in these measurements one of the usual variables is held constant. 

In many materials, the mechanical response can show both elastic and viscous types of behavior the combination is known as viscoelasticity. In elastic solids, the strain and stress are considered to occur simultaneously, whereas viscosity leads to time-dependent strain effects. Viscoelastic effects are exhibited in many different forms and for a variety of structural reasons. For example, the thermoelastic effect was shown earlier to give rise to a delayed strain, though recovery of the strain was complete on unloading. This delayed elasticity is termed anelastic-ity and can result from various time-dependent mechanisms (internal friction). Figure 5.9 shows an example of the behavior that occurs for a material that has a combination of elastic and anelastic behavior. The material is subjected to a constant stress for a time, t. The elastic strain occurs instantaneously but, then, an additional time-dependent strain appears. On unloading, the elastic strain is recovered immediately but the anelastic strain takes some time before it disappears. Viscoelasticity is also important in creep but, in this case, the time-dependent strain becomes permanent (Fig. 5.10). In other cases, a strain can be applied to a material and a viscous flow process allows stress relaxation (Fig. 5.11). 

It should be remembered that the moduli and compliances under discussion are functions of frequency. The quantities E, D etc. should thus be written E (a>), D (a>), and so forth. The frequency dependence of these quantities is governed by the same distribution of relaxation or retardation times as is stress relaxation, creep or other time-dependent mechanical phenomena. Single relaxation or retardation times cannot depict the frequency dependence of the dynamic mechanical behavior of polymers. 

Both the mechanism and the time dependence of relaxations, lamellar thickening, annealing, and plastic deformation need to be accounted for. 

Amorphous solid dispersions are prepared primarily with amorphous and/or semicrystalline materials, and therefore, the mechanical behavior of the extrudate is generally viscoelastic in nature. The materials viscoelasticity implies a strain-rate dependence of the mechanical response and time-dependent mechanical behavior such as creep and stress relaxation. For example, in cases of high strain rates, these materials tend to be more brittle than under slower strain rates where viscous flow and other molecular relaxations can dissipate the energy without fracture. Thus, high strain rates are beneficial for particle size reduction operations. 

Therefore the logarithmic creep law employs an averaged relaxation spectrum for all elapsed time (i.e., only one time-dependent mechanism is assumed) as shown in Fig. 2.16, which may cause difficulty under real, complex situations such as the long term behavior of rock. On the other hand the power law (2.292) gives... 

The isothermal time dependence of relaxation and fluctuation due to molecular motions in liquids at equilibrium usually cannot be described by the simple linear exponential function exp(-t/r), where t is the relaxation time. This fact is well known, especially for polymers, from measurements of the time or frequency dependence of the response of the equilibrium liquid to external stimuli such as in mechanical [6], dielectric [7, 33], and light-scattering [15, 34] measurements, and nuclear-magnetic-resonance spectroscopy [14]. The correlation or relaxation function measured usually decays slower than the exponential function and this feature is often referred to as non-exponential decay or non-exponentiality. Since the same molecular motions are responsible for structural recovery, certainly we can expect that the time dependence of the structural-relaxation function under non-equilibrium conditions is also non-exponential. An experiment by Kovacs on structural relaxation involving a more complicated thermal history showed that the structural-relaxation function even far from equilibrium is non-exponential. For example (Fig. 2.7), poly(vinyl acetate) is first subjected to a down-quench from Tq = 40 °C to 10 °C, and then, holding the temperature constant, the sample... 

Upon subsequent exposure to ambient environments in several humidity and temperature-controlled chambers these curvatures varied with time due to the two aforementioned contradictory time-dependent mechanisms. While the time-dependent moisture diffusion process serves as a stress-inducing mechanism, the time-dependent relaxation acts to reduce the level of those stresses. Note that relaxation depends on moisture content through the shift factor an m). 

As a first example, we shall consider the dipole-dipole mechanism. If two spins I and S in the same molecule interact by a dipolar interaction, the perturbation Hamiltonian is time dependent because the vector joining the two spins is characterized by a random motion (Euler s angles 0 and (j> are time dependent). When I and S are in different molecules, becomes time dependent. If 6 and (j) alone are time dependent, the relaxation mechanism is purely intramolecular. If is time dependent, the mechanism is intermolecular. In the first case relaxation is due to molecular rotation, in the second case it corresponds to translation. The theoretical treatment gives different results if the two spins I and S are identical or not (2, p. 291). For two identical spins, at a fixed r distance (intramolecular interaction), and R2 are given by eqs. 17 and 18. Whereas R ... 

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

Using the time-dependent aspect of state diagrams, Roos (2003) illustrated the effects of temperature, water activity, or water content on relaxation times and relative rates of mechanical changes in amorphous systems (Figure 36). This diagram can be considered as a type of mobility map, where mobility increases (relaxation time decreases) as temperature and/or water content/activity increases. Le Meste et al. (2002) suggested the establishment of mobility maps for food materials showing characteristic relaxation times for different types of molecular motions as a function of temperature and water content. 

The time dependent properties creep and stress relaxation can be considered as tests to monitor degradation or as degradation tests that add mechanical stress. 

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