Anelastic Relaxation

Rheology - The study of the flow of liquids and deformation of solids. Rheology addresses such phenomena as creep, stress relaxation, anelasticity, nonlinear stress deformation, and viscosity. 

G.E. Duvall, Propagation of Plane Shock Waves in a Stress-Relaxing Medium, in Stress Waves in Anelastic Solids (edited by H. Kolsky and W. Prager), Springer-Verlag, Berlin, 1964, pp. 20-32. 

Nowick, A.S. and Berry, B.S. (1972) Anelastic Relaxations in Crystalline Solids (Academic Press, New York). 

The effects of anelasticity are seen clearly in the strain-time schematic plot of Fig. 7. The time independent strain, 0, occurs immediately as the stress is apphed. The additional strain, o, is associated with stress-induced anelasticity. The total strain, -i- o, is approached exponentially. When the stress is removed the time independent strain is recovered immediately, while the anelastic component relaxes exponentially with a relaxation time constant, k. 

We also compared two sets of specimens that had been drawn to the same extent. Both were allowed to relax for 48 h after stretching, one set clamped at its specified elongation, the other allowed to relax freely, undamped. The dichroism results show a large difference between the recoverable (elastic+anelastic) deformation and the total deformation for both XL and NXL phases (see Ligures 3 and 4 of Davidson and Gounder ). 

Measurements of <5 yield direct information about the magnitude of the energy dissipation and the phase angle. 0 measures the fractional energy loss per cycle due to the anelasticity and is often termed the internal friction. According to the discussion above, 8 will be a function of the frequency, to should approach zero at both low and high frequencies and will have a maximum at some intermediate frequency. The maximum occurs at a frequency that is the reciprocal of the relaxation time for the re-population of the point defects. 

Examining the forms of e(0) and e(oc) and comparing the results with Eqs. 8.64-8.66 shows that aEaaas2 = Su and asa asi = Sr — Su Also, the anelastic relaxation occurs exponentially, in agreement with the results in Exercise 8.5, and the relaxation time corresponds to r = asi/ao- Equation 8.90 then takes the simpler form... 

The maximum damping (anelasticity) occurs when the applied angular frequency is tuned to the relaxation time of the anelastic process so that uit = 1. Also, 5(u>) approaches zero at both high and low frequencies, as anticipated. 

The preceding analysis provides a powerful method for determining the diffusivities of species that produce an anelastic relaxation, such as the split-dumbbell interstitial point defects. A torsional pendulum can be used to find the frequency, u>p, corresponding to the Debye peak. The relaxation time is then calculated using the relation r = 1/ojp, and the diffusivity is obtained from the known relationships among the relaxation time, the jump frequency, and the diffusivity. For the split-dumbbell interstitials, the relaxation time is related to the jump frequency by Eq. 8.63, and the expression for the diffusivity (i.e., D = ra2/12), is derived in Exercise 8.6. Therefore, D = a2/18r. This method has been used to determine the diffusivities of a wide variety of interstitial species, particularly at low temperatures, where the jump frequency is low but still measurable through use of a torsion pendulum. A particularly important example is the determination of the diffusivity of C in b.c.c. Fe, which is taken up in Exercise 8.22. 

Solution. Using a torsion pendulum, find the anelastic relaxation time, r, by measuring the frequency of the Debye peak, cup, and applying the relation cupr = 1. Having r, the relationship between r and the C atom jump frequency F is found by using the procedure to find this relationship for the split-dumbbell interstitial point defects in Exercise 8.5. Assume the stress cycle shown in Fig. 8.16 and consider the anelastic relaxation that occurs just after the stress is removed. A C atom in a type 1 site can jump into two possible nearest-neighbor type 2 sites or two possible type 3 sites. Therefore,... 

Nowick, A. S., Berry, B. S. Anelastic relaxation in crystalline solids. Academic Press, New York 1972... 

Nowick, A. S. and Heller, W. R. (1965) Dielectric and anelastic relaxation of crystals containing point defects. Adv. Phys. 14, 101-66. 

PTFE is known to undergo two crystalline first-order transitions at 19° and 30° and three viscoelastic relaxations have been observed. The temperature dependence of the relaxations (a,8,Y) as determined by dielectric and anelastic measurements is shown in the relaxation map of Figure 10 along with the temperatures of the crystalline phase transitions (vertical dotted lines) Also indicated in Figure 10 are the results of rotational diffusion rates obtained in this study. 

The relaxation times shown in Figure 9 indicate that from -128° to 300° only two minima are observed. These two minima are associated with the Y and 6 relaxations previously observed in anelastic and dielectric measurements. We see no evidence of any other relaxations on the time scale of T p and T ... 

A mobile population of hydrogen has also been observed by anelastic spectroscopy measurements that were carried out by Palombo et al. Heating NaAlIij doped with 2 mol% TiCl3 to 436 K introduces a thermally activated relaxation process with a frequency of 1 kHz at 70 K. This denotes the formation of a point defect with a very high mobility ( 5 x lO jumps/s at 70 K). The relaxation involves the reorientation of H around Ti. 

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). 

If the standard linear solid (SLS) is unloaded from a constant stress, the spring (modulus ,) closes immediately and the elastic strain is removed. The anelastic strain then decays to zero as the second spring closes the dashpot, i.e., there is complete recovery. Under the action of a constant strain, the SLS model will also show stress relaxation but, in this case, the time constant, Tf =rf /(E +E2). In applying a constant stress to the SLS model, the strain can be considered to lag behind the stress, both on loading and unloading. This lag concept is also very important in considering the effect of a dynamic stress or strain. 

It is useful to consider the behavior of the SLS as a function of dimensionless frequency (o)T, as shown in Fig. 5.16. At low frequencies, the dashpot has sufficient time to open and close and elastic behavior is obtained (relaxed modulus). The stress and strain are in phase. At very high frequencies, there is no time for the anelastic strain to develop. The stress and strain are again in phase but the modulus is higher, with only the spring of modulus , opening and closing (unrelaxed modulus). For intermediate frequencies, a lag develops and... 

N.G. McCrum, B.E. Read, and G. Williams, Molecular Theories of Relaxation in Anelastic and Dielectric Effects in Polymeric Solids, Wiley, New York, chap. 5 (1967). 

Num] Internal friction, anelastic and magnetic relaxation <2.2 mass% Cr, < 0.3 mass% N, heat treatment... 

Furthermore, characterization of the molecular dynamics must be carried out in the solid state if the objective is to understand the dynamic structure in the solid state (47). This information can be related to other relaxation methods such as anelastic and dielectric relaxation to develop an understanding of a variety of properties such as toughness, permeability, secondary/tertiary structure, and structure/property/processing relationships. 

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