Stress relaxation isotherms

Fig. 11. Stress relaxation isotherms for a polyisobutylene amorphous polymer. After To-bolsky (25).

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

Fig. 5.1 Idealized representation of the transient change in fiber and matrix stress that occurs during the isothermal tensile creep and creep recovery of a fiber-reinforced ceramic (the loading and unloading transients have been exaggerated for clarity). It is assumed that the fibers have a much higher creep resistance than the matrix. The matrix stress reaches a maximum at the end of the initial loading transient. After full application of the creep load, the matrix stress relaxes and the fiber stress increases. Upon specimen unloading, elastic contraction of the composite occurs, followed by a time-dependent decrease in fiber stress and increase in matrix stress. Overall, creep tends to increase the difference in stress between constituents and recovery tends to minimize the difference in stress. After Wu and Holmes.15...

An hysteresis phenomenon is observed when the plateau pressure determined for an absorption isotherm is higher than the plateau pressure measured at the same temperature for the desorption process. Hysteresis is caused by the large stresses associated with the metal to hydride transformation which give rise to internal defects such as dislocations and stacking faults. Hysteresis decreases with increasing temperature as thermally activated stress relaxation processes set in. It is in general important to eliminate or at least minimize hysteresis for most applications. 

The practical timescale for most stress relaxation measurements ranges from 10 to 10 s but a wider range of temperamre is desirable. Such a range can be covered relatively easily by making use of the observation, first made by Leaderman, that for viscoelastic materials time is equivalent to temperature. A composite isothermal eurve eovering the required extensive time scale can then be constracted from data eolleeted at different temperatures. 

In order to fit isothermal data (for example the linear polyethylene data for 15°C, see Figure 4.4) it is necessary to fit to the data the three adjustable parameters of the Zener model, which are J, /r, and t. Since the Zener model is supposed equalfy valid for creep, stress relaxation, and dynamic response, these same parameters should then fit (for linear polyethylene at IS C) G(t), G (tu) and J ([Pg.146]

M. L. Cerrada and G. B. McKenna, Stress Relaxation of Pol3Kethylene Naphthalate) Isothermal, Isochronal and Isostructural Responses Macromolecules 33, 3065-3076 (2000). 

Figure 15.2(c) shows a schematic representation of the process for measuring stress relaxation in a viscoelastic material. Using an isothermal step method, the sample is allowed... 

An interesting treatment has been proposed by Hopkins in which the temperature is changed according to a prescribed function of time during a transient viscoelastic experiment (creep or stress relaxation). By comparing the results, J(t) or (7(0. with those obtained in a second experiment under conventional isothermal conditions, the factor ay can be calculated over the temperature range concerned. In another modification, termed thermally stimulated creep, a strain is imposed, the temperature is suddenly lowered, and the stress is removed with subsequent temperature increase at a constant rate, the creep recovery is followed. ... 

Guo, Y. and Bradshaw, R.D. Isothermal physical aging characterization of polyether-ether-ketone (PEEK) and polyphenylene sulfide (PPS) films by creep and stress relaxation, Mechanics of Time-Dependent Materials, 2007, 11(1), p. 61-89. 

Automatic running die-swell measurements can be made by using laser scanning. This can be carried out in a temperature-controlled chamber at the die exit for isothermal die-swell. Together with stress-relaxation experiments (by stopping the piston descent), this can provide information on the viscoelastic nature of the material. High die-swell and long relaxation times would indicate a more elastic material. Many rheometers also include automatic MFR calculations on thermoplastics. 

Other tasks performed under this project consisted of isothermal evaluation of the Min-K material in compression as a function of temperature (including - assessment of size and geometric effect on the distribution of compressive strength of Min-K, determination of distribution of compressive strength of Min-K as a function of temperature, and statistical analysis of monotonic compressive strength results), isothermal stress relaxation testing, and associated modeling efforts. Results from these activities can be found in a previously published ORNL Technical Report. ... 

Figure 3. Schematic of discretization of temperature gradient stress relaxation sample into isothermal...

In another approach, the stress relaxation behavior was analyzed using the finite element program ANSYS by modeling the isothermal, time-dependent behavior of Min-K using Equation 2,... 

However, it was found that the temperature distribution was not radialy isothermal and the results of this analysis predicted more stress relaxation than what had been determined experimentally. Even after specitrten temperatures were adjusted to more closely match the actual lest temperature distribution observcxl during an actual gradient stress relaxation test, results were still found to over predict the stress relaxation behavior. Therefore, this modeling approach was abandoned for the longterm prediction of the stress relaxation behavior. 

Furthermore, the magnitude of the relaxation modulus is a function of temperature to more fuUy characterize the viscoelastic behavior of a polymer, isothermal stress relaxation measurements must be conducted over a range of temperatures. Figure 15.6 is a schematic log ,(f)-versus-log time plot for a polymer that exhibits viscoelastic behavior. Curves generated at a variety of temperatures are included. Key features of this plot are that (1) the magnitude of EXt) decreases with time (corresponding to the decay of stress. Equation 15.1), and (2) the curves are displaced to lower EXt) levels with increasing temperature. 

Initial Crack Generated Isothermally. Slow, controlled crack propagation is possible for most polymers above 160 K. Stress relaxation at the crack tips is determined by the stress-relaxation behavior at the given temperature. This can be verified when measuring Kic in a temperature range including a relaxation peak (17). The temperature dependence of Kic correlates well with that of the relaxation peak when both measurements are based on the same time scale of mechanical deformations Kic is directly related to the relaxation spectrum. 

In observing the time dependent changes in birefringence and stress-optical coefficient, for elongated samples at 25 C, it was found that the rate of crystallization of high trans SBR s was very much faster, some 10 times more rapid, than that for NR (8). This is consistent with the reported rates of isothermal crystallization for NR (2.5 hours at -26°C) and for 807. trans-1,4 polybutadiene (0.3 hours at -3°C) in the relaxed state (12). 

Models of the isothermal mechanism can be constructed using a balance equation (1) for the area of active surface per unit volume of a solid sample, with a term added which describes the propagation of this surface into the nonfractured matrix. The term requires that a certain effective transfer coefficient (analogous to the diffusion coefficient) should be introduced. To a first approximation, it can be written as D = vl = v2r, where v is the velocity of sound in the sample, / is the length of the free run of a crack for the time r, and t is the time of mechanical unloading (or the characteristic relaxation time of stresses in the real solid matrix of a reactant sample). It seems impossible to... 

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