Hardening exponent

A plot of log o against log s should thus yield a straight line whose slope is n and which makes an intercept equal to log fej on the log o axis (at s = 1). Thus the constant kj represents the true stress at unit true strain and is termed the strength coefficient. The exponent n is known as the strain hardening exponent. 

This relationship implies that the value of the true strain at which plastic instability sets in, i.e., necking starts to occur, is equal to the strain hardening exponent. 

Using the data listed in Table 1 Kj, -values are obtained which are a little higher than the values measured for these polymers under mode I-conditions. The factor n was estimated as 5 10 , that means the work hardening exponent n is near zero. 

For a power-hardening material that obeys the stress-strain relationship, a = ke", where n is the strain-hardening exponent, the slope of the stress-strain curve is given by Eqn. (5.8) ... 

Figure 6.16. Sensitivity of model prediction to inverse strain-hardening exponent N [3].

Hardness values, indentation moduli, strain hardening exponents and viscoelastic properties can be measured with the instrumented indentation test, also the fracture toughness of very brittle polymers as well as the influence of residual stresses. If needed and a suitable device provided measurements can be done with high spatial resolution and with very small indentation depths. A special application of the testing devices is the characterization of the elastic behaviour of miniaturized components or the realization of micro compression tests, i.e. using the machines like a small universal testing machine. 

Integration of this equation from e = 0 to s = e and correspondingly from So = 0 to o = So for two cases with strain-hardening exponents Al = 0.1 and 0.4, both for an initial imperfection factor tj = 0.005, gives results shown in Figs. 10.4(a) and (b) for the ratio s/so as a function of the increasing strain eo/N in the perfect bar (Hutchinson and Neal 1977). These results illustrate the process of strain localization in the bar and allow us to assess the influence of the strain-rate sensitivity. 

Material Young s modulus, E GPa 10 psi Yield strength MPa ksi Tensile strength MPa ksi Uniform elongation (%) Total elongation (%) Strain hardening exponent in) Average normal anisotropy (r, ) Planar anisotropy (Ar) Strain rate sensitivity (m)... 

Ref [193]. Reproduced with permission from The American Ceramic Society (b) Quantitative analysis of the stress values in the process zone near the crack tip at distance r from the tip. A detailed analysis yields a hardening exponent of 0.6 for the soft ferroelectric PZT (PIC 151). From Ref [192]. 

Metal 0.2% Offset Yield Strength (kg/mm ) Ultimate Tensile (kg/mm ) Uniform Elongation (%) Reduction in area (%) Strain Hardening Exponent Strain Rate (Sec.- ) Strain Rate Sensitivity (e) Anneal Temp.. (K) Ref. ... 

Fig. 8.14. Temperature dependence of tensile strain-hardening exponent and strain-rate sensitivity of yttrium from Koepke et al. (1%7).

Temperature dependence of the strain hardening exponent, n, in the equation increasing temperature to about 600 K where it begins to drop rapidly, perhaps as a consequence of recrystallization during testing. As mentioned above, the strain rate sensitivity defined by (A In cr/A In e)r,f exhibits a distinct minimum with negative values near 475 K. Whether the step behavior in the 550 to 675 K region is physically real or just a manifestation of data scatter is not clear. 

Both the strain rate sensitivity and strain hardening exponent of cerium (initially in the y phase) exhibit manifestations of the various transformations as shown in fig. 8.21. There is a gradual rise in the strain hardening exponent as the temperature decreases to the y to /3 M[Pg.628]

Fig. 8.21. Temperature dependence of the strain-rate sensitivity and the strain-hardening exponent (in tension) of cerium which was in the y phase at room temperature prior to testing. Cross-hatched bars depict temperatures at which the indicated phase changes occur. From Owen and Scott (1976).

Strong temperature dependence of the strain hardening exponent shown in fig. 8.24. 

Fig. 8.27. Temperature dependence of the strain-hardening exponent and the strain-rate sensitivity of neodymium. From Owen and Scott (1975).

Furthermore, the strain hardening exponent, fig. 8.42, dropped suddenly at 250 K as a result of elongation produced during the p to a transformation. The strain rate sensitivity, fig. 8.42, exhibited a distinct minimum centered at about 350 K. Whether this is associated with the p to a transformation or is related to another phenomenon is not known. 

Table 3.1-13 Typical values of strain-hardening exponent n and degree of anisotropy r for some aluminium-base materials based on data from various sources [1.9] n.a. - not available...

Strain hardening exponent Ability of a material to harden during cold working... 

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