Critical applied stress

Changes in modulus also affect fracture behaviour. The relaticmship between the two is most clearly seen in the case of brittle fracture. For a wide, thick plate containing an edge crack of length a, the critical applied stress at fracture Oc is related to the Young s modulus , the Poisson s ratio v, and the fracture surface energy Gjc of the material by the Griffith equation ... 

FIG U RE 10.9 Critical applied stress, ac, for cavitation in bonded blocks (see Figure 10.8) versus Young s modulus, E, of the elastomer. (From Gent and Findley (1958).)... 

FIGURE 11 Critical applied stress C7c for cavitation near rigid inclusions (see Fig. 10) versus Young s modulus E of the elastomer. (From Oberth and Bruenner [40].)... 

If Kc is defined as the critical stress intensity, then when K is greater than or equal to crack propagation will occur. can be expressed in terms of the crack length, a the critical applied stress, cr and a geometric... 

It is clear that ab will be extremely large for inclusions of small radius r, even if the level of adhesion, represented by Ga, is relatively small, only of the order of magnitude of van der Waals attractions. For example, when E is assumed to be 2 MPa, representative of soft elastomers, and Ga is given the relatively low value of 10 J/m, then the critical applied stress for detachment is predicted to reach a magnitude similar to E when the radius of the inclusion is reduced to about 20 ixm, even if the initially debonded zone is as large as feasible, 6 — 45°. These considerations appear to account for some features of reinforcement of elastomers by particulate fillers Fillers (140). 

GghI (1970). Gent (124) proposed a model in which the hydrostatic tensile stress at an inclusion or local heterogeneity increases the free volume and therefore effectively reduces the Tg of the material. At a sufficiently high stress concentration, the reduction in Tg is sufficient to reduce the local Tg to the test temperature. The reduced yield stress of the material in this mbber-like phase and the hydrostatic tensile stress then leads to cavitation and craze initiation. Implicit in this free-volume approach is that an imposed hydrostatic pressure will tend to prevent the formation of crazes in accordance with experimental observation. The criterion is summarized in the equation for the critical applied stress for initiation,... 

Figure 10. Critical applied stress of debonding at the pole position vs. radius of curvature r, for rja= 1.

The diagram provides the mean critical applied stress for TPE-PMMA y,critical y.critical 1.25 MPa and for TPE-PC =2.0 MPa. 

FIG. 20-70 The influence of moisture as a percentage of sample saturation S on granule deformabihty. Here, deformation strain (AL/L) is measured as a function of applied stress, with the peak stress and strain denoted by tensile strength and critical strain (AL/L) of the material. Dicalcium phosphate with a 15 wt % binding solution of PVP/PVA Kolhdon VAG4. [Holm et al., Powder Tech., 43, 213 (1.9S.5J,] With land permission from Elsevier Science SA, Lausanne, Switzerland. 

Viscoelastic creep data are usually presented in one of two ways. In the first, the total strain experienced by the material under the applied stress is plotted as a function of time. Families of such curves may be presented at each temperature of interest, each curve representing the creep behavior of the material at a different level of applied stress. Below a critical stress, viscoelastic materials may exhibit linear viscoelasticity that is, the total strain at a given time is proportional to the applied stress. Above this critical stress, the creep rate becomes disproportionately faster. In the second, the apparent creep modulus is plotted as a function of time. 

The viscoelastic creep modulus may be determined at a given temperature by dividing the constant applied stress by the total strain prevailing at a particular time. Since the creep strain increases with time, the viscoelastic creep modulus must decrease with time (Fig. 2-23). Below its critical stress for linear viscoelasticity, the viscoelastic creep modulus versus time curve for a material is independent of the applied stress. In other words, the family of strain versus time curves for a material at a given temperature and several levels of applied stress may be collapsed to a single viscoelastic creep-modulus-time-curve if the highest applied stress is less than the critical value. 

The decrease in K, with crack depth for fracture of IG-11 graphite presents an interesting dilemma. The utility of fracture mechanics is that equivalent values of K should represent an equivalent crack tip mechanical state and a singular critical value of K should define the failure criterion. Recall Eq. 2 where K is defined as the first term of the series solution for the crack tip stress field, ay, normal to the crack plane. It was noted that this solution must be modified at the crack tip and at the far field. The maximum value of oy should be limited to and that the far field stress should decrease only to the applied stress at increasing distance from the crack tip. The nominal fracture stress for IG-11 specimens with artificial flaws ranged from 28 to 100% of aLTS. 

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