Envelope, failure

The simplified failure envelopes are not derived from physical theories of failure in which the actual physical processes that cause failure on a microscopic level are integrated to obtain a failure theory. We, instead, deal with phenomenological theories in which we ignore the actual failure mechanisms and concentrate on the gross macroscopic events of failure. Phenomenological theories are based on curve-fitting, so they are failure criteria and not theories of any kind (the term theory implies a formal derivation process). 

The simplified failure envelopes differ little from the concept of yield surfaces in the theory of plasticity. Both the failure envelopes (or surfaces) and the yield surfaces (or envelopes) represent the end of linear elastic behavior under a multiaxial stress state. The limits of linear elastic... 

FIGURE 5.15 Failure envelope of various mixes A, natural rubber-polyethylene (NR-PE) vul-canizate (peroxide cured) , NR-PE vulcanizate (sulfur cured) , NR-PE vulcanizate with CPE as compati-bilizer V, EPDM-PE vulcanizate o, EPDM-PP vulcanizate (sulfur cured) NR-ENR-PE -PE. (Erom Roy Choudhury, N. and Bhowmick, A.K., J. Mat. Sci., 25, 161, 1990. With permission from Chapman HaU.)... 

The uniaxial failure envelope developed by Smith (95) is one of the most useful devices for the simple failure characterization of many viscoelastic materials. This envelope normally consists of a log-log plot of temperature-reduced failure stress vs. the strain at break. Figure 22 is a schematic of the Smith failure envelope. Such curves may be generated by plotting the rupture stress and strain values from tests conducted over a range of temperatures and strain rates. The rupture locus moves counterclockwise around the envelope as the temperature is lowered or the strain rate is increased. Constant strain, constant strain rate, and constant load tests on amorphous unfilled polymers (96) have shown the general path independence of the failure envelope. Studies by Smith (97) and Fishman (29) have shown a path dependence of the rupture envelope, however, for solid propellants. 

A combination of an energy criterion and the failure envelope has been proposed by Darwell, Parker, and Leeming (22) for various doublebase propellants. Total work to failure was taken from the area beneath the stress-strain curve, but the biaxial failure envelope deviated from uniaxial behavior depending on the particular propellant formulation. Jones and Knauss (46) have similarly shown the dependence of failure properties on the stress state of composite rubber-based propellants. 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

Smith,T.L., Frederick, . E. Ultimate tensile properties of elastomers. IV. Dependence of the failure envelope, maximum extensibility, and equilibrium stress-strain curve on network characteristics. J. Appl. Phys. 36,2996-3005 (1965). 

Figure 5. Failure envelope for filled SBR vulcanizates. Filled points refer to 25° C and 20 inches/min.

The comparative effect of the polystyrene and poly-2,6-dichlorosty-rene fillers on the tensile strength of a polybutadiene vulcanizate is shown in Figure 6. Despite the large difference in Tg values for these fillers, there is no difference in their effect on the vulcanizate. This is illustrated further by the failure envelope plot shown in Figure 7, where the data points for the two fillers, at equal volume fraction, appear to coincide quite well. The fact that all the points fall on the same envelope is a good indication of the constant crosslink density for these vulcanizates. Thus, the similarity in effect of these two fillers appears to be more related to their similar modulus values. 

Figure 7. Failure envelope for PBD with poly-2,6-dichlorostyrene filler...

Figure 10. Failure envelope for Teflon-filled PBD. Filled points refer to 2 inches/min at 0° and —I5°C.

Figure 8.4. Mohr circle and Mohr-Coulomb failure envelope.

The Mohr-Coulomb failure criterion can be recognized as an upper bound for the stress combination on any plane in the material. Consider points A, B, and C in Fig. 8.4. Point A represents a state of stresses on a plane along which failure will not occur. On the other hand, failure will occur along a plane if the state of stresses on that plane plots a point on the failure envelope, like point B. The state of stresses represented by point C cannot exist since it lies above the failure envelope. Since the Mohr-Coulomb failure envelope characterizes the state of stresses under which the material starts to slide, it is usually referred to as the yield locus, YL. 

The dashed line connects all failure points, in ductile as well as in brittle failure this line is called the failure envelope or fracture envelope... 

Curves of stress (divided by absolute temperature) versus log time-to-break at various temperatures can be made to coincide by introducing the temperature-dependent shift factor flT. Application of the same shift factor causes the curves of the elongation at the break br versus the logarithm of time-to-break at various temperatures to coincide. A direct consequence is that all tensile strengths (divided by absolute temperature), when plotted against elongation at break, fall on a common failure envelope, independent of the temperature of testing. Fig. 13.84 shows the behaviour of Viton B elastomer. 

Fig. 13.84c, known as the Smith failure envelope, is of great importance because of its independence of the time scale. Moreover, investigations of Smith, and Landel and Fedors (1963,1967) proved that the failure envelope is independent of the path, so that the same envelope is generated in stress relaxation, creep and constant-rate experiments. As such it serves a very useful failure criterion. Landel and Fedors (1967) showed that a further generalisation is obtained if the data are reduced to ve, i.e. the number of elastically active network chains (EANCs). The latter is related to the modulus by... 

FIG. 13.84 Generalised ultimate parameters of an elastomer (after Smith, 1962,1964). (A) Logarithmic plot of stress-at-break (a r27i/T) versus reduced time-to-break (fb/dj) for Viton B vulcanisate. Reference temperature for oT is 313 K (40 °C) (B) logarithmic plot of ultimate strain (gbr) versus reduced time to break (fbr/oj) for Viton B vulcanisate reference temperature for oT is 313 K (40 °C) (C) failure envelope for Viton B vulcanisate. 

Stress-Strain-Time Diagrams, Including Failure Envelopes, for High-Density Polyethylenes of Different Molecular Weight... 

Figure 2. Failure envelope for Sample A. Log percent elongation vs. log stress (engineering) for various isochrones obtained from creep data. Line a corresponds to necking, line p to the fully necked condition, and line y to fracture.

Fig. 19. Failure envelopes for gum and carbon black-filled SBR. The carbon black...

This is the well-known failure envelope of T. L. Smith (20/, 202). 

In the Bueehe-Halpin theory the necessity of a strong filler-rubber bond follows naturally from the requirement of a low creep compliance. On the other hand the hysteresis criterion of failure, Eq. (32), does not make the need for filler-rubber adhesion immediately obvious. It is clear, however, that Hb cannot exceed Ub. In absence of a strong filler-rubber bond, the stress will never attain a high value the only way for Ub to become large would be for eb to increase considerably. There is no reason, however, why under these conditions eb should be much greater than in the unfilled rubber at the same test conditions and, in any case, it will be limited by the so-called ultimate elongation . This is the maximum value of eh on the failure envelope and is a fundamental property of polymeric networks. The ultimate extension ratio is given by theory (2/7) as the square root of the number of statistical links per network chain, n,... 

There are four main types of models for predicting failure under monotonic loading. The first type is stress-based failure models. A stress-based failure model, in its most general form, can be written/(ct ) > 0. Since stress is a tensor quantity, the failure model becomes a function of the six independent stress components. Thanks to the difficulty and the experimental effort needed to determine the six-dimensional failure envelope in stress space, the prob-... 

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