The Failure Envelope

FIGURE 10.23 Failure envelope for an SBR vulcanizate. (From Smith (1963).) 

These conditions obtain at high rates of strain and low temperatures, when the material responds in a glasslike way. 

The principal advantages of the failure envelope representation of data seem to be twofold. First, it clearly indicates the maximum possible breaking elongation,, max5 for the material. This is found to be well correlated with the degree of crosslinking, specifically with the molecular weight between crosslinks, M, as predicted by elasticity theory  

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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... 

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. 

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). 

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. 

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... 

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... 

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,... 

Projections of the Three-Dimensional System. To simplify the three-dimensional system, its projections into the various planes can be used. In particular, the projection of the break points is important because they define the limitations sealants have in practice. Their projection in the log stress-log strain plane is the failure envelope ( ) shown for polysulfide sealant in Figure 9. The outer limit of the envelope is well defined and is drawn in with a dashed line, but the inner one disappears in the scatter. For the silicone sealant. Figure 10 gives the failure envelope where both the upper and lower limits are well defined. 

The failure envelope is used in the literature to characterize polymers because it is independent of time and temperature, but its usefulness is limited with sealants. From the point of view of sealant performance, the projection of the failure points into the log strain-log time plane is the most important characterization it is the strain that is imposed on the sealant by the movement of the joint and the stress develops as a consequence of the imposed strain. Consequently, the design of a sealed joint is usually based on an estimate of strain, not of stress, and the sealant is chosen according to its movement capability, that is, the per cent movement the sealant can take without failure in a yearly movement. Stress has to be considered only in those rare cases where the substrate is a fragile, porous material whose tensile strength approaches that of sealants. In this case, a sealant with the lowest strength possible has to be chosen. 

The time dependence of the strain at break is very different for the two types of sealant. The break points of the silicone data can be fitted by a straight line, and confidence limits at various levels can be drawn on the plot (Figure 11). The break points of the two-part polysulfide sealant form a broad band, the upper and lower limits of which are drawn qualitatively. The upper limit is better defined than the lower one (as for the failure envelope). Because of the difference between the plots for silicone and polysulfide sealants the further simplification of characterization is different for the two types of sealant. 

The failure envelope /(CTi (J3) = 0 is defined from point to i by the Mohr-Coulomb failure criterion /, = 0 with... 

The strain to break 6, may be plotted against the stress to break in the master curve to yield a failure envelope, shown schematically in Figure 1.25 (Scott, 1967). The failure envelope is independent of temperature, time to break, and strain rate. It is a universal curve independent (at least ideally) of the type of rupture test. If is further divided by the crosslink density, the resulting failure envelope is also approximately independent of both the degree of crosslinking and the chemical structure of the elastomer. The latter... 

Figure 1.25. The failure envelope for a poly-(styrene-co-butadiene) rubber. Area A to the left of the curve indicates stable levels of stress and strain. Values in region B will cause rupture. Arrow indicates direction of lower temperatures or higher strain rates.

Statement, surprising but true for noncrystallizing elastomers, follows from the equation of state for rubber elasticity and the strength of the carbon-carbon bond. The failure envelope is indeed a powerful tool in dealing with ultimate properties. 

P(3) The lamina shall be deemed to have failed for any combination of strains lying outside the failure envelope. 

P(3) In the absence of strength test or manufacturers data for a laminate, the failure envelope shall be assembled using the failure envelopes of the constitutive laminae as illustrated in Figure 4.14 in the EUROCOMP Handbook. 

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