Bueche-Halpin

The limited viscoelastic extensibility of rubber strands determines according to the theory of Bueche and Halpin [69] the fracture of elastomers. The authors assume 

The same concept of filament deformation was independently employed by Bartenev et al. [77] to the opening and growth of crazes (silver cracks) in PMMA. 

The molecular theories treated in this chapter have considered the thermally activated breakage of elements or filaments as a source of crack initiation and macroscopic failure. It is the purpose of the following chapters to investigate the mechanical strength of primary bonds and of molecular chains and to study the occurrence of chain breakages. With that information available it will be possible to resume the discussion on the nature of the elements used in molecular theories of fracture. 

Bethea, B. S. Duran, T. L. Boullion Statistical Methods for Engineers and Scientists, New York Marcel Dekker 1975. 

Hald Statistical Theory with Engineering Applications, New York Wiley 1952. 

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The Bueche-Halpin theory accounts well for the principal features of the tensile strength of unfilled rubbers. Because of the direct connection between ab and a viscoelastic function, time-temperature superposition of the strength follows naturally. Halpin (216) also found experimentally that ab was apparently the same function of the reduced time to break, tb/aT, whether the rupture experiment was carried out at constant stress, constant extension or constant rate of extension. 

At this stage, it is not possible to calculate the values of (rt Id) for a loaded vulcanizate, since the Bueche-Halpin theory deals with a particular macromolecule, and must be extended to a distribution of molecular lengths in a matrix, and to the lengths between crosslinks and carbon black particles. 

In the Bueche-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 Hf, cannot exceed Uj,. In absence of a strong filler-rubber bond, the stress will never attain a high value the only way for to become large would be for to increase considerably. There is no reason, however, why under these conditions 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 Ef, 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. 

Whereas all of the above theories [54—68] refer in one way or another to the thermally activated breakage of one kind of molecular bonds as primary fracture event, the Bueche-Halpin theory for the tensile strength of gum elastomers [69] employs the idea that the viscoelastic straining of rubber filaments together with the degree of their ultimate elongation determine the kinetics of crack propagation in an elastomeric solid. 

Bueche and Halpin (126, 203, 215-217) have developed a fracture theory for amorphous rubbers. Their model pictures rupture as the result of the propagation of tears or cracks within the material. The growth of a tear is viewed as a process in which molecular chains at the tip of the tear stretch viseoelastieally, under the influence of a high stress concentration, until they rupture. The failure process is a non-equilibrium one, developing with time and involving consecutive rupture of molecular chains. The principal result of the theory is embodied in the equation... 

Rupture of a tensile test piece may be regarded as catastrophic tearing at the tip of a chance flaw. The success of the WLF reduction principle for fracture energy, G, in tearing thus implies that it will also hold for tensile rupture properties. Indeed, a/, and may be calculated from the appropriate value of G at each rate and temperature, using relations analogous to Eqs. (10.6) and (10.7). The rate of extension at the crack tip will, however, be much greater than the rate of extension of the whole test piece, and this discrepancy in rates must be taken into account (Bueche and Halpin, 1964). 

Another viscoelastic aspect of failure that Halpin and Bueche investigated involved time-to-break experiments. When dead loads are applied to polymeric materials, failure does not occur either immediately or never, but after finite lengths of time. Figure 10.17 compares a gum SBR with HAF-reinforced material. Each curve exhibits a portion in which failure... 

Figure 10.16. Comparison of failure envelopes for an SBR rubber as a function of reinforcement concentration. ( ) 30 HAF (X) 15 HAF (O) 0 HAF. (Halpin and Bueche, 1964.)...

Figure 10.17. Comparison of master time-to-break curves at -10°C. SBR vulcanizate containing (a) 0, (b) 15, (c) 30 parts HAF black. (Halpin and Bueche, 1964.)...

These workers also showed that the apparent energy of activation of the failure process could be calculated assuming an Arrhenius mechanism. As illustrated in Table 10.3, addition of reinforcing filler raises the apparent activation energy of the viscoelastic failure processes. Halpin and Bueche ascribe the enhanced reinforcement to those processes that spread the viscoelastic motions of the filler-rubber complex over a much wider time scale, and concluded that the lower strength observed at elevated temperatures was due to the increased rate at which viscoelastic response to deformation... 

Tschoegl s result is especially interesting in the light of a recent proposal by Shuttleworth (1968, 1969) that equilibrium polymer-filler debonding is responsible for decreased tensile strength at elevated temperatures. This is contrary to the viscoelastic mechanism of high-temperature failure of Halpin and Bueche (1964), which was developed in an earlier section. A possible resolution of the relative importance of the two proposed mechanisms could lie in the application of Tschoegl s experiment to carbon black- or silica-reinforced materials. 

A large number of equations have been proposed to predict a variety of mechanical properties of filled plastics. The principles underlying the derivations require detailed and lengthy study to be understood, and regrettably there is no room here to discuss the rational basis for the predictive equations mentioned we shall simply indicate the kind of predictions that have emerged and moreover we shall confine the discussion to one property the modulus. The original references can be found in any review of the effect of fillers on mechanical properties, such as the theses by Wainwright and Phipps, mentioned elsewhere in this article. Contributors include Nielsen, Paul, Narkis, Ishai, Bueche, Sato and Furukawa, Halpin, Chow etc. 

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