Statistical crack mechanics

One scenario in which thin shear layers and large shear strains can occur in plastic-bonded explosives is at the interface between grains sliding past one another or along closed cracks within a grain. These processes have been postulated by Dienes [48] as the dominant dissipative mechanism for generating hot spots in plastic-bonded explosives, and have been incorporated into the Statistical Crack Mechanics (SCRAM) model for high-explosive response. 

Fracture Model. A powerful fracture model based on Statistical Crack Mechanics (SCM) is being developed at Los Alamos ((>). In this model, the rock is treated as an elastic material containing a distribution of penny-shaped flaws and cracks of various sizes and orientations. Plasticity near crack tips is taken into account through its effect on the fracture toughness. 

The objective of this section is to describe the probability of failure of a ceramic component analytically, using statistical fracture mechanics. Sim-plifyingly, we assume that defects with a certain defect size are distributed homogeneously in the material and that crack propagation at only one of them will cause complete failure. Initially, we will also assume a constant stress a within the component. 

Polymer modification at the macroscopic level (either as a material subjected to mechanical processing or as a running object) consists of initiating the destructive phenomena at microdefects—that is, at submicroscopic cracks, statistically distributed on the surface or within the body of the stressed material. These cracks become centers where a detachment of intermolecular bonds occurs. This process might be called a mechanical disaggregation, the opposite of aggregation, a term that expresses (in this context) the assembly of various structural elements into polymers. 

Early explanations about the effect of mechanical energy on the reactivity of solids are the hot-spot-model [23] and the magma-plasma-model [8]. The generation of hot-spof may be used to explain the initiation of a self-sustained reaction such as explosion, deflagration, or decomposition. Temperatures of over 1000 K on surfaces of about 1 pm2 for KM to 10-3 s can be created. These temperatures can also be found near the tip of a propagating crack [24]. Typically nonequilibrium thermodynamics are used to describe these phenomena. The magma-plasma-model allows for local nonequilibrium states on the solid surface during impact however, due to the very short time scale of 1(H s of these states only statistical thermodynamics can describe the behavior. 

Methylcyclopentane is a primary product from a reaction involving tertiary-to-tertiary cracking while cyclohexane is not. Therefore, the high ratios of methylcyclopentane to cyclohexane shown in Table III are consistent with the proposed mechanism. Statistical considerations similar to those discussed in the preceding paragraph account for the lower ratios of cyclopentanes to cyclohexanes in the C7 and C8 products. 

Lor Lorenz, H., Steinhauser, D., Kliippel, M. Morphology and micro-mechanics of filled elastomer blends Impact on dynamic crack propagation. In Grellmann, W., Heinrich, G., Kaliske, M., Kliippel, M., Schneider, K., Vilgis, T. (eds.) Fracture mechanics and statistical mechanics of reinforced elastomeric blends. Lecture Notes in Applied and Computational Mechanics, Vol. 70, Springer, Berlin, 2013, 81-128. 

Given the absence of statistics for occurred events, the only way to estimate the failure probability of nuclear vessels is by an analytical way on the basis of the probabilistic distribution of the involved parameters and of the available fracture mechanics models. The relevant parameters include toughness of the material, the number of cracks initially present in the component, the probability that they are detected during the pre-operational and in-service tests, the fatigue crack growth rate, etc. 

A detailed study of the strength and lifetime under constant stress of single PpPTA filaments using Weibull statistics and an exponentional kinetic breakdown model was carried out by Wu et al. [207], They found that filaments failed due to transverse crack propagation after very short creep times, but that after long creep times the failure mechanism was splitting and fibrillation. Activation energies of the failure process amounted to 340 kJ/mol, which seems to indicate rupture of the C — N bond in the chain backbone. 

Despite the use of identical specimens, the mechanical properties of ceramics show considerable scatter in the measured results. The main reason for the scatter in the values measured is a consequence of the presence, size and distribution of cracks in ceramics. A mean value must be determined via statistical evaluation. The most commonly used statistical approach for describing experimental data is Gaussian normal distribution. In ceramics, however, the use of the Weibull distribution is preferable, reviewed below. 

Constant strain bending tests have the advantage of being very simple. They do not require the use of heavy mechanical equipment. By varying the deformation, the tensile stress applied to the convex surface is modified, which allows one, in principle, to study the influence of the stress on the time to failure. However, the stress is not uniform over the entire thickness of the sample the zone adjacent to the convex surface undergoes tension, and that adjacent to the concave surface is subjected to compression. Bending tests therefore are limited to qualitative studies of 5CC susceptibility. Because the crack initiation time is not well reproducible, a statistical interpretation of the results may be necessary. 

Pits that reach a critical depth can act as crack initiation sites if they lead to a higher local stress intensity. The crack initiation time in this case corresponds to the incubation time of pits of a critical size. Alternatively, precipitation reactions at the grain boundaries can render an alloy sensitive to intergranular corrosion. The preferentially corroded grain boundary then serves as initiation site of a crack. Inclusions, preexisting microcracks, or other structural defects are also likely crack initiation sites. The crack initiation time, in this case, is defined as the time required for a crack to reach a detectable size. Crack initiation may also be the result of hydrogen formed by a corrosion reaction that may cause embrittlement of the metal or of successive ruptures of a passive film or tarnish layer, but these mechanisms are more important for the propagation than the initiation of cracks. Because of the multitude of possible crack initiation mechanisms, and because of the statistical nature of the phenomenon, it is not possible to predict the crack initiation time from first principles. 

---