Crack steady

The small slope of the stage II section of the crack-growth rate versus K curve is attributed to corrosion-related, diffusion-controlled processes in the crack. Steady-state diffusion mechanisms are required to account for the fact that the crack growth rate is essentially constant... 

Fig. 18. Comparison of finite-element and asymptotic solutions for energy release rate a = 30.7 MPa). O fea. the finite-element solution Casym. the small-scale cracking solution the long crack, steady-state solution.

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

In the steady-state regime, the crack growth rate is described by... 

The selectivity and activity of the catalyst matrix will continue to improve. The emphasis on bottoms cracking and steady reduction in the reaction residence time demands an increase in the quantity of active matrix. 

The work done in the deformation zone can be estimated assuming the deformation zone propagates together with the crack and maintains its steady state displacement profile [86]... 

Our treatment of chain reactions has been confined to relatively simple situations where the number of participating species and their possible reactions have been sharply bounded. Most free-radical reactions of industrial importance involve many more species. The set of possible reactions is unbounded in polymerizations, and it is perhaps bounded but very large in processes such as naptha cracking and combustion. Perhaps the elementary reactions can be postulated, but the rate constants are generally unknown. The quasi-steady hypothesis provides a functional form for the rate equations that can be used to fit experimental data. 

Each breakdown is accompanied by some sound effect and is followed by a steady degradation of properties.284 It can also lead to a complete destruction of the oxide with visible fissures and cracks.286 The particular behavior observed depends on a large number of factors (electrolyte concentration,287 defect concentration in the oxide,288 etc.). The breakdown of thin-film systems (M-O-M and M-O-S structures) as a rule leads to irreversible damage of oxide dielectric properties.289... 

It is noted that the time required for reaching the steady state of hydrogen diffusion is stress, specimen geometry, and dimension dependent, and cannot be used to compare with the permeation data as discussed earlier, which was measured from stress- and crack-free thin membrane specimens. 

Figure 4. Plot of the full-field solution for the normalized hydrostatic stress

Figure 6. Normalized NILS hydrogen concentration CL / C at steady state vs. normalized distance R/b from the crack tip for the full-field (crack depth wh=0.2) and MBL (domain size L=h-a) solutions under zero hydrogen flux conditions on the OD surface and remote boundary, respectively. The parameter b denotes the crack tip opening displacement for each case. The inset shows the concentrations near the crack tip.

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

Figure 9. MBL formulation results plotted against normalized domain size L lb. under zero concentration boundary condition on the remote boundary while the crack faces are in equilibrium with 15 MPa hydrogen gas (a) non-dimensionalized effective time to steady state = >t lb (b) peak values of the normalized hydrogen concentration in NILS at / =/ (effective time to steady state).

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