Fracture calculations

Source ICI Technical Service Note PP127 Propathene for Pipework  

The relationship between hoop stress and pressure is given by P(D-s) 

Since maximum realizable pressure in the plant is 1.7 bar, when entire system is full, the pipe cannot have burst in a simple manner, because this would require 4.8 bar, maintained for one year continuously, x 1.3 FOS = 6.2 bar. 

fracture mechanics expressions are used to find the pressure required to propagate the crack initiated in the faulty weld, and also the total energy required to give a crack of the observed length. 

For a crack to run in the brittle manner in a pipe it is necessary for the energy stored in the wall of the pressurized pipe to exceed that required to fracture a unit length of pipe. 

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It is important to note that we assume the random fracture approximation (RPA) is applicable. This assumption has certain implications, the most important of which is that it bypasses the real evolutionary details of the highly complex process of the lattice bond stress distribution a) creating bond rupture events, which influence other bond rupture events, redistribution of 0(microvoid formation, propagation, coalescence, etc., and finally, macroscopic failure. We have made real lattice fracture calculations by computer simulations but typically, the lattice size is not large enough to be within percolation criteria before the calculations become excessive. However, the fractal nature of the distributed damage clusters is always evident and the RPA, while providing an easy solution to an extremely complex process, remains physically realistic. 

Figure 20.3 shows how SiC>2 concentration in the fluid varies along the fracture, calculated assuming different traversal times Af. For slow flow rates (values of Af of about 1000 years or longer), the fluid has enough reaction time to remain near equilibrium with quartz. When flow is more rapid, however, the fluid maintains much of its silica content, quickly becoming supersaturated with respect to quartz. Farther along the fracture, the fluid also becomes supersaturated with respect to cristobalite, and at traversal times of less than about 100 years, with respect to amorphous silica. 

Maximum energy to fracture calculated by //2 Experimentally measured in this work. 

The normal stress at the beginning of fracture. Calculated from the load at the beginning of fracture during a tension test, and the original cross-sectional area of the specimen. 

Failure in a polypropylene vessel, 295-304 bubble collapse, 303 failure incident, 297 fracture calculations, 299-304 investigation of failure, 298 plant layout, 295 Family moulds, 14 Fibre reinforcement, 160 Fibreglass Ltd, 128 Fibrelam aircraft flooring, 211-20... 

The problem of the orientation of fibres in cement matrices was considered first by Morton (1979) where it was shown that the work of a fracture may be considerably increased when fibres are not aligned with the direction of the principal tensile strain. These results have been based on previously published experimental works by Hing and Groves (1972), Harris et al. (1972) and Morton and Groves (1974). They have shown that the work of fracture, calculated as the amount of work of external load absorbed by the element, is the most important magnitude to be considered in the design of brittle matrix composites. This approach was developed in a proposal of formulae for energy calculation (Brandt 1982, 1984), and later in the solution of a... 

The true stress at fracture calculated using the area based on the fracture surface results in a value that IS 80°i of that found using the diameter prior to fracture... 

Models for transport distinguish between the unsaturated zone and the saturated zone, that below the water table. There the underground water moves slowly through the sod or rock according to porosity and gradient, or the extent of fractures. A retardation effect slows the motion of contaminant by large factors in the case of heavy metals. For low level waste, a variety of dose calculations are made for direct and indirect human body uptake of water. Performance assessment methodology is described in Reference 22. 

A partial answer to the first question has been provided by a theoretical treatment (1,2) that examines the conditions under which a matrix crack will deflect along the iaterface betweea the matrix and the reinforcement. This fracture—mechanics analysis links the condition for crack deflection to both the relative fracture resistance of the iaterface and the bridge and to the relative elastic mismatch between the reinforcement and the matrix. The calculations iadicate that, for any elastic mismatch, iaterface failure will occur whea the fracture resistance of the bridge is at least four times greater than that of the iaterface. For specific degrees of elastic mismatch, this coaditioa can be a conservative lower estimate. This condition provides a guide for iaterfacial desiga of ceramic matrix composites. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

In the numerical calculations, an elastic-perfectly-plastic ductile rod stretching at a uniform strain rate of e = lO s was treated. A flow stress of 100 MPa and a density of 2700 kg/m were assumed. A one-millimeter square cross section and a fracture energy of = 0.02 J were used. These properties are consistent with the measured behavior of soft aluminim in experimental expanding ring studies of Grady and Benson (1983). Incipient fractures were introduced into the rod randomly in both position and time. Fractures grow... 

At the conclusion of the calculation, a fragment size distribution as well as fragment number is provided. A cumulative number distribution is shown in Fig. 8.22 and compared with aluminum ring data acquired at = lO s (Grady and Benson, 1983). With the assumed fracture site nucleation law, the calculated distribution appears to agree reasonably well with the data. The calculation better predicts the tails of the distribution which have trends which deviate from strict exponential behavior as was noted in the previous section. 

The present discussion of continuum modeling of dynamic fracture is not an exhaustive review. Rather, it points out the variety of approaches which have been, and are still being, pursued to provide methods for calculating dynamic fracture phenomena. Such work is still quite active and considerable effort... 

Numerical simulation of a complex dynamic fracture application can be illustrated by calculations of impact induced damage in a ceramic cylinder. The computer model used was originally developed for oil shale explosive fragmentation (Grady and Kipp, 1980), with various extended applications considered by Boade et al. (1981) and Chen et al. (1983). In this model, stress and strain are related through... 

Plastic) strain after fracture, or tensile ductility. The broken pieces are put together and measured, and Cf calculated from (/ - IqI/Iq, where / is the length of the assembled pieces. 

Two wooden beams are butt-jointed using an epoxy adhesive (Fig. A1.3). The adhesive was stirred before application, entraining air bubbles which, under pressure in forming the joint, deform to flat, penny-shaped discs of diameter 2fl = 2 mm. If the beam has the dimensions shown, and epoxy has a fracture toughness of 0.5 MN mT , calculate the maximum load F that the beam can support. Assume K = cT Tra for the disc-shaped bubbles. 

Thermal Gradients may be measured or calculated by means of heat flow formulas, etc. After they are established it is likely to be found from the formula that for most cyclic heating conditions the tolerable temperature gradient is exceeded. This means that some plastic flow will result (for a ductile alloy) or that fracture will occur. Fortunately, most engineering alloys have some ductility. However, if the cycles are repeated and flow occurs on each cycle, the ductility can become exhausted and cracking will then result. At this point it should be recognized that conventional room temperature tensile properties may have little or no relation to the properties that control behavior at the higher temperatures. 

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