J integral

Here C is a closed curve encircling the crack tip, w = Jaij deij is the energy density, u the displacement vector and n an outwardly pointing normal vector on C as shown in figure 5.21. The derivation of the J integral and detailed explanations can be found in appendix D. If no crack is present, equation (5.35) yields J = 0. Strictly speaking, equation (5.35) is applicable only for elastic 

The choice of the path C is arbitrary as long as it encloses the crack tip. In practice - for instance when doing finite element simulations - it is usually wise to put it not too close to the crack tip so that it runs only through elastically deformed regions. 

Equations (5.10) and (5.35) are equivalent. Therefore, for linear-elastic materials and a state of plane stress, the equation 

We now want to show the equivalence of the J integral and the energy release rate (equation (5.10)) by calculating the energy balance. This is elaborated further in appendix D.6. 

For an infinite plate, crack propagation by da in positive xi direction can also be considered as a shift of the integration region A in negative Xi direction, resulting in 

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Fig. 6. Diagram of load versus load-point displacement illustrating a graphical determination of the J-integral [48].

Fig. 1. The microscopic enlanglemenl slruciure, e.g, at an interface or in the bulk, is related to the measured macroscopic fracture energy G, via the VP theory of breaking connectivity in the embedded plastic zone (EPZ) at the crack tip. The VP theory determines

Nevertheless, the formal A/ scaling has spawned approaches which reduce the dependence to A/. This may be achieved by fitting the electron density to a linear combination of functions, and using the fitted density in evaluating the J integrals in the Coulomb term. 

The density fitting functions may or may not be the same as those used in expanding the orbitals. The fitting constants a are chosen so that the Coulomb energy arising from the difference between the exact and fitted densities is minimized, subject to the constraint of charge conservation. The J integrals then become... 

A more universal fracture characteristic for use with ductile materials is the J integral . This is similar to CTOD but relates a volume integral to a surface integral and is independent of the path of the integral it can be classed as a material property. The J integral can also be used to predict critical stress levels for known crack lengths or vice versa. 

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Figure 10. Effect of Hydrogen, Tritium and Decay Helium on the J-Integral (J) vs. Change-in-Crack-Length (da) Behavior for Conventionally Forged Type 21-6-9 Stainless Steels.

Figure 11. Effect of Decay Helium Content on the J-Integral Fracture Toughness Values of Conventionally Forged and High-Energy-Rate-Forged Type 21-6-9 Stainless Steel.

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