4 So, a connection between the global quantity "energy release rate" G and the crack front field "stress intensity factor" K was given. Nevertheless the problem is that the equation (2.6) includes 1 j 1 i 0 ij, u und n σ which refer to the different states of time. This makes the equation (2.6) difficult to use for complicated problems. 2.3 The J-Integral The path-independent J-integral ∫ = −σ A ij i,1 j 1 J (wn u n )dA (2.8) was introduced by Cherepanov (1967) [11] und Rice (1968a) [7] into the fracture mechanics to determine some specific problems. The J-integral is not only applicable to linear elastic material but also used for hyper-elastic material. Specifically, Rice (1968b) [8] and Budiansky and Rice (1973) [9] have derived the relationship ∫ = ∂ ∂Π − 0A 1 wn dA a , (2.9) and by consideration of the traction-free crack surface the equation (2.9) is finally equal to the Jintegral J a = ∂ ∂Π − . (2.10) So the following Eq. (2.11) (1 ) E K E K E K J G 2 III 2 II 2 I +ν + ′ + ′ = = (2.11) is obtained, which is valid for linear elastic material, where w is the strain energy density, A0 the crack front surface to the state of time t, A the area refers to any path in body, ij σ the stress tensor, iu the displacement vector and in the normal and 1 ,1 ( ) ( )/ x =∂ ∂ . It is seen above that for linear elastic material, the J-integral is both equal to the Griffith's energy release rate Eq. (2.10) and supplies the same result as from the Eq. (2.7) provided by Irwin. It is also applicable to elastic-plastic material. Thus, the J-integral became important in fracture mechanics and particularly for elastic-plastic material. However, it can be recognized from the equations (2.6) and (2.8), that on the one hand the two equations Eq. (2.6) and Eq. (2.8) seem to be quite different, although they provide identical results (Eq. (2.7) and Eq. (2.11)), and on the other hand that the two integrals refer to different crack front surfaces, whereby the equation (2.6) to the crack front surface is linked to the time t t +Δ , and the equation (2.8) is linked to the time t. Due to the different formulations of the above theories we can provide the following questions: • Do the Griffith's energy release rate G, the Irwin's I-integral expression of the crack closure energy and the Rice's J-integral really describe the same fact? • Is there a uniform rule from which the different equations as Eq. (2.6) and Eq. (2.8) can be derived? • Does new knowledge hide behind this difference? To answer these questions requires us to conduct a thorough analysis in the next chapter.
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