10 • the new vector quantity Tk has an integral form, • it is path-independent, • it describes the distortion of the crack front surface. 5. Two New Quantities Sk and Tk for Elastic-Plastic Material Behavior Now we will extend the new found quantities Sk and Tk for linear elastic material behavior into elastic-plastic material behavior with power law hardening. So let us write the strain energy density w for this material as follows ij ij n 1 n w σ ε + ≈ , (5.1) where n is the hardening parameter of material. In the above equation the part of elastic strain energy density is neglected, since the value of this part is very small in comparison with the plastic part. It is shown that the Eq. (5.1) is a more generalized form of the strain energy density, which is valid not only for elastic-plastic material behavior but with n=1 also valid for linear elastic material behavior. Therefore, we can derive some quantities for this material as done in the last chapter 3 and 4. In analogy to Eq. (3.11) we get the new generalized I*-integral for elastic-plastic material behavior ∫ σ Δ + = Δ → 1A 1 j 1 i 0 ij a 0 * u n dA n 1 n a 1 I lim . (5.2) In the same way we can still receive in analogy to Eq. (4.10) ∫ ∫ σ + σ − + = σ + = A j ij i,k ij,k i A ij,k i j k u )n dA n 1 1 u n 1 n ( u n dA n 1 n S 0 (5.3) and finally in analogy to Eq. (4.13) ∫ ε ε ∂ σ + = 0A qmlljk m ij i q k u n dA n 1 n T . (5.4) The integrals of the new found quantities Sk and Tk in Eq. (5.3) and (5.4) for this material behavior are also path-independent. 6. Crack Driving Energy and Crack Force In the above considerations, we have examined the basic equation (3.3) and could first derive the previously existing fracture mechanics parameters such as the generalized Irwin's I*-integral expression of the crack closure energy and the generalized Rice's Jk-integral. In particular we found the both new quantities Sk and Tk, where Sk describes the crack front extension and Tk the crack front distortion. Now, we can write the generalized energy release rate (4.11) as follows k k k (S T )l da d = + ∏ − . (6.1) By substituting k k kP S T = + and by setting k = 1 for a special case "pure mode I", we obtain a very important result from the equation (6.1):
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