13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- cyclic plastic-adaptation is accomplished completely. On the way of the cyclic plastic-adaptation process, as it is shown by the segment gh, each maximum values of the principal stress become equal to that of the nominal stress path, respectively. This is an important matter which should be mentioned specially; for, it means that, if the cyclic stress shown by the segment gh is applied to the un-notched material, the cyclic plastic-adaptation process at the notch root can be reproduced at the surface layer of the un-notched material. Therefore, it can be said that the segment gh corresponds to the equivalent cyclic stress condition between the notched and un-notched specimens. In the present study, the stress ratio RN ∗ is newly defined for multi-axial stress condition other than the usual nominal stress ratio RN . The RN ∗ is formulated as a ratio of the algebraic sum of the x/y-coordinate value for each of the peak point p and q of the rectangle paqb with the line segment ab as a diagonal line (henceforth, the basic equations are shown with the three-dimensional form); RN ∗ = σ1maxN −Δσ1N ( ) + σ2maxN −Δσ2N ( ) + σ3maxN −Δσ3N ( ) σ1maxN + σ2maxN + σ3maxN (4) Next, the equivalent cyclic stress ratio REQ is formulated as the ratio of the algebraic sum of the x/y-coordinate values for each of the peak points p and r of the rectangle pgrh with the line segment gh as the diagonal line, and the expression is moreover simplified by using Eq. (4); REQ = σ1maxN − Kt eqΔσ1N ( ) + σ2maxN − Kt eqΔσ2N ( ) + σ3maxN − Kt eqΔσ3N ( ) σ1maxN + σ2maxN + σ3maxN = RN ∗ − K t eq −1 ( ) 1− RN ∗ ( ) (5) Last, the equivalent mean stress σmeanEQ is formulated as the algebraic sum of the x/y-coordinate values of the middle points m of the rectangle pgrh with the line segment gh as a diagonal line; σmeanEQ = σ1maxN + σ2maxN + σ3maxN ( ) − Kt eq Δσ1N +Δσ2N +Δσ3N ( ) 2 = σ1maxN + σ2maxN + σ3maxN ( ) − Kt eq σ1aN + σ2aN + σ3aN ( ) (6) where, σiaN (i=1-3) are the principal stress amplitude of the nominal condition. Saying again, REQ and σmeanEQ behave just like a hydrostatic stress ratio and hydrostatic mean stress in the process of the cyclic plastic-adaptation, regardless whether the stress components constituting them synchronizes or not, respectively. For the un-notch condition of Kt eq =1, σmeanEQ of Eq. (6) is coincident with the mean hydrostatic stress that Sines introduced into his criterion [5]. 2.3. Applicability of the REQ-ratio to fatigue strength diagramming In order to prove the applicability of the equivalent cyclic stress ratio REQ and the equivalent mean stress σmeanEQ , the fatigue strength of the specimen containing a comparatively large size notch (from the reason why influence of the notch size effect is little) is plotted on the diagram where the abscissa shows REQ and σmeanEQ and the ordinate does the notch root stress range ΔσNR. Fig. 3 shows the fatigue test result concerning the notched and un-notched round-bar specimen of SM400 (low carbon structural steel) subject to cyclic axial loading under mean stress of tension side [1] and suject to rotating bending. Fig. 3(a) and (b) represent the REQ-based and σmeanEQ -based fatigue strength diagram, respectively. The depth of the circumferential notch t is 3 mm for axial cyclic loading test and 1.5 mm for rotating bending test. The notch root radius ρ is changed in the range
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