PSI - Issue 19

Ho Sung Kim / Procedia Structural Integrity 19 (2019) 472–481

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Author name / Structural Integrity Procedia 00 (2019) 000–000

Any radial line emanating from the origin at a given R in CFL diagram (Fig. 2a) is given by � = � � � � � � ���� .

(31)

For T-T segment, a point (e.g. point a in Fig. 2a) on the radial line intersected by the fatigue CFL line (PRCFL) for given N f and R can be obtained by equating Equation (31) to Equation (13) such that, � = � (��) (���) ����� �(��) (���) . (32a) Similarly, for T-C loading, C-T loading, and C-C loading, the PRCFLs are obtained by replacing I (TT) with I (TC) , I (CT) , and I (CC) respectively for given N f and R . Accordingly, a chosen number of cycles ( N f = N fc ) for T-T and T-C loadings with a given R for � is found from Equations (3) and (29) with ��� = 2 � /(1 − ) ; �� = (� �� ) �� �� (���� ) (���) �� �� � /(���) � �� � ��� − 1� + � (33a) (33b) These two equations may be directly used for finding for a given R by numerically solving the equation for a chosen �� , and then can be determined according to Equation (29). A method determining α and β using Equation (33) will be hereafter referred to as one-point method . This method may be efficient but the computing precision should be sometimes considered for accuracy because of the process involving both high powered numbers and small numbers. Another method for finding , which will be hereafter referred to as two-point method , can be found if we want to avoid the weakness of one-point method. This method is to utilise the linearity of the damage rate (Fig. 2b) without solving the equation numerically. It may start by choosing appropriate two numbers of cycles i.e. N f = N f1 (e.g 1000cycles) and N f2 (> N f1 e.g 10 5 cycles) for two points a and b as shown schematically for T-T loading in Fig. 2. The two points in CFL diagram (Fig. 2a) correspond to σ max1 and σ max2 ( σ max1 > σ max2 ) respectively in a diagram of log dD f /dN f versus log σ max (Fig. 2b), which may be obtained from [note ��� = 2 � /(1 − ) ], ��� = �� ����� � (���) , (34) for a given R at which β is to be found. Once σ max1 and σ max2 are obtained, another set of corresponding two values of ΔD f /ΔN f (i.e. ΔD f1 /ΔN f and ΔD f2 /ΔN f ) can be obtained (Fig. 2b) using �� � �� � = �(�� � �� � � � � ) �� � . (35) where � = ( �� + � � � ) − ( �� + � � � ) or � = ( �� + � � � ) − ( �� + � � � ) , �� = ��(�) − ��(�) or �� = ��(�) − ��(�) , ��(�) = 1 − ����(�) / �� at ��(�) = �� + � � � or ��(�) = 1 − ���� (�) / �� at ��(�) = �� + � � � , ��(�) = 1 − ���� (�) / �� at ��(�) (= �� − � � � ) or ��(�) = 1 − ���� (�) / �� at ��(�) (= �� − � � � ) . In numerical calculation, � = �� /100 or � = �� /100 may be sufficiently accurate. From the diagram (Fig. 2b), we find, and for C-T and C-C loadings [note ��� = 2 � /(1 − ) ]; �� = |� �� | �� �� (���� ) (���) �� �� � �/(���) � �� � ��� − 1� + � .