PSI - Issue 7

388 6

Hans-Jakob Schindler / Procedia Structural Integrity 7 (2017) 383–390 H.-J. Schindler / Structural Integrity Procedia 00 (2017) 000–000

E m K K σ ⋅ ⋅ − 2 max

2 min

(13)

δ δ δ ∆ = − =

max

min

f

is expected to include some of these effects of local plasticity, thus to be less sensitive to local contact problems. Cast into the general form of eq. (2), inserting (13) and (9) in (8) leads to

2

n

− 2

  

 ⋅ ∆  

R R (1 ) 1 − − 2

dN da

for R < 0.7

(14)

n

C

K

=

⋅

0

p

2

From (14), S(R) as defined in (2) is identified to be

2

n

2 −

  

  

2

1

R R

− −

for R < 0.7

(15a)

( )

S R

=

2

(1 )

and, from the condition of continuity at R = 0.7,

2

n

2 −

( ) 5.67 =

for R > 0.7

(15b)

S R

In Fig. 3, eqs. (15a-b) are compared to S(R) as given by (3). For R < 0.7 the agreement is nearly perfect, which is remarkable, considering the fact that the two curves are derived independent and on completely different backgrounds. Thus, they confirm each other mutually. For R > 0.7, where the deviation between the two is larger, published experimental data are rare, so it is difficult to judge whether (15b) or (3) is more realistic. 4. Prediction of da/dN-curves As far as the main aim of this investigation is concerned – to capture the influence of R on da/dN in a mathematical equation – eqs. (15a-b) represent the final result. They enable da/dN -curves to be extended such that the R-effect can be accounted for. Furthermore, the presented derivation can be extended readily to predict da/dN analytically. In principle, C p0 in (14) could be determined analogously to the derivation of (12). However, since the phenomenon of crack closure interferes for R < 0.7, and since (12) agrees well with experimental data, it is preferable to determine C P0 in (14) from the condition of continuity of da/dN at the transition from (12) to (14) at R ≈ 0.7. This condition yields

2

n

2 −

2

n

−

1    ⋅ −

  

2 max

2

K

0.088

(16)

C

=

0

P

2

2

2

n

157

−

K

K

⋅

⋅

σ

f

i

i

Using (16) in (14) enables da/dN to be estimated without any da/dN tests, provided n is roughly known, which usually is the case. However, regarding the simplifying assumptions used in the derivation of (16) on one hand and the complexity of the involved physical processes on the other, the prediction cannot be expected to be very accurate, but at least in the correct order of magnitude. Therefore, it is suitable to introduced a non-dimensional calibration factor, C nd , to adjust the predictions to possibly available experimental or specified data. Furthermore, K i can be replaced by the more common and often known material property K Ic . Thus: