PSI - Issue 52

Ding Zhou et al. / Procedia Structural Integrity 52 (2024) 430–437 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

433

4

To determine T 0 , the measured toughness of the 0.18T specimen has to be calculated first. Brittle toughness being intrinsically specimen size dependent, the toughness of 0.18T specimen must then be adjusted to an equivalent 1T value (see below). Fracture toughness K Jc was estimated based on the elastic-plastic theory following the recommendations of ASTM-E1921 (2022). K Jc was obtained from the value J-integral at the onset of cleavage fracture J c calculated from the load-displacement curves: ʹ ͳ  = − (2) where E is Young’s modulus and ν is Poisson’s ratio, and J c consists of the elastic component of the J-integral, J e , and the plastic component, J p : ‡ ’ = + (3) Constraint loss is addressed in the ASTM-E1921 standard by defining a specimen maximum measuring capacity with Eq. (4): ( ) Ͳ̴ ʹ ͳ ›• Ž‹‹– „   = − (4) where b 0 is the initial ligament length and sys the yield stress. According the ASTM-E1921 standard, M = 30 and this value should be sufficient to ensure that a high level of constraint is maintained along the crack front length. In addition, the standard provides a toughness size adjustment if specimen sizes different from 1T are used. This correction accounts for the statistical size effect associated with the crack front length and reads: ʹ ͳȀͶ ͳ ‹ ʹ ‹ ͳ ሺ ሻ ƒ†Œ—•–‡† −   = + −       with K min = 20 MPa m 1/2 (5) The standard also assumes that the cumulative failure probability of a dataset at a given temperature follows Eq. (6) if K Jc < K Jc_limit . 1/2 . For values greater than the limit, K Jc > K Jc_limit , it is assumed that loss of constraint affects the measured K Jc by increasing it to an apparent toughness, which in turn is reflected in a deviation of the data from Eq. (6). However, a value above the limit still carries useful information: the toughness of the specimen was at least equal or greater than the limit because before reaching the limit it did not lose constraint and did not break. The standard combines the equations above to determine T 0 by means of the maximum likelihood method. This derives to Eq. (7), which is solved for T 0 by iteration. ( ) 4 min 0 min 1 exp  − − Jc P K K K K K K −             −  (6) The temperature dependence in Eq. (6) appears in K 0 . K min is taken as constant and equal to 20 MPam

N 

N 

4

(

20) exp(0.019(

))

K

T T −

−

exp(0.019(

))

T T −

(7)

( )

0

Jc i

i

0

i

0

−

=

i 

5

11 77exp(0.019( +

))

T T −

(11 77exp(0.019( +

)))

T T −

0

i

0

i

1

1

i

i

=

=

N = is the number of tested specimens.

•

• K Jc(i) = K Jc(i) if K Jc(i) < K Jc_limit , or K Jc(i) = K Jc_limit if K Jc(i) > K Jc_limit . •  i = 1.0 if K Jc < K Jc_limit ), and  i = if if K Jc > K Jc_limit .