PSI - Issue 2_A

Grzegorz Lesiuk et al. / Procedia Structural Integrity 2 (2016) 3218–3225 Lesiuk et. al/ Structural Integrity Procedia 00 (2016) 000–000

3221

4

2 0 ˆ ( ) v F B B F B F Q    1

2

(4)

In this figure,  ˆ ( ) v F L estimated linear function of a linear regime of the recorded hysteresis loop,  ˆ ( ) v F Q estimated quadratic function of a non-linear regime of the recorded hysteresis loop. From linear regression the constants A 0 and A 1 will be easily obtained. The values B 0 , B 1 , B 2 are unknown and they should respect the following conditions from the common F k – knee value identification: ˆ ( ) ˆ ( ) v F v F L Q  (5)

( ˆ ( )) Q

dF d v F

ˆ ( ) v F L



(6)

(for F=F

k )

According to the above conditions, we obtain:

k k B B F B F A A F B B F A A F 1 0 2 2 1 0 1 0 2 1 2        k k

(7)

k

The constants B 0 , B 1 , B 2 should be optimal from the mathematical point of view. In order to validity their optimal values, we can find the mentioned constants minimalizing the residual sum of squares  (RSS) defined as:

  

  

  Q for P P v F v for P P v F v 2 2 ( ˆ ( ) ) ( ˆ ( ) )   i i i

1

  N i 1

k







(8)

.

2

(

)

 v v max

min

L

i

i

i

k

The value of F k which corresponds to the minimal value  will be treated as closure load F cl . V max and V min are the maximal and minimal values of COD during each cycle of loading. The described method is easy to automatization during the experiment with a guarantee of the optimal value F cl corresponding to the minimal  . However, if the raw data are strongly influenced by the noise (or wrong tuning of the recorded signals) in that case, the algorithm may choose a wrong value of identified closure load point.

Table 1. Chemical composition of analyzed steel Chemical element C [%]

Mn [%]

Si [%]

P [%]

S [%]

PP – steel (1863)

0.08 0.06 0.09

0.025

0.15 0.17 0.02 N/A

0.245 0.198

0.015 0.025

RS – steel 1850-1900 BC– steel 1850-1900

0.1 0.2

0.03 <0.6

0.03

Typical puddle iron based on Czapliński et al. (2009) Typical mild 19 th century rimmed steel based on Czapliński et al. (2009)

<0.08

<0.4

variable

0.02-0.15

0.2-0.5

Variable

0.03-0.06

0.02-0.15