PSI - Issue 47

Ezio Cadoni et al. / Procedia Structural Integrity 47 (2023) 630–635 Author name / Structural Integrity Procedia 00 (2023) 000–000

632

3

Fig. 2. Comparison between input and output load for 5 and 15mm gauge length.

C 0 L sp.

t

δ input ( t ) − δ output ( t ) L sp.

eng ( t ) =

0 [ I ( t ) − R ( t ) − T ( t )] dt

(4)

=

C 0 L sp .

˙ eng ( t ) =

[ I ( t ) − R ( t ) − T ( t )]

(5)

where A sp. is the area of specimen gauge length cross-section and L sp . is the length of specimen gauge-length. For the three gauge lengths analysed, the travel time of the wave through the specimen is small in comparison to the duration of the test. Thus, the specimen can be considered as being in load equilibrium at its ends and in a uniform stress state created by the many wave reflections taking place at the ends of the specimen. The condition of force equilibrium at both ends of the specimens is expressed by: F input ( t ) = F output ( t ) or I ( t ) + R ( t ) = T ( t ) (6) Leading to the following simplified relationships for the stress, strain and strain-rate in the specimen:

L sp .

t

A bar A sp .

2 · C 0

2 · C 0 L sp .

σ eng ( t ) = E bar ·

T ( t ) and eng ( t ) = −

0 R ( t ) dt and ˙ eng ( t ) = −

R ( t )

(7)

The simplified relationships (7) can be used for the determination of the stress-strain-strain rate curves after verifi cation of the condition (6) of force equilibrium at the ends of the specimen. Such verification can be done by checking that at each time t of the test the algebraic sum of the incident pulse I ( t ), of the reflected pulse R ( t ), and of transmitted pulse T ( t ), is equal to zero. Figure 1b shows that equilibrium is reached in the first half of the elastic field.

3. E ff ect of the gauge length of the specimens

In order to determine the gauge length dependancy three di ff erent lengths have been considered (5, 10 and 15 mm) as shown in Fig. 1a. By imposing the same preloading level in the pretensioned bar and by using the equations 7 the stress versus strain curves shown in Fig. 3a are obtained Because of the increasing gauge length the resulting strain-rate obtained by using equation 7 is not the same. In fact, observing Fig. 3b it can be seen the comparison of the stress and strain-rate versus time curves. The values measured for the three gauge length were 460, 285 and 185 s − 1 for 5, 10 and 15 mm, respectively. Standard ISO 26203-1 (2018) defines the average strain-rate as the value of the strain-rate obtained by averaging between strains of 1% (0.01) and 10% (0.1) as:

0 . 10 − 0 . 01 t 10 − t 1

(8)

˙ ave =

where t 10 and t 1 are the time at a strain of 10% and 1%, respectively. Following equation 8 a value of 566 s − 1 is obtained while the average value of the strain-rate in the same interval is 588 s − 1 , a tinydi ff erence due to the single value read at the two strain level. To obtain the target strain-rate of 600 s − 1 , for longer gauge length case, it is necessary to increase