PSI - Issue 5

Kaveh Samadian et al. / Procedia Structural Integrity 5 (2017) 1245–1252 Kaveh Samadian/ Structural Integrity Procedia 00 (2017) 000 – 000

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4

Since equivalent plastic strain is a strain tensor invariant, it can be assumed that the coordinate system is oriented along the direction of principal strains (denoted as ε p 1 , ε p 2 and ε p 3 ) and then Eq. (1) can be written as follows:

2 3 2 (( ) ( ) ( ) ) 3 P P P      2 2 1 2

p eq 



(2)

Since experimental strain measurements such as DIC do not allow to distinguish between plastic and total strain, the contribution of elasticity to total strain is neglected in this study. In other words, Eq. (2) is used for equivalent total (rather than plastic) strain. Similar assumptions have been adopted in other studies supported by full-field strain analysis, e.g. Fagerholt et al. (2012); Hertelé et al. (2016). Finally, by assuming incompressibility and substituting ε 3 with – ( ε 1 + ε 2 ), the following relation is obtained for equivalent strain  eq .

4 (( ) ( )

2

2   2 

)





1 

1 2  

(3)

eq

3

In conclusion, Eq. (3) allows to calculate equivalent strain on the basis of principal surface strains, which can be readily measured by means of DIC.

3.2. Algorithm to determine and characterize bands of maximum equivalent strain

The procedure to calculate equivalent strain was applied within the area of interest, discretized into a grid of points at which principal strain values were extracted. Then, by comparing equivalent strain values of all grid points within a vertical line perpendicular to the load direction, the point of maximum equivalent strain in each line was obtained (Figure 2). Connecting these points for adjacent vertical lines creates a trajectory that follows the band of maximum equivalent strain. This process was carried out by employing a devoted code in MATLAB ® .

α 1 α 2

Strain band’s angle analysis

Midpoint

Concentration of equivalent plastic strain

Extracting nodes equivalent plastic strain from a grid

Connection of maximum points on the grid and dividing them .

Curvature analysis

Fig. 2. Graphical summary of procedure to obtain and characterize bands of maximum equivalent strain connecting notch tips.

In this study, strain bands are depicted in the coordinate system of the un-deformed specimen (i.e., the grid itself does not move), for the sake of simplicity. Subsequently, bands of maximum strain connecting notch tips were divided into two separate bands at specimen mid width, as if the double edge notched specimen acts as two tangent single edge notched specimens. Then, each band was mathematically described by means of two techniques: linear regression analysis with the aim to determine the average angles of the bands with respect to the longitudinal direction ( α 1 and