PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 1237–1244 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000 5 material used in this work under goes a “linear elastic - linear plastic” hardening behaviour, the hardening parameter can be defined as (Owen & Hinton 1980), = 0 1 − ( 0 0 ⁄ ) (14) where 0 and 0 signify the elastic and the tangential modulus in the reference direction, respectively. In the perspective of the RPIMmeshless approach, the material behaviour is modelled in the form of an incremental relation between the incremental stress vector and the strain increment. In order to force the stress to return to the yield surface, the “backward - Euler” procedure (Crisfield 1991) is considered. This work also considers a variation of the Newton-Raphson non-linear solution method - initial stiffness method combined with an incremental solution (KT0) - in order to solve the non-linear equations (Crisfield 1991). Within this approach, the stiffness matrix is calculated only once: in the first iteration of the first load increment. The non-linear solution algorithm is schematically shown in Box 1. 1241

Initial Stiffness matrix: 0 Force vector: f Tolerance value: tol

Input:

Increment : i

Iteration: n i) = 0−1 ii) = iii) ( ) ≤ 0

Displacement field

→ ∆ = ( ) > 0 → ∆ = ) ( ) ≥ → = ( ) ( ) <

Stress field

Check the yield function = −1 ∆

iv) = − ∫ ∆ Ω Ω

v) (

Residual force

Go to the next iteration n+1

Go to the next increment i+1

End of the test.

Box 1. Non-linear solution algorithm.

2. Experimental procedure A bi-failure specimen was used and loaded by a standard servo-hydraulic system, 10 ( ) MTS-812 tensile machine and a frequency of = 5 ( ) , the specimen geometry is illustrated in Fig. 1-a. The specimen thickness is = 0.8 ( ) . The sample was coated in white color and sprinkled with black dots to create an arbitrary template for DIC examination. The specimen is linked to the tensile machine by specific rods passing through holes positioned in the extremes of the specimen. A computer system operated to receive images from two synchronized DIC cameras and the load cell. The cameras were thus calibrated using a 12-by-9 calibration pattern, 2.5 (mm). It is adequate to rely on the 2D DIC acquisition. The material properties for Dual-phase steel DP600 are presented in Table 1. The experimental test was performed and captured data was reserved in the DIC software. To process the gathered data, in the DIC processing software, a central section was considered and two discrete points with the following coordinates, 1 = (12.5,7.10) and 2 = (12.5, 31.86) were thus identified. The vertical displacement field was therefore determined on the selected points identified as 2 and 1 . The problem domain and selected points are presented in Fig. 2-a. The reaction force received from load cell in terms of the vertical displacement variation ( 21 = 2 − 1 ) was obtained as Fig. 2-b shows.