PSI - Issue 8

Matteo Loffredo / Procedia Structural Integrity 8 (2018) 265–275 M. Lo ff redo / Structural Integrity Procedia 00 (2017) 000–000

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updated kinematic hardening constants that, for ensuring the concurrence of σ − uniaxial curves for various p , shall be function of p itself (an example of dependency is shown in Sec.4).

IF [ Ω n ≥ 1]

(5)

σ y n + 1 = σ y n

C (U) 1 C (U) 2

(U) 1 n d p − C

(U) 2 ( α − β ) d p α ( p ) − β =

(1 − exp( C (U)

2 ( p − p )) IF [ Ω n ≥ 1]

(6)

d α = C

β n = α n − 1 IF [ Ω n Ω n − 1 ]

(7)

4. Tuning of the model with experimental data

In this section an example of the identification of model parameters defined in Sec.3 is proposed. The identification has been performed using data coming from cycles of type 1 (Figg.5 and 6), since specimens used for cycle 2, having been subjected to several closed loop loading, may have experienced a form of cyclic softening. Then data of Fig.5 for cycle of type 1 have been processed, removing elastic strain subsequent to the saw tooth shaped unloading, and converting them to true stress-true strain (Fig.7). The material exhibits an abrupt tensile yielding followed by a low level softening (less than 5% of σ (L) y ), then the material has been assumed, for sake of simplicity (the aim is describing properly the unloading which impacts mostly the residual stress in an autofrettaged component), to be globally elastic perfectly plastic in the loading phase. This is equivalent to impose that kinematic hardening and isotropic softening are compensating themselves in the loading part (Eq.8).

− C (L) 1

(L) 1

(L) 2

(L) 2

= H

C

= H

(8)

The identification of H (L) y was defined at the condition when the not linearity produce a discrepancy of 0 . 005% in the strain for all unloading curves. This resulted in the the yielding line of Fig.7, basing on which H (L) 1 and H (L) 2 have been evaluated by least squares using Eq.3. For the unloading phase C (U) 1 and C (U) 2 are function of p . Eq.9 shows the dependencies assumed for C (U) 1 and C (U) 2 in the case of elastic-perfectly plastic loading phase used in the present paper. The first dependency states that , given the exponential trend of the unloading σ − curve, the hardening e ff ect shall expire in an strain interval multiple of p . The second dependeny states that, for all unloading curves, the asymptotic stress level is common. 1 and H (L) 2 has been perfomed on the basis of the unloading σ − curve. σ (U)

γ 1 p + γ 2

C (U) 2

C (U) 1

(U) 2

= ( γ 3 − 2 σ y ( p )) C

(9)

=

Then firstly C (U) 1 (U) 2 have been identified separately for each unloading curve (then for each p ) by least squares using Eq.6 (uniaxial). Then the constants γ i of Eq.9 have been tuned basing on this set of data by least squares. and C