PSI - Issue 61

Zhichao Wei et al. / Procedia Structural Integrity 61 (2024) 26–33 Z. Wei et al. / Structural Integrity Procedia 00 (2024) 000–000

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3. Theoretical framework

The proposed material model is briefly summarized in this section. Please refer to (Wei et al., 2022, 2023, 2024) for further details. The plastic material parameters, describing the isotropic and kinematic hardening rules, are identified using the uniaxial monotonic and cyclic tension-compression tests. The original experimental data and fitting strat egy are described in Wei et al. (2022). Furthermore, the isotropic and kinematic softening parameters are inversely calibrated by analyzing the micro-simulations with an initial micro-void.

3.1. Plasticity

As highlighted by Wei et al. (2022) for the investigated aluminum alloy EN AW 6082-T6, the compressive yield stress di ff ers from the tensile one by comparing the monotonic uniaxial compression and tension tests. Furthermore, the Bauschinger e ff ect is notably observed in both uniaxial tension-compression and one-axial shear cyclic tests. Thus, the Drucker-Prager yield criterion incorporated with the combined hardening law

=

1 2 dev( ¯ T − ¯ α ) · dev( ¯ T − ¯ α ) − ¯ c (1 − a ¯ c

f pl

tr( ¯ T − ¯ α )) = 0

(2)

is used to model the plastic deformations in the undamaged configurations, where ¯ T denotes the e ff ective Kirchho ff stress tensor, ¯ α represents the e ff ective back stress tensor, ¯ c characterizes the current e ff ective yield stress, and a / ¯ c = 32TPa − 1 is the constant hydrostatic coe ffi cient, which can be determined by analyzing the uniaxial monotonic tensile and compressive experimental data (Wei et al., 2022). The double Voce strain-hardening law ¯ c = c 0 + Q 1 (1 − e − p 1 γ ) + Q 2 ξ (1 − e − p 2 γ ) (3) describes the change in the shape of the current yield surface, where c 0 denotes the initial yield stress, and Q 1 = 74 . 93MPa, Q 2 = 21 . 32MPa, p 1 = 8 . 96 and p 2 = 676 . 01 are material constants. Additionally, ξ characterizes the di ff erent hardening behavior at the onset of plasticity under a wide range of strain states, see Wei et al. (2022, 2023). Moreover, the modified non-linear Chaboche kinematic hardening law (Chaboche and Rousselier, 1983; Voyiadjis et al., 2013; Wei et al., 2022) is given by ˙¯ α 1 = b 1 χ ˙¯ H pl − b 2 χ ˙ γ ¯ α 1 , ˙¯ α 2 = b 3 ˙¯ H pl − b 4 ˙ γ ¯ α 2 , and ˙¯ α 3 = b 5 ˙¯ H pl − (1 − cos 2 θ ) b 6 ˙ γ ¯ α 3 , (4) and it is introduced to predict the transformation of the current yield surface under revere loading condition, where b 1 = 61250MPa, b 2 = 1750, b 3 = 895MPa, b 4 = 15, b 5 = 115 MPa, and b 6 = 7 . 5 are material constants, ˙¯ H pl denotes the rate of e ff ective plastic strain tensor, χ = 0 . 8 e − 300 γ + 0 . 2 represents an exponential Decay function for the cumulative equivalent plastic strain γ , and θ is angle parameter, see (Wei et al., 2022) for further details.

3.2. Damage

The stress-state-dependent damage condition incorporating the combined softening rule

= ˆ α tr( T − α ) + ˆ β 1 2

f da

dev( T − α ) · dev( T − α ) − ˜ σ = 0

(5)

is introduced, where T and α are the Kirchho ff stress tensor and the damage back stress tensor, respectively. ˜ σ repre sents the current equivalent softening stress, as well as ˆ α and ˆ β are the stress-state-dependent coe ffi cients. Furthermore, the damage strain evolution equation in damaged configurations is given by ˙ H da = ˙ µ (˜ α 1 √ 3 1 + ˜ β ˜ N ) , (6)