PSI - Issue 42

Lucas Mangas Araújo et al. / Procedia Structural Integrity 42 (2022) 1591–1599 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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2.2. Gao-Based Model with Mixed Kinematic Hardening In this contribution, a Gao-based model with mixed hardening is proposed. One opts for a mixed hardening approach because it not only comprises all the phenomena studied but also allows the particularization to one specific hardening type (isotropic and kinematic) by simply setting to zero some material parameters. Furthermore, the model is presented in small deformation context. One proposes a yield function based on Gao’s equivalent stress: (3) In which , σ and ε̅ are the relative stress tensor, the uniaxial yield stress and accumulated plastic strain respectively. It is worth observing that in this case, neither nor are deviatoric due to model pressure sensitivity. Therefore, may be expressed as the sum of a spherical η and deviatoric components: (4) (5) which is a 4-parameter equation, namely σ 0 , , ∞ and . The backstress tensor evolution ̇ is assumed to follow Armstrong-Frederick kinematic hardening law: (6) where and denote the kinematic hardening modulus and saturation coefficient respectively. 2.3. State Update Procedure The discretization of the mathematical model leads to a problem that can be summarized as follows: given the total strain increment ∆ and the value of internal variables (∗) at the time instant (i.e. at the beginning of the increment), one desires to compute the updated internal variables (∗) +1 at the end of the increment (i.e. at +1 ). The following linearized system of equations needs to be solved: The hardening law chosen is the relation proposed by Kleinermann and Ponthot [6]: