PSI - Issue 35

Andreas Seupel et al. / Procedia Structural Integrity 35 (2022) 10–17 A. Seupel et al. / Procedia Structural Integrity 00 (2021) 000–000

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2

some aluminum alloys (Clausen et al. (2004)) are well known. In contrast to the aforementioned materials, highly alloyed TRIP-steels exhibit a curve-crossing -phenomenon, i.e., the yield stress is initially elevated with increasing strain rate and falls below the static hardening curve at a certain strain far before the ultimate stress is reached. The underlying mechanical tests are performed at environmental temperatures where the formation of α -martensite prominently occurs in case of quasi-static, isothermal loading. The main assumption of the present paper is that the strong temperature dependence of α -martensite evolution, its influence on work hardening and the interaction with self heating are the reasons for the reported strain rate sensitivity. A phenomenological modeling approach is applied in order to capture the characteristic material response of a particular highly alloyed TRIP-steel at moderate technical strain rates in the range from ˙ t = 4 · 10 − 4 s − 1 to 1 s − 1 . Some recent simulation studies have already addressed similar problems. But the investigations are restricted to single loading states (Andrade-Campos et al. (2005); Pru¨ger et al. (2014); Seupel et al. (2020); Jia et al. (2020)) or do not take a realistic martensite evolution into account (Andrade-Campos et al. (2005); Jia et al. (2020)). Firstly, the model is calibrated against quasi-static tensile and compression tests at di ff erent temperatures. Secondly, thermomechanically coupled FE-simulations at various loading rates are conducted using the full setups of the considered tests in order to incorporate the influence of transient heat conduction. Finally, the model predictions are assessed with help of experimentally recorded stress-strain-curves as well as martensite and local temperature evolutions.

2. Model

As mentioned above a reasonable model for highly alloyed TRIP-steels should take into account (i) rate sensitivity, (ii) martensite evolution and (iii) asymmetric strain hardening as functions of temperature ϑ . The proposed plasticity model is based on an Eulerian description starting from an additive split of the deformation rate D into an elastic and a viscoplastic part, D el and D pl , respectively:

D = D el + D pl .

(1)

A thermo-hypoelastic relation between Cauchy-stress σ and elastic deformation rate D el is assumed

σ = C : D el − α inst δ ˙ ϑ ,

(2)

where the logarithmic rate of the Cauchy stress σ is utilized and δ means the identity tensor of second order. The isotropic elasticity tensor C and the coe ffi cient α inst of instantaneous thermal expansion are assumed to be the same for austenite and α -martensite as well as temperature independent in the considered range. This formulation ensures the existence of a corresponding hyperelastic relation for nearly incompressible behavior, see Bruhns (2020). Viscoplastic behavior is governed by the Norton-type potential Φ and an associated flow rule

σ eq σ y

˙ ε 0 σ y m + 1

( m + 1)

Φ ( σ ) =

(3)

,

˙ ε eq =

= ˙ ε 0

σ eq σ y

m 3 S

2 3

∂ Φ ∂ σ

= ˙ ε eq N ,

D pl : D pl .

D pl =

(4)

2 σ eq

Therein, the dependency on the von Mises stress σ eq = √ 3 / 2 S : S = √ 3 J 2 yields plastic incompressibility, where the stress deviator is introduced as S = σ − 1 / 3tr ( σ ) δ . The Norton-law contains the exponent m and the reference strain