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

12

3

rate ˙ ε 0 as parameters. Isotropic strain hardening is described with the yield stress

σ y ( r , z , ϑ ) = σ 0 ( ϑ ) + Z 1 m z ( z ) + H ( r )

(5)

which consists of three contributions: A temperature dependent initial yield stress σ 0 ( ϑ ), an empirical mixture rule m z ( z ) depending on martensitic volume fraction z and a conventional strain hardening part H ( r ) depending on the hardening variable r : σ 0 = Σ 0 exp ( c 1 ( ϑ − c 2 )) , m z = exp Z 2 z Z 3 − 1 , H =   H 0 r q r ≤ r c H 0 r q c + H inf 1 − exp − H 0 qr q − 1 c H inf ( r − r c ) else. (6) The strain driven evolution of the volume fraction of shear bands f sb and α -martensite z , respectively, are formulated according to the model of Olson and Cohen (1975):

˙ f sb = α oc (1 − f sb ) ˙ ε eq ˙ z = (1 − z ) β oc n oc f

(7) (8)

˙ f sb .

( n oc − 1) sb

This multi-axial formulation of the Olson-Cohen-model depends on the equivalent plastic strain rate ˙ ε eq given by Eq. (4). In order to capture the influence of temperature and stress state on martensite evolution the parameters α oc and β oc are introduced as functions of ϑ and stress triaxiality h = 1 / 3tr ( σ ) /σ eq , whereas n oc is a constant. The detailed formulations of α oc and β oc can be found in (Seupel et al., 2020, Section 22.3.3.3). The asymmetry in strain hardening is implicitly evoked due to the stress state dependent martensite evolution a ff ecting the mixture rule in the hardening function. However, the experimental tendencies are not fully captured. Hence, a Lode-angle influence on the evolution of the hardening variable r is incorporated as proposed by Seupel and Kuna (2017):

˙ r = L (cos 3 φ, ϑ, z ) ˙ ε eq .

(9)

The hardening variable is proportional to the equivalent plastic strain ˙ ε eq . The impact of the Lode-parameter cos 3 φ

3 √ 3 2

J 3

1 3

tr ( S · S · S )

cos 3 φ =

(10)

J 3 =

,

3 2

J

2

and therewith the magnitude of the asymmetry is highly influenced by the underlying deformation mechanisms (tem perature dependent, martensite content) which is modeled by the pre-factor

b 0 exp   −

2  

exp   −

2  

ϑ b1

z b1

z − z b0

1 2

ϑ − ϑ b0

(1 − cos 3 φ ) .

L (cos 3 φ, ϑ, z ) = 1 +

(11)