13th International Conference on Fracture June 16–21, 2013,Beijing, China -3- martensite transformation [12]. The potential function is defined as f = σe −k =0, (6) where σe denotes the effective Mises stress for kinematic hardening, σij = 3 2 sij − αij ( ) sij − αij ( ) , k is the material resistance against plastic deformation. The backstress is assumed to be αij = 2 3 cL εij p −c NL αij εp, (7) in the nonlinear kinematic hardening model. According to the associated flow rule the plastic strain rate is proportional to the derivative of the potential function, εij p = λ ∂f ∂σij = 3 2 λ sij − αij σe , (8) where the plastic multiplier λ≥0 and the deviatoric stress sij = σij − σmδij . The equivalent plastic strain is defined as εp = εp ∫ dt = λdt ∫ . The plasticity model contains both kinematic hardening and isotropic hardening. Whereas the kinematic hardening takes the conventional form [11], the isotropic hardening is affected by martensite transformation and expressed by the deformation resistance, k = H ε εp +H χ χ, (9) that is, the deformation resistance is linear proportional to plastic strain rate and martensite transformation rate. Both plastic strain hardening modulus, Hε, and phase transformation hardening modulus, Hχ, are functions of stress and strain states. Taking the combined hardening model [11] into account, the material resistance under absence of phase transformation can be written as kε =k0 +H0 1−exp −βε p ( ) ⎡ ⎣ ⎤ ⎦, (10) with the model parameters k0 as initial yield stress of the material, H0 as plastic hardening factor and β hardening exponent. Consequently, the hardening due to plastic deformation can be expressed as Hε = βH0 exp −βε p ( ), (11) The additional phase transformation hardening modulus, Hχ, has to be determined based on experimental observation. Investigation of Beese [10] confirmed a constant modulus Hχ provides sufficient accurate results for steels. Based on the Santacreu model the isotropic hardening can be expressed into k = H ε +Hχ χmax − χ ( )mD Dε p ( )m−1 ⎡ ⎣ ⎤ ⎦ εp, (12) with Hε defined in Eq. (11). Under uniaxial loading condition, the equations can be simplified due to σe = σ− αand εp = εp where σ and εp are the first principal stress/plastic strain, respectively. The stress-plastic strain curve under monotonic tensile testing can be expressed as σy = α+k, (13) where the backstress can be evaluated from α= 2cL 3cNL 1−exp −cNL ε p ( ) ⎡ ⎣ ⎤ ⎦ (14) and the isotropic hardening stress is expressed as ( ) ( ) ( ) 0 0 max 1 exp 1 exp m p p k k H H D χ βε χ ε ⎡ ⎤ ⎡ ⎤ = + − − + − − ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ (15)
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