PSI - Issue 32

N.V. Boychenko et al. / Procedia Structural Integrity 32 (2021) 326–333 Boychenko N.V./ Structural Integrity Procedia 00 (2019) 000 – 000

328

3

2.2. Conventional mechanism-based strain gradient plasticity theory

Strain gradient plasticity (SGP) theories describe stress fields taking into account the dislocation density distributions and scale effects, since stresses depend not only on strains, but also on the strain gradient. The plasticity zone in the vicinity of crack tip is small and contains high strain gradients, which leads to a significantly higher stress levels comparing to the classical plasticity theories (Shlyannikov et al. (2021)). Currently, the strain gradient theories that enjoyed great popularity in the literature are the following: Mechanism-based strain gradient plasticity (MSG) proposed by Gao et al. (1999, 2000); Strain Gradient Plasticity(SGP) by Fleck and Hutchinson (1997, 2001); unified model of Gudmundson (2004) and Distortion gradient plasticity (DGP) by Gurtin (2004). In present paper CMSG theory developed by Huang et al. (2004) was used. As it mentioned above the CMSG theory does not include high-order terms; the strain gradient effect is realized through an additional plastic module. According to the CMSG theory, the relationship between true stresses and strains under uniaxial tension is described in following form:   N N y p p ref y y E f E                      (4) and the nondimensional function of plastic strain f(ε p ) is determined from the uniaxial stress – strain curve, which can be written as a power law relation:       N p p y f E      (6) In equations (4 – 6), σ y is the yield stress, N is the strain hardening exponent, which varies within ( 0

m

    

    

m

   

   

e 

e 

p

(7)

  





  2 p



p

f

l   



flow

ref

m

   

   



  

3

e

2         kk ij ij K ij

ij 

 



,

(8)

  

  

2



e

flow

  2 P





2

P

2 18

here is the intrinsic material length in the strain gradient plasticity based on parameters of elasticity (shear modulus  ), and atomic spacing (Burgers vector b ),  is an empirical coefficient ranging from 0.3 to 0.5,  e is the effective stress,  P is the effective plastic strain gradient, ij   is the deviatoric strain rate, and m is the rate-sensitivity exponent. flow ref f l       is flow stress, y l b    