PSI - Issue 39

D. Fedotova et al. / Procedia Structural Integrity 39 (2022) 419–431 Author name / Structural Integrity Procedia 00 (2019) 000–000

421

3

and Shlyannikov et al. (2020, 2021) also quantified the relationship between the properties of the material and the characteristic size at which the gradient effects noticeably increase the true stresses in the crack tip region. Studies conducted within the framework of phenomenological theories and theories based on dislocation mechanisms have shown that deformations near the crack tip contribute to local hardening and lead to much higher stress levels compared to the classical theory of plasticity. 2. Constitutive equations of conventional mechanism-based strain gradient plasticity theory In this study uses a simplified formulation of the conventional theory of gradient plasticity (CMSGP), developed by Huang et al. (2004). This simplified model is very popular due to its simpler numerical implementation associated with higher-order models. The effects of plastic deformation gradients are manifested through an additional modulus of plasticity, which does not require taking into account stresses and boundary conditions of a higher order. Thus, this model can be implemented numerically as a custom material using standard finite element formulations. CMSG plasticity theory, which does not include high-order terms, is based on the Taylor dislocation model, where the plastic deformation gradient appears only in the constitutional model, and the equilibrium equations and boundary conditions are the same as in traditional continuum theories. According to the CMSG theory, the strain-stress relation under simple uniaxial tension is described by the equation:

N

N

y   E          

y E 

  

  p

p

f      





(1)

ref

y

where  ref is a reference stress of uniaxial tension:   N ref y y E    

(2)

The nondimensional function of plastic strain f is determined by the uniaxial stress-strain curve and for most ductile materials can be written as a power law relation:       N p p y f E      (3) In equations (1-3),  y – yield strength of the material, N – is the plastic work hardening exponent, which varies within the limits of 0 < N <1. The flow stress in CMSG theory is as follows:   2 P P flow ref f l       (4) where l is the intrinsic material length in the strain gradient plasticity based on parameters of elasticity (  ), plasticity (  ref ) and p  - plastic strain gradient; atomic spacing (Burger’s vector b ) for face-centered-cubic polycrystals metals:   2 2 18 y l b     (5) it is worth noting that if the internal characteristic size l is significantly smaller than the characteristic size of the plastic deformation associated with GNDs, then l p  becomes negligible, such that the flow stress degenerates to  ref f ( � � , as in conventional plasticity. For metallic materials, the IML - is indeed on the order of microns, as it is