PSI - Issue 39

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

420

2

1. Introduction In recent decades, more and more attention of scientists and specialists has been focused on the behavior of metals at the microlevel, where the effects of gradient plasticity are manifested on small scales with respect to the structure of the material, which leads to a sharp local increase in true stresses. The problem is that classical theories of plasticity of continuum mechanics are unable to evaluate these micro-scale effects, since their constitutional models do not contain an internal parameter of characteristic length. A wide range of microlevel experiments, such as torsion of copper wire Fleck et al. (1994), Guo et al. (2017) or bending and limited shear of thin foil Stölken and Evans (1998), Mu et al. (2014) have shown that metals under loading above the yield strength exhibit strong dimensional effects when the characteristic length scale is on the order of one micron. In this case, plastic deformation gradients occur either due to inhomogeneous deformation of the material, or due to the method of loading. Thus, the need was formulated to develop a new generally accepted theory to substantiate large-scale effects and to link destruction at the micro level with atomistic processes of destruction in plastic materials.

Nomenclature a

crack length specimen width Young`s modulus Poisson`s ratio

strain hardening exponent specimen thickness intrinsic material length

n

w E

t l

ν

applied force

F

yield stress

plastic stress intensity factor

Ϭ y

K p

K 1 , K 2 elastic stress intensity factors for mode I/II equivalent elastic stress intensity factor K eqv

applied force angle effective stress

α

Ϭ e

The first theories of gradient plasticity were proposed by Fleck and Hutchinson (1993, 1997, 2001) for numerical accounting of scale effects. This theory corresponds to the mathematical structure of the theory of elasticity, taking into account the terms of high orders with deformation gradients. Out of the need to correlate the dimensions of deformations and deformation gradients, a new parameter of intrinsic material length l , was introduced, which is introduced into the plasticity of the deformation gradient and is associated with the density of dislocations. This parameter is considered as the intrinsic length of the material depending on the microstructure of the material, the size of which varies from a tenth of a micron to ten microns. The theory of Nix and Gao (1998) partially clarified the meaning of the parameter of the intrinsic material length l , introduced by Fleck and Hutchinson (1993), and also pointed out the need to supplement gradient theories of plasticity with experimental laws based on the analysis of dominant deformation mechanisms. The Nix and Gao analysis is based on the Taylor dislocation model (1938), which interacts between shear strength and dislocation density in the material. Later, Gao et al. (1999) supplemented the formulation with a more detailed analysis, which was called mechanism-based strain-gradient plasticity (MSG). In MSG theory, the total dislocation density is calculated through the sum of statistical and geometric components, and the effects of strain gradients themselves become significant when these densities have the same order of magnitude. Smaller samples lead to the presence of a stronger strain gradient, and, consequently, to higher densities of geometrically necessary dislocations, since the flow stresses depend on the total dislocation density. MSG theory is a direct attempt to establish a connection between continuum mechanics and the atomistic structure of a material. In 2004, Huang et al. presented a simplified theory by Gao et al. (1999), having justified and excluded from its members of high orders associated with rotational components, and this theory was called the Conventional mechanism-based strain gradient plasticity. This theory of lower orders, as well as MSG, is based on the Taylor dislocation model. In CMSGP theory, the plastic deformation gradient appears only in the constitutional model of the behavior of the medium, and the equilibrium equations and boundary conditions coincide with the traditional theories of continuum mechanics. Recently, Martínez-Pañeda (2015, 2017) implemented this lower-order scheme for estimating gradient effects, since it does not experience convergence problems in numerically solving complex problems of fracture mechanics, unlike the higher-order model (MSG). Recently, Martínez-Pañeda et al. (2016, 2019)