The flow theory of mechanism-based strain gradient plasticity

Abstract

The flow theory of mechanism-based strain gradient (MSG) plasticity is established in this paper following the same multiscale, hierarchical framework for the deformation theory of MSG plasticity in order to connect with the Taylor model in dislocation mechanics. We have used the flow theory of MSG plasticity to study micro-indentation hardness experiments. The difference between deformation and flow theories is vanishingly small, and both agree well with experimental hardness data. We have also used the flow theory of MSG plasticity to investigate stress fields around a stationary mode-I crack tip as well as around a steady state, quasi-statically growing crack tip. At a distance to crack tip much larger than dislocation spacings such that continuum plasticity still applies, the stress level around a stationary crack tip in MSG plasticity is significantly higher than that in classical plasticity. The same conclusion is also established for a steady state, quasi-statically growing crack tip, though only the flow theory can be used because of unloading during crack propagation. This significant stress increase due to strain gradient effect provides a means to explain the experimentally observed cleavage fracture in ductile materials

Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

“…The result based on this parameter ratio only produces a negligible difference to the result based on a realistic value of the ratio, l e =l, from 10 À3 to 10 À2 [33,34]. Additionally, the behaviors of the present steady-state crack growth problem can compare with those of the stationary crack problem predicted by Qiu et al [44]. Fig.…”

“…The MSG flow theory has been derived by Qiu et al [44] following the derivation line of the MSG deformational theory [33,34]. The procedures can be summarized as follows: (1) Consider that a micro-scale cell, with the cell size l e and with the geometrically necessary dislocation density q G , is homogenized to an equivalent solid characterized by the elastic-plastic theory plus a hardening law based on the Taylor model [45].…”

“…r 0 is a reference stress; e p is accumulative plastic strain; b is the Burgers vector; a is an empirical coefficient in the Taylor model and is taken the value around 0.3 for the conventional metals; K ¼ E=3ð1 À 2mÞ and l ¼ E=2ð1 þ mÞ are the elastic bulk modulus and shear modulus, respectively; l e is the cell size, and from analysis of Gao et al [33], Huang et al [37] and Qiu et al [44],…”

“…Two vastly different cell sizes, l e =l ¼ 10 À8 and 10 À2 , have led to only a small difference of the corresponding results. Therefore, one can take the ratio l e =l as zero for simplicity [37,44]. However, for the present steady-state problem solved by adopting the iteration scheme as formulated in Eq.…”

Decrease of Nano-hardness at Ultra-low Indentation Depths in Copper Single Crystal

“…An interesting method to simulate the lattice rotation and an inhomogeneous distribution of the plastic zone under the indenter tip based on the molecular dynamics was presented in [20]. The indentation size effect was also explained using flow theory of mechanism based strain gradient plasticity [21]. In [22] it has been reported that for indentations performed with spherical tips having different radii, the size effect manifested as an increase of stiffness of load-penetration curve (in scaled P/R 2 -h/R reference system) when the tip radius decreases.…”