On dissipative gradient effect in higher-order strain gradient plasticity: the modelling of surface passivation

Abstract

The phenomenological flow theory of higher-order strain gradient plasticity proposed by Fleck and Hutchinson (J. Mech. Phys. Solids, 2001) and then improved by Fleck and Willis (J. Mech. Phys. Solids, 2009) is used to investigate the surface-passivation problem and micro-scale plasticity. An extremum principle is stated for the theory involving one material length scale. To solve the initial boundary value problem, a numerical scheme based on the framework of variational constitutive updates is developed for the strain gradient plasticity theory. The main idea is that, in each incremental time step, the value of the effective plastic strain is obtained through the variation of a functional in regard to effective plastic strain, provided the displacement or deformation gradient. Numerical results for elasto-plastic foils under tension and bending, thin wires under torsion, are given by using the minimum principle and the numerical scheme. Implications for the role of dissipative gradient effect are explored for three non-proportional loading conditions: (1) stretch-passivation problem, (2) bending-passivation problem, and (3) torsion-passivation problem. The results indicate that, within the Fleck–Hutchinson–Willis theory, the dissipative length scale controls the strengthening size effect, i.e. the increase of initial yielding strength, while the surface passivation gives rise to an increase of strain hardening rate.

Graphic abstract

Appendix

Three material length scales can be introduced to the Fleck–Hutchinson–Willis theory as seen in Eq. (10). We restrict attention to the simplest model, containing a single material length scale. Following Danas et al. [62], we introduce a particular choice for the relative magnitude of the length scales $\ell_{I}$ by allowing

$$\ell_{1} = \sqrt {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}} \ell ,\;\;\ell_{2} = \sqrt {{1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-0pt} 6}} \ell \;\;{\text{and}}\;\;\ell_{3} = {\ell \mathord{\left/ {\vphantom {\ell 2}} \right. \kern-0pt} 2}.$$

(44)

With this choice, one can write the length scales $\ell_{I}$ in terms of a single length scale $\ell$.

For the cases of foil tension and bending, the generalized effective plastic strain rate Eq. (10) is written as

$$\dot{E}_{\text{P}} = \sqrt {\dot{\varepsilon }_{\text{P}}^{2} + \left( {\tfrac{1}{2}\ell_{1}^{2} + \tfrac{8}{3}\ell_{3}^{2} } \right)\dot{\varepsilon }_{\text{P}}^{{{\prime }2}} } ,$$

(45)

where the second material length parameter $\ell_{2}$ does not enter the expression. By (44), Eq. (45) is reduced to be

$$\dot{E}_{\text{P}} = \sqrt {\dot{\varepsilon }_{\text{P}}^{2} + \ell^{2} \dot{\varepsilon }_{\text{P}}^{{{\prime }2}} } .$$

(46)

In Eq. (22), the coefficients read

$$\omega_{1} = 1, \, \omega_{2} = \ell^{2} .$$

(47)

For the case of wire torsion, the generalized effective plastic strain rate Eq. (10) is expressed as

$$\dot{E}_{\text{P}} = \sqrt {c_{1} \dot{\varepsilon }_{\text{P}}^{2} + c_{2} \dot{\varepsilon }_{\text{P}} \dot{\varepsilon }_{\text{P}}^{{\prime }} + c_{3} \dot{\varepsilon }_{\text{P}}^{{{\prime }2}} } ,$$

(48)

with

$$\left\{ \begin{aligned} & c_{1} = 1 + \left( {\tfrac{1}{2}\ell_{1}^{2} + 4\ell_{2}^{2} } \right)r^{ - 2} \hfill \\ & c_{2} = \left( { - \ell_{1}^{2} + 4\ell_{2}^{2} } \right)r^{ - 1} \hfill \\ & c_{3} = \left( {\tfrac{1}{2}\ell_{1}^{2} + 4\ell_{2}^{2} } \right) \hfill \\ \end{aligned} \right.,$$

(49)

where $\ell_{3}$ is not involved in the expression. By (44), Eq. (48) is reduced to be

$$\dot{E}_{\text{P}} = \sqrt {\left( {1 + \ell^{2} r^{ - 2} } \right)\dot{\varepsilon }_{\text{P}}^{2} + \ell^{2} \dot{\varepsilon }_{\text{P}}^{{{\prime }2}} } .$$

(50)

In Eq. (22), the coefficients become

$$\omega_{1} = 1 + \ell^{2} r^{ - 2} , \, \omega_{2} = \ell^{2} .$$

(51)