A Brief Note on the Nix–Gao Strain Gradient Plasticity Theory

A Borokinni,Dabiao Liu

doi:10.3390/met8090708

Abstract

The mathematical nature of the flow rule for the strain gradient plasticity theory proposed by Nix and Gao (W.D. Nix and H. Gao, J Mech Phys Solids 46(3), 411(1998)) is discussed based on the paradigm developed by Gurtin and Anand (M.E. Gurtin and L. Anand, J Mech Phys Solids 57 (3), 405 (2009)). It is shown that, when investigated on the basis of Gurtin–Anand theory, the Nix–Gao flow rule is a combination of constitutive equations for microstresses, balance law, and a constraint. As an accessory, we demonstrate that the strain gradient term introduced in the model is energetic. The results are obtained by combining a virtual-power principle of Fleck and Hutchinson, and the free-energy imbalance under isothermal conditions.

Highlights

Many experiments at small scales, including nano/microindentation [1], torsion of thin metallic wires [2,3,4,5], and bending of thin foils [6,7], have clearly demonstrated a strong size-dependent strengthening associated with non-uniform plastic deformation
In order to elucidate whether the plastic-strain gradient term involved is energetic or dissipative, the general flow rule established by Gurtin and Anand [34] is adopted, and a form of quadratic defect energy is assumed
It has been shown that the Nix–Gao theory—in the light of Gurtin and Anand [34]—is a combination of (i) constitutive relations (see Equation (31)) for the microstresses, (ii) balance law given by Equation (30), and (iii) an additional constraint given by Equation (36)

Summary

A Brief Note on the Nix–Gao Strain Gradient

Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China Received: 25 August 2018; Accepted: 6 September 2018; Published: 10 September 2018

Introduction

Kinematic Relations

Virtual-Power Principle and Force Balances

Mathematical Nature of the Nix–Gao Flow Rule

Conclusions