A finite deformation theory of strain gradient plasticity

doi:10.1016/S0022-5096(01)00020-5

Journal of the Mechanics and Physics of Solids

Volume 50, Issue 1, January 2002, Pages 81-99

https://doi.org/10.1016/S0022-5096(01)00020-5 Get rights and content

Abstract

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.

Introduction

Recent experiments have repeatedly shown that the material behaviour displays strong size dependence at the micron or sub-micron scales. For example, in micro-indentation and nano-indentation hardness experiments, the measured indentation hardness increases by a factor of 2 or even 3 as the depth of indentation decreases to microns or sub-microns (Nix, 1989; De Guzman et al., 1993; Stelmashenko et al., 1993; Atkinson, 1995; Ma and Clarke, 1995; Poole et al., 1996; McElhaney et al., 1998; Suresh et al., 1999). Fleck et al. (1994) have observed in micro-torsion of thin copper wires that the scaled shear strength increases by a factor of 3 as the wire diameter decreases from 170 to $12 μm$ . Stolken and Evans (1998) have found similar strength increase in micro-bending of thin nickel foils as the foil thickness decreases from 50 to $12.5 μm$ . In particle-reinforced metal-matrix composites, Lloyd (1994) has observed substantial increase in work hardening as the particle diameter is reduced from 16 to $7.5 μm$ at a fixed particle volume fraction of 15%. Nan and Clarke (1996) have made similar observations. Classical plasticity theories, however, cannot explain this size dependence observed at the micron or sub-micron scale because their constitutive models possess no internal material lengths. At the micron scale, however, there are still hundreds of dislocations such that there should be a continuum plasticity theory (but not classical plasticity) that can describe the collective behavior of these dislocations.

For this reason, strain gradient plasticity theories have been developed and are intended for applications to materials and structures whose dimension controlling plastic deformation falls roughly within a range from a tenth of a micron to $10 μm$ (e.g., Fleck and Hutchinson 1993, Fleck and Hutchinson 1997; Fleck et al.,1994; Gao et al., 1999; Acharya and Bassani, 2000; Acharya and Beaudoi, 2000; Huang 2000a, Huang 2000b; Dai and Parks, 2001). Strain gradients have been introduced in the constitutive model. From dimensional considerations, the internal material length parameters have been introduced to scale the strain gradients and these length parameters are determined to be on the order of microns or sub-microns either from the aforementioned micro-scale experiments of indentation, torsion and bending, or from the dislocation models (Nix and Gao, 1998).

The higher-order stresses serve as the work conjugates of strain gradients in the theories of Fleck and Hutchinson (1997), Gao et al. (1999), and Huang 2000a, Huang 2000b. These theories have shown reasonable agreement with the micro-scale experiments. However, they are limited to infinitesimal deformation, while the aforementioned micro-scale experiments involve rather large strains and rotations. For example, the surface strain in the micro-torsion experiment exceeded 100% (Fleck et al., 1994), while the strain near the indenter tip was also very large. Moreover, finite deformation is important in the analysis of some micro-scale phenomena where strain gradient effects are expected to be significant, such as plastic flow localization and crack tip fields.

The purpose of this paper is to develop a finite deformation theory of strain gradient plasticity. We begin with a review of the infinitesimal deformation theories in Section 2, including the phenomenological theory of strain gradient plasticity (Fleck and Hutchinson, 1997) and the mechanism-based strain gradient (MSG) plasticity theory derived from the Taylor (1938) dislocation model. We then generalize them to finite deformation theories in the reference configuration, and study the size effect in micro-indentation hardness experiments.

Section snippets

Review of strain gradient plasticity theories: infinitesimal deformation

The theories of strain gradient plasticity for infinitesimal deformation (Fleck and Hutchinson, 1997; Gao et al., 1999; Huang 2000a, Huang 2000b) are summarized in this section.

Strain gradient tensor

Let $X_{I}, X_{II}, X_{III}$ be the Cartesian Lagrangian coordinates in the reference (initial) configuration $R$ , and x₁,x₂,x₃ the Eulerian coordinates in the current configuration r. The corresponding gradient operators in the reference and current configurations $R$ and r are denoted by $∇ <$ and $∇ >$ , respectively, $ϕ ∇ < = ∂ ϕ ∂ X_{A} e_{A}, ∇ < ϕ= e_{A} ∂ ϕ ∂ X_{A},ϕ ∇ > = ∂ ϕ ∂ x_{i} e_{i}, ∇ > ϕ= e_{i} ∂ ϕ ∂ x_{i},$ where ϕ denotes any scalar or tensor fields, $e_{A} (A= I,II,III)$ and $e_{i} (i=1,2,3)$ are unit vectors along coordinate axes X_A and x_i, respectively. The operators

Constitutive law of strain gradient plasticity

The finite-deformation constitutive relations are given for the Fleck–Hutchinson strain gradient plasticity (Fleck and Hutchinson, 1997) and MSG plasticity theories (Gao et al., 1999; Huang 2000a, Huang 2000b) in the reference configuration. Therefore, the constitutive laws automatically meet the requirement of frame indifference.

Strain rate and strain gradient rate tensors

As shown in the next section, we need the strain rate and strain gradient rate tensors to derive the equilibrium equations in finite deformation from the principle of virtual work. The rate of deformation tensor $d$ is defined as the symmetrization of velocity gradient $v ∇ >$ , $d = 12 (v ∇ > + ∇ > v),d_{ij} = 12 (v_{i,j} +v_{j,i}).$ It is well known in continuum mechanics that $d (or E ̇)$ is the covariant push-forward (or pull-back) of the rate of Green strain $E ̇ (or d)$ , $d = F − T · E ̇ · F −1 =(F − T F − T)∗∗ E ̇, d_{ij} = ∂ X_{A} ∂ x_{i} ∂ X_{B} ∂ x_{j} E ̇_{AB},$ $E ̇ = F^{T} · d · F =(F^{T} F^{T}$

Principle of virtual work: equilibrium equations and boundary conditions

The equilibrium equations and boundary conditions are established from the principle of virtual work in this section. The solid, which occupies the region $V$ in the reference configuration $R$ , deforms to the region v in the current configuration r. Similarly, the surface $A$ deforms to a, and the edge $C$ (if present) formed by the intersection of two smooth surface segments $A^{(1)}$ and $A^{(2)}$ deforms to the edge c formed by the corresponding deformed surface segments a⁽¹⁾ and a⁽²⁾. Let $N$ and $n$ be the

Finite deformation modeling of the micro-indentation experiment

Micro-indentation is a widely used experimental method to probe mechanical properties of materials at micron or sub-micron scales. These experiments have repeatedly shown that the indentation hardness depends strongly on the depth of indentation, the smaller the harder. Such size effect has been attributed to strain gradient hardening associated with the geometrically necessary dislocations. Accordingly, the theories of strain gradient plasticity (Fleck and Hutchinson 1993, Fleck and Hutchinson

Summary

We have developed the finite deformation theories of strain gradient plasticity by generalizing the infinitesimal deformation theories of Fleck and Hutchinson (1997), Gao et al. (1999) and Huang 2000a, Huang 2000b. All kinematic relations and the constitutive laws are given in the reference configuration, and therefore meet the requirement of frame indifference. Based on the principle of virtual work, the equilibrium equations and traction boundary conditions are established in the current

Acknowledgements

KCH acknowledges the support from the Ministry of Education, China. YH acknowledges NSF (grant CMS-0084980 and a supplement to grant CMS-9896285 from NSF International Program). HG acknowledges NSF (grant CMS-9979717). The support from NSFC is also acknowledged.