A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

A physically based gradient plasticity theory

The intent of this work is to derive a physically motivated mathematical form for the gradient plasticity that can be used to interpret the size effects observed experimentally. The step of translating from the dislocation-based mechanics to a continuum formulation is explored. This paper addresses a possible, yet simple, link between the Taylor’s model of dislocation hardening and the strain gradient plasticity. Evolution equations for the densities of statistically stored dislocations and geometrically necessary dislocations are used to establish this linkage. The dislocation processes of generation, motion, immobilization, recovery, and annihilation are considered in which the geometric obstacles contribute to the storage of statistical dislocations. As a result, a physically sound relation for the material length scale parameter is obtained as a function of the course of plastic deformation, grain size, and a set of macroscopic and microscopic physical parameters. Comparisons are made of this theory with experiments on micro-torsion, micro-bending, and micro-indentation size effects.     

Post-necking behaviour modelled by a gradient dependent plasticity theory

The delay of the onset of localization and the post-necking behaviour for stretched thin sheets are determined by three-dimensional effects. Thus, a 2-D finite element analysis based on a local plasticity theory will give a physically unrealistic mesh dependent solution. This, in spite of the fact that the stress state, is essentially two-dimensional. By incorporating a length scale with relation to the thickness of the sheet, it is demonstrated how a 2-D finite element analysis based on a gradient dependent plasticity theory can give a good approximation of the post-necking behaviour. This is illustrated by numerical comparison of results from a full 3-D finite element analysis, with results from a 2-D finite element model based on a finite strain version of a gradient dependent J₂-flow theory. Some numerical problems in the modeling will be discussed briefly.     

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation condition (implying that no long range energy interactions are allowed between the body and the exterior environment) is used to derive the higher order boundary conditions, as well as to ascertain a principle of the plastic power redistribution in which the energy residual plays the role of redistributor and guarantees that the actual plastic dissipation satisfies the second thermodynamics principle. The (nonlocal) Clausius-Duhem inequality, into which the long range stress enters aside the Cauchy stress, is used to derive the thermodynamic restrictions on the constitutive equations, which include the state equations and the dissipation inequality. Consistent with the latter inequality, the evolution laws are formulated for rate-independent models. These are shown to exhibit multiple size effects, namely (energetic) size effects on the hardening rate, as well as combined (dissipative) size effects on both the yield strength (intrinsic resistance to the onset of plastic strain) and the flow strength (resistance exhibited during plastic flow). A friction analogy is proposed as an aid for a better understanding of these two kinds of strengthening effects. The relevant boundary-value rate problem is addressed, for which a solution uniqueness theorem and a minimum variational principle are provided. Comparisons with other existing gradient theories are presented. The dissipation redistribution mechanism is illustrated by means of a simple shear model.     

A nonlocal strain gradient plasticity theory for finite deformations

Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of PDEs and related higher order boundary conditions (including those associated with the moving elastic/plastic boundary). Two fourth-order projection tensor operators, functions of the elastic and plastic strain states, are shown to relate the skew-symmetric parts of the Mandel stress and back stress to the related symmetric parts. Consistent with the thermodynamic restrictions therein derived, the flow laws for rate-independent associative plasticity are formulated in a six-dimensional tensor space in terms of symmetric parts of Mandel stresses and related work-conjugate generalized plastic strain rates. A simple shear problem application is presented for illustrative purposes.     

A unified thermodynamic theory of elasticity and plasticity

A thermodynamics is developed for a unified theory of elasticity and plasticity in infinitesmal strain. The constitutive equations which relate stress and strain deviators are rate type differential equations. When they satisfy a Lipschitz condition, uniqueness for the initial value problem dictates that the stress and strain will be related through elastic relations. Failure of the Lipschitz condition occurs when a von Mises yield condition is achieved: Plastic yield then occurs and the deviator relations turn into the Prandtl-Reuss equations. The plastic yield solution is stable during loading and unstable during unloading. The requirement that the solution followed during unloading be stable dictates entry into an elastic regime. Appropriate thermodynamic functions are constructed. It then appears that stress deviator (not strain deviator) is a viable state variable, and the thermodynamic relations are constructed in terms of a Gibbs function. The energy balance leads to satisfaction of the Clausius-Duhem inequality (and thus the second law of thermodynamics) in an elastic regime because it is shown that in an elastic regime entropy production is caused only by heat flux. During yield, the proper method of differentiating yields entropy production terms in addition to those arising from heat flux. These terms are positive during loading, whence it is concluded that the requirement that a stable solution be followed leads to satisfaction of the Clausius-Duhem inequality during plastic as well as elastic behavior.     

A conventional theory of mechanism-based strain gradient plasticity

There exist two frameworks of strain gradient plasticity theories to model size effects observed at the micron and sub-micron scales in experiments. The first framework involves the higher-order stress and therefore requires extra boundary conditions, such as the theory of mechanism-based strain gradient (MSG) plasticity [J Mech Phys Solids 47 (1999) 1239; J Mech Phys Solids 48 (2000) 99; J Mater Res 15 (2000) 1786] established from the Taylor dislocation model. The other framework does not involve the higher-order stress, and the strain gradient effect come into play via the incremental plastic moduli. A conventional theory of mechanism-based strain gradient plasticity is established in this paper. It is also based on the Taylor dislocation model, but it does not involve the higher-order stress and therefore falls into the second strain gradient plasticity framework that preserves the structure of conventional plasticity theories. The plastic strain gradient appears only in the constitutive model, and the equilibrium equations and boundary conditions are the same as the conventional continuum theories. It is shown that the difference between this theory and the higher-order MSG plasticity theory based on the same dislocation model is only significant within a thin boundary layer of the solid.     

A finite deformation theory of strain gradient plasticity

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.     

A finite deformation second gradient theory of plasticity

Extending the previous work by Chambon et al. [2] to the finite deformation regime, a local second gradient theory of plasticity for isotropic materials with microstructure is developed based on the multiplicative decomposition of the deformation gradient, the additive decomposition of the second deformation gradient and the principle of maximum dissipation.     

On large-strain finite element solutions of higher-order gradient crystal plasticity

A finite-strain higher-order gradient crystal plasticity model accounting for the backstress effect originating from the existence of geometrically necessary dislocations (GNDs) is applied to plane strain finite element analysis. Different element types are tested to seek out an element formulation that is reliable and useful for solving problems involving severe plastic deformation. In the present finite element formulation, the GND density rates are chosen to be additional nodal degrees of freedom. Different orders of shape functions are employed for the interpolation of displacement rates and GND density rates. Their effects on solutions are examined in detail by considering three boundary value problems: a simple shear of a constrained layer (a film), a compression problem with loading surfaces impenetrable to dislocations, and a tension problem involving shear band formation. In all the cases, the formulation in which eight-node elements with reduced integration and four-node elements with full integration are used respectively for displacement rates and the GND density rates gives reasonable solutions. In addition to the discussion on the choice of finite elements, detailed behavior in gradient-dependent solids, such as the accumulation of GND density and the distribution of backstress on each slip system, is investigated by utilizing the reliable computational results obtained.     

On the formulations of higher-order strain gradient crystal plasticity models

Recently, several higher-order extensions to the crystal plasticity theory have been proposed to incorporate effects of material length scales that were missing links in the conventional continuum mechanics. The extended theories are classified into work-conjugate and non-work-conjugate types. A common feature of the former is that existence of higher-order stresses work-conjugate to gradients of plastic strain is presumed and an extended principle of virtual work involving such an additional virtual work contribution is formulated. Meanwhile, in the latter type, the higher-order stress quantities do not appear explicitly. Instead, rates of crystallographic slip are influenced by back stresses that arise in response to spatial gradients of the geometrically necessary dislocation densities. The work-conjugate type and the non-work-conjugate type of theories have different theoretical backgrounds and very unlike mathematical representations. Nevertheless, both types of theories predict the same kind of material length scale effects. We have recently shown that there exists some equivalency between the two approaches in the special situation of two-dimensional single slip under small deformation. In this paper, the discussion is extended to a more general situation, i.e. the context of multiple and three-dimensional slip deformations.     

A simple theory of strain gradient plasticity based on stress-induced anisotropy of defect diffusion

A new strain gradient plasticity theory is formulated to accommodate more than one material length parameter. The theory is an extension of the classical J₂ flow theory of metal plasticity to the micron scale. Distinctive features of the proposed theory as compared to other existing theories are the simplicities of mathematical formulation, numerical implementation and physical interpretation.     

An evaluation of higher-order plasticity theories for predicting size effects and localisation

Conventional plasticity theories are unable to capture the observed increase in strength of metallic structures with diminishing size. They also give rise to ill-posed boundary value problems at the onset of material softening. In order to overcome both deficiencies, a range of higher-order plasticity theories have been formulated in the literature. The purpose of this paper is to compare existing higher-order theories for the prediction of a size effect and the handling of localisation effects. To this end, size effect predictions for foils in bending are compared with existing experimental data. Furthermore, a study of one-dimensional harmonic incremental solutions from a uniform reference state allows one to assess the nature of material localisation as predicted by these competing higher-order theories. These analyses show that only one of the theories considered—the Fleck–Hutchinson strain gradient plasticity theory based upon the Toupin–Mindlin strain gradient framework [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361]—allows one to describe both phenomena. The other theories show either nonphysical size effects or a pathologically localised post-peak response.     

An alternative treatment of phenomenological higher-order strain-gradient plasticity theory

Phenomenological higher-order strain-gradient plasticity is here presented through a formulation inspired by previous work for strain-gradient crystal plasticity. A physical interpretation of the phenomenological yield condition that involves an effect of second gradient of the equivalent plastic strain is discussed, applying a dislocation theory-based consideration. Then, a differential equation for the equivalent plastic strain-gradient is introduced as an additional governing equation. Its weak form makes it possible to deduce and impose extra boundary conditions for the equivalent plastic strain. A connection between the present treatment and strain-gradient theories based on an extended virtual work principle is discussed. Furthermore, a numerical implementation and analysis of constrained simple shear of a thin strip are presented.     

A finite-deformation,gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations

This paper develops a finite-deformation, gradient theory of single crystal plasticity. The theory is based on a system of microscopic force balances, one balance for each slip system, derived from the principle of virtual power, and a mechanical version of the second law that includes, via the microscopic forces, work performed during plastic flow. When combined with thermodynamically consistent constitutive relations the microscopic force balances become flow rules for the individual slip systems. Because these flow rules are in the form of partial differential equations requiring boundary conditions, they are nonlocal. The chief new ingredient in the theory is a free energy dependent on (geometrically necessary) edge and screw dislocation-densities as introduced in Gurtin [Gurtin, 2006. The Burgers vector and the flow of screw and edge dislocations in finite-deformation plasticity. Journal of Mechanics and Physics of Solids 54, 1882].