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 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 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 simplified nonlinear theory of the generalized plane strain

A finite deformation theory of plane strain is formulated for transversely isotropic, homogeneous bodies with nonlinear stress-strain law. A new set of simplified field equations, which is valid in the case of some deviations from Hooke's law, is derived systematically with the help of the method of order estimation. For illustration purposes, a circular hole in a body under generalized plane strain is considered, together with the solution of an example problem by perturbation techniques.     

A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations

We propose a deformation theory of strain gradient crystal plasticity that accounts for the density of geometrically necessary dislocations by including, as an independent kinematic variable, Nye's dislocation density tensor [1953. Acta Metallurgica 1, 153-162]. This is accomplished in the same fashion as proposed by Gurtin and co-workers (see, for instance, Gurtin and Needleman [2005. J. Mech. Phys. Solids 53, 1-31]) in the context of a flow theory of crystal plasticity, by introducing the so-called defect energy. Moreover, in order to better describe the strengthening accompanied by diminishing size, we propose that the classical part of the plastic potential may be dependent on both the plastic slip vector and its gradient; for single crystals, this also makes it easier to deal with the “higher-order” boundary conditions. We develop both the kinematic formulation and its static dual and apply the theory to the simple shear of a constrained strip (example already exploited in Shu et al. [2001. J. Mech. Phys. Solids 49, 1361-1395], Bittencourt et al. [2003. J. Mech. Phys. Solids 51, 281-310], Niordson and Hutchinson [2003. Euro J. Mech. Phys. Solids 22, 771-778], Evers et al. [2004. J. Mech. Phys. Solids 52, 2379-2401], and Anand et al. [2005. J. Mech. Phys. Solids 53, 1789-1826]) to investigate what sort of behaviour the new model predicts. The availability of the total potential energy functional and its static dual allows us to easily solve this simple boundary value problem by resorting to the Ritz method.     

An exact solution for a model of pressure-dependent plasticity in an un-steady plane strain process

An incompressible material obeying a pressure-dependent yield condition is confined between two planar plates which are inclined at an angle 2α. The plates intersect in a hinged line and the angle α slowly decreases from an initial value. An initial/boundary value problem for the flow of the material is formulated and solved for the stress and the velocity fields, the solution being in closed form. The material is assumed to obey a special case of the double slip and rotation model, which generalises the classical plastic potential model and is also a variant of the double shearing model. The solution for the velocity field may exhibit sliding or sticking at the plates. Solutions which exhibit sticking may have a rigidly rotating zone in the region adjacent to the plates. It is shown that sliding occurs when the value of α is less than a certain critical value α_c ; that sticking occurs without a rigid zone if α exceeds or equals α_c but is less than a second critical value α₀; and that sticking with a rigid zone adjacent to the plates occurs if α exceeds α₀. The values of α_c and α₀ coincide for a certain range of model parameters. Solutions which exhibit sliding are singular. Qualitative features of the solution found are compared with those of the solution for the classical plastic potential model.     

Singular rigid/plastic solutions in anisotropic plasticity under plane strain conditions

Assuming a rigid plastic material model with arbitrary smooth yield criterion, it is shown that the plane strain solutions are singular in the vicinity of maximum friction surfaces. In particular, some components of the strain rate tensor and thus the equivalent strain rate approach infinity. It is also shown that the exact asymptotic representation of the solution near maximum friction surfaces depends on the shape of the yield contour in the Mohr stress plane.     

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.     

A finite-deformation theory of plasticity

Based on a multiplicative decomposition of local deformation into elastic and plastic deformations general constitutive equations of elastic-plastic materials are proposed. Two alternative approaches are discussed: one in which the elastic deformation is used as an independent variable, and the other in which the stress is one of the independent variables. The appropriate material symmetries are defined, and it is shown that the plastic spin is absent in the theory of isotropic materials. Analysis of a simple extension is given as an example.     

A simple theory of plasticity

An internal-variable model of rate-independent plastic behavior, based on loading-unloading irreversibility, is proposed. The model is compatible with thermodynamics and assumes no yield or loading function, stability postulate or specific nature of the internal variables. It is shown that current theories of plasticity are restricted forms of the proposed theory.     

A hyperbolic theory of plasticity

The finite deformation model which was adopted for investigation was restricted so that it would exhibit non-analytic behaviour. Specifically, discontinuities in the first order partial derivatives of stress and strain were permitted. Mathematical difficulties limited the investigation to the solution of the problems of generalized plane strain and axial symmetry. For both of these problems, it was determined that the analysis yielded stress-strain laws of the flow type. An example was included to illustrate the application of the theory to both of these problems.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

Hypoplasticity theory for granular materials—I: Two-dimensional plane strain exact solutions

Numerous theories have been developed in order to gain a better understanding of the behaviour of granular materials. One such theory, originally developed by the Institute for Soil and Rock Mechanics at the University of Karlsruhe over the last decade or so, is the rational continuum mechanical theory termed hypoplasticity. This theory involves a constitutive law for which the stress-rate is a properly invariant isotropic tensorial function of the stress- and strain-rate tensors, but possesses a non-differentiable dependence on the strain-rate tensor. From a practical perspective, it would be highly desirable to determine simple solutions of hypoplasticity applying to a range of fundamental problems, such as gravity flow in a two-dimensional wedge-shaped hopper. Although this is the original motivation of this study, the complexity of the theory appears to preclude the determination of simple analytical solutions, such as the classical solution of Jenike applying to the Coulomb-Mohr granular solid. In this paper, we undertake a mathematical investigation to determine solutions for two-dimensional steady quasi-static plane strain compressible gravity flow for hypoplastic granular materials. For certain special cases we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solution is not a trivial matter. Three-dimensional axially symmetric solutions analogous to those given in this present study are presented in a companion paper, part II.     

A theory for grain boundaries with strain-gradient plasticity

In this work, the effect of the material microstructural interface between two materials (i.e., grain boundary in polycrystalls) is adopted into a thermodynamic-based higher order strain gradient plasticity framework. The developed grain boundary flow rule accounts for the energy storage at the grain boundary due to the dislocation pile up as well as energy dissipation caused by the dislocation transfer through the grain boundary. The theory is developed based on the decomposition of the thermodynamic conjugate forces into energetic and dissipative counterparts which provides the constitutive equations to have both energetic and dissipative gradient length scales for the grain and grain boundary. The numerical solution for the proposed framework is also presented here within the finite element context. The material parameters of the gradient framework are also calibrated using an extensive set of micro-scale experimental measurements of thin metal films over a wide range of size and temperature of the samples.     

A Physical theory of asymmetric plasticity

Experiments have shown the strong rotation in plastic deformation, which is caused by the disclination, specific arrangement of dislocation and inhomogeneity of the gliding motion of the defects in the microscopic scale. Based on the microscopic mechanism of the rotational plastic deformation, the conservation equation satisfied by the defects motion (dislocation and disclination) has been developed in this paper. Then the diffusion motion of the defects are reduced based on the asymmetric theory of continuum mechanics. By utilizing the maximization procedure for the micro plastic work and a scale-invariance argument, various models of Cosserat-type plasticity are obtained in this manner.     

A general theory for nonlocal softening plasticity of integral-type

This paper deals with a comparison of several models, proposed in the literature, of softening plasticity with internal variables regularized by nonlocal averaging of integral type.     

A microfriction constitutive theory for cyclic plasticity of metals

A constitutive theory of plasticity is presented which models the gradual strain softening in the initial plastic range that is exhibited by most real metals. The theory is an attempt to model the successive movements of many dislocations in a crystal lattice. Some forms of the final equations correspond closely to early empirical expressions.The loading-unloading behaviour of the model is compared with the results of a cyclic loading experiment on a sample of high strength steel at elevated temperature. It is shown that the model simulates the cyclic plastic behaviour of the metal very accurately so that it is possible to represent the phenomenon of strain softening in repeated cyclic plasticity.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.