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.     

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.     

On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation

The purpose of this work is the unified formulation and generalization of selected models for extended, gradient, or “higher-order” crystal plasticity via the application of a recently developed rate variational approach to the formulation of continuum thermodynamic models for history-dependent, inelastic systems. The investigation here includes models which were not originally formulated in a thermodynamic or “work-conjugate” fashion. The approach is based on the formulation of rate potentials for each model whose form is determined by (i) energetic processes via the free energy, (ii) kinetic processes via the dissipation potential, and (iii) the form of the evolution relations for the internal-variable-like quantities upon which the free energy and dissipation potential depend. For the case of extended crystal plasticity, these latter quantities include for example the inelastic local deformation, or dislocation densities. The stationarity conditions of the corresponding rate functional then yield volumetric and surficial balance-like field relations determining in the current context for example the form of momentum balance or that of the generalized glide-system flow rule. With the help of this approach, we derive thermodynamically consistent forms of specific models for extended crystal plasticity. Since most of these were formulated for small deformation, we also investigate their generalization to large deformation with the help of, e.g., form invariance. Among other things, the current rate variational approach implies that, beyond the form of the free energy itself, it is form of the evolution relations for the dislocation densities which is important in determining whether or not higher-order model quantities like the glide-system back stress can be formulated in a thermodynamic fashion.     

Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2

A crystal plasticity finite element code is developed to model lattice strains and texture evolution of HCP crystals. The code is implemented to model elastic and plastic deformation considering slip and twinning based plastic deformation. The model accounts for twinning reorientation and growth. Twinning, as well as slip, is considered to follow a rate dependent formulation. The results of the simulations are compared to previously published in situ neutron diffraction data. Experimental results of the evolution of the texture and lattice strains under uniaxial tension/compression loading along the rolling, transverse, and normal direction of a piece of rolled Zircaloy-2 are compared with model predictions. The rate dependent formulation introduced is capable of correctly capturing the influence of slip and twinning deformation on lattice strains as well as texture evolution.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void

The effects of void size and hardening in a hexagonal close-packed single crystal containing a cylindrical void loaded by a far-field equibiaxial tensile stress under plane strain conditions are studied. The crystal has three in-plane slip systems oriented at the angle 60° with respect to one another. Finite element simulations are performed using a strain gradient crystal plasticity formulation with an intrinsic length scale parameter in a non-local strain gradient constitutive framework. For a vanishing length scale parameter the non-local formulation reduces to a local crystal plasticity formulation. The stress and deformation fields obtained with a local non-hardening constitutive formulation are compared to those obtained from a local hardening formulation and to those from a non-local formulation. Compared to the case of the non-hardening local constitutive formulation, it is shown that a local theory with hardening has only minor effects on the deformation field around the void, whereas a significant difference is obtained with the non-local constitutive relation. Finally, it is shown that the applied stress state required to activate plastic deformation at the void is up to three times higher for smaller void sizes than for larger void sizes in the non-local material.     

A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin

In the small deformation range, we consider crystal and isotropic “higher-order” theories of strain gradient plasticity, in which two different types of size effects are accounted for: (i) that dissipative, entering the model through the definition of an effective measure of plastic deformation peculiar of the isotropic hardening function and (ii) that energetic, included by defining the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., [Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]). In order to compare the two modellings, we recast both of them into a unified deformation theory framework and apply them to a simple boundary value problem for which we can exploit the Γ-convergence results of [Bardella, L., Giacomini, A., 2008. Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity. J. Mech. Phys. Solids 56 (9), 2906–2934], in which the crystal model is made isotropic by imposing that any direction be a possible slip system. We show that the isotropic modelling can satisfactorily approximate the behaviour described by the isotropic limit obtained from the crystal modelling if the former constitutively involves the plastic spin, as in the theory put forward in Section 12 of [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568]. The analysis suggests a criterium for choosing the material parameter governing the plastic spin dependence into the relevant Gurtin model.     

Strain gradient effects on microscopic strain field in a metal matrix composite

The purpose of this paper is to demonstrate the improved modeling accuracy of a finite-deformation strain gradient crystal plasticity formulation over its classical counterpart by conducting a joint experimental and numerical investigation of the microscopic details of the deformation of a whisker-reinforced metal-matrix composite. The lattice rotation distribution around whiskers is obtained in thin foils using a TEM technique and is then correlated with numerical predictions based on finite element analyses of a unit-cell of a single crystal matrix containing a rigid whisker. The matrix material is first characterized by a classical, scale-independent crystal plasticity theory. It is found that the classical theory predicts a lattice rotation distribution with a spatial gradient much higher than experimentally measured. A strain gradient crystal plasticity formulation is then applied to model the matrix. The strain gradient formulation accounts for both strain hardening and strain gradient hardening. The deformation thus predicted exhibits a strong dependence on the size of the whisker. For a constitutive length scale comparable to the whisker diameter, the spatial gradient of the lattice rotation is several times lower than that predicted by the classical theory, and hence correlates significantly better with the experimental results.     

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.     

A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier

A phenomenological, flow theory version of gradient plasticity for isotropic and anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379-1406]. Both energetic and dissipative stresses are considered in order to develop a kinematic hardening theory, which in the absence of gradient terms reduces to conventional J₂ flow theory with kinematic hardening. The dissipative stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition with associated plastic flow. The theory includes interfacial terms: elastic energy is stored and plastic work is dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate stress quantities at the interface. Uniqueness and extremum principles are constructed for the solution of boundary value problems, for both the rate-dependent and the rate-independent cases. In the absence of strain gradient and interface effects, the minimum principles reduce to the classical extremum principles for a kinematically hardening elasto-plastic solid. A rigid-hardening version of the theory is also stated and the resulting theory gives rise to an extension to the classical limit load theorems. This has particular appeal as previous trial fields for limit load analysis can be used to generate immediately size-dependent bounds on limit loads.     

Dislocation-based micropolar single crystal plasticity: Comparison of multi- and single criterion theories

Two new formulations of micropolar single crystal plasticity are presented within a geometrically linear setting. The construction of yield criteria and flow rules for generalized continuum theories with higher-order stresses can be done in one of two ways: (i) a single criterion can be introduced in terms of a combined equivalent stress and inelastic rate or (ii) or individual criteria can be specified for each conjugate stress/inelastic kinematic rate pair, a so-called multi-criterion theory. Both single and multi-criterion theories are developed and discussed within the context of dislocation-based constitutive models. Parallels and distinctions are made between the proposed theories and some of the alternative generalized crystal plasticity models that can be found in the literature. Parametric numerical simulations of a constrained thin film subjected to simple shear are conducted via finite element analysis using a simplified 2-D version of the fully 3-D theory to highlight the influence of specific model components on the resulting deformation under both loading and unloading conditions. The deformation behavior is quantified in terms of the average stress-strain response and the local shear strain and geometrically necessary dislocation density distributions. It is demonstrated that micropolar single crystal plasticity can qualitatively capture the same range of behaviors as slip gradient-based models, while offering a simpler numerical implementation and without introducing plastic slip rates as generalized traction-conjugate velocities subject to an additional microforce balance.     

Couple stresses in crystalline solids: origins from plastic slip gradients, dislocation core distortions, and three-body interatomic potentials

In the presence of plastic slip gradients, compatibility requires gradients in elastic rotation and stretch tensors. In a crystal lattice the gradient in elastic rotation can be related to bond angle changes at cores of so-called geometrically necessary dislocations. The corresponding continuum strain energy density can be obtained from an interatomic potential that includes two- and three-body terms. The three-body terms induce restoring moments that lead to a couple stress tensor in the continuum limit. The resulting stress and couple stress jointly satisfy a balance law. Boundary conditions are obtained upon stress, couple stress, strain and strain gradient tensors. This higher-order continuum theory was formulated by Toupin (Arch. Ration. Mech. Anal. 11 (1962) 385). Toupin's theory has been extended in this work to incorporate constitutive relations for the stress and couple stress under multiplicative elastoplasticity. The higher-order continuum theory is exploited to solve a boundary value problem of relevance to single crystal and polycrystalline nano-devices. It is demonstrated that certain slip-dominated deformation mechanisms increase the compliance of nanostructures in bending-dominated situations. The significance of these ideas in the context of continuum plasticity models is also dwelt upon.     

On the evolution of lattice deformation in austenitic stainless steels—The role of work hardening at finite strains

In this work, a three dimensional crystal plasticity-based finite element model is presented to examine the micromechanical behaviour of austenitic stainless steels. The model accounts for realistic polycrystal micromorphology, the kinematics of crystallographic slip, lattice rotation, slip interaction (latent hardening) and geometric distortion at finite deformation. We utilise the model to predict the microscopic lattice strain evolution of austenitic stainless steels during uniaxial tension at ambient temperature with validation through in situ neutron diffraction measurements. Overall, the predicted lattice strains are in very good agreement with those measured in both longitudinal and transverse directions (parallel and perpendicular to the tensile loading axis, respectively). The information provided by the model suggests that the observed nonlinear response in the transverse {200} grain family is associated with a competitive bimodal evolution of strain during inelastic deformation. The results associated with latent hardening effects at the microscale also indicate that in situ neutron diffraction measurements in conjunction with macroscopic uniaxial tensile data may be used to calibrate crystal plasticity models for the prediction of the inelastic material deformation response.     

Effects of microscopic boundary conditions on plastic deformations of small-sized single crystals

The finite deformation version of the higher-order gradient crystal plasticity model proposed by the authors is applied to solve plane strain boundary value problems, in order to obtain an understanding of the effect of the higher-order boundary conditions. Numerical solutions are carried out for uniaxial plane strain compression of a single crystal block and for uniform pure bending of a single crystal foil. The compressed block has loading surfaces that are penetrable or impenetrable to dislocations. This allows for a study of the two types of higher-order boundaries available, and a significant effect of higher-order boundary conditions on the overall deformation mode of the block is observed. The bent foil has free surfaces through which dislocations can go out of the material, and we observe a strong size-dependent mechanical response resulting from the surface condition assumed.     

Fracture of bicrystal metal/ceramic interfaces: A study via the mechanism-based strain gradient crystal plasticity theory

Two continuum mechanical models of crystal plasticity theory namely, conventional crystal plasticity theory and mechanism-based crystal plasticity theory, are used to perform a comparative study of stresses that are reached at and ahead of the crack tip of a bicrystal niobium/alumina specimen. Finite element analyses are done for a stationary crack tip and growing cracks using a cohesive modelling approach. Using mechanism-based strain gradient crystal plasticity theory the stresses reached ahead of the crack tip are found to be two times larger than the stresses obtained from conventional crystal plasticity theory. Results also show that strain gradient effects strongly depend on the intrinsic material length to the size of plastic zone ratio (l/R₀). It is found that the larger the (l/R₀) ratio, the higher the stresses reached using mechanism-based strain gradient crystal plasticity theory. An insight into the role of cohesive strength and work of adhesion in macroscopic fracture is also presented which can be used by experimentalists to design better bimaterials by varying cohesive strength and work of adhesion.     

A mathematical basis for strain-gradient plasticity theory—Part I: Scalar plastic multiplier

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems.     

A new finite element method for strain gradient theories and applications to fracture analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J₂ deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.     

Numerical modeling of formability of extruded magnesium alloy tubes

In this paper, a constitutive framework based on a rate-dependent crystal plasticity theory is employed to simulate the large strain deformation phenomena in hexagonal closed-packed (HCP) metals such as magnesium. The new framework is incorporated into in-house codes. Simulations are performed using the new crystal plasticity model in which crystallographic slip and deformation twinning are the principal deformation mechanisms. Simulations of various stress states (uniaxial tension, uniaxial compression and the so-called ring hoop tension test) for the magnesium alloy AM30 are performed and the results are compared with experimental observations of specimens deformed at 200 °C. Numerical simulations of forming limit diagrams (FLDs) are also performed using the Marciniak–Kuczynski (M–K) approach. With this formulation, the effects of crystallographic slip and deformation twinning on the FLD can be assessed.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I

A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the positive-definite tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations.Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.