Intra-granular plastic slip heterogeneities: Discrete vs. Mean Field approaches

In this paper, we derive the mechanical fields (internal stresses, elastic energy) arising from the presence of an inelastic distortion field representing a typical intra-granular “microstructure” as the one observed during the plastification of metallic polycrystals. This “microstructure” is due to the formation of discrete intra-granular plastic slip heterogeneities characterized by at least two internal lengths: the first one is the individual grain size which represents a stochastic parameter inherent to the processing route (prior working, annealing), and, the second one is the spatial distance between active slip lines or slip bands associated with inhomogeneous plastic slip in the interior of grains. These internal lengths can be observed and measured using conventional experimental techniques (EBSD, TEM, AFM). The micro-mechanical modeling of the mechanical fields associated with plastic slip events inside grains is performed with two different assumptions. The first one is based on the well-known Eshelby’s problem of plastic inclusion where only the grain diameter is considered as internal length scale. This classical method considers homogeneous plastic distortion in the grain and leads to a uniform and grain size independent total strain field in the grain. The second method accounts for a non-uniform plastic distortion in the grain characterized by its discrete nature and the two aforementioned internal lengths. Both methods consider grains as spherical inclusions with a given diameter embedded in a homogeneous medium. For the second method, plastic slip is constrained by grain boundaries seen as impenetrable obstacles to dislocations. Thus, plastic strain is embodied by distributions of discrete circular glide loops. After writing the field equations and the free energy of the medium, a micro-mechanical formulation based on the Fourier transform method is developed. It is then found that in contrast with the mean-field approach, the internal stress fields as well as the elastic energy corresponding to different dislocation configurations depend on internal lengths associated to the deformed medium. Different possible configurations associated with intra-granular plastic flow due to circular glide dislocation loops are analyzed. Finally, the results are discussed with respect to the grain size dependence of the flow strength and the Bauschinger effect for plastically deforming polycrystals and perspectives to develop new micro–macro transition schemes accounting for internal length scales are sketched out.     

Phase-field slip-line theory of plasticity

A variational approach to determine the deformation of an ideally plastic substance is proposed by solving a sequence of energy minimization problems under proper conditions to account for the irreversible character of plasticity. The flow is driven by the local transformation of elastic strain energy into plastic work on slip surfaces, once that a certain energetic barrier for slip activation has been overcome. The distinction of the elastic strain energy into spherical and deviatoric parts is used to incorporate in the model the idea of von Mises plasticity and isochoric plastic strain. This is a “phase field model” because the matching condition at the slip interfaces is substituted by the evolution of an auxiliary phase field that, similar to a damage field, is unitary on the elastic phase and null on the yielded phase. The slip lines diffuse in bands, whose width depends upon a material length-scale parameter.Numerical experiments on representative problems in plane strain give solutions with noteworthy similarities with the results from classical slip-line field theory, but the proposed model is much richer because, accounting for elastic deformations, it can describe the formation of slip bands at the local level, which can nucleate, propagate, widen and diffuse by varying the boundary conditions. In particular, the solution for a long pipe under internal pressure is very different from the one obtainable from the classical macroscopic theory of plasticity. For this case, the location of the plastic bands may be an insight to explain the premature failures that are sometimes encountered during the manufacturing process. This practical example enhances the importance of this new theory based on the mathematical sciences.     

Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).     

Sub-Grain Scale Digital Image Correlation by Electron Microscopy for Polycrystalline Materials during Elastic and Plastic Deformation

Damage during loading of polycrystalline metallic alloys is localized at or below the scale of individual grains. Quantitative assessment of the heterogeneous strain fields at the grain scale is necessary to understand the relationship between microstructure and elastic and plastic deformation. In the present study, digital image correlation (DIC) is used to measure the strains at the sub-grain level in a polycrystalline nickel-base superalloy where plasticity is localized into physical slip bands. Parameters to minimize noise given a set speckle pattern (introduced by chemical etching) when performing DIC in a scanning electron microscope (SEM) were adapted for measurements in both plastic and elastic regimes. A methodology for the optimization of the SEM and DIC parameters necessary for the minimization of the variability in strain measurements at high spatial resolutions is presented. The implications for detecting the early stages of damage development are discussed.     

The effect of contact conditions and material properties on the elasticity terminus of a spherical contact

The fundamental problem of elastic–plastic normally loaded contact between a deformable sphere and a rigid flat is analyzed under perfect slip and full stick conditions for a wide range of the sphere mechanical properties. The effect of these properties on failure inception is investigated by finding the critical interference and normal loading as well as the location of the first plastic yield or brittle failure. The analysis is based on the analytical Hertz solution under frictionless slip condition and on a numerical solution under stick condition. The failure inception is determined by using either the von Mises criterion of plastic yield or the maximum tensile stress criterion of brittle failure. For small values of the Poisson’s ratio the behavior in stick, when high tangential stresses prevail in the contact interface, is much different than in slip. For high values of the Poisson’s ratio the tangential stresses under stick condition are low and the behavior of the failure inception in stick and slip is similar.     

Physics-based modeling for partial slip behavior of spherical contacts

A physics-based modeling approach for partial slip behavior of a spherical contact is proposed. In this approach, elastic and elastic–plastic normal preload and preload-dependent friction coefficient models are integrated into the Cattaneo–Mindlin partial slip solution. Partial slip responses to cyclic tangential loading (fretting loops) obtained by this approach are favorably compared with experiments and finite element results from the literature. In addition to load-deformation curves, tangential stiffness of the contact and energy dissipation per fretting cycle predictions of the models are also provided. Finally, the critical assumptions of elastically similar bodies, smooth contact surface and negligible adhesion, and limitations of this physics-based modeling approach are discussed.     

A numerical study of crack tip constraint in ductile single crystals

In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids.     

On invariance properties of an extended energy balance

Gradient plasticity theories are of utmost importance for accounting for size effects in metals, especially on the grain scale. Today, there are several methods used to derive the governing equations for the additional degrees of freedom in gradient plasticity theories. Here, the equivalence between an extended principle of virtual power and an extended energy balance is shown. The energy balance of a Boltzmann continuum is supplemented by contributions based on a scalar-valued degree of freedom. It is considered to be invariant with respect to a change of observer. This yields unambiguously the existence of a corresponding micro-stress vector, which is presumed from the outset in the context of an extended principle of virtual power. A thermodynamically consistent nonlocal evolution equation for the additional, scalar-valued degree of freedom is obtained by evaluation of the dissipation inequality in terms of the Clausius–Duhem inequality. Partitioning the nonlocal flow rule yields a partial differential equation, often referred to as micro-force balance. The approach presented is applied to derive a slip gradient crystal plasticity theory regarding single slip. Finally, the distribution of the plastic slip is exemplified with respect to a laminate material consisting of an elastic and an elastoplastic phase.     

Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels

Micro-macro scale transition theories were developed to model the inelastic behaviour of polycrystals starting from the local behaviour of the grains. The anisotropy of the plastic behaviour of polycrystalline metals was essentially explained by taking into account the crystallographic textures. Issues like plastic heterogeneities due to grain size dispersion, involving the Hall-Petch mechanism at the grain scale, were often not taken into account, and, only the role of a mean grain size was investigated in the literature. Here, both sources of plastic heterogeneities are studied using: (i) experimental data from EBSD measurements and tensile tests, and, (ii) a self-consistent model devoted to elastic-viscoplastic heterogeneous materials. The results of the model are applied to two different industrial IF steels with similar global orientation distributions functions but different mean grain sizes and grain size distributions. The coupled role of grain size distributions and crystallographic textures on the overall tensile behaviour, local stresses and strains, stored energy and overall plastic anisotropy (Lankford coefficients) is deeply analyzed by considering different other possible correlations between crystallographic orientations and grain sizes from the measured data.     

Scaling laws of nanoporous gold under uniaxial compression: Effects of structural disorder on the solid fraction,elastic Poisson's ratio,Young's modulus and yield strength

In this work the relationship between the structural disorder and the macroscopic mechanical behavior of nanoporous gold under uniaxial compression was investigated, using the finite element method. A recently proposed model based on a microstructure consisting of four-coordinated spherical nodes interconnected by cylindrical struts, whose node positions are randomly displaced from the lattice points of a diamond cubic lattice, was extended. This was done by including the increased density as result of the introduced structural disorder. Scaling equations for the elastic Poisson's ratio, the Young's modulus and the yield strength were determined as functions of the structural disorder and the solid fraction. The extended model was applied to identify the elastic–plastic behavior of the solid phase of nanoporous gold. It was found, that the elastic Poisson's ratio provides a robust basis for the calibration of the structural disorder. Based on this approach, a systematic study of the size effect on the yield strength was performed and the results were compared to experimental data provided in literature. An excellent agreement with recently published results for polymer infiltrated samples of nanoporous gold with varying ligament size was found.     

Plastic size effect analysis of lamellar composites using a discrete dislocation plasticity approach

Plastic size effect analysis of lamellar composites consisting of elastic and elastic-plastic layers is performed using a discrete dislocation plasticity approach, which is based on applying periodic homogenization to the superposition method for discrete dislocation plasticity. In this approach, the decomposition of displacements into macro and perturbed components circumvents the calculation of superposing displacement fields induced by dislocations in an infinitely homogeneous medium, resulting in two periodic boundary value problems specialized for the analysis of representative volume elements. The present approach is verified by analyzing a model lamellar composite that includes edge dislocations fixed at interfaces. The plastic size effects due to dislocation pile-ups at interfaces are also analyzed. The analysis shows that, strain hardening in elastic-plastic layers arises depending on two factors, namely the thickness and stiffness of elastic layers; and the gap between slip planes in adjacent elastic-plastic layers. In the case where the thickness of elastic layers is several dozen nm, strain hardening in elastic-plastic layers is restrained as the gap of the slip planes decreases. This particular effect is attributed to the long range stress due to pile-ups in adjacent elastic-plastic layers.     

Giant faults in deformed Gum Metal

We have experimentally characterized and theoretically described plastic flow localization in Gum Metal, a special titanium alloy with high strength, low Young’s modulus, excellent cold workability and low resistance to shear in certain crystallographic planes. The electron transmission microscopy experiments demonstrate that plastic flow is localized in giant faults – macroscopic planar defects carrying very large plastic strains (thousand percent or more) – in deformed Gum Metal. Also, regions with highly inhomogeneous elastic strains and varying crystal lattice orientation are experimentally observed in the vicinity of giant faults. A theoretical model is suggested describing the generation of giant faults as a process resulting from generation and evolution of nanodisturbances (nanoscopic planar areas of local shear) in Gum Metal. It is shown that giant faults can effectively nucleate and evolve in Gum Metal, and their intersection with grain boundaries produces both elastic strain accumulation and inhomogeneities of crystal lattice orientation. This behavior of giant faults is expected to be essential for excellent cold ductility of high-strength Gum Metal.     

Plane-strain discrete dislocation plasticity incorporating anisotropic elasticity

The increasing application of plane-strain testing at the (sub-) micron length scale of materials that comprise elastically anisotropic cubic crystals has motivated the development of an anisotropic two-dimensional discrete dislocation plasticity (2D DDP) method. The method relies on the observation that plane-strain plastic deformation of cubic crystals is possible in specific orientations when described in terms of edge dislocations on three effective slip systems. The displacement and stress fields of such dislocations in an unbounded anisotropic crystal are recapitulated, and we propose modified constitutive rules for the discrete dislocation dynamics of anisotropic single crystals. Subsequently, to handle polycrystalline problems, we follow an idea of O’Day and Curtin (J. Appl. Mech. 71 (2004) 805–815) and treat each grain as a plastic domain, and adopt superposition to determine the overall response. This method allows for a computationally efficient analysis of micro-scale size effects. As an application, we study freestanding thin copper films under plane-strain tension. First, the computational framework is validated for the special case of isotropic thin films modeled by means of a standard 2D DDP method. Next, predictions of size dependent plastic behavior in anisotropic columnar-grained thin films with varying thickness/grain size are presented and compared with the isotropic results.     

A micromechanical theory of grain-size dependence in metal plasticity

The effect of grain-size on the elastoplastic behavior of metals is investigated from the micromechanics standpoint. First, based on the observations that dislocation pile-ups, formation of cell structures, and other inelastic activities influenced by the presence of grain boundary actually take place transcrystallinely, a grain-size dependent constitutive equation is proposed for the slip deformation of slip systems. By means of a modified Hill's self-consistent relation the local stress of a grain is calculated, and used in conjunction with this constitutive equation to evaluate the plastic strain of each constituent grain. The grain-size effect on the plastic flow of polycrystals then can be determined by an averaging process. To check the validity of the proposed theory it was finally applied to predict the stress-strain curves and flow stresses of a copper at various grain-sizes. The obtained results were found to be in good agreement with experimental data.     

Slip Bands and Stress Oscillations in Bars

A model is proposed that is able to describe the phenomena of strain localization and serrated deformation, peculiar to many materials, but particularly evident in mild steel. A nonlinear elastic bar is conceived like the assemblage of thin filaments, whose energy is supposed to be a nonconvex function of the relative displacement of each filament's extremities. The model explains the onset of elastic, plastic and strain-hardening phases in the bar, interpreting the formation of slip bands and stress oscillations.     

AN ANALYSIS OF THE EFFECTS OF SYMMETRICAL TILT GRAIN BOUNDARIES ON THE PLASTIC DEFORMATION

A numerical investigation on the simple polycrystals containing threesymmetrical tilt grain boundaries(GBs)is carried out within the framework of crystalplasticity which precisely considers the finite deformation and finite lattice rotation aswell as elastic anisotropy.The calculated results show that the slip geometry and theredistribution of stresses arising from the anisotropy and boundary constraint play animportant role in the plastic deformation in the simple polycrystals.The stress levelalong GB is sensitive to the load level and misorientation,and the stresses along GB aredistributed nonuniformly.The GB may exhibit a softening or strengthening feature,which depends on the misorientation angle.The localized deformation bands usuallydevelop accompanying the GB plastic deformation,the impingement of the localizedband on the GB may result in another localized deformation band.The yield stresseswith different misorientation angles are favorably compared with the experimentalresults.     

A substructure mixtures model for the cyclic plasticity of single slip oriented nickel single crystal at low plastic strain amplitudes

The cyclic plasticity behavior of nickel single crystals oriented for single slip is characterized by uniaxial, symmetric, tension–compression, strain controlled tests carried out at constant plastic strain amplitudes ranging from 5(10⁻⁵) to 1(10⁻³). Annealed single crystals are cycled in this manner to post-cyclic saturation and microstructural characterizations, including transmission electron microscopy and optical micrographs of specimen surface replicas are used to verify and evaluate dislocation substructures. Stress–strain and microstructure data are used to construct a mixtures model that couples cyclic plasticity models for three substructures as well as a model for reverse magnetostriction (Villari effect) that is a significant component of inelastic strain at the lower plastic strain amplitudes. The model is used to correlate the stress–plastic strain hysteresis loop responses over the range of plastic strain amplitudes and from cumulative plastic strains from 0.3 to post-cyclic saturation. Complex evolution of substructure plastic strain amplitudes toward their so-called intrinsic values upon the formation of persistent slip bands is modeled. Additionally, bulk Young’s modulus is found to vary significantly with plastic strain amplitude and cumulative plastic strain. A correlation of this behavior is included.     

New inroads in an old subject: Plasticity, from around the atomic to the macroscopic scale

Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a travelling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton-Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor-dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes-Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.