Plasticity size effects in tension and compression of single crystals

The effect of size and loading conditions on the tension and compression stress-strain response of micron-sized planar crystals is investigated using discrete dislocation plasticity. The crystals are taken to have a single active slip system and both small-strain and finite-strain analyses are carried out. When rotation of the tensile axis is constrained, the build-up of geometrically necessary dislocations results in a weak size dependence but a strong Bauschinger effect. On the other hand, when rotation of the tensile axis is unconstrained, there is a strong size dependence, with the flow strength increasing with decreasing specimen size, and a negligible Bauschinger effect. Below a certain specimen size, the flow strength of the crystals is set by the nucleation strength of the initially present Frank-Read sources. The main features of the size dependence are the same for the small-strain and finite-strain analyses. However, the predicted hardening rates differ and the finite-strain analyses give rise to some tension-compression asymmetry.     

The role of the weakest-link mechanism in controlling the plasticity of micropillars

We present a computational study on the effects of sample size on the strength and plastic flow characteristics of micropillars under compression loading. We conduct three-dimensional simulations using the parametric dislocation dynamics coupled with the boundary element method. Two different loading techniques are performed. The plastic flow characteristics as well as the stress-strain behavior of simulated micropillars are shown to be in general agreement with experimental observations. The flow strength versus the diameter of the micropillar follows a power law with an exponent equal to -0.69. A stronger correlation is observed between the flow strength and the average length of activated dislocation sources. This relationship is again a power law, with an exponent -0.85. Simulation results with and without the activation of cross-slip are compared. Discontinuous hardening is observed when cross-slip is included. Experimentally observed size effects on plastic flow and work-hardening are consistent with a “weakest-link activation mechanism”.     

Size effects under homogeneous deformation of single crystals: A discrete dislocation analysis

Mechanism-based discrete dislocation plasticity is used to investigate the effect of size on micron scale crystal plasticity under conditions of macroscopically homogeneous deformation. Long-range interactions among dislocations are naturally incorporated through elasticity. Constitutive rules are used which account for key short-range dislocation interactions. These include junction formation and dynamic source and obstacle creation. Two-dimensional calculations are carried out which can handle high dislocation densities and large strains up to 0.1. The focus is laid on the effect of dimensional constraints on plastic flow and hardening processes. Specimen dimensions ranging from hundreds of nanometers to tens of microns are considered. Our findings show a strong size-dependence of flow strength and work-hardening rate at the micron scale. Taylor-like hardening is shown to be insufficient as a rationale for the flow stress scaling with specimen dimensions. The predicted size effect is associated with the emergence, at sufficient resolution, of a signed dislocation density. Heuristic correlations between macroscopic flow stress and macroscopic measures of dislocation density are sought. Most accurate among those is a correlation based on two state variables: the total dislocation density and an effective, scale-dependent measure of signed density.     

Plastic deformation of freestanding thin films: Experiments and modeling

Experimental measurements and computational results for the evolution of plastic deformation in freestanding thin films are compared. In the experiments, the stress-strain response of two sets of Cu films is determined in the plane-strain bulge test. One set of samples consists of electroplated Cu films, while the other set is sputter-deposited. Unpassivated films, films passivated on one side and films passivated on both sides are considered. The calculations are carried out within a two-dimensional plane strain framework with the dislocations modeled as line singularities in an isotropic elastic solid. The film is modeled by a unit cell consisting of eight grains, each of which has three slip systems. The film is initially free of dislocations which then nucleate from a specified distribution of Frank-Read sources. The grain boundaries and any film-passivation layer interfaces are taken to be impenetrable to dislocations. Both the experiments and the computations show: (i) a flow strength for the passivated films that is greater than for the unpassivated films and (ii) hysteresis and a Bauschinger effect that increases with increasing pre-strain for passivated films, while for unpassivated films hysteresis and a Bauschinger effect are small or absent. Furthermore, the experimental measurements and computational results for the 0.2% offset yield strength stress, and the evolution of hysteresis and of the Bauschinger effect are in good quantitative agreement.     

Finite versus small strain discrete dislocation analysis of cantilever bending of single crystals

Plastic size effects in single crystals are investi-gated by using finite strain and small strain discrete dislo-cation plasticity to analyse the response of cantilever beam specimens. Crystals with both one and two active slip sys-tems are analysed, as well as specimens with different beam aspect ratios. Over the range of specimen sizes analysed here, the bending stress versus applied tip displacement response has a strong hardening plastic component. This hardening rate increases with decreasing specimen size. The hardening rates are slightly lower when the finite strain discrete disloca-tion plasticity (DDP) formulation is employed as curving of the slip planes is accounted for in the finite strain formulation. This relaxes the back-stresses in the dislocation pile-ups and thereby reduces the hardening rate. Our calculations show that in line with the pure bending case, the bending stress in cantilever bending displays a plastic size dependence. How-ever, unlike pure bending, the bending flow strength of the larger aspect ratio cantilever beams is appreciably smaller. This is attributed to the fact that for the same applied bend-ing stress, longer beams have lower shear forces acting upon them and this results in a lower density of statistically stored dislocations.     

Study of size effects in thin films by means of a crystal plasticity theory based on DiFT

In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide given by Yefimov et al. [2004b. A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity simulations. J. Mech. Phys. Solids 52, 279-300]. In this dislocation field theory (DiFT) the transport equations for both the total dislocation density and geometrically necessary dislocation (GND) density on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested itself in the form of a back stress in addition to the external shear and the self-consistent internal stress. We here present the study of size effects in single crystalline thin films with symmetric double slip using the novel continuum theory. Two boundary value problems are analyzed: (1) stress relaxation in thin films on substrates subject to thermal loading, and (2) simple shear in constrained films. In these problems, earlier discrete dislocation simulations had shown that size effects are born out of layers of dislocations developing near constrained interfaces. These boundary layers depend on slip orientations and applied loading but are insensitive to the film thickness. We investigate the stress response to changes in controlled parameters in both problems. Comparisons with previous discrete dislocation simulations are discussed.     

Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension

The membrane deflection experiment developed by Espinosa and co-workers was used to examine size effects on mechanical properties of free-standing polycrystalline FCC thin films. We present stress-strain curves obtained on films 0.2, 0.3, 0.5 and thick including specimen widths of 2.5, 5.0, 10.0 and for each thickness. Elastic modulus was consistently measured in the range of 53- for Au, 125- for Cu and 65- for Al. Several size effects were observed including yield stress variations with membrane width and film thickness in pure tension. The yield stress of the membranes was found to increase as membrane width and thickness decreased. It was also observed that thickness plays a major role in deformation behavior and fracture of polycrystalline FCC metals. A strengthening size scale of one over film thickness was identified. In the case of Au free-standing films, a major transition in the material inelastic response occurs when thickness is changed from 1 to . In this transition, the yield stress more than doubled when film thickness was decreased, with the thick specimen exhibiting a more brittle-like failure and the thick specimen exhibiting a strain softening behavior. Similar plasticity size effects were observed in Cu and Al. Scanning electron microscopy performed on Au films revealed that the number of grains through the thickness essentially halved, from approximately 5 to 2, as thickness decreased. It is postulated that this feature affects the number of dislocations sources, active slip systems, and dislocation motion paths leading to the observed strengthening. This statistical effect is corroborated by the stress-strain data in the sense that data scatter increases with increase in thickness, i.e., plasticity activity.The size effects here reported are the first of their kind in the sense that the measurements were performed on free-standing polycrystalline FCC thin films subjected to macroscopic homogeneous axial deformation, i.e., in the absence of deformation gradients, in contrast to nanoindentation, beam deflection, and torsion, where deformation gradients occur. To the best of our understanding, continuum plasticity models in their current form cannot capture the observed size scale effects.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

Size effects on void growth in single crystals with distributed voids

The effect of void size on void growth in single crystals with uniformly distributed cylindrical voids is studied numerically using a finite deformation strain gradient crystal plasticity theory with an intrinsic length parameter. A plane strain cell model is analyzed for a single crystal with three in-plane slip systems. It is observed that small voids allow much larger overall stress levels than larger voids for all the stress triaxialities considered. The amount of void growth is found to be suppressed for smaller voids at low stress triaxialities. Significant differences are observed in the distribution of slips and on the shape of the deformed voids for different void sizes. Furthermore, the orientation of the crystalline lattice is found to have a pronounced effect on the results, especially for the smaller void sizes.     

Processing induced size effects in plastic yielding upon miniaturisation

Size effects in metals have received considerable attention in literature in the last decades. For preparing specimens dedicated processing techniques, such as laser-cutting, micro-milling, turning, etc., are used. Most of these processing methods intrinsically damage crystals just below the worked surface. In macroscopic applications, the effect on the overall mechanical behaviour can safely be neglected in most cases. Upon miniaturisation, however, the influence of the affected region becomes more important and may induce a processing induced size effect, which is far from negligible. Processing induced size effects are analysed by carefully characterising the plastic yielding in uniaxial tension of rectangular, -thick aluminium sheet specimens, with a well-defined homogeneous microstructure containing through-thickness grains. The specimens are processed to different widths by three independent machining techniques: (1) laser-cutting, (2) mechanical cutting, and (3) extensive grinding from a larger width. These independent techniques all result in a distinct processing induced size effect upon miniaturisation, i.e. an increase of up to 200% in yield stress for a decrease from about 12 to 3 grains over the specimen width. Using a simple Taylor averaging model, it is shown that the yield stress in the affected edge region increased to 210–350% of its initial (or bulk) value. In addition, it is found that even a prolonged anneal near the melting temperature can only partially remove the processing induced size effect. The results clearly demonstrate that processing induced size effects have to be considered in the design of miniaturised devices and parts as well as in scientific research relying on the testing of manufactured small-scale test specimens.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

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.     

Analytical solution of plane constrained shear problem for single crystals within continuum dislocation theory

Berdichevsky and Le have recently found the analytical solution of the anti-plane constrained shear problem within the continuum dislocation theory (CMT, Contin. Mech. Thermodyn. 18:455–467, 2007). Interesting features of this solution are the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. In this paper an analytical solution of the plane constrained shear problem for single crystals exhibiting similar features is obtained and the comparison with the discrete dislocation simulation is provided. Dedicated to the memory of George Herrmann     

A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes

We present a self-consistent formulation of 3-D parametric dislocation dynamics (PDD) with the boundary element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. The error in the image force on a screw dislocation approaching a free surface is shown to increase as the dislocation approaches the surface, but is nevertheless controllable. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000×2000 (in units of lattice parameter), and 64 quadrature points. The Eshelby twist angle in a finite-length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Finally, large scale 3-D simulation results of single slip behavior in cylindrical microcrystals are presented. Plastic flow characteristics and the stress-strain behavior of cylindrical microcrystals under compression are shown to be in agreement with experimental observations. It is shown that the mean length of dislocations trapped at the surface is the dominant factor in determining the size effects on hardening of single crystals. The influence of surface image fields on the flow stress is finally explored. It is shown that the flow stress is reduced by as much as 20% for small single crystals of size less than .     

Micro-pillar plasticity: 2.5D mesoscopic simulations

The plastic behavior of micro- and nano-scale crystalline pillars is investigated under nominally uniform compression. The transition from forest hardening to exhaustion hardening dominated behavior is shown to emerge from discrete dislocation dynamics simulations upon reduction in the initial source density. The analyses provide new insight into the scaling of flow stress with specimen size and also highlight the connection between individual dislocation mechanisms, collective phenomena and overall behavior.     

Discrete dislocation plasticity analysis of the wedge indentation of films

The plane strain indentation of single crystal films on a rigid substrate by a rigid wedge indenter is analyzed using discrete dislocation plasticity. The crystals have three slip systems at ±35.3^° and 90^° with respect to the indentation direction. The analyses are carried out for three values of the film thickness, 2, 10 and , and with the dislocations all of edge character modeled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Over the range of indentation depths considered, the indentation pressure for the 10 and thick films decreases with increasing contact size and attains a contact size-independent value for contact lengths . On the other hand, for the films, the indentation pressure first decreases with increasing contact size and subsequently increases as the plastic zone reaches the rigid substrate. For the 10 and thick films sink-in occurs around the indenter, while pile-up occurs in the film when the plastic zone reaches the substrate. Comparisons are made with predictions obtained from other formulations: (i) the contact size-independent indentation pressure is compared with that given by continuum crystal plasticity; (ii) the scaling of the indentation pressure with indentation depth is compared with the relation proposed by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 43, 411-423]; and (iii) the computed contact area is compared with that obtained from the estimation procedure of Oliver and Pharr [1992. An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564-1583].     

Continuum modeling of size effects on the composition and stresses in nanoparticles of ionic solids with application to ceria

Owing to its broad potential applications, nanostructured ceria has been subject of intense investigation in the past few decades. Experiments have demonstrated that various material properties of the nanostructured ionic solids including ceria vary with the feature size. Here, we present a theoretical study of the size effects on the composition, defect concentrations and stresses in free-standing nanoparticles of nonstoichiometric ionic solids. To this end, a continuum model is developed which accounts for the highly nonlinear coupling between mechanical, chemical and electrical driving forces, and their effects on the thermodynamic equilibrium of the defect species. It is demonstrated that the model, once applied to the case of ceria, predicts size-dependent defect concentrations and surface stresses. It is further shown that the theoretical predictions of the size effects on the composition and lattice parameter are in good agreement with the experimental observations.     

Relations between a micro-mechanical model and a damage model for ductile failure in shear

Gurson type constitutive models that account for void growth to coalescence are not able to describe ductile fracture in simple shear, where there is no hydrostatic tension in the material. But recent micro-mechanical studies have shown that in shear the voids are flattened out to micro-cracks, which rotate and elongate until interaction with neighbouring micro-cracks gives coalescence. Thus, the failure mechanism is very different from that under tensile loading. Also, the Gurson model has recently been extended to describe failure in shear, by adding a damage term to the expression for the growth of the void volume fraction, and it has been shown that this extended model can represent experimental observations. Here, numerical studies are carried out to compare predictions of the shear-extended Gurson model with the shear failures predicted by the micro-mechanical cell model. Both models show a strong dependence on the level of hydrostatic tension. Even though the reason for this pressure dependence is different in the two models, as the shear-extended Gurson model does not describe voids flattening out and the associated failure mechanism by micro-cracks interacting with neighbouring micro-cracks, it is shown that the trends of the predictions are in good agreement.     

Plasticity of metal wires in torsion: Molecular dynamics and dislocation dynamics simulations

The orientation dependent plasticity in metal nanowires is investigated using molecular dynamics and dislocation dynamics simulations. Molecular dynamics simulations show that the orientation of single crystal metal wires controls the mechanisms of plastic deformation. For wires oriented along , dislocations nucleate along the axis of the wire, making the deformation homogeneous. These wires also maintain most of their strength after yield. In contrast, wires oriented along and directions deform through the formation of twist boundaries and tend not to recover when high angle twist boundaries are formed. The stability of the dislocation structures observed in molecular dynamics simulations are investigated using analytical and dislocation dynamics models.     

Strain gradient effects on cyclic plasticity

Size effects on the cyclic shear response are studied numerically using a recent higher order strain gradient visco-plasticity theory accounting for both dissipative and energetic gradient hardening. Numerical investigations of the response under cyclic pure shear and shear of a finite slab between rigid platens have been carried out, using the finite element method. It is shown for elastic-perfectly plastic solids how dissipative gradient effects lead to increased yield strength, whereas energetic gradient contributions lead to increased hardening as well as a Bauschinger effect. For linearly hardening materials it is quantified how dissipative and energetic gradient effects promote hardening above that of conventional predictions. Usually, increased hardening is attributed to energetic gradient effects, but here it is found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening. Furthermore, it is shown that dissipative gradient effects can lead to both an increase and a decrease in the dissipation per load cycle depending on the magnitude of the dissipative length parameter, whereas energetic gradient effects lead to decreasing dissipation for increasing energetic length parameter. For dissipative gradient effects it is found that dissipation has a maximum value for some none zero value of the material length parameter, which depends on the magnitude of the deformation cycles.