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].     

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.     

Distribution of dislocation source length and the size dependent yield strength in freestanding thin films

A method is proposed to estimate the size-dependent yield strength of columnar-grained freestanding thin films. The estimate relies on assuming a distribution of the size of Frank-Read sources, which is then translated into a log-normal distribution of the source strength, depending on film thickness, grain size and theoretical strength of the material, augmented with a single fit parameter. Two-dimensional discrete dislocation plasticity (DDP) simulations are carried out for two sets of Cu films and the fit parameter is determined from independent experiments. Subsequent DDP predictions of the stress-strain curves in comparison with the corresponding experimental data show excellent agreement of initial yield strength and hardening rate for films of varying film thickness and grain size. Having thus demonstrated the power of the proposed strength distribution, it is shown that the mode of this distribution governs the most effective source strength. This is then used to suggest a method to estimate the yield strength of thin films as a function of film thickness and grain size. Simple maps are presented that are in very good agreement with recent experimental results for Cu thin films.     

Compliant interfaces: A mechanism for relaxation of dislocation pile-ups in a sheared single crystal

Discrete dislocation plasticity models and strain-gradient plasticity theories are used to investigate the role of interfaces in the elastic–plastic response of a sheared single crystal. The upper and lower faces of a single crystal are bonded to rigid adherends via interfaces of finite thickness. The sandwich system is subjected to simple shear, and the effect of thickness of crystal layer and of interfaces upon the overall response are explored. When the interface has a modulus less than that of the bulk material, both the predicted plastic size effect and the Bauschinger effect are considerably reduced. This is due to the relaxation of the dislocation stress field by the relatively compliant surface layer. On the other hand, when the interface has a modulus equal to that of the bulk material a strong size effect in hardening as well as a significant reverse plasticity are observed in small specimens. These effects are attributed to the energy stored in the elastic fields of the geometrically necessary dislocations (GNDs).     

Bounds on the hydrostatic plastic strength of voided polycrystals and implications for linear-comparison homogenization techniques

A linear-comparison homogenization technique and its relaxed version are used to compute bounds of the Hashin–Shtrikman and the self-consistent types for the hydrostatic strength of ideally plastic voided polycrystals. Closed-form analytical results are derived for isotropic aggregates of various cubic symmetries (fcc, bcc, ionic). The impact of the variational relaxation on the bounds is found to be significantly larger than that previously observed in fully dense polycrystals. So much so that, quite surprisingly, relaxed self-consistent bounds are found to be weaker than non-relaxed Hashin–Shtrikman bounds in some of the material systems considered.     

Analysis of grain size effects on transformation-induced plasticity based on a discrete dislocation-transformation model

There is much interest recently in the possibility of combining two strengthening effects, namely the reduction of grain size (Hall-Petch effect) and the transformation-induced plasticity effect (strengthening due to a martensitic transformation). The present work is concerned with the analysis of the combination of these two effects using a discrete dislocation-transformation model. The transformation-induced plasticity mechanism is studied for aggregates of grains of ferrite and austenite of different sizes. The discrete model allows to simulate the behavior at sub-grain length scales, capturing the complex interaction between pile-ups at grain boundaries and the evolution of the microstructure due to transformation. The simulations indicate that, as the average grain size decreases, the relative strengthening due to the formation of martensite is significantly reduced and that the overall strengthening is mostly due to a Hall-Petch effect. This finding suggests that strengthening by the transformation-induced plasticity mechanism is ineffective in the presence of fine-grained microstructures.     

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”.     

Role of discrete intra-granular slip bands on the strain-hardening of polycrystals

This work investigates a new micromechanical modeling of polycrystal plasticity, accounting slip bands for physical plastic heterogeneities considered as periodically distributed within grains. These intra-granular plastic heterogeneities are modeled by parallel flat ellipsoidal sub-domains, each of them may have a distinct uniform plastic slip. To capture the morphology of slip bands occurring in plastically deforming polycrystals, these interacting sub-domains are considered as oblate spheroids periodically distributed and constrained by spherical grain boundaries. In this paper, we focus the study on the influences of internal length scale parameters related to grain size, spatial period and thickness of slip bands on the overall material’s behavior. In a first part, the Gibbs free energy accounting for elastic interactions between plastic heterogeneities is calculated thanks to the Green function’s method in the case of an isolated spherical grain with plastic strain occurring only in slip bands embedded in an infinite elastic matrix. In a second part, the influence of discrete periodic distributions of intra-granular slip bands on the polycrystal’s behavior is investigated considering an aggregate with random crystallographic orientations. When the spatial period of slip bands is on the same order as the grain radius, the polycrystal’s mechanical behavior is found strongly dependent on the ratio between the spatial period of slip bands and the grain size, as well as the ratio between the slip band thickness and the grain size, which cannot be captured by classic length scale independent Eshelby-based micromechanics.     

Three-dimensional analysis of scale dependence of sub-micron polycrystals due to configuration entropy

The authors proposed a plausible explanation for the deviation of experimental data for sub-micron polycrystals from the Hall-Petch relation by introducing the configuration entropy. The present paper extends the previous two-dimensional analysis to the three-dimensional case. The statistical distribution of dislocation lengths within a spherical grain and the bow-out of dislocations are considered. According to Ashby's model, analyses are pursued for the statistically stored dislocations and geometrically necessary dislocations, respectively. It is confirmed that the configuration entropy model can predict the abnormal Hall-Petch dependence for grain sizes in the sub-micron range. The project supported by the National Natural Science Foundation of China (19891180-01)     

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.     

Discrete dislocation dynamics modelling of mechanical deformation of nickel-based single crystal superalloys

Discrete dislocation dynamics (DDD) has been used to model the deformation of nickel-based single crystal superalloys with a high volume fraction of precipitates at high temperature. A representative volume cell (RVC), comprising of both the precipitate and the matrix phase, was employed in the simulation where a periodic boundary condition was applied. The dislocation Frank-Read sources were randomly assigned with an initial density on the 12 octahedral slip systems in the matrix channel. Precipitate shearing by superdislocations was modelled using a back force model, and the coherency stress was considered by applying an initial internal stress field. Strain-controlled loading was applied to the RVC in the [0 0 1] direction. In addition to dislocation structure and density evolution, global stress-strain responses were also modelled considering the influence of precipitate shearing, precipitate morphology, internal microstructure scale (channel width and precipitate size) and coherency stress. A three-stage stress-strain response observed in the experiments was modelled when precipitate shearing by superdislocations was considered. The polarised dislocation structure deposited on the precipitate/matrix interface was found to be the dominant strain hardening mechanism. Internal microstructure size, precipitate shape and arrangement can significantly affect the deformation of the single crystal superalloy by changing the constraint effect and dislocation mobility. The coherency stress field has a negligible influence on the stress-strain response, at least for cuboidal precipitates considered in the simulation. Preliminary work was also carried out to simulate the cyclic deformation in a single crystal Ni-based superalloy using the DDD model, although no cyclic hardening or softening was captured due to the lack of precipitate shearing and dislocation cross slip for the applied strain considered.     

A two-dimensional dislocation dynamics model of the plastic deformation of polycrystalline metals

Two-dimensional dislocation dynamics (2D-DD) simulations under fully periodic boundary conditions are employed to study the relation between microstructure and strength of a material. The material is modeled as an elastic continuum that contains a defect microstructure consisting of a preexisting dislocation population, dislocation sources, and grain boundaries. The mechanical response of such a material is tested by uniaxially loading it up to a certain stress and allowing it to relax until the strain rate falls below a threshold. The total plastic strain obtained for a certain stress level yields the quasi-static stress-strain curve of the material. Besides assuming Frank-Read-like dislocation sources, we also investigate the influence of a pre-existing dislocation density on the flow stress of the model material. Our results show that - despite its inherent simplifications - the 2D-DD model yields material behavior that is consistent with the classical theories of Taylor and Hall-Petch. Consequently, if set up in a proper way, these models are suited to study plastic deformation of polycrystalline materials.     

Effect of crystalline grain structures on the mechanical properties of twinning-induced plasticity steel

In order to improve the mechanical properties of twinning-induced plasticity steel, the grain morphology was tailored by different solidification technologies combined with deformation and heat treatment processing routes. Three typical grain morphologies, i.e., equiaxed, columnar as well as equiaxed/columnar grains were formed, and their mechanical behaviors were comparatively studied. Among the three materials, the equiaxed grain material exhibited the highest strength but the lowest plasticity. Depending on the grain size, the smaller the grain size, the higher the strength, but the lower the elongation. The columnar grain material possessed the most excellent plasticity but the weakest strength.These properties presented a non-monotonic dependence on the dendrite spacing, and the moderate spacing resulted in the optimum combination of strength and plasticity. The equiaxed/columnar grain coexisted material showed interesting properties, i.e., the strength and plasticity were just between those of single grain-shaped materials. The three materials also presented different strain hardening behaviors particularly in the uniform deformation stage. The equiaxed grain material showed a constant strain hardening rate, while the columnar grain and equiaxed/columnar grain materials showed a progressively increasing rate with increasing the true strain.     

Effects of size on the strength and deformation mechanism in Zr-based metallic glasses

We report results of uniaxial compression tests on Zr₃₅Ti₃₀Co₆Be₂₉ metallic glass nano-pillars with diameters ranging from ∼1.6 μm to ∼100 nm. The tested pillars have nearly vertical sidewalls, with the tapering angle lower than ∼1° (diameter >200 nm) or ∼2° (diameter ∼100 nm), and with a flat pillar top to minimize the artifacts due to imperfect geometry. We report that highly-localized-to-homogeneous deformation mode change occurs at 100 nm diameter, without any change in the yield strength. We also find that yield strength depends on size only down to 800 nm, below which it remains at its maximum value of 2.6 GPa. Quantitative Weibull analysis suggests that the increase in strength cannot be solely attributed to the lower probability of having weak flaws in small samples - most likely there is an additional influence of the sample size on the plastic deformation mechanism.     

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.     

Shape and size effects of ceria nanoparticles on the impact strength of ceria/epoxy resin composites

Ceria nanoparticles with various shapes (rods, cubes, and plates) and sizes were controllably synthesized and then introduced into epoxy resin. Subsequently, we investigated correlations between the shape and size of ceria nanostructures and the mechanical performance of composites. The samples were characterized by transmission electron microscopy, scanning electron microscopy, and X-ray diffraction. Compared with commercial ceria filled composites, the composites made with morphology-controlled ceria nano...     

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.     

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 .