Size-dependent Yield Strength Research Articles

Twinning, slip and size effect of phase-transforming ferroelectric nanopillars

Ferroelectric materials are widely used in energy applications due to their field-driven multiferroic properties. The stress-induced phase transformation plays an important role in the functionality over repeated and consecutive operation cycles, especially at the micro/nanoscales. Here we report a systematic in-situ uniaxial compression tests on cuboidal Barium titanate (BaTiO3) nanopillars with size varying from 100 nm to 3000 nm, by which we explore the stress-induced transformation and its interplay with plastic deformation. We confirm the superelasticity achieved in pillars by martensitic phase transformation from tetragonal to orthorhombic. There exists a critical size, 330 nm, for the yield stress. Above 330 nm, martensitic phase transformation aids slip along the plane with a low Schmid factor, in turn, the pseudo-compatible twins form within the shear band. The scaling exponent of size-dependent yield strength is found to be exactly 1. For nanopillars smaller than 330 nm, no twins form, only slips with large Schmid factors are activated, and size effect vanishes. All pillars with sizes from 100 nm to 300 nm achieve the theoretical yield limit around 9 GPa. Our experimental results uncover the interplay between twins and slips in BaTiO3 nanopillars, which pave the way for the optimization of microstructure design of ferroelectric materials for microelectronic applications at small scales.

High-pressure studies of size dependent yield strength in rhenium diboride nanocrystals.

The superhard ReB2 system is the hardest pure phase diboride synthesized to date. Previously, we have demonstrated the synthesis of nano-ReB2 and the use of this nanostructured material for texture analysis using high-pressure radial diffraction. Here, we investigate the size dependence of hardness in the nano-ReB2 system using nanocrystalline ReB2 with a range of grain sizes (20-60 nm). Using high-pressure X-ray diffraction, we characterize the mechanical properties of these materials, including bulk modulus, lattice strain, yield strength, and texture. In agreement with the Hall-Petch effect, the yield strength increases with decreasing size, with the 20 nm ReB2 exhibiting a significantly higher yield strength than any of the larger grained materials or bulk ReB2. Texture analysis on the high pressure diffraction data shows a maximum along the [0001] direction, which indicates that plastic deformation is primarily controlled by the basal slip system. At the highest pressure (55 GPa), the 20 nm ReB2 shows suppression of other slip systems observed in larger ReB2 samples, in agreement with its high yield strength. This behavior, likely arises from an increased grain boundary concentration in the smaller nanoparticles. Overall, these results highlight that even superhard materials can be made more mechanically robust using nanoscale grain size effects.

Theoretical insight of strengthening and hardening behavior in ultrafine-grained metals under high pressure

The mechanism-based theoretical models are presented to provide the theoretical explanations for strengthening and pressure-induced hardening behaviors in ultrafine-grained metals under high pressure. The grain boundary deformation model is extended to construct the relationship among the critical stress for grain boundary deformation, pressure and the grain size. The pressure-dependent critical twinning stress is derived on the basis of dislocation theory to describe the pressure-induced hardening behavior. The classic Hall-Petch for the grain size-dependent yield strength is modified through involving the contribution of partial dislocations. The simulation results demonstrate that the proposed models provide good descriptions of the strengthening and hardening behaviors in ultrafine-grained metals with respect to the high pressure. The critical grain size for the transition of deformation mechanisms is sensitive to the pressure and the grain boundary thickness. The predicted grain size-dependent yield strength and flow stress under high pressure are agreeable well with the experimental tests. These findings could shed some lights into understanding the plastic deformations of ultrafine-grained metals under high pressure.

Unified Model for Size-Dependent to Size-Independent Transition in Yield Strength of Crystalline Metallic Materials.

Size-dependent yield strength is a common feature observed in miniaturized crystalline metallic samples, and plenty of studies have been conducted in experiments and numerical simulations to explore the underlying mechanism. However, the transition in yield strength from bulklike to size-affected behavior has received less attention. Here a unified theoretical model is proposed to probe the yield strength of crystalline metallic materials with sample size from nanoscale to macroscale. We show that the transition in yield strength versus size can be fully explained by the competition between the stresses required for dislocation source activation and dislocation motion, which is regulated by dislocation density, irradiation defect, grain boundary, and so on. Based on various grain boundary densities, the extended Hall-Petch relation, incorporated into the unified model, captures the reverse size effect for polycrystalline samples. The proposed model predictions agree well with reported experimental measurements of various specimens, including the prestrained nickel, irradiated copper, ultrafine grain tungsten, and so on.

Shear localization and size-dependent strength of YCd6 quasicrystal approximant at the micrometer length scale

Mechanical properties of materials are strongly dependent of their atomic arrangement as well as the sample dimension, particularly at the micrometer length scale. In this study, we investigated the small-scale mechanical properties of single-crystalline YCd6, which is a rational approximant of the icosahedral Y-Cd quasicrystal. In situ microcompression tests revealed that shear localization always occurs on {101} planes, but the shear direction is not constrained to any particular crystallographic directions. Furthermore, the yield strengths show the size dependence with a power law exponent of 0.4. Shear localization on {101} planes and size-dependent yield strength are explained in terms of a large interplanar spacing between {101} planes and the energetics of shear localization process, respectively. The mechanical behavior of the icosahedral Y-Cd quasicrystal is also compared to understand the influence of translational symmetry on the shear localization process in both YCd6 and Y-Cd quasicrystal micropillars. The results of this study will provide an important insight in a fundamental understanding of shear localization process in novel complex intermetallic compounds.

Effect of surface energy upon size-dependent yield strength of single-crystalline hollow micro- and nanopillars

The physical nature of the size-dependent yield strength of micro- and nanopillars is still an open issue, where the surface energy may be one factor that influences the size effect of these pillars, especially for hollow pillars. In this study, the influence of the surface energy and surface elasticity upon the size-dependent yield strength of single-crystalline hollow micro- and nanopillars is considered quantitatively. An analytical expression of the size-dependent yield strength is established that considers the surface energy and the inner defects by the principle of minimum potential energy. The predicted yield strength of our model as a function of the pillar size is more precise than that of the well-known single arm source model when comparing the predictions with the experimental results of pillars made of Al or Ni. According to the surface elasticity, the surface energy can be determined by the thickness of the hollow pillars, the surface modulus and the surface strain, where the latter two are related to the external damages.

Hertz beyond belief

We examine the validity of Hertz's linear elastic theory for central collisions of spheres using a viscoelastic model. This model explains why Hertz's theory is accurate in predicting the collision time and maximum contact area even when 40% of the kinetic energy is lost due to viscous dissipation. The main reason is that both the collision time and maximum contact area have a very weak dependence on the impact velocity. Moreover, we show that colliding objects exhibit an apparent size-dependent yield strength, which results from larger objects dissipating less energy at a given impact velocity.

A gradient plasticity grain boundary yield theory

A strain gradient crystal plasticity theory is presented that accounts for the resistance of grain boundaries against plastic flow based on an interface yield condition. This theory incorporates the previously presented numerically efficient visco-plastic treatment by the gradient of an equivalent plastic strain ∇γeq in Wulfinghoff and Böhlke (2012). The finite element implementation is discussed and the three-dimensional numerical model is fitted to experimental data of polycrystalline copper micro-tensile tests. The size dependent yield strength is reproduced notably well.

Coupled Interfacial Energy and Temperature Effects on Size-Dependent Yield Strength and Strain Hardening of Small Metallic Volumes

Plasticity in heterogeneous metallic materials with small volumes is governed by the interactions of dislocations at interfaces. In particular, interfaces of a material confined in a small volume can strongly affect the mechanical properties of micro and nanosystems. In this paper, the framework of higher-order strain gradient plasticity theory with interfacial energy effect is used to investigate the coupling of interfacial energy with temperature and how it affects the initial yield strength (i.e., onset of plasticity) and the strain hardening rates of confined small metallic volumes. It is postulated that the interfacial energy decreases as temperature increases such that size effect decreases as temperature increases. As an application, the size effect of thermal loading of a film-substrate system is investigated. It is shown that the temperature at which the film starts to yield plastically is size-dependent, which is attributed to the size-dependent yield strength. Furthermore, the flow stress is more temperature sensitive as the size decreases.

Source-controlled yield and hardening of Cu(1 0 0) studied by in situ transmission electron microscopy

In the present work we investigate the mechanical properties of multiple slip oriented single crystal Cu(100) compression samples to shed light on size-dependent yield and hardening behavior at small-scales. Samples with diameters ranging from 90nm to 1700nm were fabricated using focused ion beam milling and tested in situ in a transmission electron microscope. The results demonstrate a dislocation source-limited size-dependent yield strength, as evidenced by size-dependent changes in the deformation morphology. Moreover, we report size dependency and strain dependency in the hardening behavior at these dimensions, where higher hardening is observed for smaller samples and at lower strains. This is explained by the source-limited nature of plasticity in small dimensions, which we demonstrate affects not just yield but also the hardening behavior in the nanopillars.

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.

A Dimensional Analysis Model for Geometric Size Effects of Single Crystals

Due to the relatively high surface-to-volume ratio, the surface effect can be significant for micro/nano-scale materials. This paper focuses on geometric size-dependent strength mechanisms of micro/nano-scale metal single crystals. A dimensional analysis model relating surface energy with the geometric size-dependent yield strength is presented and compared with results of microscale uniaxial compression tests on Ni and Au single crystals. The results indicate this model can predict the geometric size effects on the yield strength of micro/nano-scale metal single crystals.

Atomistic simulation on size-dependent yield strength and defects evolution of metal nanowires

A simple derivation based on continuum mechanics is given, which shows the surface stress is critical for yield strength at ultra-small scales. Molecular dynamics (MD) simulations with modified embedded atom method (MEAM) are employed to investigate the mechanical behaviors of single-crystalline metal nanowires under tensile loading. The calculated yield strengths increasing with the decrease of the cross-sectional area of the nanowires are in accordance with the theoretical prediction. Reorientation induced by stacking faults is observed at the nanowire edge. In addition, the mechanism of yielding is discussed in details based on the snapshots of defects evolution. The nanowires in different crystallographic orientations behave differently in stretching deformation. This study on the plastic properties of metal nanowires will be helpful to further understanding of the mechanical properties of nanomaterials.

A size-dependent crystal plasticity finite-element model for creep and load shedding in polycrystalline titanium alloys

A rate-dependent anisotropic elastic-crystal plasticity based finite-element (FE) model with size-dependent yield strength is developed for polycrystalline Ti-6242. The initial slip system deformation resistances in the crystal plasticity relations are expressed as Hall–Petch type relations, where the grain size, lath size and colony size are chosen as characteristic lengths depending on the nature of slip. The FE model incorporates accurate phase volume fractions and orientation distributions that are statistically equivalent to those observed in orientation imaging microscopy (OIM) maps of the microstructure. The model is validated with experimental results on constant strain rate and creep tests. A relationship between the macroscopic flow stress and grain size and lath size for two-phase Ti-6242 is proposed. The effect of grain morphology on creep-induced load shedding and localization is discussed.

Size-dependent yield strength and surface energies of thin films

The strain gradient plasticity theory recently proposed by Gudmundson [P. Gudmundson, J. Mech. Phys. Solids 52 (2004) 1379–1406] is used to analyse the behaviour of a thin film on an elastic substrate. Boundary conditions for the film–substrate interface are introduced via a surface energy that depends on the plastic strain state at the interface. Finite element results show a strong dependence on the surface energy. If the surface energy is small, no size effects appear. On the other hand, if a stiff interface is simulated, corresponding to a large surface energy, a thickness dependence of the yield strength is found. The application of several alternative strain gradient models would predict a thickness dependent hardening, but strictly not a size dependence of the yield strength. The presently predicted thickness dependence on yield strength and hardening is supported by experimental results.

Size-dependent yield strength of thin films

Biaxial strain and pure shear of a thin film are analysed using a strain gradient plasticity theory presented by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406]. Constitutive equations are formulated based on the assumption that the free energy only depends on the elastic strain and that the dissipation is influenced by the plastic strain gradients. The three material length scale parameters controlling the gradient effects in a general case are here represented by a single one. Boundary conditions for plastic strains are formulated in terms of a surface energy that represents dislocation buildup at an elastic/plastic interface. This implies constrained plastic flow at the interface and it enables the simulation of interfaces with different constitutive properties. The surface energy is also controlled by a single length scale parameter, which together with the material length scale defines a particular material. Numerical results reveal that a boundary layer is developed in the film for both biaxial and shear loading, giving rise to size effects. The size effects are strongly connected to the buildup of surface energy at the interface. If the interface length scale is small, the size effect vanishes. For a stiffer interface, corresponding to a non-vanishing surface energy at the interface, the yield strength is found to scale with the inverse of film thickness. Numerical predictions by the theory are compared to different experimental data and to dislocation dynamics simulations. Estimates of material length scale parameters are presented.