Size dependent mechanical behaviour of tantalum

The size dependence of deformation of Ta was studied using compression tests of focused ion beam (FIB) machined microcolumns. Columns with diameters between 0.5 and 8 μm with 〈1 1 1〉 and 〈1 0 0〉 orientations along the column axis were tested. By comparing results of bcc Ta columns with results from previous experiments on fcc metals it was found that Ta shows significantly higher normalized yield stresses in combination with a weaker sample size dependence. The differences between bcc and fcc metals can be attributed to the different dislocation behaviour of bcc metals, especially to the lower mobility of screw dislocations.     

Influence of single crystal orientation on homogeneous dislocation nucleation under uniaxial loading

Atomistic simulations are used to investigate how the stress required for homogeneous nucleation of partial dislocations in single crystal copper under uniaxial loading changes as a function of crystallographic orientation. Molecular dynamics is employed based on an embedded-atom method potential for Cu at 10 and 300 K. Results indicate that non-Schmid parameters are important for describing the calculated dislocation nucleation behavior for single crystal orientations under tension and compression. A continuum relationship is presented that incorporates Schmid and non-Schmid terms to correlate the nucleation stress over all tensile axis orientations within the stereographic triangle. Simulations investigating the temperature dependence of homogeneous dislocation nucleation yield activation volumes of ≈0.5- and activation energies of . For uniaxial compression, full dislocation loop nucleation is observed, in contrast to uniaxial tension. One of the main differences between uniaxial tension and compression is how the applied stress is resolved normal to the slip plane on which dislocations nucleate—in tension, this normal stress is tensile, and in compression, it is compressive. Last, the tension-compression asymmetry is examined as a function of loading axis orientation. Orientations with a high resolved stress normal to the slip plane on which dislocations nucleate have a larger tension-compression asymmetry with respect to dislocation nucleation than those orientations with a low resolved normal stress. The significance of this research is that the resolved stress normal to the slip plane on which dislocations nucleate plays an important role in partial (and full) dislocation loop nucleation in FCC Cu single crystals.     

Cyclic behavior of extruded magnesium: Experimental, microstructural and numerical approach

The present study aims at determining the influence of cyclic straining on the behavior of pure extruded magnesium. For this purpose, tensile, compressive and cyclic tests are performed (small plastic strains are applied (Δε^p/2 = 0.1% and 0.4%). Deformation mechanisms (slip and twin systems) have been observed by TEM and the different critical resolved shear stress (CRSS) have been determined. Based on microscopic observations, a crystal-plasticity-based constitutive model has been developed. The asymmetry between tensile and compressive loadings mainly results from the activation of hard slip systems in tension (such as 〈a〉 pyramidal and prismatic and 〈c + a〉 pyramidal glides) and twinning in compression. It is shown that basal slip is very easy to activate even for small Schmid factors. Numerical simulations reveal that untwinning in tension subsequent to compression must be considered to correctly fit the experimental S-shaped hysteresis curves. TEM observations indicate also intense secondary slips or twins inside the mother twins under cyclic conditions, so that twinning in compression and dislocation glide in tension are affected by cycling. The polycrystalline model allows to predict slip activities and twin volume fraction evolutions.     

Numerical simulations of necking during tensile deformation of aluminum single crystals

Finite element simulations are used to study strain localization during uniaxial tensile straining of a single crystal with properties representative of pure Al. The crystal is modeled using a constitutive equation incorporating self- and latent-hardening. The simulations are used to investigate the influence of the initial orientation of the loading axis relative to the crystal, as well as the hardening and strain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails by diffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain to localization is a decreasing function of m for m < 100, and is relatively insensitive to m for m > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactly parallel to) a high symmetry direction such as [1 0 0] or [1 1 1] and the variation of the strain to localization with orientation is highly sensitive to the strain rate exponent and latent-hardening behavior of the crystal. This behavior can be explained in terms of changes in the active slip systems as the initial orientation of the crystal is varied.     

Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry

This paper is devoted to modeling the effects of the tension-compression asymmetry of the matrix on yielding of the void-matrix aggregate. The matrix plastic behavior is described by the Cazacu et al. [2006. Orthotropic yield criterion for hexagonal closed packed metals. Int. J. Plasticity 22, 1171-1194] isotropic yield criterion, which captures strength differential effects. Using an upper-bound approach, a new analytic isotropic plastic potential for a random distribution of spherical voids is obtained. The derived analytic potential is sensitive to the third invariant of the stress deviator and displays tension-compression asymmetry. In the case when the matrix material has the same yield in tension and compression, it reduces to Gurson's [1977. Continuum theory of ductile rupture by void nucleation and growth: Part I: Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. Trans. ASME Ser. H 99, 2-15.] criterion. Furthermore, the proposed criterion predicts the exact solution of a hollow sphere loaded in hydrostatic tension or compression. The accuracy of the proposed analytical criterion is assessed through comparisons with finite-element cell calculations.     

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J₂J₃ class model, where J₂ and J₃ are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.     

Orientation effects in nanoindentation of single crystal copper

Numerical simulations and experimental results of nanoindentation on single crystal copper in three crystallographic orientations [(1 0 0), (0 1 1) and (1 1 1)] using a spherical indenter (3.4 μm radius) were reported. The simulations were conducted using a commercial finite element code (ABAQUS) with a user-defined subroutine (VUMAT) that incorporates large deformation crystal plasticity constitutive model. This model can take full account of the crystallographic slip as well as the orientation effects during nanoindentation. Distributions of the out-of-plane displacements and shear stresses as well as shear strains were obtained for indentation depths of up to 310 nm. The experimental studies were conducted using an MTS Nano Indenter (XP) system from which the load–displacement relationships were obtained while the surface topography as well as the surface profile along a line scan of indents were obtained using a Digital Instruments (Dimension 3100) atomic force microscope (AFM). The top views of the indent pile-up patterns under the spherical indenter show two-fold, three-fold, and four-fold symmetries for the (0 1 1), (1 1 1), and (1 0 0) orientations, respectively. Attempt was made to relate the anisotropic nature of the surface topographies around the indents in different crystallographic orientations of the single crystal copper specimens with the active slip systems and local texture variations. A reasonably good agreement had been obtained on several aspects of nanoindentation between the experimental and numerical results reported in this investigation as well as similar results reported in the literature. Thus, material properties of single crystal copper can be determined based on an appropriate numerical modeling of the nanoindentation on three crystallographic orientations.     

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.     

Cyclic thermomechanical behavior of a polycrystalline pseudoelastic shape memory alloy

In this work, 3D finite element modeling is employed to examine the thermomechanical behavior of a polycrystalline Ni-Ti shape memory alloy in the pseudoelastic regime. It is shown that the tension-compression asymmetry during uniaxial cyclic loading is due to a preferred orientation of the crystallographic texture. In pure shear loading, the thermomechanical behavior exhibits symmetry in both senses of shear, due to the fiber texture of the specimen bar stock. It is also shown that the apparent strain rate-dependence is due to thermomechanical coupling with latent heat generation/absorption during phase transformation.     

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

Multi-axial behavior of shape-memory alloys undergoing martensitic reorientation and detwinning

Recently, a rate-independent, finite-deformation-based crystal mechanics constitutive model for martensitic reorientation and detwinning in shape-memory alloys has been developed by Thamburaja [Thamburaja, P., 2005. Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys. Journal of the Mechanics and Physics of Solids 53, 825–856] and implemented in the ABAQUS/Explicit [Abaqus reference manuals. 2005. Providence, RI] finite-element program. In this work, we show that the aforementioned model is able to quantitatively predict the experimental response of an initially textured and martensitic polycrystalline Ti–Ni rod under a variety of uniaxial and multi-axial stress states. By fitting the material parameters in the model to the stress–strain response in simple tension, the constitutive model predicts the stress–strain curves for experiments conducted under simple compression, torsion, proportional-loading tension–torsion, and path-change tension–torsion loading conditions to good accord. Furthermore the constitutive model also reproduces the force–displacement response for an indentation experiment to reasonable accuracy.     

Multiscale modeling of the plasticity in an aluminum single crystal

This paper describes a numerical, hierarchical multiscale modeling methodology involving two distinct bridges over three different length scales that predicts the work hardening of face centered cubic crystals in the absence of physical experiments. This methodology builds a clear bridging approach connecting nano-, micro- and meso-scales. In this methodology, molecular dynamics simulations (nanoscale) are performed to generate mobilities for dislocations. A discrete dislocations numerical tool (microscale) then uses the mobility data obtained from the molecular dynamics simulations to determine the work hardening. The second bridge occurs as the material parameters in a slip system hardening law employed in crystal plasticity models (mesoscale) are determined by the dislocation dynamics simulation results. The material parameters are computed using a correlation procedure based on both the functional form of the hardening law and the internal elastic stress/plastic shear strain fields computed from discrete dislocations. This multiscale bridging methodology was validated by using a crystal plasticity model to predict the mechanical response of an aluminum single crystal deformed under uniaxial compressive loading along the [4 2 1] direction. The computed strain-stress response agrees well with the experimental data.     

Influence of anisotropic elasticity on the mechanical properties of fivefold twinned nanowires

Previous atomistic simulations and experiments have shown an increased Young's modulus and yield strength of fivefold twinned (FT) face-centered cubic metal nanowires (NWs) when compared to single crystalline (SC) NWs of the same orientation. Here we report the results of atomistic simulations of SC and FT Ag, Al, Au, Cu and Ni NWs with diameters between 2 and 50 nm under tension and compression. The simulations show that the differences in Young's modulus between SC and FT NWs are correlated with the elastic anisotropy of the metal, with Al showing a decreased Young's modulus. We develop a simple analytical model based on disclination theory and constraint anisotropic elasticity to explain the trend in the difference of Young's modulus between SC and FT NWs. Taking into account the role of surface stresses and the elastic properties of twin boundaries allows to account for the observed size effect in Young's modulus. The model furthermore explains the different relative yield strengths in tension and compression as well as the material and loading dependent failure mechanisms in FTNWs.     

A model for plasticity kinetics and its role in simulating the dynamic behavior of Fe at high strain rates

The recent diagnostic capability of the Omega laser to study solid-solid phase transitions at pressures greater than 10 GPa and at strain rates exceeding 10⁷ s⁻¹ has also provided valuable information on the dynamic elastic-plastic behavior of materials. We have found, for example, that plasticity kinetics modifies the effective loading and thermodynamic paths of the material. In this paper we derive a kinetics equation for the time-dependent plastic response of the material to dynamic loading, and describe the model’s implementation in a radiation-hydrodynamics computer code. This model for plasticity kinetics incorporates the Gilman model for dislocation multiplication and saturation. We discuss the application of this model to the simulation of experimental velocity interferometry data for experiments on Omega in which Fe was shock compressed to pressures beyond the α-to-ε phase transition pressure. The kinetics model is shown to fit the data reasonably well in this high strain rate regime and further allows quantification of the relative contributions of dislocation multiplication and drag. The sensitivity of the observed signatures to the kinetics model parameters is presented.     

Tension–compression asymmetry of phosphor bronze for electronic parts and its effect on bending behavior

In-plane tension and compression experiments on copper alloy sheets (phosphor bronze) and 6000 series aluminum alloy sheets (AA6016-T4) were conducted using a specially designed testing apparatus. The apparatus is equipped with comb-type dies so that stress–strain curves of a sheet specimen subjected to tension followed by compression, and vice versa, can be measured without buckling of the specimen, as well as those for monotonic tension and compression. A difference was observed in the flow stresses between tension and compression for the as-received copper alloy, but not for the aluminum alloy. Moreover, stress reversal tests, such as tension followed by compression and compression followed by tension, were carried out in order to measure the Bauschinger effect. In the second part of the experiment, bending moment–curvature diagrams were measured for the as-received and pre-stretched specimens. The bending moment–curvature diagrams were compared with those calculated using the stress–strain curves obtained from the tension–compression tests, and were in good agreement with those calculated with the tension–compression asymmetry and the Bauschinger effect correctly reproduced.     

Strain hardening in bent copper foils

A series of systematic tensile and microbend tests were conducted on copper foil specimens with different thicknesses. The specimens were made of a copper foil having almost unidirectional crystal orientations that was considered to be nearly single-crystal. In order to investigate the effects of slip system interactions, two different crystal orientations relative to the tensile direction were considered in the tests: one is close to coplanar double-slip orientation, and the other is close to the ideal cube orientation (the tensile direction nearly coincides to [0 0 1]) that yields multi-planar multi-slip deformation. We extended the microbend test method to include the reversal of bending, and we attempted to divide the total amount of strain-hardening into isotropic and kinematic hardening components. In the tensile tests, no systematic tendency of size dependence was observed. In the microbend tests, size-dependent kinematic hardening behavior was observed for both the crystal orientations, while size dependence of isotropic hardening was observed only for the multi-planar multi-slip case. We introduce an extended crystal plasticity model that accounts for the effects of the geometrically necessary dislocations (GNDs), which correspond to the spatial gradients of crystallographic slips. Through numerical simulations performed using the model, the origin of the size-dependent behavior observed in the microbend tests is discussed.     

Lattice orientation effects on void growth and coalescence in fcc single crystals

Void growth and coalescence in fcc single crystals were studied using crystal plasticity under uniaxial and biaxial loading conditions and various orientations of the crystalline lattice. A 2D plane strain unit cell with one and two cylindrical voids was employed using three-dimensional 12 potentially active slip systems. The results were compared to five representative orientations of the tensile axis on the stereographic triangle. For uniaxial tension conditions, the void volume fraction increase under the applied load is strongly dependent on the crystallographic orientation with respect to the tensile axis. For some orientations of the tensile axis, such as [1 0 0] or [1 1 0], the voids exhibited a growth rate twice as fast compared with other orientations ([1 0 0], [2 1 1]). Void growth and coalescence simulations under uniaxial loading indicated that during deformation along some orientations with asymmetry of the slip systems, the voids experienced rotation and shape distortion, due mainly to lattice reorientation. Coalescence effects are shown to diminish the influence of lattice orientation on the void volume fraction increase, but noteworthy differences are still present. Under biaxial loading conditions, practically all differences in the void volume fraction for different orientations of the tensile axes during void growth vanish. These results lead to the conclusion that at microstructural length scales in regions under intense biaxiality/triaxiality conditions, such as crack tip or notched regions, the plastic anisotropy due to the initial lattice orientation has only a minor role in influencing the void growth rate. In such situations, void growth and coalescence are mainly determined by the stress triaxiality, the magnitude of accumulated strain, and the spatial localization of such plastic strains.     

The role of texture in tension–compression asymmetry in polycrystalline NiTi

The purpose of the present study is to thoroughly understand the stress–strain behavior of polycrystalline NiTi deformed under tension versus compression. To do this, a micro-mechanical model is used which incorporates single crystal constitutive relationships and experimentally measured polycrystalline texture into the self-consistent formulation. For the first time it is quantitatively demonstrated that texture measurements coupled with a micro-mechanical model can accurately predict tension/compression asymmetry in NiTi shape memory alloys. The predicted critical transformation stress levels and transformation stress–strain slopes under both tensile and compressive loading are consistent with experimental results. For textured polycrystalline NiTi deformed under tension it is demonstrated that the martensite evolution is very abrupt, consistent with the Luders type deformation experimentally observed. The abrupt transformation under tension is attributed to the fact that the majority of the grains are oriented along the [111] crystallographic direction, which is soft under tensile loading. Since single crystals of the [111] orientation are hard under compression it is also demonstrated that under compression the martensite in textured polycrystalline NiTi evolves relatively slower.