Simulated microstructure-sensitive extreme value probabilities for high cycle fatigue of duplex Ti-6Al-4V

A newly developed microstructure-sensitive extreme value probabilistic framework to characterize the performance/variability for damage evolution processes is exercised to compare the driving forces for fatigue crack formation (nucleation and early growth) at room temperature for four different microstructure variants of a duplex Ti-6Al-4V alloy. The aforementioned probabilistic framework links certain extreme value fatigue response parameters with microstructure attributes at fatigue critical sites through the use of marked correlation functions. By applying this framework to study the driving forces for fatigue crack formation in these microstructure variants of Ti-6Al-4V, these microstructures can be ranked in terms of relative high cycle fatigue (HCF) performance and the correlated microstructure attributes that have the most influence on the predicted fatigue response can be identified. Nonlocal fatigue indicator parameters (FIPs) based on the cyclic plastic strain averaged over domains on the length scale of the microstructure attributes (e.g., grains, phases) are used to estimate the driving force(s) for fatigue crack formation at the grain scale. By simulating multiple statistical volume elements (SVEs) using crystal plasticity constitutive relations, extreme value distributions of the predicted driving forces for fatigue crack formation are estimated using these FIPs. This strategy of using multiple SVEs contrasts with simulation based on a single representative volume element (RVE), which is often untenably large when considering extreme value responses. The simulations demonstrate that microstructures with smaller relative primary α grain sizes and lower volume fractions of the primary α grains tend to exhibit less variability and smaller magnitudes of the driving forces for fatigue crack formation. The extreme value FIPs are predicted to most likely occur at clusters of primary α grains oriented for easy basal slip. Additionally, surrounding grains/phases with soft orientation shed load to less favorably oriented primary α grains, producing extreme value FIPs.     

Slip effects on unsteady stagnation-point flow and heat transfer over a shrinking sheet

This paper investigates the unsteady boundary layer stagnation-point flow and heat transfer over a linearly shrinking sheet in the presence of velocity and thermal slips. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically using fourth order Runge-Kutta method with shooting technique. The numerical results show that multiple solutions exist for certain range of the ratio of shrinking velocity to the free stream velocity (i.e., α) which again depend on the unsteadiness parameter β and the velocity slip parameter (i.e., δ). An enhancement of the velocity slip parameter δ causes more increment in the existence range of similarity solution. Fluid velocity at a point increases increases with the increase in the value of the velocity slip parameter δ, resulting in a decrease in the temperature field. The effects of the velocity and thermal slip parameters, unsteadiness parameter (β) and the velocity ratio parameter (α) on the velocity and temperature distributions are computed, analyzed and discussed. The reported results are in good agreement with the available published results in the literature.     

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.     

Plasticity in polycrystalline fretting fatigue contacts

Plastic deformation at the scale of microstructure plays an important role in fretting fatigue failure of metals under cyclic loading. In this study, crystal viscoplasticity theory with a planar triple slip idealization is employed to represent crystallographic plasticity in two-dimensional fretting analyses of Ti-6Al-4V. Subsurface deformation maps, fretting maps, and shakedown maps are constructed based on application of J₂ plasticity theory for the polycrystalline substrate. Comparisons are then made with polycrystal viscoplasticity simulations, the latter suggesting that plastic ratchetting plays a significant role in the fretting fatigue process.     

An exact solution for a model of pressure-dependent plasticity in an un-steady plane strain process

An incompressible material obeying a pressure-dependent yield condition is confined between two planar plates which are inclined at an angle 2α. The plates intersect in a hinged line and the angle α slowly decreases from an initial value. An initial/boundary value problem for the flow of the material is formulated and solved for the stress and the velocity fields, the solution being in closed form. The material is assumed to obey a special case of the double slip and rotation model, which generalises the classical plastic potential model and is also a variant of the double shearing model. The solution for the velocity field may exhibit sliding or sticking at the plates. Solutions which exhibit sticking may have a rigidly rotating zone in the region adjacent to the plates. It is shown that sliding occurs when the value of α is less than a certain critical value α_c ; that sticking occurs without a rigid zone if α exceeds or equals α_c but is less than a second critical value α₀; and that sticking with a rigid zone adjacent to the plates occurs if α exceeds α₀. The values of α_c and α₀ coincide for a certain range of model parameters. Solutions which exhibit sliding are singular. Qualitative features of the solution found are compared with those of the solution for the classical plastic potential model.     

Predicting EXAFS signals from shock compressed iron by use of molecular dynamics simulations

Simulated EXAFS signals from ab initio models and configurational averaging of molecular dynamics (MD) data are compared for α-Fe, and configurationally averaged MD EXAFS signals are compared with experimental data for iron shock compressed to pressures above the α–ε transition pressure. It is shown that molecular dynamics potentials and ab initio models capable of recreating similar vibrational density of states lead to EXAFS signals in good mutual agreement. The effects of the classical nature of the phonon distribution in the MD and the anharmonicity of the potential give rise to noticeable differences between ab initio models and configurational averaging of MD data. However, the greatest influence on the spectra is the form of the potential itself. We discuss the importance of these effects in simulating EXAFS spectra for shock compressed polycrystalline iron. It is shown that EXAFS is an insensitive probe for determining the nature of the close packed product phase in this system.     

The hydrodynamic wall effect for a disperse system

The flow of a highly dilute suspension of spheres (radius α) between two parallel ridid planes (distance L) in slow shearing motion is studied. Even for the limiting situation, (α/L) small but finite, there is a layer-one sphere diameter thick—immediately adjacent to the wall in which bulk quantities are so complicated functionals of the parameters of the microstructure that evaluating them seems out of the question. Nevertheless, it is still simple to obtain average bulk quantifies (e.g. apparent viscosity) and even the evaluation of local bulk quantities far away from the wall poses no problem. The reason being that the customary continuum constitutive equation for the bulk stress can be utilized, although a slip velocity has to supplement it. This applies to any disperse system and can be applied to different flows, too. For the spherical suspension at hand an explicit expression for this slip velocity is obtained.     

Continuity of Drag and Domain Stability in the Low Mach Number Limits

We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving with a velocity prescribed on the boundary of a large channel containing the body. We assume that the Mach number is proportional to a small parameter ε and that the general boundary of the body contains small asperities of amplitude proportional to ε ^α for a certain α?>?0 and suppose the Navier’s slip condition on this rough boundary. We show that time averages of the drag functional converge, as ε → 0, to the corresponding time averages of the drag for the limit system, whereas the limit system is turning out to be the incompressible Navier–Stokes system with no-slip condition on the smooth limit body.     

Determination of Ti-6242 α and β slip properties using micro-pillar test and computational crystal plasticity

The properties and behaviour of an α−β colony Ti-6242 alloy have been investigated at 20 °C utilising coupled micro-pillar stress relaxation tests and computational crystal plasticity. The β-phase slip strength and intrinsic slip system strain rate sensitivity have been determined, and the β-phase shown to have stronger rate sensitivity than that for the α phase. Close agreement of experimental observations and crystal plasticity predictions of micro-pillar elastic-plastic response, stress relaxation, slip activation in both α and β-phases, and strain localisation within the α−β pillars with differing test strain rate, β morphology, and crystal orientations is achieved, supporting the validity of the properties extracted. The β-lath thickness is found to affect slip transfer across the α−β−α colony, but not to significantly change the nature of the slip localisation when compared to pure α-phase pillars with the same crystallographic orientation. These results are considered in relation to rate-dependent deformation, such as dwell fatigue, in complex multiphase titanium alloys.     

A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

This study develops a gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. The theory is based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems. The field equations consist of the yield conditions coupled to the standard macroscopic force balance; these are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. As an aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach-Koehler force on a single dislocation.     

Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V₀ We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V₀ to change from stable to unstable with a decrease in the spring stiffness k below a critical value k_cr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V₀ is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.     

Dynamic anti-plane sliding of dissimilar anisotropic linear elastic solids

The stability of steady, dynamic, anti-plane slipping at a planar interface between two dissimilar anisotropic linear elastic solids is studied. The solids are assumed to possess a plane of symmetry normal to the slip direction, so that in-plane displacements and normal stress changes on the slip plane do not occur. Friction at the interface is assumed to follow a rate and state-dependent law with velocity weakening behavior in the steady state. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along the interface is studied. The critical wavenumber magnitude, |k|_cr, above which there is stability and the corresponding phase velocity, c, of the neutrally stable mode are obtained from the stability analysis. Numerical plots showing the dependence of |k|_cr and c on the unperturbed sliding velocity, V_o, are provided for various bi-material combinations of practical interest.     

An analysis of nanoindentation in linearly elastic solids

The conventional method to extract elastic properties in the nanoindentation of linearly elastic solids relies primarily on Sneddon’s solution (1948). The underlying assumptions behind Sneddon’s derivation, namely, (1) an infinitely large incompressible specimen; (2) an infinitely sharp indenter tip, are generally violated in nanoindentation. As such, correction factors are commonly introduced to achieve accurate measurements. However, little is known regarding the relationship between the correction factors and how they affect the overall accuracy. This paper first proposes a criterion for the specimen’s geometry to comply with the first assumption, and modifies Sneddon’s elastic relation to account for the finite tip radius effect. The relationship between the finite tip radius and compressibility of the specimen is then examined and a composite correction factor that involves both factors, derived. The correction factor is found to be a function of indentation depth and a critical depth is derived beyond which, the arbitrary finite tip radius effect is insignificant. Techniques to identify the radius of curvature of the indenter and to decouple the elastic constants (E and ν) for linear elastic materials are proposed. Finally, experimental results on nanoindentation of natural latex are reported and discussed in light of the proposed modified relation and techniques.     

An analysis of nanoindentation in elasto-plastic solids

This paper examines the accuracy of the extracted elastic properties using the nanoindentation technique on elasto-plastic materials. The application of the correction factor evaluated in the linearly elastic case [Poon, B., Rittel, D., Ravichandran, G., 2008. An analysis of nanoindentation in linearly elastic solids. Int. J. Solids Structures 45 (24), 6018–6033.] on elastic–plastic materials is critically examined. It is then established that the accurate determination of the projected area of contact is found to be crucial for the accurate determination of elastic material properties. The conventional methods for the accurate determination of contact area are generally limited to ratios of Young’s modulus over yield stress, E/σ_y < 30 for elastic-perfectly plastic materials, which is too stringent for most materials. Thus, a new electrical resistance method is proposed to measure directly the projected contact area. Using numerical simulations, it was found that with the accurate determination of A, the error associated with the extracted elastic material properties is reduced by more than 50% in some cases. Using the newly proposed procedure, the error is also found to be independent of E/σ_y and the tip radius, ρ, and it is only a function of Poisson’s ratio, ν. This suggests that the errors might be due to the residual stresses at the plastic imprint that were found to depend on ν as well.     

Numerical modeling and experimental investigation of gas–liquid slug formation in a microchannel T-junction

Gas–liquid two-phase flow in a microfluidic T-junction with nearly square microchannels of 113 μm hydraulic diameter was investigated experimentally and numerically. Air and water superficial velocities were 0.018–0.791 m/s and 0.042–0.757 m/s, respectively. Three-dimensional modeling was performed with computational fluid dynamics (CFD) software FLUENT and the volume of fluid (VOF) model. Slug flow (snapping/breaking/jetting) and stratified flow were observed experimentally. Numerically predicted void fraction followed a linear relationship with the homogeneous void fraction, while experimental values depended on the superficial velocity ratio U_G/U_L. Higher experimental velocity slip caused by gas inlet pressure build-up and oscillation caused deviation from numerical predictions. Velocity slip was found to depend on the cross-sectional area coverage of the gas slug, the formation of a liquid film and the presence of liquid at the channel corners. Numerical modeling was found to require improvement to treat the contact angle and contact line slip, and could benefit from the use of a dynamic boundary condition to simulate the compressible gas phase inlet reservoir.     

Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires

Superelastic deformation of thin Ni-Ti wires containing various nanograined microstructures was investigated by tensile cyclic loading with in situ evaluation of electric resistivity. Defects created by the superelastic cycling in these wires were analyzed by transmission electron microscopy. The role of dislocation slip in superelastic deformation is discussed. Ni-Ti wires having finest microstructures (grain diameter <100 nm) are highly resistant against dislocation slip, while those with fully recrystallized microstructure and grain size exceeding 200 nm are prone to dislocation slip. The density of the observed dislocation defects increases significantly with increasing grain size. The upper plateau stress of the superelastic stress-strain curves is largely grain size independent from 10 up to 1000 nm. It is hence claimed that the Hall-Petch relationship fails for the stress-induced martensitic transformation in this grain size range. It is proposed that dislocation slip taking place during superelastic cycling is responsible for the accumulated irreversible strains, cyclic instability and degradation of functional properties. No residual martensite phase was found in the microstructures of superelastically cycled wires by TEM and results of the in situ electric resistance measurements during straining also indirectly suggest that none or very little martensite phase remains in the studied cycled superelastic wires after unloading. The accumulation of dislocation defects, however, does not prevent the superelasticity. It only affects the shape of the stress-strain response, makes it unstable upon cycling and changes the deformation mode from localized to homogeneous. The activity of dislocation slip during superelastic deformation of Ni-Ti increases with increasing test temperature and ultimately destroys the superelasticity as the plateau stress approaches the yield stress for slip. Deformation twins in the austenite phase ({1 1 4} compound twins) were frequently found in cycled wires having largest grain size. It is proposed that they formed in the highly deformed B19′ martensite phase during forward loading and are retained in austenite after unloading. Such twinning would represent an additional deformation mechanism of Ni-Ti yielding residual irrecoverable strains.     

Effect of γ − α′ phase transformation on plastic adaptation to cyclic loads at cryogenic temperatures

Metastable, type FCC metals and alloys are often applied at extremely low temperatures because of their excellent ductility over the whole temperature range practically down to the absolute zero. These materials (like stainless steels) are frequently characterised by the low stacking fault energy and undergo at low temperatures the plastic strain induced transformation from the parent phase “γ” to the secondary phase “α′”. The phase transformation process consists in creation of two-phase continuum, where the parent phase coexists with the inclusions of secondary phase in thermodynamic equilibrium. The evolution of material micro-structure induces strain hardening related to interaction of dislocations with the inclusions and to increase of equivalent tangent stiffness as a result of evolving proportions of both phases, each characterised by different stiffness. The corresponding hardening model is based on micromechanics and on the Hill concept (1965) supplemented by Mori and Tanaka (1973) homogenisation scheme. Identification of parameters of the constitutive model has been carried out for 304L and 316L stainless steels, based on the available experimental data. The model has been used to describe phase transformation in rectangular beams, circular rods and thin-walled shells subjected to cyclic loads at cryogenic temperatures. Moreover, non-proportional loading paths were studied. A new feature of structures made of metastable materials has been observed. As soon as the γ ? α′ phase transformation begins, the evolution of material micro-structure accelerates the process of adaptation of structural member to cyclic loads and enhances therefore its fatigue life when compared to classical elastic–plastic structures.     

Measurement of Residual Elastic Strains in a Titanium Alloy Using High Energy Synchrotron X-Ray Diffraction

Residual elastic strains in a bent bar of titanium alloy Ti-6Al-4V were measured using high energy diffraction on station 16.3 at SRS Daresbury. Using a single bounce Laue crystal monochromator, diffraction peaks were collected for reflections (00.2), (10.1), (10.2) and (11.0) from the hcp alpha phase of the titanium alloy. Reference values of the lattice spacing for each of the reflections were found from the diffraction pattern collected from a stress-free sampling volume. The residual elastic strain values calculated on the basis of each reflection were then computed and plotted as a function of position across the bent bar. The average macroscopic residual elastic strain was computed using an averaging procedure taking into account the multiplicity of each reflection. Energy dispersive white beam diffraction from the same bent bar was used to collect diffraction patterns over the range of lattice spacings between 0.8 and 2.2 Å. Detector calibration was carried out using the procedure described in Liu et al. (2005) and detailed interpretation of the energy dispersive profiles was carried out allowing the identification of average residual elastic strains in the two principal phases present in the titanium alloy considered, the α-Ti hcp and the β-Ti bcc phases. Peak-specific residual strain profiles computed on the basis of monochromatic measurements show significant differences reflecting the variation in the elastic and plastic properties with grain orientation, i.e., crystal anisotropy. Using the contrast between the elastic and plastic properties of different directions within the α-Ti hcp lattice, the difference between residual elastic strains measured for (00.2) and (11.0) reflections was plotted, as well as the ‘difference strain’ between (00.2) and (10.1) reflections. These profiles show a good qualitative correlation with the plastic strain profile introduced by inelastic bending that was computed from the analysis of Pawley refinement of the energy-dispersive diffraction measurements.     

A finite-deformation,gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations

This paper develops a gradient theory of single-crystal plasticity based on a system of microscopic force balances, one balance for each slip system, derived from the principle of virtual power, and a mechanical version of the second law that includes, via the microscopic forces, work performed during plastic flow. When combined with thermodynamically consistent constitutive relations the microscopic force balances become nonlocal flow rules for the individual slip systems in the form of partial differential equations requiring boundary conditions. Central ingredients in the theory are geometrically necessary edge and screw dislocations together with a free energy that accounts for work hardening through a dependence on the accumulation of geometrically necessary dislocations.