Crystallographic failure analysis of film near cooling hole under temperature gradient of nickel-based single crystal superalloys

In this paper, a unit cell model with a film cooling hole has been set up to analyze the crystallographic stress characterization and failure behavior under temperature gradient of nickel-base single crystallographic superalloys (SC). The aim of this work is to study the failure behavior of SC blades with film cooling. The distribution of cooling air pressure on the hole side surface and the distribution of the temperature around the hole are obtained from the fluid analysis. The result of the temperature distribution is then transferred to the finite element model (cell model) by the interpolation method. The cell model is analyzed by the crystallographic rate dependent finite element method (FEM). Special attention is put on the influence of temperature gradient. The influence of the loading boundaries, i.e. displacement loading and stress loading, on the stress characterization around hole is also taken into consideration. The results show that temperature gradient hole has much influence on the stress characterization. Different types of loading boundaries result in different types of stress and strain distributions. There is clear stress concentration near the hole under displacement loading, while there is clear strain concentration under stress loading. The failure characterization has been studied by the strain energy density criterion. It is shown that the temperature gradient has influence on the failure behavior.     

Low cycle fatigue behavior of single crystal superalloy with temperature gradient

In present study, low cycle fatigue model based on rate dependent constitutive formulation with kinematic hardening and fatigue damage rule is introduced to investigate the fatigue behavior of single crystal superalloys with temperature gradient. Low cycle fatigue tests with uniform temperature and with temperature gradient are carried out to investigate the influence of temperature gradient. The microstructure reveals that the slip deformation is a principal mechanism of low cycle fatigue with temperature gradient. Complex stress experiments of low cycle fatigue are carried out to verify the life prediction rule. Bauschinger effect, ratcheting effect and fatigue damage of single crystal superalloy at different temperatures are studied. Special attention is placed on the simulation of low cycle fatigue behavior with temperature gradient. The simulation results show that the ratcheting effect at high temperature is more remarkable than that at low temperature due to the influence of temperature gradient, which results in the difference of damage evolution between high temperature zone and low temperature zone. Comparison between experiments and simulations with temperature gradient proves that the predicted model based on the damage of the second cycle is reasonable.     

The key role of dislocation dissociation in the plastic behaviour of single crystal nickel-based superalloy with low stacking fault energy: Three-dimensional discrete dislocation dynamics modelling

To model the deformation of single crystal nickel based superalloys (SCNBS) with low stacking fault energy (SFE), three-dimensional discrete dislocation dynamics (3D-DDD) is extended by incorporating dislocation dissociation mechanism. The present 3D-DDD simulations show that, consistent with the existing TEM observation, the leading partial can enter the matrix channel efficiently while the trailing partial can hardly glide into it when the dislocation dissociation is taken into account. To determine whether the dislocation dissociation can occur or not, a critical percolation stress (CPS) based criterion is suggested. According to this CPS criterion, for SCNBS there exists a critical matrix channel width. When the channel width is lower than this critical value, the dislocation tends to dissociate into an extended configuration and vice versa. To clarify the influence of dislocation dissociation on CPS, the classical Orowan formula is improved by incorporating the SFE. Moreover, the present 3D-DDD simulations also show that the yielding stress of SCNBSs with low SFE may be overestimated up to 30% if the dislocation dissociation is ignored. With dislocation dissociation being considered, the size effect due to the width of γ matrix channel and the length of γ′ precipitates on the stress–strain responses of SCNBS can be enhanced remarkably. In addition, due to the strong constraint effect by the two-phase microstructure in SCNBS, the configuration of formed junctions is quite different from that in single phase crystals such as Cu. The present results not only provide clear understanding of the two-phase microstructure levelled microplastic mechanisms in SCNBSs with low SFE, but also help to develop new continuum-levelled constitutive laws for SCNBSs.     

Effect of secondary orientation on notch-tip plasticity in superalloy single crystals

Numerical and experimental evolutions of slip fields in notched Ni-Base Single Crystal superalloy tensile specimens are presented as a function of secondary crystallographic orientation. The numerical predictions based on three-dimensional anisotropic elasticity and crystal plasticity are compared with experimental observations. The results illustrate the strong dependence of the slip patterns and the plastic zone size and shape on the secondary orientation of notches, which can have important consequences on crack initiation. Specific orientations or non-symmetric notch geometries lead to non-symmetric patterns on both sides of the sample. The computations show that strongly different plastic zones are expected in the core of the sample and at free surfaces. The ability of the anisotropic elastic model to anticipate the plastic domains, based on identifying dominant slip systems, is confirmed by the crystal plasticity computations, at low load levels. An important observation is that kink shear banding is a real deformation mode operating at crack tips and notches in high strength nickel-based single crystal superalloys for specific orientations.     

Bending of single crystal microcantilever beams of cube orientation: Finite element model and experiments

The aim of this work is to investigate the microstructure evolution, stress-strain response and strain hardening behavior of microscale beams. For that purpose, two single crystal cantilever beams in the size dependent regime were manufactured by ion beam milling and beams were bent with an indenter device. A crystal plasticity material model for large deformations was implemented in a finite element framework to further investigate the effect of boundary constraints. Simulations were performed using bulk material properties of single crystal copper without any special treatment for the strain gradients. The difference between the slopes of the experimental and the simulated force displacement curves suggested negligible amount of strain gradient hardening compared to the statistical hardening mechanisms.     

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.     

A crystallographic model for the orientation dependence of low cyclic fatigue property of a nickel-base single crystal superalloy

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A dislocation density-based single crystal constitutive equation

Single crystal constitutive equations based on dislocation density (SCCE-D) were developed from Orowan’s strengthening equation and simple geometric relationships of the operating slip systems. The flow resistance on a slip plane was computed using the Burger’s vector, line direction, and density of the dislocations on all other slip planes, with no adjustable parameters. That is, the latent/self-hardening matrix was determined by the crystallography of the slip systems alone. The multiplication of dislocations on each slip system incorporated standard 3-parameter dislocation density evolution equations applied to each slip system independently; this is the only phenomenological aspect of the SCCE-D model. In contrast, the most widely used single crystal constitutive equations for texture analysis (SCCE-T) feature 4 or more adjustable parameters that are usually back-fit from a polycrystal flow curve. In order to compare the accuracy of the two approaches to reproduce single crystal behavior, tensile tests of single crystals oriented for single slip were simulated using crystal plasticity finite element modeling. Best-fit parameters (3 for SCCE-D, 4 for SCCE-T) were determined using either multiple or single slip stress–strain curves for copper and iron from the literature. Both approaches reproduced the data used for fitting accurately. Tensile tests of copper and iron single crystals oriented to favor the remaining combinations of slip systems were then simulated using each model (i.e. multiple slip cases for equations fit to single slip, and vice versa). In spite of fewer fit parameters, the SCCE-D predicted the flow stresses with a standard deviation of 14 MPa, less than one half that for the SCCE-T conventional equations: 31 MPa. Polycrystalline texture simulations were conducted to compare predictions of the two models. The predicted polycrystal flow curves differed considerably, but the differences in texture evolution were insensitive to the type of constitutive equations. The SCCE-D method provides an improved representation of single-crystal plastic response with fewer adjustable parameters, better accuracy, and better predictivity than the constitutive equations most widely used for texture analysis (SCCE-T).     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

Absolute and convective instability of a liquid sheet with transverse temperature gradient

The spatial–temporal instability behavior of a viscous liquid sheet with temperature difference between the two surfaces was investigated theoretically. The practical situation motivating this investigation is liquid sheet heated by ambient gas, usually encountered in industrial heat transfer and liquid propellant rocket engines. The existing dispersion relation was used, to explore the spatial–temporal instability of viscous liquid sheets with a nonuniform temperature profile, by setting both the wave number and frequency complex. A parametric study was performed in both sinuous and varicose modes to test the influence of dimensionless numbers on the transition between absolute and convective instability of the flow. For a small value of liquid Weber number, or a great value of gas-to-liquid density ratio, the flow was found to be absolutely unstable. The absolute instability was enhanced by increasing the liquid viscosity. It was found that variation of the Marangoni number hardly influenced the absolute instability of the sinuous mode of oscillations; however it slightly affected the absolute instability in the varicose mode.     

A continuum model for intermittent deformation of single crystal micropillars

Strain bursts are often observed during compression tests of single crystal micropillars. In this work, we formulate a new continuum model that accounts for the strain bursts within the framework of crystal plasticity. The strain bursts are separated from the loading stage (nearly elastic loading) by introducing a dimensionless constant in the continuum model, and are detected by load serrations. The boundary conditions in the context of micropillar compression are studied and they are shown to be changing and unpredictable as plastic deformation proceeds. To evaluate the validity of our model, finite element simulations of the uniaxial compression tests on nickel micropillars are performed. Our simulations produce clearly visible strain bursts during the plastic flow and the produced intermittent flows are comparable with the experimental observations. For the bulk crystal, a series of strain bursts is identified in the course of plastic flow, despite an apparently smooth stress–strain response. We also show that the intermittent flow is intensified in the micrometer-scale due to both increasing numbers of the successive strain bursts and increasing amplitude of the strain burst, when the specimen size decreases. Finally, we show that the occurrences of the strain bursts are always associated with negative values of the second-order work.     

Cylindrical void in a rigid-ideally plastic single crystal III: Hexagonal close-packed crystal

The analytical solution is derived for the plane strain stress field around a cylindrical void in a hexagonal close-packed single crystal with three in-plane slip systems oriented at the angle π/3 with respect to one another. The critical resolved shear stress on each slip system is assumed to be equal. The crystal is loaded by both internal pressure and a far-field equibiaxial compressive stress. The deformation field takes the form of angular sectors, called slip sectors, within which only one slip system is active; the boundaries between different sectors are radial lines. The stress fields are derived by enforcing equilibrium and a rigid, ideally plastic constitutive relationship, in the spirit of anisotropic slip line theory. The results show that each slip sector is divided into smaller regions denoted as stress sectors and the stress state valid within each stress sector is derived. It is shown that stresses are unique and are continuous within stress sectors and across stress sector boundaries, but the gradient of stresses is not continuous across the boundaries between stress sectors. The solution shows self-similarity in that the stresses over the entire domain can be determined from the stresses within a small region adjacent to the void by invoking certain scaling and symmetry properties. In addition, the stress state exhibits periodicity along logarithmic spirals which emanate from the void. The results predict that the mean value of in-plane pressure required to activate plastic deformation around a void in a single crystal can be higher than that necessary for a void in an isotropic material and is sensitive to the orientation of the slip systems relative to the void.     

On the effect of temperature gradient on the stability of circular tubes made of hyperelastic entropic material

Rubber-like materials are very applicable in almost all fields of industries, but due to their large deformation characteristic, they can exhibit a variety of instabilities. Accordingly, many researchers have been motivated to investigate the effects of different parameters on the stability of hyperelastic cylindrical tubes under finite deformation, while the effects of temperature gradient have not been considered. In this paper, the effects of temperature variation on the stability and thermo-mechanical behavior of the cylindrical tubes made of the entropic materials such as rubber-like materials and elastomers are investigated via an effective strain energy density function. To this purpose, an Ogden-type strain energy density with only integer powers is applied in order to determine an analytical solution, not involving the integral form, for the stress distribution through the wall thickness of cylindrical tubes at finite deformation thermoelasticity. This problem is examined in two cases including (i) a thick-walled cylindrical tube under internal pressure and uniform variation of temperature and (ii) a thick-walled cylindrical tube under internal pressure and temperature gradient, simultaneously. It was observed that the positive temperature gradients in comparison with environment temperatures improve the stability of the circular tubes made of the entropic materials.     

Experimental study on radial temperature gradient effect of a Taylor-Couette flow with axial wall slits

The flow between two concentric cylinders with the inner cylinder rotating and an imposed radial temperature gradient was studied using a digital particle image velocimetry method. The flow transition process under both a positive and negative temperature gradient with four different models of a stationary outer cylinder without and with differing numbers of slits (6, 9 and 18) was studied. The results showed that the buoyant force due to the temperature gradient clearly generated a helical flow when the rotating Reynolds number was small. For the plain and 6-slit models, the transition to a turbulent Taylor vortex flow was not affected by the temperature gradient considered in this study. In addition, the transition process of a larger number of slits (9-, 18-slit models) was accelerated due to the slit wall. As the temperature gradient became larger, the critical Reynolds number of the transition process decreased.     

A micro mechanical study on failure initiation of dual phase steels under tension using single crystal plasticity model

Both experimental and numerical methods were employed to investigate the mechanism of failure in dual phase steels. The tensile test was interrupted in different steps to capture the mechanism of void initiation and void growth during material failure. The results can be considered as a first report for the commercial DP800 steel. Numerical simulations, which were carried out using the real micro-structure, are able to predict the void initiation in the material. In addition, through the numerical simulation a new understanding of the deformation localization was gained. Deformation localization, which causes severely deformed regions in the material, is most probably the main source of rupture in the final stages of the failure. In the SEM micrographs of the material after failure some voids are observable which can validate the results obtained by the simulation.     

Size effects on void growth in single crystals with distributed voids

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

Plane constrained shear of single crystal strip with two active slip systems

Within continuum dislocation theory the plane constrained shear of a single crystal strip with two active slip systems is considered. An analytical solution is found for symmetric double slip which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effects. Comparison with discrete dislocation simulations shows good agreement between the discrete and continuum approaches. Numerical procedures in the general case of non-symmetric double slip are proposed.     

Effect of temperature gradient on Marangoni condensation heat transfer for ethanol–water mixtures

This paper experimentally studied the effect of macroscopic temperature gradient on Marangoni condensation of ethanol–water vapor mixtures under a wide range of concentrations. For each concentration, the experiment was performed at different velocities and pressures. An oblique copper block was employed to create surface temperature gradient. The results indicated that local heat flux was varied along transversal condensation surface, which was caused by surface temperature gradient. This difference in heat flux might be attributed to the variation of condensate thickness on condensation surface. In addition, a mean heat transfer coefficient was derived along transversal condensation surface. For low ethanol concentration (0.5%, 1%), the coefficient kept a high value over a relative wide range of vapor-to-surface temperature difference (<10 K) and could be augmented as much as 15% as compared with literature under similar experimental condition. Moreover, the mean heat transfer coefficient generally increased with increasing velocity or pressure for all concentrations of the ethanol–water mixtures.     

Creep,plasticity, and fatigue of single crystal superalloy

Single crystal components in gas turbine engines are subject to such extreme temperatures and stresses that life prediction becomes highly inaccurate resulting in components that can only be shown to meet their requirements through experience. Reliable life prediction methodologies are required both for design and life management. In order to address this issue we have developed a thermo-viscoplastic constitutive model for single crystal materials. Our incremental large strain formulation additively decomposes the inelastic strain rate into components along the octahedral and cubic slip planes. We have developed a crystallographic-based creep constitutive model able to predict sigmoidal creep behavior of Ni base superalloys. Inelastic shear rate along each slip system is expressed as a sum of a time dependent creep component and a rate independent plastic component. We develop a new robust, computationally efficient rate-independent crystal plasticity approach and combined it with creep flow rule calibrated for Ni-based superalloys. The transient variation of each of the inelastic components includes a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The complete formulation accurately predicts both monotonic and cyclic tests at different crystallographic orientations for constant and variable temperature conditions (low cycle fatigue (LCF) and thermo-mechanical fatigue (TMF) tests). Based on the test and modeling results we formulate a new life prediction criterion suitable for both LCF and TMF conditions.     

An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming

Up to now, several computational methods have been proposed for crystal plasticity models. The main objective of these computational methods has been to overcome the problem with the non-uniqueness of active slip systems during the plastic deformation of a single crystal. Crystal plasticity models based on a single crystal yield function have been proposed as alternative algorithms to overcome this problem. But the problem with these models is that they use a highly non-linear yield function for the crystal, which makes them computationally expensive. In this paper, a computational method is proposed that would modify a single crystal yield function in order to make it computationally efficient. Also to better capture experimental data, a new parameter is introduced into the single crystal yield function to make it more flexible. For verification, this crystal plasticity model was directly applied for the simulation of hydroforming of an extruded aluminum tube under complex strain paths. It was found that the current model is considerably faster than the previous crystal plasticity model based on a power-law type single crystal yield surface. Due to its computational efficiency, the current crystal plasticity model can also be used to calculate the anisotropy coefficients of phenomenological yield functions.