Dwell fatigue crack nucleation model based on crystal plasticity finite element simulations of polycrystalline titanium alloys

In this paper a crystal plasticity-based crack nucleation model is developed for polycrystalline microstructures undergoing cyclic dwell loading. The fatigue crack nucleation model is developed for dual-phase titanium alloys admitting room temperature creep phenomenon. It is a non-local model that accounts for the cumulative effect of slip on multiple slip systems, and involves evolving mixed-mode stresses in the grain along with dislocation pileups in contiguous grains. Rate dependent, highly anisotropic behavior causes significant localized stress concentration that increases with loading cycles. The crystal plasticity finite element (CPFE) model uses rate and size-dependent anisotropic elasto-crystal plasticity constitutive model to account for these effects. Stress rise in the hard grain is a consequence of time-dependent load shedding in adjacent soft grains, and is the main cause of crack nucleation in the polycrystalline titanium microstructure. CPFE simulation results are post-processed to provide inputs to the crack nucleation model. The nucleation model is calibrated and satisfactorily validated using data available from acoustic microscopy experiments for monitoring crack evolution in dwell fatigue experiments.     

A multivariant micromechanical model for SMAs Part 2. Polycrystal model

An averaging scheme is developed to simulate the behavior of a polycrystalline shape memory alloy (SMA) specimen using the Multivariant Micromechanics approach. An untextured polycrystalline specimen is assumed to be formed by a number of randomly oriented single crystal grains. The previously developed Multivariant technique is used to model the response of each single crystal grain subjected to its stress field seen in the polycrystalline sample. Using spherical grains, the Eshelby–Kröner approach is used to formulate the interaction between grains and to determine the stress state in each individual grain. This model successfully captures the basic features of SMA polycrystalline response to loading and temperature. In addition, comparison is made to recent experimental data with fully triaxial load states. Reasonable qualitative agreement is obtained and some issues related to crystallography of the material model are addressed.     

Modeling deformation behavior of polymers with viscoplasticity theory based on overstress

The nonlinear strain rate sensitivity, multiple creep and recovery behavior of polyphenylene oxide (PPO), which were explored through strain rate-controlled experiments at ambient temperature by Khan [The deformation behavior of solid polymers and modeling with the viscoplasticity theory based overstress, Ph.D. Thesis, Rensselaer Polytechnic Institute, New York], are modeled using the modified viscoplasticity theory based on overstress (VBO). In addition, VBO used by Krempl and Ho [An overstress model for solid polymer deformation behavior applied to Nylon 66, ASTM STP 1357, 2000, p. 118] and the classical VBO are used to demonstrate the improved modeling capabilities of VBO for solid polymer deformation. The unified model (VBO) has two tensor valued state variables, the equilibrium and kinematic stresses and two scalar valued states variables, drag and isotropic stresses. The simulations include monotonic loading and unloading at various strain rates, multiple creep and recovery at zero stress. Since creep behavior has been found to be profoundly influenced by the level of the stress, the tests are performed at different stresses above and below the yield point. Numerical results are compared to experimental data. It is shown that nonlinear rate sensitivity, nonlinear unloading, creep and recovery at zero stress can be reproduced using the modified viscoplasticity theory based on overstress.     

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.     

High temperature creep behaviour of an FeAl intermetallic strengthened by nanoscale oxide particles

The creep behaviour of an FeAl intermetallic strengthened by nanosized oxide particles has been examined at temperatures of 700–825 °C. For all temperatures the strain rate shows a power law dependence on the applied stress. At the lowest temperature and with the highest stresses there is evidence of a threshold stress produced by the difficulty of overcoming the particle barriers, while for higher temperatures as well as at low stresses there is no threshold stress and creep appears to be controlled by general climb. The fine oxide particles produce good strengthening at low temperatures but are more readily overcome at high temperatures due to their very small size and limited attractive relaxation force. Despite such fall in creep strength, this material remains one of the strongest iron aluminides to the temperature range evaluated.     

Creep constitutive modeling of an aluminum alloy under multiaxial and cyclic loading

A constitutive model is developed to characterize creep response of polycrystalline metals. The model is based on the effective stress concept and back stress is utilized as an internal variable. A memory aspect is incorporated in the model to account for the previous maximal stress as a result of dislocation related micromechanisms. The model is used to predict the creep potential of an aluminum alloy under multiaxial and cyclic loading. The predicted results are compared with the available experimental results on an aluminum alloy (2618-T61). The associated six basic model parameters are evaluated by using an optimization technique, called ‘Box Algorithm’. These model parameters are then used to predict some unoptimized experimental data sets. The influence of multiaxial and cyclic loadings is investigated in detail for the selected aluminum alloy. Overall, excellent correlations are observed.     

A thermo-viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials

A temperature-dependent viscodamage model is proposed and coupled to the temperature-dependent Schapery’s nonlinear viscoelasticity and the temperature-dependent Perzyna’s viscoplasticity constitutive model presented in Abu Al-Rub et al., 2009, Huang et al., in press in order to model the nonlinear constitutive behavior of asphalt mixes. The thermo-viscodamage model is formulated to be a function of temperature, total effective strain, and the damage driving force which is expressed in terms of the stress invariants of the effective stress in the undamaged configuration. This expression for the damage force allows for the distinction between the influence of compression and extension loading conditions on damage nucleation and growth. A systematic procedure for obtaining the thermo-viscodamage model parameters using creep test data at different stress levels and different temperatures is presented. The recursive-iterative and radial return algorithms are used for the numerical implementation of the nonlinear viscoelasticity and viscoplasticity models, respectively, whereas the viscodamage model is implemented using the effective (undamaged) configuration concept. Numerical algorithms are implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The model is then calibrated and verified by comparing the model predictions with experimental data that include creep-recovery, creep, and uniaxial constant strain rate tests over a range of temperatures, stress levels, and strain rates. It is shown that the presented constitutive model is capable of predicting the nonlinear behavior of asphaltic mixes under different loading conditions.     

A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior

A model for high temperature creep of single crystal superalloys is developed, which includes constitutive laws for nonlocal damage and viscoplasticity. It is based on a variational formulation, employing potentials for free energy, and dissipation originating from plasticity and damage. Evolution equations for plastic strain and damage variables are derived from the well-established minimum principle for the dissipation potential. The model is capable of describing the different stages of creep in a unified way. Plastic deformation in superalloys incorporates the evolution of dislocation densities of the different phases present. It results in a time dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage. Herein, the nonlocal one is included in order to model strain localization as well as to remove mesh dependence of finite element calculations. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown.     

Microstructure-sensitive modeling of polycrystalline IN 100

A rate dependent, microstructure-sensitive crystal plasticity model is formulated for correlating the mechanical behavior of a polycrystalline Ni-base superalloy IN 100 at 650 °C. This model has the capability to capture first order effects on the stress–strain response due to (a) grain size, (b) γ′ precipitate size distribution, and (c) γ′ precipitate volume fraction. Experimental fatigue data with variable strain rates are used to calibrate the model for several distinct IN 100 microstructures (grain size, precipitate size distributions and volume fractions) obtained from thermomechanical processing. Physically based hardening laws are employed to evolve the dislocation densities for each slip system, taking into consideration the dislocation interaction mechanisms.     

The evolution of crystalline stresses of a polycrystalline metal during cyclic loading

The presence of a positive average applied stress during cyclic uniaxial loading leads to a reduction in fatigue life of metallic parts. The metals are typically polycrystalline, with stresses varying from crystal to crystal due to differences in lattice orientation and slip system strength. Simulations enable us to better understand how polycrystals behave under cyclic loading and how the changing stress over many cycles influences fatigue life. Specifically, uniaxial cyclic simulations of pre-strained HY100 steel were conducted using an elastic viscoplastic continuum slip model employing a Taylor hypothesis. Stress-controlled loading conditions were employed to mimic fatigue tests on cold-bent bar specimens for three different load levels. The macroscopic axial strains and the crystal axial stresses were monitored during the cycles. The stress–strain response for the first cycle was used to determine the load input for the material point simulations. The peak values of crystal axial stress were found to evolve continuously with the number of loading cycles. It was found that the stress change in a crystal is influenced not only by its own orientation but also by the orientations of the other crystals in the aggregate. Furthermore, the distribution of crystal stresses after thousands of cycles at a lower stress amplitude closely resembled the distribution after tens of cycles at a larger stress amplitude.     

Measurements of the contact angle between R134a and both aluminum and copper surfaces

Measurements of quasi-static advancing contact angles of refrigerant R134a on copper and aluminum surfaces are reported over a temperature range from 0 °C to 80 °C. The metal surfaces tested were aluminum (alloy 3003) and copper (alloy 101) plates. Measurements were done using a direct optical observation technique where the liquid meniscus at the surface of a vertical plate was captured using a high magnification camera system. The contact angle of solid–liquid interface was deduced by enhancing and manipulating the digital image using solid modeling software by drawing a tangent line to the meniscus at the intersection location of the solid, liquid and vapor. Values of the contact angle were found to vary between 8.3° and 5.6° for aluminum and between 5.1° and 6.5° for copper when the temperature rose from 0 °C to 80 °C. Maximum standard deviation amongst the measured values of contact angles was 1.3°.     

Effect of creep prestrain on subsequent plastic deformation

The effect of creep prestrain on subsequent plastic deformation is experimentally investigated. The experiments are performed by subjecting thin-walled tubular specimens of stainless steel SUS 304 after creep prestraining to combined axial load and torsion at room temperature to 600°C. The stress-strain relations subsequent to creep prestrain are determined under combined stress state with and without temperature changes in prestraining and subsequent plastic straining. On the experimental results, the plastic hardening effects by creep prestrain are discussed under various temperature conditions. The subsequent stress-strain relations are compared with the calculated results on the equi-plastic strain surfaces.     

The viscoplastic behavior of hastelloy-X single crystal

A viscoplastic constitutive model for Hastelloy-X single crystal material is developed based on crystallographic slip theory. The constitutive model was constructed for use in a viscoplastic self-consistent model for isotropic Hastelloy-X polycrystalline material, which has been described in a recent publication. It is found that, by using the slip geometry known from the metallurgical literature, the anisotropic response can be accurately predicted. The model was verified by using tension and torsion data taken at 982°C (1800°F). The constitutive model used on each slip system is a simple unified visoplastic power law model in which weak latent interaction effects are taken into account. The drag stress evolution equations for the octahedral system are written in a hardening/recovery format in which both hardening and recovery depend on separate latent interaction effects between the octahedral crystallographic slip systems. The strain rate behavior of the single crystal material is well correlated by the constitutive model in uniaxial and torsion tests, but it is necessary to include latent information effects between the octahedral slip systems in order to obtain the best possible representation of biaxial cyclic strain rate behavior. Finally, it was observed that the single crystal exhibited dynamic strain aging at 871°C (1600°F). Similar dynamic strain aging occurs at 649°C (1200°F) in the polycrystalline version of the alloy.     

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

Metastable flow in capillary tubes: An experimental evaluation

This work presents the results of an experimental study with pure refrigerants R-134a and R-600a and refrigerant–oil mixtures flowing through capillary tubes in order to analyse the metastable flow. A large number of experiments were carried out to verify the influence of several variables on the underpressure of vaporization, mainly the inlet subcooling, internal diameter and inlet pressure. Capillary tubes with internal diameter of 0.69 mm and 0.82 mm were tested for condensation temperatures between 40 °C and 50 °C and subcooling degrees between 3 °C and 12 °C. Measurements for oil concentrations of 1% and 3% were conducted and compared with those for pure refrigerant R-134a. The oil influence on the metastable flow was tested and the effect on the underpressure of vaporization is addressed for lower oil concentrations.     

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.     

A simple endochronic transient creep model of metals with applications to variable temperature creep

Similar to the theory of endochronic plasticity, modified by Valanis in 1980, a simple endochronic transient creep model of metals is proposed by using a definition of intrinsic time ζ, measured within the creep strain tensor space, whose metric tensor is treated as a simple power form of creep strain-rate sensitive material function. The resulting constitutive equation of creep (Endocreep) contains only three material constants whose values can be determined completely by a simple creep test. An incremental form involving isothermally constant creep stress, with or without jump, and constant stress with temperature jump, are then formulated.In the applications of Endocreep on 304SS under variable temperature creep, data of simple creep tests, provided by Ohashi et al. at 650°C, Ohno et al. at 600°C, Findley and Cho at 593°C–649°C, are employed to determine material constants. The computational results in the simulation of creep tests under step-up and step-down temperature with constant axial stress are found in very good agreement with data provided by Findley and Cho. However, the results reveal that the model is too simple to deal with the recovery response of unloading. Beside this deficiency the model and its computational method proposed have a potential in the future FEM creep analysis of general thermomechanical loading history.     

Cyclic behavior of unidirectional and cross-ply titanium matrix composites

Relatively simple and efficient micromechanical models are used to obtain the uniaxial response of SCS-6/Timetal 21S with [0]₄ and [0/90]_s laminates when subjected to isothermal and thermomechanical fatigue (TMF) loadings. Features of the modeling that are required to obtain the accurate deformation behavior for this class of materials under these loadings are highlighted. To this end, a comparison is made between the concentric cylinder model and the uniaxial stress model for representing the [0] laminate. The axial stresses from the two models are very similar under mechanical loading. The greatest differences appear under thermal loading alone. The differences on the composite response between a time-independent elastic-plastic and a viscoplastic matrix constitutive model are also examined. The latter is based on the Bodner-Partom unified constitutive model. The [0/90] laminate is treated by adding a parallel element with smeared [90] ply properties to the [0] model and invoking axial strain compatibility as well as stress equilibrium. The proposed constitutive law for the [90] ply includes both matrix viscoplasticity and fiber/matrix separation damage and is based on damage mechanics concepts. The effect of cyclic frequency on TMF behavior is examined. The in-phase TMF life is shown to be very sensitive to frequency due to the relaxation of matrix stress and the attendant increase in fiber stress.     

Crystal plasticity modeling of slip activity in Ti–6Al–4V under high cycle fatigue loading

Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a softening law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.     

Multiaxial creep and cyclic plasticity in nickel-base superalloy C263

Physically-based constitutive equations for uniaxial creep deformation in nickel alloy C263 [Acta Mater. 50 (2002) 2917] have been generalised for multiaxial stress states using conventional von Mises type assumptions. A range of biaxial creep tests have been carried out on nickel alloy C263 in order to investigate the stress state sensitivity of creep damage evolution. The sensitivity has been quantified in C263 and embodied within the creep constitutive equations for this material. The equations have been implemented into finite element code. The resulting computed creep behaviour for a range of stress state compares well with experimental results. Creep tests have been carried out on double notched bar specimens over a range of nominal stress. The effect of the notches is to introduce multiaxial stress states local to the notches which influences creep damage evolution. Finite element models of the double notch bar specimens have been developed and used to test the ability of the model to predict correctly, or otherwise, the creep rupture lifetimes of components in which multiaxial stress states exist. Reasonable comparisons with experimental results are achieved. The γ^′ solvus temperature of C263 is about 925 °C, so that thermo-mechanical fatigue (TMF) loading in which the temperature exceeds the solvus leads to the dissolution of the γ^′ precipitate, and a resulting solution treated material. The cyclic plasticity and creep behaviour of the solution treated material is quite different to that of the material with standard heat treatment. A time-independent cyclic plasticity model with kinematic and isotropic hardening has been developed for solution treated and standard heat treated nickel-base superalloy C263. It has been combined with the physically-based creep model to provide constitutive equations for TMF in C263 over the temperature range 20–950 °C, capable of predicting deformation and life in creep cavitation-dominated TMF failure.