Further development of generalized constitutive relations for metal deformation

A set of constitutive equations is presented to describe the history dependent plastic deformation behavior of anisotropic metals under multiaxial loading conditions. The primary variables that characterize the material behavior with increasing deformation are the effective flow strength,k; the residual or back stress vectorα _i ; and the anisotropy matrix,M _tj . The strain rate is given in terms of an equivalent plastic strain rate, the gradient of a plastic potential, which incorporates key material variables and the stress state. All the material parameters have been determined for a recrystallized Zircaloy-2 from 25 to 450 °C and a solution treated 304 stainless steel from 25 to 650 °C based on experiments that included monotonic, load reversal, and load relaxation tests. The analytical model has been used to simulate the deformation behavior of both metals under a variety of testing conditions, including cyclic and plane strain loading conditions. The general form of the constitutive relation is shown to be consistent with experimental data obtained for various material orientations under different loading conditions.     

<Emphasis Type="Italic">In Situ</Emphasis> Observation of Strain Evolution in Cp-Ti Over Multiple Length Scales

The strain evolution in polycrystalline CP-Ti strip under tension was studied in situ and at two length scales using Synchrotron X-ray diffraction. To establish the bulk material behavior, experiments were performed at the Australian Synchrotron facility. Because of the relatively large grain size, discontinuous “spotty” Debye ring patterns were observed, and a peak fitting algorithm was developed to determine the individual spot positions with the necessary precision for strain determination. The crystallographic directional dependence of strain anisotropy during the loading cycle was determined. Strain anisotropy and yielding of individual crystallographic planes prior to the macroscopic yield point were further clarified by in situ loading experiments performed at the Advanced Light Source (ALS). The deviatoric strain accumulation and plastic response were mapped on a grain-by-grain basis. The onset of microscopic yielding in the grains was identified and correlated with the relative orientation of the grains with respect to the loading direction.     

<Emphasis Type="Italic">In-Situ</Emphasis> Neutron Diffraction Studies of Micromechanical Behavior in a Friction Stir Welded AA7475-T761

An in-situ neutron diffraction technique was used to investigate the lattice strain distributions and micromechanical behavior in a friction stir welded (FSW) sheet of AA7475-T761. The neutron diffraction experiments were performed on the spectrometer for material research, STRESS-SPEC, at FRM II (Garching, Germany). The lattice strain profiles around the weld center were measured as a function of the applied strain during the tensile loading and unloading. The anisotropic elastic and plastic properties of the FSW aluminum alloy were simulated by elasto-plastic self-consistent (EPSC) model to predict the anisotropic deformation behaviors involving the grain-to-grain interactions. Material parameters used for describing the constitutive laws of each test position were determined from the measured lattice strain distributions for different diffraction hkl planes as well as the macroscopic stress-strain curve of the FSW aluminum alloy. A good agreement between experimental results and numerical simulations was obtained. The present investigations provided a reliable prediction of the anisotropic micromechanical behavior of the FSW aluminum alloy during tensile deformation.     

The effect of hydrogen on the fracture toughness of alloy X-750

The effect of hydrogen on the fracture toughness behavior of a nickel-base superalloy, Alloy X-750, in the solutionized and aged condition was investigated. Notched bend specimens were tested to determine if the fracture process was stress or strain controlled. The fracture was observed to initiate at a distance between the location of maximum stress and maximum strain, suggesting that fracture required both a critical stress and strain. The effect of hydrogen was further investigated and modeled using fracture toughness testing and fractographic examination. The fracture toughness of the non-charged specimen was 147 MPa√m. Charging with hydrogen decreased the fracture toughness, K _Ic , to 52 MPa√m at a rapid loading rate and further decreased the toughness to 42 MPa√m for a slow loading rate. This is consistent with the rate-limiting step for the embrittlement process being hydrogen diffusion. The fracture morphology for the hydrogen-charged specimens was intergranular ductile dimple, while the fracture morphology of noncharged specimens was a mixture of large transgranular dimples and fine intergranular dimples. The intergranular failure mechanism in Alloy X-750 was a microvoid initiation process at grain boundary carbides followed by void growth and coalescence. One role of hydrogen was to reduce the void initiation strain for the fine intergranular carbides. Hydrogen may have also increased the rate of void growth. The conditions ahead of a crack satisfy the critical stress criterion at a much lower applied stress intensity factor than for the critical fracture strain criterion. A model based on a critical fracture strain criterion is shown to predict the fracture behavior.     

Deformation and failure of Zr<Subscript>57</Subscript>Nb<Subscript>5</Subscript>Al<Subscript>10</Subscript>Cu<Subscript>15.4</Subscript>Ni<Subscript>12.6</Subscript>/W particle composites under quasi-static and dynamic compression

We have investigated the mechanical behavior of a composite material consisting of a Zr₅₇Nb₅Al₁₀Cu_15.4Ni_12.6 metallic glass matrix with 60 vol pct tungsten particles under uniaxial compression over a range of strain rates from 10⁻⁴ to 10⁴ s⁻¹. In contrast to the behavior of single-phase metallic glasses, the failure strength of the composite increases with increasing strain rate. The composite shows substantially greater plastic deformation than the unreinforced glass under both quasi-static and dynamic loading. Under quasi-static loading, the composite specimens do not fail even at nominal plastic strains in excess of 30 pct. Under dynamic loading, fracture of the composite specimens is induced by shear bands at plastic strains of approximately 20 to 30 pct. We observed evidence of shear localization in the composite on two distinct length scales. Multiple shear bands with thicknesses less than 1 μm form under both quasi-static and dynamic loading. The large plastic deformation developed in the composite specimens is due to the ability of the tungsten particles both to initiate these shear bands and to restrict their propagation. In addition, the dynamic specimens also show shear bands with thicknesses on the order of 50 μm; the tungsten particles inside these shear bands are extensively deformed. We propose that thermal softening of the tungsten particles results in a lowered constraint for shear band development, leading to earlier failure under dynamic loading.     

The influence of texture,strain rate,temperature, and chemistry on the mechanical behavior of hafnium

The compressive mechanical behavior of highly textured, polycrystalline hafnium has been examined as a function of texture, strain rate, temperature, and material chemistry. The microstructural and substructural evolution in Hf was also examined as a function of texture. Decreasing temperature, increasing strain rate, and increasing impurity concentrations were found to increase the yield-stress and work-hardening rates, as well as increase the amount of twinning in Hf. Crystallographic texture was found to exhibit the most marked effect on the mechanical behavior of Hf. Differences in the orientation of the c-axis with respect to the loading direction were found to affect the yield stress, work-hardening behavior, and anisotropy of the tested specimen, with the highest yield stresses and rates of work hardening and the lowest anisotropies in specimens compressed along the c-axis. The amount of deformation twinning and the slip systems activated during deformation were seen to vary based on texture and are shown to correlate well with the observed yield stresses and work-hardening behaviors.     

Modeling the Stress-Strain Relations of Sand in Cyclic Plane Strain Loading

A modeling procedure to simulate stress-strain relations of sand subjected to cyclic loading is proposed. Results from drained plane strain compression, extension, and cyclic loading tests on Toyoura sand are analyzed. The monotonic loading behavior is simulated by the generalized hyperbolic equation to use as the skeleton curves in the simulation of cyclic behavior. To construct hysteretic stress-strain curves based on the skeleton curves, the Masing’s rule is generalized to the proportional rule consisting of the internal and external rules. The drag rule is then introduced to simulate cyclic stress-strain behavior in which the stress amplitude increases at a decreasing rate during cyclic loading with a constant strain amplitude. It is assumed that any plastic shear strain increment taking place in a certain direction drags the whole skeleton curve for loading in the opposite direction towards the direction of the concerned shear strain increment. The measured cyclic stress-strain behavior is well simulated by the proposed method.     

Yielding behavior of prestrained interstitial-free steel and 70/30 brass

The yielding behavior of interstitial-free (IF) steel and 70/30 brass prestrained in plane strain tension and subsequently strained in uniaxial tension has been investigated experimentally. Upon reloading in uniaxial tension, brass exhibited a negative transient (decrease in flow stress) and steel exhibited a positive transient (increase in flow stress). When the yield stress is defined by the offset method, the positive transient is difficult to model using conventional yield theories as elastic deformation is thought to occur outside the original yield or loading surface. In this work, the yield point was defined using the axial strainvs transverse strain curve as measured with biaxial resistance strain gages. The curve has an initially linear elastic portion; the slope then gradually changes until the linear plastic slope is reached. The intersection of the elastic and plastic slopes is defined as the yield point. Using this alternate definition, the yielding behavior of the prestrained metals was investigated. The yield stress for both prestrained brass and steel was found to be lower than the expected monotonic stress. Compared to previous research based on a traditional definition of yield point, this result is unexpected in prestrained steel and shows that yielding does occur inside the loading surface. The positive transient may, therefore, be modeled using conventional yield theories provided that the yield surface is defined using this alternate technique.     

Isotropic and kinematic hardening in a dispersion-strengthened aluminum alloy

The room temperature mechanical behavior of a dispersion strengthened aluminum alloy was examined in tension, compression, and in fully reversed loading. The alloy, 8009, is characterized by a high volume fraction of 50–100 nm dispersoid (25%) and 0.5 mm grain size. In tension, 8009 exhibits low strain to UTS and large post uniform elongation; in compression, near steady state deformation is observed after 2–3% strain. The Bauschinger effect was quantified as a function of prestrain in the forward direction. The experimental reverse loading curves were compared to those expected for ideal isotropic hardening and ideal K1 type kinematic hardening. The alloy exhibits nearly pure kinematic hardening of the K1 type. Based on the microstructure and the fully reversed loading behavior, the monotonic deformation behavior is explained.     

The shape memory effect and superelasticity in two-phase polycrystalline α/β brasses

The shape memory effect (SME), superelasticity (SE), and cyclic deformation behavior of two-phase α/β brasses have been investigated at various temperatures, using tensile tests andin situ optical microscopic observations. The morphology and characteristics of the (thermoelastic) martensitic transformation and the mechanism of the SME are similar to those for single-phase β-brass, but the amount of irrecoverable strain is larger in the two-phase alloys due to plastic deformation of the α particles. After unloading and heating, the slipbands in the discrete a particles remain, whereas the martensite almost disappears; thus, the higher the volume fraction of α particles, the larger the amount of irrecoverable strain. The deformation behavior of alloy A at temperatures above the martensite start (M_s) temperature (with 26 pct α phase) is dominated by deformation of the α phase, so complete SE cannot be obtained after cyclic deformation, both at room temperature and at -40 °C. While in alloy B (containing 15 pct α phase), the deformation behavior is dominated by the formation of stress-induced martensite (SIM). The α particles are deformed before SIM formation on loading at room temperature, but on the contrary, SIM forms before the α particles are deformed on loading at -40 °C (>M_s). Complete SE can be obtained in alloy B after cyclic deformation at room temperature to a given strain but does not occur at -40 °C because the a particles are deformed along with the growth of pre-existing SIM under larger strain during cycling at this temperature.     

The effect of hydrogen on the multiaxial stress-strain behavior of titanium tubing

The influence of internal hydrogen on the multiaxial stress-strain behavior of commercially pure titanium has been studied. Thin-walled tubing specimens containing either 20 or 1070 ppm hydrogen have been tested at constant stress ratios in combined tension and internal pressure. The addition of hydrogen lowers the yield strength for all loading paths but has no significant effect on the strain hardening behavior at strains ε ≥ 0.02. Thus, the hydrogen embrittlement of titanium under plain strain or equibiaxial loading is not a consequence of changes of flow behavior. The yielding behavior of this anisotropic material is described well by Hill’s quadratic yield criterion. As measured mechanically and by pole figure analysis, the plastic anisotropy changes with deformation in a manner which depends on stress state. Hill’s criterion and the associated flow rule do not describe the multiaxial flow behavior well because of their inability to account for changes of texture which depend on multiaxial stress path. Hence, a strain dependent, texture-induced strengthening effect in equibiaxial tension is observed, this effect having the form of an enhanced strain hardening rate. Formerly with Michigan Technological University     

Bauschinger effect in haynes 230 alloy: Influence of strain rate and temperature

Quasistatic and dynamic Bauschinger behavior in HAYNES 230 alloy is examined. At low strain rate (10⁻³/s), theas- received 230 alloy does not show a drop in flow stress,i.e., noBauschinger effect is displayed. At high strain rate (10³/s), a drop in flow stress of 240 MPa was observed upon stress reversal. In contrast, theprecipitation- strengthened condition exhibited a Bauschinger effect in both low and high strain rate stress-reversal experiments. The magnitude of the Bauschinger effect was found to increase with increasing strain rate, forward strain, and decreasing temperature. The substructure evolution accompanying the forward loading cycles was investigated by transmission electron microscopy and is related to the back stresses that developed. The increased Bauschinger stress drop observed at high strain rate and/or low temperature was correlated to an increased degree of planar slip under these conditions. This article is based on a presentation made in the symposium “Dynamic Behavior of Materials,” presented at the 1994 Fall Meeting of TMS/ASM in Rosemont, Illinois, October 3-5, 1994, under the auspices of the TMS-SMD Mechanical Metallurgy Committee and the ASM-MSD Flow and Fracture Committee.     

Constitutive Behavior of Commercial Grade ZEK100 Magnesium Alloy Sheet over a Wide Range of Strain Rates

The constitutive behavior of a rare-earth magnesium alloy ZEK100 rolled sheet is studied at room temperature over a wide range of strain rates. This alloy displays a weakened basal texture compared to conventional AZ31B sheet which leads to increased ductility; however, a strong orientation dependency persists. An interesting feature of the ZEK100 behavior is twinning at first yield under transverse direction (TD) tensile loading that is not seen in AZ31B. The subsequent work hardening behavior is shown to be stronger in the TD when compared to the rolling and 45 deg directions. One particularly striking feature of this alloy is a significant dependency of the strain rate sensitivity on orientation. The yield strength under compressive loading in all directions and under tensile loading in the TD direction is controlled by twinning and is rate insensitive. In contrast, the yield strength under rolling direction tensile loading is controlled by non-basal slip and is strongly rate sensitive. The cause of the in-plane anisotropy in terms of both strength and strain rate sensitivity is attributed to the initial crystallographic texture and operative deformation mechanisms as confirmed by measurements of deformed texture. Rate-sensitive constitutive fits are provided of the tensile stress–strain curves to the Zerilli–Armstrong[1] hcp material model and of the compressive response to a new constitutive equation due to Kurukuri et al.[2]     

Rate-Independent and Rate-Dependent Models for Hysteretic Behavior of Elastomers

Rate-independent and rate-dependent models are presented for the hysteretic shear stress-strain behavior of elastomeric damping materials. A rate-independent hysteretic model, called the general asymptote and power function (GAPF) model, is presented that simulates different types of hysteretic behavior depending on the selected asymptote function. A rate-dependent hysteretic model, formed from a parallel combination of the GAPF model and a dashpot, is also presented which simulates loading frequency dependent behavior in addition to strain amplitude dependent behavior. Closed-form expressions for the shear stress as a function of shear strain are provided for each model. The models are calibrated for three different damping materials, and good correlation between experimental and analytical hysteretic behavior is observed. The models are investigated under variable cyclic loading. To prevent unrealistic stress values (overshooting) after a small strain reversal followed by reloading, a sequential asymptote model is introduced, based on the GAPF model. The hysteretic models were incorporated into a finite-element program within an elastomeric damper element, and the results of nonlinear time history analyses of a building structure with elastomeric dampers under simulated earthquake loading are presented to illustrate behavior of the hysteretic models under several loading histories.     

Indentation power-law creep of high-purity indium

Using a variety of depth-sensing indentation techniques, the creep response of high-purity indium, from room temperature to 75 °C, was measured. The dependence of the hardness on the variables of indentation strain rate (stress exponent for creep (n)) and temperature (apparent activation energy for creep (Q)) and the existence of a steady-state behavior in an indentation test with a Berkovich indenter were investigated. It was shown for the first time that the indentation strain rate (-este-/h) could be held constant during an experiment using a Berkovich indenter, by maintaining the loading rate divided by the load (-este-/P) constant. The apparent activation energy for indentation creep was found to be 78 kJ/mol, in accord with the activation energy for self-diffusion in the material. Finally, by performing -este-/P change experiments, it was shown that a steady-state path independent of hardness could be reached in an indentation test with a geometrically similar indenter.     

Atomic-Scale Study of Plastic-Yield Criterion in Nanocrystalline Cu at High Strain Rates

Large-scale molecular dynamics (MD) simulations are used to understand the macroscopic yield behavior of nanocrystalline Cu with an average grain size of 6 nm at high strain rates. The MD simulations at strain rates varying from 10⁹ s⁻¹ to 8 × 10⁹ s⁻¹ suggest an asymmetry in the flow stress values in tension and compression, with the nanocrystalline metal being stronger in compression than in tension. The tension-compression strength asymmetry is very small at 10⁹ s⁻¹, but increases with increasing strain rate. The calculated yield stresses and flow stresses under combined biaxial loading conditions (X-Y) gives a locus of points that can be described with a traditional ellipse. An asymmetry parameter is introduced that allows for the incorporation of the small tension-compression asymmetry. The biaxial yield surface (X-Y) is calculated for different values of stress in the Z direction, the superposition of which gives a full three-dimensional (3-D) yield surface. The 3-D yield surface shows a cylinder that is symmetric around the hydrostatic axis. These results suggest that a von Mises-type yield criterion can be used to understand the macroscopic deformation behavior of nanocrystalline Cu with a grain size in the inverse Hall–Petch regime at high strain rates.     

Modeling Liquefaction by a Multimechanism Model

An anisotropic constitutive model was recently presented for describing the stress–strain behavior of granular materials with considerations for the initial and induced anisotropy. The model was developed within the framework of a microstructural theory known as the sliding–rolling theory. The resulting model falls within the definition of multimechanism models. The model was shown to satisfactorily represent the drained and undrained behaviors under monotonic loading. The framework used in the model allows extension to describe the behavior under cyclic loading, which is the subject of the present paper. Specifically, the model is further developed for representing the undrained behavior of granular materials under one- and two-way cyclic loading, some of which cause liquefaction resulting in large strain accumulations and the others lead to limited pore pressure and strain accumulations. The validity of the model is verified using triaxial data on Nevada sand.     

The yielding, plastic flow, and fracture behavior of ultra-high molecular weight polyethylene used in total joint replacements

The yielding, plastic flow, and fracture behavior of UHMWPE plays an important role in wear and failure mechanisms of total joint replacement components. The primary objective of this study was to compare the yielding, plastic flow, and fracture behavior of two implantable grades of UHMWPE (GUR 1120 vs 4150 HP). The first part of this work explored the hypothesis that up to the polymer yield point, the monotonic loading behavior of UHMWPE displays similar true stress strain behavior in tension and compression. Uniaxial tension and compression tests were conducted to compare the equivalent true stress vs strain response of UHMWPE up to 0.12 true strain. During monotonic loading, the equivalent true stress strain behavior was similar in tension and compression up to the yield point. However, investigation of the unloading behavior and permanent plastic deformations showed that classical deviatoric rate independent plasticity theory may dramatically overpredict the permanent strains in UHMWPE. A secondary goal of this study was to determine the ultimate true stress and strain for UHMWPE and to characterize the fracture surfaces after failure. Using a fracture mechanics approach, the critical flaw sizes were used in combination with the true ultimate stresses to predict the fracture toughness of the two resins. A custom video-based strain measurement system was developed and validated to characterize the true stress-strain behavior up to failure and to verify the accuracy of the incompressibility assumption in calculating the true stress-strains up to failure. In a detailed uncertainty analysis, theoretical expressions were derived for the relative uncertainty in digital video-based estimates of nominal strain, true strain, homogeneous stress, and true stress. Although the yielding behavior of the two UHMWPE resins was similar, the hardening and plastic flow behavior clearly discriminated between the GUR 1120 and 4150 HP. A statistically significant difference between the fracture toughness of the two resins was also evident. The long-term goal of this research is to provide detailed true stress strain data for UHMWPE under uniaxial tension and compression for future numerical simulations and comparison with more complex multiaxial loading conditions.