Fictive Stress Research Articles

A Modified Fictive Stress Model for Zr55Al10Ni5Cu30 Bulk Metallic Glass in Supercooled Liquid Region by Introducing a Time Relaxation Factor

Linear and nonlinear viscoelastic behaviors of a Zr55Al10Ni5Cu30 bulk metallic glass are investigated through experiments and described by the fictive stress model. Systematic deviations between the predicted stress-strain curves by fictive stress model and by the experimental results were found. In order to describe the flow stress curves of the Zr55Al10Ni5Cu30 BMG at different temperatures and strain rates in the supercooled liquid region more precisely, the fictive stress model was modified. The parameters of the model were optimized by the genetic algorithm, and a time relaxation factor Z' was introduced. The comparisons of the predicted compressive stress-strain curves and extrusion load-punch stroke curve by the modified fictive stress model with the experimental data show good agreements.

On structural relaxation and viscous work heating during non-Newtonian viscous flow in a Zr 55Al 10Ni 5Cu 30 bulk metallic glass

Both mechanical (stress, strain, and viscosity) and thermal (specimen temperature) behaviors during nonlinear viscoelastic processes, i.e., softening and the following recovery phenomena, which were observed experimentally during constant load compressions in a Zr 55Al 10Ni 5Cu 30 bulk metallic glass at the glass transition region, were calculated using the fictive stress model. Although the model contains several approximations for simplification, the simulated results fit well with the experimental curves. This model has good potential as a tool for designing an adequate near-net shaping condition for glassy materials in the supercooled liquid state to avoid devitrification during processing.

A Fictive Stress Model Calculation of Creep Deformation of Zr-based Bulk Metallic Glass in the Glass Transition Region under a Constant Load

A model calculation based on the fictive stress is conducted on the creep deformation of a Zr55Al10Ni5Cu30 bulk metallic glass near the glass transition temperature under a constant applied load. The model successfully reproduces qualitatively the main features associated with creep viscoelastic behavior at various initial applied stresses, σ0. Whenever the value of σ0 exceeds the critical flow stress, σc (=80 MPa), the flow stress curve, log σ–log t, shows a dip due to stress-induced softening, and the stress dependent viscosity, η(σ), initially decreases. It attains a minimum value, ηmin(σmin), as the glass attains an equilibrium structure corresponding to the imposed stress, σ. For stress, σ, less than the viscosity minimum stress, the η(σ) curves all decrease and tend to merge together, and may be fit with a master curve established previously. This reveals that the model curves depend mainly on the choice of structural relaxation time, λfic, and little on the value of Young's modulus used in a simple Maxwell model calculation. We found a systematic deviation between the model and experimental curves, such that the model tends to overestimate the structural softening in case with smaller initial applied stress. The cause of this deviation is unclear and is a subject for future studies.

Fictive stress model based finite element analysis for bulk metallic glasses at an elevated temperature

It is important to manufacture micromachine parts and simulate the deformation behavior of bulk amorphous alloys in a superccoled liquid region. For these purposes, a correct constitutive model that can reproduce viscosity results is essential for good predictive capability. In this paper, finite element analyses of nonlinear behaviour in bulk metallic glasses during die compression in the supercooled liquid region have been carried out based on the fictive stress model in conjunction with the Maxwell viscoelastic model. The friction effect between the work piece and the die played an important role in inhomogeneous deformation.

Finite element analysis of compressive deformation of bulk metallic glasses

Finite element analysis of a stress-overshoot phenomenon in a Pd-based bulk metallic glass was carried out using a fictive stress model in conjunction with the Maxwell viscoelastic model under uniaxial compression. The overshoot results from non-linear viscoelasticity in the glass transition region, under deformation at constant strain rate. The friction effect between a cylindrical workpiece and a die was shown to produce mainly three stress-concentration sites, which play an important role in creating free volume. These sites give rise to a redistribution of the structural disordering state inside the workpiece, and lead to a change of the average stress blunt. Once inhomogeneous deformation occurs during the development of the stress peak at the initial stages under high strain rates (5 × 10 −3–2.2 × 10 −2 s −1), the effective stress distribution within the workpiece becomes narrow and the deformation homogeneity is recovered in the ensuing steady state flow.

벌크비정질합금의 미세성형 : 구성모델과 적용

Microforming can be a good application for bulk metallic glasses. It is important to simulate the deformation behaviour of the bulk metallic glasses in a supercooled liquid region for manufacturing micromachine parts. For these purposes, a correct constitutive model which can reproduce viscosity results is essential for good predicting capability. In this paper, we studied deformation behaviour of the bulk metallic glasses using the finite element method in conjunction with the fictive stress constitutive model which can describe non-Newtonian as well as Newtonian behaviour. A combination of kinetic equation which describes the mechanical response of the bulk metallic glasses at a given temperature and evolution equations fur internal variables provide the constitutive equation of the fictive stress model. The internal variables are associated with fictive stress and relation time. The model has a modular structure and can be adjusted to describe a particular type of microforming process. Implementation of the model into the MARC software has shown its versatility and good predictive capability.

Microforming of Bulk Metallic Glasses: Constitutive Modelling and Applications

Microforming can be a good application for bulk metallic glasses. It is important to simulate the deformation behaviour of the bulk metallic glasses in a supercooled liquid region for manufacturing micromachine parts. For these purposes, a correct constitutive model which can reproduce viscosity results is essential for good predicting capability. In this paper, we studied deformation behaviour of the bulk metallic glasses using the finite element method in conjunction with the fictive stress constitutive model which can describe non-Newtonian as well as Newtonian behaviour. A combination of kinetic equation which describes the mechanical response of the bulk metallic glasses at a given temperature and evolution equations for internal variables provide the constitutive equation of the fictive stress model. The internal variables are associated with fictive stress and relation time. The model has a modular structure and can be adjusted to describe a particular type of microforming process. Implementation of the model into the MARC software has shown its versatility and good predictive capability.

Influence of <i>In-Situ</i> Nanoprecipitation on Constant Load Deformation in the Glass Transition Region of a Cu<SUB>60</SUB>Zr<SUB>30</SUB>Ti<SUB>10</SUB> Bulk Metallic Glass

Nonlinear viscoelasticity influenced by an in situ nanocrystallization under constant load compression at the glass transition temperature, Tg, is investigated with a Cu 6 0 Zr 3 0 Ti 1 0 (at %) bulk metallic glass (BMG). The experimental curves showed the characteristics, softening and subsequent recovering phenomena which have been observed in a 'stable' Zr-based BMG and also hardening through whole deformation process due to the in situ nanoprecipitation in the glass matrix. These characteristics are fairly reproduced by the fictive stress model extended by the Johnson-Mehl-Avrami (JMA) equation and mixture rule for respectively expressing influences of the Newtonian viscosity and of Young's modulus and critical stress on volume fraction of the nanoprecipitates.

In Situ Observation of Stress-Induced Structural Disorder and Fictive Stress in a Zr55Al10Ni5Cu30 Glassy Alloy

The heat evolution of stress-induced structural disorder, ΔHs(t), in a Zr55Al10Ni5Cu30 glassy alloy during creep deformation under a constant load at the glass transition region (Tg=680 K) is investigated by an in situ measurement of temperature change of a deforming sample. For an initial stress, σ0, greater than a critical stress, σc=80 MPa, all ΔHs(t) curves grow gradually attaining a maximum then decrease with further deformation. The peak in ΔHs indicates the attainment of an equilibrium flow structure. The ΔHs curves plotted against the logarithm of normalized viscosity (ratio of viscosity, η, to the Newtonian viscosity, ηN) tend to merge and increase linearly with a slope of -500 J/mol, as established previously for the steady-state flow state. The present result implies that the same relationship between the structural disorder and viscosity is applicable in the transient flow state as well as in the steady-state flow state. There exists a fictive stress, or steady-state flow stress, which indirectly represents a glass structure. It is also shown that the fictive stress, σf, can describe the free volume, vf, as well as the η and ΔHs of a glass experiencing various stress deformations.

Deformation behavior of the Zr 41.2Ti 13.8Cu 12.5Ni 10Be 22.5 bulk metallic glass over a wide range of strain-rates and temperatures

The stress-strain relations for the Zr 41.2Ti 13.8Cu 12.5Ni 10Be 22.5 bulk metallic glass (Vitreloy 1) over a broad range of temperatures (room temperature to its supercooled liquid region) and strain rates (10 −5 to 10 3 s −1) were established in uniaxial compression using both quasi-static and dynamic Kolsky (split Hopkinson) pressure bar loading systems. Relaxation and jump in strain rate experiments were conducted to further understand the time dependent behavior of Vitreloy 1. The material exhibited superplastic flow above its glass transition temperature (623 K) and strain rates of up to 1 s −1. The viscosity in the homogeneous deformation regime was found to decrease dramatically with increasing strain rate. A fictive stress model was used to describe the basic deformation features of Vitreloy 1 under constant strain-rate loading as well as multiple strain-rate loading at high temperatures.

Obsevations Ofstress-Induced Structuraldisorder and Fictive Stress in Bulk Metallic Glasses

ABSTRACTThe transformation from Newtonian to non-Newtonian viscous flow at the glass transition of Pd- and Zr-based alloy glasses has been investigated in compressive tests under either a constant strain rate, or a constant load. The transition occurs at a critical stress being nearly independent of temperature. The mechanism of the transition thus has been attributed to the stress-induced structural relaxation. This paper describes the evidence of stress-induced disorder as indicated by the change in the viscosity with stress and the evolution of specific heat of the alloy glasses subjected to non-Newtonian steady-state viscous flow. Also presented in this paper is the in-situ observation of structural disorder, by direct measurements of the temperature change of sample, in particular the soften process during a constant load deformation. The heat of evolution is then calculated, and found to scale as the logarithm of the normalized viscosity during entire deformation. This result implies that the relationship between the structural disorder, as indicated by the enthalpy change and viscosity is the same in the transient state during deformation as well as in the steady-state flow process. This is conceptionally very important in that it enables us to introduce a fictive stress which indirectly represents the glass structure.

Creep deformation and stress-induced structural disorder near Tg in a Zr55Al10Ni5Cu30 glassy alloy

Creep deformation under a constant applied load in a Zr55Al10Ni5Cu30 glassy alloy at the glass transition region is investigated. At an initial stress, σ0, less than a critical stress, σc=80 MPa, the glass shows a Newtonian flow. When σ0⩾σc, the flow viscosity, η, initially decreases and attains a minimum. It then increases as the true stress, σ, decreases with further deformation. The initial decrease in η and the attendance of viscosity minimum are due to the stress-induced structural disorder and the structural equilibration with the applied stress, respectively. For stress, σ, less than the viscosity minimum stress, the stress dependence of viscosity, η(σ), curves all tend to merge together, and is fitted well with a master curve, η(σ), established previously for the steady-state flow under constant strain-rate experiments. These results render further support to the hypothesis of stress-induced structural disorder and the concept of fictive stress.

Modeling of stress–strain curves for Pd 40Ni 10Cu 30P 20 glass alloy under constant strain-rate deformation

A model which is based on the hypothesis of stress-induced structural relaxation and the concepts of fictive stress for viscoelastic deformation of glass materials around the glass transition temperature is proposed. It expresses both linear and non-linear viscoelastic behaviors. For constant strain-rate deformation, it performs the stress-overshoot phenomenon observed in the condition of non-Newtonian flow. The model calculation reproduces fairly well the experimental results of Pd 40Ni 10Cu 30P 20 glass alloy, in particular, the development of stress-overshoot with increasing strain-rate and lowering temperature. Moreover, this model performs oscillatory behavior (stress-overshoot and -undershoot) which has been observed in many polymer solutions and glass alloy recently observed in stress–strain curves.

A Fictive Stress Model and Nonlinear Viscoelastic Behaviors in Metallic Glasses

Transition behaviors from linear to nonlinear viscoelasticity under constant strain-rate deformation near the glass transition have been investigated for Pd- and Zr-based alloy glasses. The transition occurs at critical strain-rate, and the steady-state viscosity may decrease by many orders of magnitude above the critical strain-rate. Concurrently with the transition, the growth of the stress shows a stress-overshoot; the stress increases initially attaining a maximum, then decreases and attains a steady-state flow. The transition between steady-state Newtonian and non-Newtonian flows can be analyzed by a stretched exponent relaxation function, and both the normalized viscosity and the flow stress can be represented by a master curve in terms of the product of the strain-rate and the Newtonian viscosity. These results imply that the occurrence of the transition from the Newtonian to non-Newtonian is explicitly determined by the flow stress. A model, based on the hypothesis of stress-induced structural relaxation and the concept of fictive stress for the nonlinear viscoelastic behaviors has been proposed. The model calculation has reproduced fairly well the experimental results of the Pd- and Zr-based glasses, in particular, the development of the stress-overshoot behavior. In addition, the model reveals a stress-overshoot and under-shoot oscillation at very high strain-rate. This oscillatory nonlinear behavior has been observed in many polymer solutions, and also the latest study in metallic glasses. The model calculations of other nonlinear viscoelastic behaviors, such as stress relaxation during stress growth after abrupt cessation of steady-state flow, and a stress regrowth after a brief interval of relaxation, are also presented.

Analysis of the hysteresis loop in stainless steels II. Austenitic–ferritic duplex steel and the effect of nitrogen

The hysteresis loop of austenitic and austenitic–ferritic duplex stainless steels, both with different levels of nitrogen, was analyzed using a generalised statistical theory. The plot of the second derivative yields information on the average effective stress and the probability density function of the critical internal stresses, its evolution in cyclic straining and its dependence on the strain amplitude. The probability density function in duplex steels has two peaks, which correspond to the two-phase structure. The position of these peaks plotted versus the fictive stress corresponds very well to the position of the peaks in the individual phases. The changes of the effective moduli and the evolution of the effective and internal components of the cyclic stress are reported. Nitrogen takes a strong effect on the probability density function of the critical internal stresses in austenitic steel. The suppression of the second peak formed in high amplitude straining is discussed in terms of the appearance of planar dislocation structures, which develop due to the presence of nitrogen.

A Fictive Stress Model Calculation of Nonlinear Viscoelastic Behaviors in a Zr-Based Glassy Alloy: Stress Growth and Relaxation

Nonlinear viscoelastic behaviors of a Zr55Al10Ni5Cu30 glassy alloy are investigated experimentally and by theoretical modeling, based on the hypothesis of stress-induced structural relaxation and the concept of fictive stress. The model calculation curves agree well quantitatively with the experimental results. Structural changes of the nonlinear viscoelasticity during complex-step tests are expressed directly by the fictive stress. In addition, the critical condition under which the transition from linear to nonlinear viscoelasticity occurs is clarified on the basis of this model.

A Fictive Stress Model Calculation of Stress-Overshoot: A Nonlinear Viscoelastic Behavior in Metallic Glass

A model, based on the hypothesis of stress-induced structural relaxation and a concept of fictive stress, is proposed for nonlinear viscoelastic behavior in glassy materials. A model calculation has been carried out on the stress-overshoot phenomenon observed in metallic glasses. The model calculation reproduces the experimental results of a Pd40Ni10Cu30P20 glass fairly well, in particular, the development of stress-overshoot behavior with increasing strain rate and lowered temperature. In addition, the model reveals a stress-overshoot and -undershoot oscillatory behavior which has not been reported previously in metallic glasses, but has been observed in many polymer solutions.

Transition from Linear to Nonlinear Viscoelasticity during Deformation in a Zr-based Glassy Alloy

The transition behaviors from linear to nonlinear viscoelasticity during constant strain-rate deformations of a Zr-based glassy alloy near the glass transition temperature are investigated and a model calculation based on concept of a fictive stress is performed. The experimental results show that the viscoelastic behavior of the glassy alloy is characterized by a very narrow relaxation-time distribution due to its simple atomic structure. Hence, the transition between steady-state Newtonian and non-Newtonian flows can be analyzed by a stretched exponent relaxation-function of strain rate. And the condition at which the transition occurs in the Zr-based glassy alloy is investigated with a new model proposed on the basis of the hypothesis of stress-induced structural relaxation and a concept of fictive stress that expresses the structure of the material indirectly. Stress-strain curves calculated from the model agree quantitatively well with experimental results. The calculated curves of sufficiently higher strain-rates in the nonlinear viscoelastic regime show a stress- oscillation. This has been observed in many polymers, but has not been reported previously in glassy alloys. In the Zr-based glassy alloy, the oscillation is observed as predicted by the model.

A model for non-linear viscoelastic relaxation

A model for non-linear viscoelastic relaxation is proposed based on the hypothesis of stress-induced structural relaxation and the concept of fictive stress. This model predicts both the overshoot and the short-time shift observed in the experiments on stress growth under constant strain rate. A phenomenological explanation of these phenomena is also given.

Evaluation of design values for strain limitation on the basis of cyclic fatigue data

The objective was to describe material response to cyclic deformation in the low-cycle fatigue range for wide plastic strain ranges. The test results extend the crack initiation curve of TRD 301 for consistency's sake, and verify those of KTA 3201:2 for the < 100 cycles range. Steels which are important in the construction of nuclear piping systems were subjected to cyclic tension/compression tests at room temperature and at 350°C. Materials were tested to determine the effect of strain rate. Surface roughness was not found to have any effect on failure at the selected test conditions. Notched specimens having shape factors of α K = 2.0 and 6.9 were also tested. The results of the tests on base material and welded joints were compared with the curves of TRD 301 and KTA 3201.2. It proved possible to use the correlation between crack propagation da/dn and fictive stress intensity factor K I to determine the coefficients for smooth specimens in accordance with Paris' law.