Unified Viscoplasticity Research Articles

A Method of Determining Unified Viscoplastic Constitutive Equations for Hot Forging Simulations

Constitutive equations have been used extensively to accurately describe material properties over a wide range of temperatures and strain rates in numerical simulations. In this paper, an algorithmic method of determining the constants of such constitutive equations is presented. The Genetic Algorithm implementation utilising MATLAB is described, and example fits to experimental data are presented.

Unified viscoplasticity modeling for isothermal low-cycle fatigue and fatigue-creep stress–strain responses of Haynes 230

• A constitutive model for combustor liner material Haynes 230 is developed. • A broad set of isothermal low-cycle fatigue and fatigue-creep responses are simulated. • Static recovery feature simulated stress relaxation in rate-independent regime. • Simulation of mean stress evolution needs additional modeling feature. • Strain range modeling features are essential to simulate hysteresis loop shapes. A robust cyclic viscoplasticity model is developed for simulating a broad set of isothermal, low-cycle fatigue and fatigue-creep responses of Haynes 230 (HA 230) under uniaxial loading. High temperature components experiencing thermo-mechanical fatigue failures can be designed considering their failure responses such that their fatigue life is predictable. Hence, design of high temperature components in aerospace, automobile, nuclear power, and chemical industries should be based on viscoplastic nonlinear analysis using a robust constitutive model. A unified viscoplasticity model based on the nonlinear kinematic hardening rule of Chaboche with several added features for strain-range dependence, rate-dependence, static recovery, and mean stress evolution is developed and evaluated against a broad set of HA 230 responses. Robustness of the constitutive model is demonstrated against predicting fatigue and dwell period stress relaxation responses under uniaxial strain-controlled loading for a broad temperature range of 25–982 °C and strain rate range of 1.1×10 −2 to 2.6×10 −5 /s. Parameter determination of such an advanced model is discussed showing the importance of a well thought out experimental database and thereby providing physical meaning to model parameters.

Cyclic Viscoplasticity Testing and Modeling of a Service-Aged P91 Steel

A service-aged P91 steel was used to perform an experimental program of cyclic mechanical testing in the temperature range of 400 °C–600 °C, under isothermal conditions, using both saw-tooth and dwell (inclusion of a constant strain dwell period at the maximum (tensile) strain within the cycle) waveforms. The results of this testing were used to identify the material constants for a modified Chaboche, unified viscoplasticity model, which can deal with rate-dependant cyclic effects, such as combined isotropic and kinematic hardening, and time-dependent effects, such as creep, associated with viscoplasticity. The model has been modified in order that the two-stage (nonlinear primary and linear secondary) softening which occurs within the cyclic response of the service-aged P91 material is accounted for and accurately predicted. The characterization of the cyclic viscoplasticity behavior of the service-aged P91 material at 500 °C is presented and compared to experimental stress–strain loops, cyclic softening and creep relaxation, obtained from the cyclic isothermal tests.

A Unified Viscoplastic Model for High Temperature Low Cycle Fatigue of Service-Aged P91 Steel

The finite element (FE) implementation of a hyperbolic sine unified cyclic viscoplasticity model is presented. The hyperbolic sine flow rule facilitates the identification of strain-rate independent material parameters for high temperature applications. This is important for the thermo-mechanical fatigue of power plants where a significant stress range is experienced during operational cycles and at stress concentration features, such as welds and branched connections. The material model is successfully applied to the characterisation of the high temperature low cycle fatigue behavior of a service-aged P91 material, including isotropic (cyclic) softening and nonlinear kinematic hardening effects, across a range of temperatures and strain-rates.

Pragmatic optimisation methods for determining material constants of viscoplasticity model from isothermal experimental data

A procedure to estimate material constants for the unified Chaboche viscoplasticity model from experimental data has been published elsewhere; however several critical assumptions are made to enable this and potential numerical problems can limit the effectiveness of the optimisation. Pragmatic optimisation procedures are therefore required to determine the material properties accurately and efficiently. This is made more complex by the presence of several deformation mechanisms and their interactions. Automation is critical due to the large amounts of data generated in testing. Complications that inhibit this process can arise due to factors such as experimental scatter. In this paper, a general optimisation framework is discussed and investigated using data from isothermal tests on a P91 steel at 600°C. Potential obstacles in the procedure are addressed and solutions (such as pre-optimisation experimental data ‘cleaning’) are suggested. Methods to maximise the amount of confidence a user has in a particular optimised constant set are also discussed.

Comparison of several optimisation strategies for the determination of material constants in the Chaboche visco-plasticity model

Determining representative material constant sets for models that can accurately predict the complex plasticity and creep behaviour of components undergoing cyclic loading is of great interest to many industries. The Chaboche unified visco-plasticity model is an example of a model that, with the correct modifications, shows much promise for this particular application. Methods to approximate material constant values in the Chaboche model have been well established; however, the need for optimisation of these parameters is vital due to assumptions made in the initial estimation process. Optimisation of a material constant set is conducted by fitting the predicted response to the experimental results of cyclic tests. It is expected that any experimental data set (found using the same values of test parameters such as temperature; the dependency of which is not accounted for in the original Chaboche model) should yield a single set of optimised material parameters for a given material. In practice, this may not be the case. Experimental test programs usually include multiple loading waveforms; therefore, it is often possible to optimise for separate, different sets of material constants for the same material operating under comparable conditions. Several optimisation strategies that utilise multiple sets of experimental data to form the objective functions in optimisation programs have been applied and critiqued. A procedure that evaluates objective functions derived from the multiple experimental data types simultaneously (i.e. in the same optimisation iteration) was found to give the most consistently high-quality fitting. In the present work, this is demonstrated using cyclic experimental data for a P91 steel at 600 °C. Similar strategies may be applied to many constitutive laws that require some form of optimisation to determine material constant values.

Research on WALKER Constitutive Model Tangent Modulus

In elasto-plasticity computation on materials by sub-increase finite element method, in general, it is necessary to calculate the consistent tangent modulus of elements. In this paper, based on the backward Euler integration, for an unified viscoplasticity constitutive equations, a new expression of consistent tangent modulus is derived for rate-dependent plasticity. The constitutive equations and consistent tangent modulus expression are implemented into a commercial finite element code-MARC. Numerical examples are given to verify the finite element implementation.This template explains and demonstrates how to prepare your camera-ready paper for Trans Tech Publications. The best is to read these instructions and follow the outline of this text.

Constitutive model for a new kind of metastable β titanium alloy during hot deformation

The constitutive model was developed to describe the relationship among flow stress, strain, strain rate, and deformation temperature completely, based on the characteristics of flow stress curves for a new kind of metastable β Ti2448 titanium alloy from isothermal hot compression tests, in a wide range of temperatures (1023-1123 K) and strain rates (63-0.001 s−1). During this process, the adopted hyperbolic sine function based on the unified viscoplasticity theory was used to model the flow behavior of alloy undergoing flow softening caused by dynamic recovery (DRV) at high strain rates (≥1 s−1). The standard Avrami equation was adopted to represent the softening mechanism attributed to dynamic recrystallization (DRX) at low strain rates (<1 s−1). Additionally, the material constants were determined by optimization strategy, which is a new method to solve the nonlinear constitutive equation. The stress—strain curves predicted by the developed constitutive model agree well with the experimental results, which confirms that the developed constitutive model can give an accurate estimate of the flow stress of Ti2448 titanium alloy and provide an effective method to model the flow behavior of metastable β titanium alloys during hot deformation.

Effects of the Unified Viscoplastic Formulation and Temperature Terms on the Thermomechanical Behavior of Soldering Materials

Solder materials are critical packaging compounds and due to usually weakest melting temperature among packaging constitutive materials, thus, they are frequently subjected to a multitude of physical phenomena: creep, fatigue and combined hardening effects. The complexity and interaction of such factors must be considered in suitable way in the mechanical behavior modeling using the appropriate material behavior laws. The choice of the mechanical model depends on several factors such as the complexity of constitutive equations to be integrated, the availability and suitability of implementation in the FE codes, the number of parameters to be identified, the capability of the model to represent the most common physical features of the material… Following these observations and in order to deal with these critical remarks, comparisons between the most common unified viscoplastic models should be done in the local and finite element levels for the decision upon the most efficient model. That is the aim of this paper with application to a tin based solder token as the test material.

Use of spherical indentation technique for measurement of property variations of γTiAl

Abstract

Modeling of Flow Stress for one New Kind of Metastable β Titanium Alloy at High Temperature

Based on the characteristics of the flow stress curves for one new kind of metastable Ti2448 titanium alloy from isothermal hot compression tests, the constitutive model was developed to describe the relation between flow stress and strain, strain rate, deformation temperature completely. During this process, the flow behavior of alloy at high temperature undergo flow softening caused by dynamic recovery (DRV) was modeled by the adopted hyperbolic sine function based on the unified viscoplasticity theory, the further drop in flow stress after the peak value in stress-strain curves was assumed to be caused by temperature rise and the constitutive model was modified accordingly. Additionally, the material constants were determined by optimization strategies, which is a new method to solve the nonlinear constitutive equation. The stress-strain curves predicted by the developed constitutive model well agree with experimental results, which confirms that the developed constitutive model gives an accurate estimate of the flow stress of Ti2448 titanium alloy and can provide an effect method to model the flow behavior of metastable titanium alloy at high temperature.

Constitutive model for uniaxial time-dependent ratcheting of 6061-T6 aluminum alloy

Based on the experimental results of uniaxial time-dependent ratcheting of 6061-T6 aluminum alloy, a new time-dependent cyclic visco-plastic constitutive model is proposed by employing a new non-linear kinematic hardening rule in the framework of unified visco-plasticity. A static recovery term is introduced into the kinematic hardening rule to reflect the remarkable time-dependence of ratcheting, and a nonlinear isotropic hardening rule is also included in the proposed model to reflect the cyclic softening feature of the alloy. All the material parameters used in the proposed model are assumed to be temperature-dependent. Comparison of the experimental and simulated results shows that the proposed visco-plastic constitutive model can reasonably describe the uniaxial time-dependent ratcheting of 6061-T6 aluminum alloy presented at room and elevated temperatures.

Modeling of Oxygen Diffusion Along Grain Boundaries in a Nickel-Based Superalloy

Finite element analyses of oxygen diffusion at the grain level have been carried out for a polycrystalline nickel-based superalloy, aiming to quantify the oxidation damage under surface oxidation conditions at high temperature. Grain microstructures were considered explicitly in the finite element model where the grain boundary was taken as the primary path for oxygen diffusion. The model has been used to simulate natural diffusion of oxygen at temperatures between 650∘C and 800∘C, which are controlled by the parabolic oxidation rate and oxygen diffusivity. To study the effects of mechanical stress on oxygen diffusion, a sequentially coupled deformation-diffusion analysis was carried out for a generic specimen geometry under creep loading condition using a submodeling technique. The material constitutive behavior was described by a crystal plasticity model at the grain level and a unified viscoplasticity model at the global level, respectively. The stress-assisted oxygen diffusion was driven by the gradient of hydrostatic stress in terms of pressure factor. Heterogeneous deformation presented at the grain level imposes a great influence on oxygen diffusion at 750∘C and above, leading to further penetration of oxygen into the bulk material. Increased load level and temperature enhance oxygen concentration and penetration within the material. At 700∘C and below, mechanical loading seems to have negligible influence on the oxygen penetration because of the extremely low values of oxygen diffusivity and pressure factor. In the case of an existing surface microcrack, oxygen tends to accumulate around the crack tip due to the high stress level presented near the crack tip, leading to localized material embrittlement and promotion of rapid crack propagation.

FE analysis of a notched bar under thermomechanical fatigue using a unified viscoplasticity model

Z-mat software is used to implement the multiaxial form of the Chaboche unified viscoplastic constitutive equations into the Abaqus finite element (FE) code. Complex transient stress fields within a grade P91 steel notched bar specimen are investigated in detail for different thermomechanical fatigue (TMF) loading conditions. The results of this work can be used to determine suitable test conditions for notched specimens by providing predictions of the stress-strain history local to the notch. In future, such tests will be used along with the FE analyses to develop and validate multiaxial TMF lifing models.

의탄성 형상기억합금에 대한 현상학적 구성모델

Abstract Shape memory alloys (SMAs) have the ability to recover their original shape upon thermo-mechanical loading even after large inelastic deformation. The unique feature is known as pseudoelasticity and shape memory effect caused by the crystalline structural transformation between two solid-state phases called austenite and martensite. To support the engineering application, a number of constitutive models, which can be formally classified into either micromechanics-based or phenomenological model, have been developed. Most of the constitutive models include a kinetic law governing the crystallographic transformation. The present work presents a one-dimensional, phenomenological constitutive model for SMAs in the context of the unified viscoplasticity theory. The proposed model does not incorporate the complex mechanisms of phase transformation. Instead, the effects induced by the transformation are depicted through the growth law for the back stress that is an internal state variable of the model.

Cyclic thermo-mechanical material modelling and testing of 316 stainless steel

A programme of cyclic mechanical testing of a 316 stainless steel, at temperatures of up to 600 °C under isothermal conditions, for the identification of material constitutive constants, has been carried out using a thermo-mechanical fatigue test machine (with induction coil heating). The constitutive model adopted is a modified Chaboche unified viscoplasticity model, which can deal with both cyclic effects, such as combined isotropic and kinematic hardening, and rate-dependent effects, associated with viscoplasticity. The characterisation of 316 stainless steel is presented and compared with results from tests consisting of cyclic isothermal, as well as in-phase and out-of-phase thermo-mechanical fatigue conditions, using interpolation between the isothermal material constants to predict the material behaviour under anisothermal conditions.

Determination of Material Parameters in the Chaboche Unified Viscoplasticty Model

An experimental programme of cyclic mechanical testing of a 316 stainless steel, at temperatures up to 600°C, under isothermal conditions, for the identification of material constitutive constants, has been carried out using a thermo-mechanical fatigue (TMF) test machine with induction coil heating. The constitutive model adopted is a modified Chaboche unified viscoplasticity model, which can deal with both cyclic effects, such as combined isotropic and kinematic hardening, and rate-dependent effects, associated with viscoplasticity. The characterisation of 316 stainless steel is presented and compared to results from cyclic isothermal tests. A least squares optimisation algorithm has been developed and implemented for determining the material constants in order to further improve the general fit of the model to experimental data, using the initially obtained material constants as the starting point in this optimisation process. The model predictions using both the initial and optimised material constants are compared to experimental data.

Determination of material properties in the Chaboche unified viscoplasticity model

An experimental programme of cyclic mechanical testing of a 316 stainless steel, at temperatures up to 600°C, under isothermal conditions, for the identification of material constitutive constants, has been carried out using a thermo-mechanical fatigue (TMF) test machine with induction coil heating. The constitutive model adopted is a modified Chaboche unified viscoplasticity model, which can deal with both cyclic effects, such as combined isotropic and kinematic hardening, and rate-dependent effects, associated with viscoplasticity. The characterization of 316 stainless steel is presented and compared with results from cyclic isothermal tests. A least-squares optimization algorithm has been developed and implemented for determining the material constants in order to further improve the general fit of the model to experimental data, using the initially obtained material constants as the starting point in this optimization process. The model predictions using both the initial and optimized material constants are compared to experimental data.

Development of a unified viscoplasticity constitutive model based on classical plasticity theory

The traditional unified viscoplasticity constitutive model can be only applied to metal materials. The study of the unified constitutive theory for metal materials has discovered the correlation between the classical plasticity theory and the unified viscoplasticity constitutive model, thus leading to the concepts of the classic plastic potential and yield surface in the unified constitutive model. Moreover, this research has given the continuous expression of the classical plastic multiplier and presented the corresponding constructive method, which extends its physical significance and lays down a good foundation for the application of the unified constitutive theory to the material analysis in more fields. This paper also introduces the unified constitutive model for metal materials and geo-materials. The numerical simulation indicates that the construction should be both reasonable and practical.

Inelastic behavior and numerical analysis in twin-roll casting process of AZ31 alloy

In this paper, Anand’s model, a unified visco-plasticity constitutive model, was employed to simulate the highly nonlinear behavior in the twin-roll casting process. Anand model’s parameters were regressed based on compression tests at various temperatures and strain rates for magnesium alloy AZ31. To calculate the thermal mechanical stresses, the thermal flow of twin-roll casting process was simulated firstly; then the stresses were calculated by the imposed thermal flow as the body load, and a small displacement load along roller’s tangential direction was imposed simultaneously in order to simulate the rolling action. The stresses and deformation results were compared with experimental data. Based on the stresses analysis and experimental tests, it reveals that separating force should be strictly controlled in the twin roll casting process in order to avoid cracks caused by thermal and deformation stresses.