Viscoplastic Constitutive Equations Research Articles

Comprehensive Numerical Simulation of Stress and Damage Fields under Thermo-Mechanical Loading for TBC-Coated Ni-Based Superalloy

A finite element analysis code was developed to accurately predict stress and damage fields in thermal barrier coatings (TBCs) systems subjected to thermo-mechanical loadings. An inelastic constitutive equation for TBCs, and a Chaboche-type viscoplastic constitutive equation for Ni-based super alloys (IN738LC) were employed to simulate high temperature creep and cyclic deformation. Simulations of the TBC/IN738LC system subjected to two types of loading, namely, a triangle-wave loading and a GT-operation loading, were performed using the developed analysis code. The results confirmed that the stress and damage fields in the TBC/IN738LC system could be simulated accurately, and provided us with credible results regarding the crack occurrence. Additionally, the analysis under the GT-operation loading conditions revealed that a peak stress generated during the start-up operation would lead to delamination of the TBC, while a peak stress at the shut-down would lead to cracking in the substrate.

Artificial neural networks and intelligent finite elements in non-linear structural mechanics

In recent years, artificial neural networks were included in the prediction of deformations of structural elements, such as pipes or tensile specimens. Following this method, classical mechanical calculations were replaced by a set of matrix multiplications by means of artificial intelligence. This was also continued in finite element approaches, wherein constitutive equations were substituted by an artificial neural network (ANN). However, little is known about predicting complex non-linear structural deformations with artificial intelligence. The aim of the present study is to make ANN accessible to complicated structural deformations. Here, shock-wave loaded plates are chosen, which lead to a boundary value problem taking geometrical and physical non-linearities into account. A wide range of strain-rates and highly dynamic deformations are covered in this type of deformation. One ANN is proposed for the entire structural model and another ANN is developed for replacing viscoplastic constitutive equations, integrated into a finite element code, leading to an intelligent finite element. All calculated results are verified by experiments with a shock tube and short-time measurement techniques.

Predicting the Microstructure of a Valve Head during the Hot Forging of Steel 21-4N

Valve microstructure is important during hot forging. Austenitic 21-4N steel is often used in exhaust valves. In this study, the microstructure evolution of the forging valve process was predicted using the internal state variables (i.e., average grain size, recrystallized fraction, and dislocation density) modus for 21-4N. First, 21-4N was subjected to hot compression tests on a Gleeble-1500D and static grain growth tests in a heating furnace. A set of uniform viscoplastic constitutive equations was established based on experimental data. Next, the determined unified constitutive equations were conducted in DEFORM-3D, and the microstructure evolution of 21-4N during forging was calculated. Finally, the simulation results of grain size evolution were validated via experiments. Results showed good consistency between the simulations and experiments. Thus, the models adequately predicted the microstructure evolution.

Unified modelling of the flow behaviour and softening mechanism of a TC6 titanium alloy during hot deformation

In this paper, the flow behaviour and globularization kinetics of a TC6 alloy were investigated by conducting hot compression tests at different temperatures (910–970 °C), strain rates (0.01–10.0 s-1) and true strains (0.223, 0.511, 0.916). Scanning electron microscopy (SEM) measurements were performed to evaluate the phase transformation and globularization fraction. Based on the experimental results, a set of internal state variable-based unified viscoplastic constitutive equations was developed to model the flow behaviour and the globularization evolution of the TC6 alloy. The model was fully determined by an optimization method based on genetic algorithm (GA). The results show that the predicted beta phase transformation curve fits the experimental data well, and that the predicted flow stress and globularization fraction of the alpha phase are in good agreement with the experimental data, even at strain rates outside the limits of the model. Furthermore, the flow stress begins to decrease at the onset of the softening mechanism, which occurs at a true strain of approximately 0.05. While an analysis of the predicted variations in the strain hardening rate and strain rate of each phase reveals that the softer beta phase accumulates more overall strain, whereas the strain accumulation in the alpha phase decrease.

Density-based constitutive modelling of P/M FGH96 for powder forging

A set of viscoplastic constitutive equations is presented in this study to predict hot compressive deformation behaviour and densification levels of powder metallurgy (P/M) FGH96 nickel-base superalloy during direct powder forging (DPF) process. The constitutive equations make use of the elliptic equivalent stress proposed in porous material models, and unify the evolution of relative density, normalised dislocation density, isotropic hardening and flow softening of the powder compact. A gradient-based optimisation technique is adopted to determine the material constants using the experimental data obtained from Gleeble isothermal uniaxial compression tests of HIPed FGH96 at different temperatures and strain rates. The developed constitutive equations are incorporated into finite element code DEFORM via user-defined subroutine for coupled thermo-mechanical DPF process modelling. The constitutive equations benefiting from the viscoplastic densification model of the calibrated Abouaf, among the six studied porous material models, compare favourably with the experimental data, while the equations integrating the porous material model of Shima and Oyane provide excellent agreement with experiments in the low density outer region of the powder compact.

Investigation on precision and performance for hot gas forming of thin-walled components of Ti2AlNb-based alloy

Ti2AlNb-based alloys have received considerable attention as potential materials to replace the nickel alloy at 600-750 °C, depending on their advantages of high specific strength, good corrosion and oxidation resistance. To realize the precision and performance control for Ti2AlNb-based alloy thin-walled components, the microstructure evolution was analyzed for setting up the unified viscoplastic constitutive equations based on the physical variables and simulating the forming process coupled between the deformation and the microstructure evolution. Through the finite element model with coupling of microstructure and mechanical parameters, the microstructure evolution and shape fabricating can be predicted at the same time, to provide the basis for the process parameters optimization and performance control. With the reasonable process parameters for hot gas forming of Ti2AlNb thin-walled components, the forming precision and performance can be controlled effectively.

High Temperature Dynamic Response of a Ti-6Al-4V Alloy: A Modified Constitutive Model for Gradual Phase Transformation

Dynamic deformation behavior of a commercial Ti-6Al-4V alloy is measured between room temperature and beyond the β-transus temperature with high thermal resolution using a rapid-heating Kolsky bar technique. The high thermal resolution allows for a thorough investigation of the dynamic thermal softening behavior of this alloy including effects related to the transformation from the initial hcp α/bcc β dual phase structure to a full β structure for improved modeling of high temperature dynamic manufacturing processes such as high-speed machining. Data are obtained at an average strain rate of 1800 s−1 from room temperature to 1177 °C, with total heating times limited to 3.5 s for all tests. Short heating times prevent thermal distortion of the Kolsky bar loading waves and can allow an investigation of non-equilibrium mechanical behavior, although no such behavior was identified in this study. Between 800 °C and 1000 °C, a progressive change in the thermal softening rate was observed that corresponded well with the equilibrium phase diagram for this alloy. The dynamic thermal softening behavior in the transformation region is incorporated via a new modification of the Johnson–Cook (J–C) viscoplastic constitutive equation. Rate sensitivity is determined at room temperature by combining Kolsky bar data with quasi-static measurements at strain rates from 7.5 × 10−5 s−1 to 0.16 s−1 and the data are fit using multi-parameter optimization to arrive at a full modified J–C model for Ti-6Al-4V to nearly 1200 °C. In its generic form, the modification factor we propose, G(T), is applicable to any material system undergoing gradual phase transformation over a range of temperatures.

A Viscoplastic Constitutive Equation from Kolsky Bar Data for Two Steels

Kolsky bar data for viscoplastic materials are usually presented as true stress versus true strain curves at nearly constant values of engineering strain rates. However, constitutive models for numerical simulations require the true axial rate of deformation which is true strain rate. We present a procedure that uses existing data with constant engineering strain rate to obtain a constitutive equation in terms of true strain rate for 304 stainless steel and 4340 Rc 45 steel.

Thermodynamics-based finite strain viscoelastic-viscoplastic model coupled with damage for asphalt material

A thermodynamics based thermo-viscoelastic-viscoplastic model coupled with damage using the finite strain framework suitable for asphalt material is proposed in this paper. A detailed procedure for model calibration and validation is presented, utilizing a set of experimental measurements such as creep-recovery, constant creep, and repeated creep-recovery tests under different loading conditions. The calibrated constitutive model is able to predict the sophisticated time- and temperature- dependent responses of asphalt material, both in tension and in compression. Moreover, a scenario case study on permanent deformation (rutting) prediction of a practical asphalt pavement structure is presented in this work. This paper presents the main features of this new constitutive model for asphalt: (1) A thermodynamics-based framework developed in the large strain context to derive the specific viscoelastic, viscoplastic and damage constitutive equations; (2) A viscoelastic dissipation potential involving deviatoric and volumetric parts, in which Prony series representations of the Lamé constants are used; (3) A modified Perzyna’s type viscoplastic formulation with non-associated flow rule adopted to simulate the inelastic deformation, using a Drucker–Prager type plastic dissipation potential; (4) A specific damage model developed for capturing the evolution disparity between tension and compression. As such, the developed model presents a robust, fully coupled and validated constitutive framework that includes the major behavioral components of asphalt materials, enabling thus an optimized simulation of predicted performance under various conditions. Further development improvements to the model in continued research efforts can be to include further environmental and physico-chemical material behavior such as ageing, healing or moisture induced damage.

A damage-coupled unified viscoplastic constitutive model for prediction of forming limits of 22MnB5 at high temperatures

Hot stamping of the boron steel 22MnB5 has been widely used to produce high strength parts in automobile industry. The forming limits of 22MnB5 sheets are governed by the damage evolution in hot stamping. The aim of this work is to formulate a constitutive model to predict the forming limits of 22MnB5 sheets under hot stamping condition according to continuum damage mechanics (CDM). The authors developed a set of damage-coupled viscoplastic constitutive equations to describe the thermal deformation behavior of 22MnB5 steel under uniaxial tensile state. The material constants of uniaxial constitutive equations were determined from the tensile testing data at a temperature range of 600–900 °C and a strain rate range of 0.01–10 s−1. The uniaxial damage-coupled constitutive equations were generalized to the plane stress state by introducing a multiaxial damage factor considering the effect of stress state on damage evolution. Hot Nakajima testing was conducted to get the forming limits of 22MnB5 sheets at different deformation temperatures and punch speeds. The material constants of multiaxial damage factor were calibrated and confirmed from the experimental data of Nakajima testing. The Nakajima test was numerically simulated by implementing the established constitutive equations via the user material subroutine VUMAT. The prediction by the numerical simulation agrees with the experimental results. Using the determined plane stress damage-coupled constitutive model, the forming limits of 22MnB5 sheets can be predicted under the hot stamping condition.

Three-dimensional elastoplastic phase-field simulation of γ′ rafting and creep deformation

A modified phase-field model coupling viscoplastic constitutive equations has been built up to simulate the creep process of nickel-base single-crystal superalloys at 1223 K/300 MPa. The kinematic and isotropic hardening effects as well as interactions of slip systems are included in the present model. Under the external tension along [001] direction, the plastic strain prefers to concentrate in the channel vertical to [001] direction, promoting γ′ precipitate to raft along the direction vertical to [001] direction. The interactions between slip systems alter the value of plastic strain and thus the stress field in inner γ channel. In turn, the stress field readjusts the plastic deformation. The simulative results and experimental data are in good agreement in the initial creep stage. In addition, this modified model gives a possibility to simulate the microstructure evolution during cycle fatigue.

Effect of Grade on Thermal–Mechanical Behavior of Steel During Initial Solidification

Thermal–mechanical analysis of solidification is important to understand crack formation, shape problems, and other aspects of casting processes. This work investigates the effect of grade on thermal–mechanical behavior during initial solidification of steels during continuous casting of a wide strand. The employed finite element model includes non-linear temperature-, phase-, and carbon content-dependent elastic–viscoplastic constitutive equations. The model is verified using an analytical solution, and a mesh convergence study is performed. Four steel grades are simulated for 30 seconds of casting without friction: ultra-low-carbon, low-carbon, peritectic, and high-carbon steel. All grades show the same general behavior. Initially, rapid cooling causes tensile stress and inelastic strain near the surface of the shell, with slight complementary compression beneath the surface, especially with lower carbon content. As the cooling rate decreases with time, the surface quickly reverses into compression, with a tensile region developing toward the solidification front. Higher stress and inelastic strain are generated in the high-carbon steel, because it contains more high-strength austenite. Stress in the δ-ferrite phase near the solidification front is always very small, owing to the low strength of this phase. This modeling methodology is a step toward designing better mold taper profiles for continuous casting of different steels.

Mechanism-based constitutive equations for superplastic forming of TA15 with equiaxed fine grain structure

Abstract In this study, a set of mechanism-based viscoplastic constitutive equations has been established to predict the viscoplastic flow of TA15 alloy sheets in superplastic forming (SPF) processes. Internal variables are introduced in these constitutive equations to represent individual physical features of the material with equiaxed fine grain structure, such as dislocation density, isotropic hardening, recrystallization and dynamic recovery. 13 material constants in the constitutive equations have been determined from experimental data at a range of temperatures and strain rates. A gradient based optimisation method was applied for the calibration of the equations. Good agreement between the computational and experimental results has been obtained. These newly determined constitutive equations can be used for product and process design through superplastic forming processes.

Formability evaluation for sheet metals under hot stamping conditions by a novel biaxial testing system and a new materials model

Hot stamping and cold die quenching has been developed in forming complex shaped structural components of metals. The aim of this study is the first attempt to develop unified viscoplastic damage constitutive equations to describe the thermo-mechanical response of the metal and to predict the formability of the metal for hot stamping applications. Effects of parameters in the damage evolution equation on the predicted forming limit curves were investigated. Test facilities and methods need to be established to obtain experimental formability data of metals in order to determine and verify constitutive equations. However, conventional experimental approaches used to determine forming limit diagrams (FLDs) of sheet metals under different linear strain paths are not applicable to hot stamping conditions due to the requirements of rapid heating and cooling processes prior to forming. A novel planar biaxial testing system was proposed before and was improved and used in this work for formability tests of aluminium alloy 6082 at various temperatures, strain rates and strain paths after heating, soaking and rapid cooling processes. The key dimensions and features of cruciform specimens adopted for the determination of forming limits under various strain paths were developed, optimised and verified based on the previous designs and the determined heating and cooling method [1]. The digital image correlation (DIC) system was adopted to record strain fields of a specimen throughout the deformation history. Material constants in constitutive equations were determined from the formability test results of AA6082 for the prediction of forming limits of alloys under hot stamping conditions. This research, for the first time, enabled forming limit data of an alloy to be generated at various temperatures, strain rates and strain paths and forming limits to be predicted under hot stamping conditions.

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.

Microstructure evolution and constitutive equations for the high-temperature deformation of 5Cr21Mn9Ni4N heat-resistant steel

The high-temperature deformation behavior of 5Cr21Mn9Ni4N heat-resistant steel was studied using a Gleeble-1500D thermal simulation test machine for high-temperature compression tests. The deformation temperatures ranged from 1273 K to 1393 K, and the strain rates ranged from 0.1 s−1–10 s−1. The height reductions were 20%, 40%, and 60%. The flow stress of 5Cr21Mn9Ni4N increased with increasing strain rate, whereas it decreased with increasing temperature. The high-temperature deformation of 5Cr21Mn9Ni4N began from the strain-hardening stage to the steady-state deformation stage. The characteristics of the stress–strain curves were determined through the interaction of work hardening, dynamic recovery, and dynamic recrystallization. The relationship between microstructure and processing parameters was analyzed. A set of unified viscoplastic constitutive equations based on changes in dislocation density, volume fraction of dynamic recrystallization, and grain size was established to predict deformation behavior and microstructure during a high-temperature working process. The average relative error between the calculated and experimental flow stress was 4.8%. This finding indicated that the constitutive equations could be used to predict the flow behavior of 5Cr21Mn9Ni4N accurately during high-temperature deformation.

Constitutive equation for the hot deformation behavior of Csf/AZ91D composites and its validity for numerical simulation

The flow stress behavior of 10vol. % short carbon fibers reinforced AZ91D composites (Csf/AZ91D) were investigated by hot compression test. The results show the flow stress reach the peak value at small strain and then decrease monotonically until the end of the large strain, which exhibits an obvious dynamic strain softening. The decrease of stress level with deformation temperature increasing or strain rate decreasing can be represented by Zener–Hollomon parameter in a hyperbolic sine equation. By considering the effect of strain on material constants, a modified viscoplastic constitutive equation was established to characterize the dependence of flow stress on the deformation temperature, strain, and strain rate. The stress-strain values calculated by the constitutive equation are in consistent with the experimental results. Applying the constitutive equation, the plastic deformation of Csf/AZ91D composites during the hot compression process were analyzed by finite element simulation. The calculated punch force-stroke curves agree well with the measured ones. The results confirmed that the established constitutive equation can accurately describe the hot plastic deformation behavior of Csf/AZ91D composites, and can be used for the finite element analysis on the hot forming process.

Microdamage-coupled inelastic deformation analysis of ceramic thermal barrier coatings subjected to tensile loading

Abstract To protect the hot parts of a gas turbine (GT) from high-temperature combustion gas, thermal barrier coatings (TBCs) are deposited on the surface of the blades and the inner surface of the combustor. Thermal stress is cyclically generated in TBC systems by the start-steady-stop operation of the GTs, resulting in TBC spallation. Recently, it has become necessary for electricity generation companies to adopt TBC life assessment technology to determine the appropriate interval to maintain the TBC. Therefore, we decided to develop a numerical code for damage-coupled inelastic deformation analysis of GT components coated with TBCs. In this study, a finite element (FE) code was developed based on the commercial FE software MARC by implementing both the complete deformation model of the inelastic constitutive equation for TBCs, which was developed by us, and the Chaboche-type viscoplastic constitutive equation for nickel-based superalloy IN738LC. The simulation of the damage process, including crack initiation and propagation in TBC systems, was conducted under tensile loading using this FE code. It was shown that the crack propagation feature simulated using the FE code completely matched the continuous observation results obtained in the high-temperature tensile test.

A study on the effect of stress state on damage evolution in hot deformation of free cutting steels using double notched bars

The effect of stress state on the initiation of damage for leaded free cutting steel has been investigated under hot rolling conditions. Double notched (DN) circumferential tension samples were designed and used to simulate damage development at different stress states and deformation conditions using a Gleeble (3800) thermal-mechanical testing system. Two DN sample geometries with varying notch profiles were used to account for different states of stress. To simulate the conditions of hot rolling the samples were tested at high temperatures (900–1200 °C) and moderate strain rates (0.1–1 s−1). After testing to failure, which normally occurs at one notch of the specimen, the unfailed notch of each sample was sectioned to examine the sites where damage occurs since the material has been captured in a state very close to failure. Two of the cases examined have shown definitive damage paths occurring from ‘outside–in’ for a sharp notch deformed at T = 900 °C and from ‘inside–out’ for a blunt notch tested at T = 1200 °C for the same strain rate of 0.1 s−1. The experimental results of the failure initiation sites were compared with computed values of the stress fields around the notch profiles, obtained from FE analysis using a set of viscoplastic constitutive equations calibrated for free cutting steel. The temperature profiles from high temperature mechanical testing were used in the FE calculations of the stress state.

THE DEVELOPMENT OF STRESS BASED CONTINUUM DAMAGE MECHANICS MODEL FOR PREDICTING THE FORMABILITY OF MAGNESIUM ALLOYS UNDER COLD/WARM STAMPING PROCESSES

This paper presents the development and application of the novel 2D-stress based continuum damage mechanics (CDM) model for prediction of the formability of magnesium alloys under cold/warm stamping conditions. The CDM model is divided into parts; firstly, a set of uniaxial viscoplastic damage constitutive equations is determined from tensile data. Secondly, a set of multiaxial viscoplastic damage constitutive equations is formulated and calibrated from the forming limit diagram (FLD) data. The experimental uniaxial tensile data for AZ31B magnesium alloy (at different deformation conditions (temperature range of 20°C to 300°C and strain rate range of 0.001 and 0.01s-1) and FLD at the temperature of 250°C were published by Wang et al., [1] and were used to formulate and calibrate the CDM model. A good agreement has been achieved between the experimental and numerical data. Using the newly developed plane-stress unified viscoplastic damage constitutive equations, the FLD of materials can be predicted at different temperatures and strain rates with complex strain path forming conditions.