Constitutive Equations for the Rate-Dependent Deformation of Metals at Elevated Temperatures

1982 ◽  
Vol 104 (1) ◽  
pp. 12-17 ◽  
Author(s):  
L. Anand
Keyword(s):  
Dynamic Recrystallization ◽  
Strain Rate ◽  
Strain Hardening ◽  
Constitutive Equations ◽  
Elevated Temperatures ◽  
Internal Variable ◽  
Internal Variables ◽  
Homologous Temperature ◽  
Static Recovery ◽  
Rate Dependent

Approximate constitutive equations are proposed for use in the analysis of the rate-dependent deformation of metals at temperatures in excess of a homologous temperature of 0.5. The constitutive equations are formulated within the scope of some recent theories of elastoviscoplasticity with internal variables, but employ only a single scalar internal variable representing an isotropic resistance to plastic flow offered by the internal microstructural state of the material. The special constitutive euqations incorporate strain hardening of the Voce type, and account for the effects of the prior histories of strain rate and temperature undergone by the material. These equations, however, do not represent the important effects of static recovery or of static and dynamic recrystallization.

scholarly journals Strain Rate Dependent Ductility and Strain Hardening in Q&P Steels

2021 ◽  
Author(s):  
Christopher B. Finfrock ◽  
Melissa M. Thrun ◽  
Diptak Bhattacharya ◽  
Trevor J. Ballard ◽  
Amy J. Clarke ◽  
...  
Keyword(s):  
Strain Rate ◽  
Strain Hardening ◽  
Rate Dependent

Investigation on Hardening and Softening Behavior of Steel after Rapid Strain Rate Changes

2016 ◽  
Vol 716 ◽  
pp. 121-128 ◽  
Author(s):  
Jens Dierdorf ◽  
Johannes Lohmar ◽  
Gerhard Hirt
Keyword(s):  
Flow Stress ◽  
Dynamic Recrystallization ◽  
Strain Rate ◽  
Metal Forming ◽  
Elevated Temperatures ◽  
Relaxation Behavior ◽  
Rate Change ◽  
Strain Rates ◽  
Flow Curves ◽  
Strain Rate Change

The design of industrial hot metal forming processes nowadays is mostly carried out using commercial Finite Element (FE) software codes. For precise FE simulations, reliable material properties are a crucial factor. In bulk metal forming, the most important material property is the materials flow stress, which determines the form filling and the necessary forming forces. At elevated temperatures, the flow stress of steels is determined by strain hardening, dynamic recovery and partly by dynamic recrystallization, which is dependent on strain rate and temperature. To simulate hot forming processes, which are often characterized by rapidly changing strain rates and temperatures, the flow stress is typically derived from flow curves, determined at arbitrary constant temperatures and strain rates only via linear interpolation. Hence, the materials instant reaction and relaxation behavior caused by rapid strain rate changes is not captured during simulation. To investigate the relevance of the relaxation behavior for FE simulations, trails with abrupt strain rate change are laid out and the effect on the material flow stress is analyzed in this paper. Additionally, the microstructure evolution due to the strain rate change is investigated. For this purpose, cylinder compression tests of an industrial case hardening steel are conducted at elevated temperatures and different strain rates. To analyze the influence of rapid strain rate changes, changes by one power of ten are performed at a strain of 0.3. As a reference, flow curves of the same material are determined at the initial and final constant strain rate. To investigate the microstructure evolution, compression samples are quenched at different stages, before and after the strain rate change. The results show that the flow curves after the strain rate change tend to approximate the flow curves measured for the final strain rate. However, directly after the strain rate change significant differences between the assumed instant flow stress and the real material behavior can be observed. Furthermore, it can be shown that the state of dynamic recrystallization at the time of the strain rate change influences the material response and relaxation behavior resulting in different slopes of the investigated flow curves after the strain rate change.

Constitutive Modeling of a Thermoplastic Olefin Over a Broad Range of Strain Rates

2006 ◽  
Vol 128 (4) ◽  
pp. 551-558 ◽  
Author(s):  
Yan Wang ◽  
Ellen M. Arruda
Keyword(s):  
Strain Rate ◽  
Strain Hardening ◽  
Three Dimensional ◽  
Elastic Response ◽  
Strain Rates ◽  
Strain Behavior ◽  
Rate Dependent ◽  
State Dependent ◽  
Deformation State ◽  
Ann Arbor

A microstructually motivated, three-dimensional, large deformation, strain rate dependent constitutive model has been developed for a semi-crystalline, blended, thermoplastic olefin (TPO) (Wang, Y., 2002, Ph.D. thesis, The University of Michigan, Ann Arbor, MI). Various experiments have been conducted to characterize the TPO and to verify the modeling approach (Wang, Y., 2002, Ph.D. thesis, The University of Michigan, Ann Arbor, MI). The model includes a quantitative rate-dependent Young’s modulus, a nonlinear viscoelastic response between initial linear elastic response and yield due to inherent microstructural irregularity, rate and temperature dependent yield with two distinctive yield mechanisms for low and high strain rates, temperature-dependent strain hardening, plastic deformation of crystalline regions, and adiabatic heating. It has been shown to accurately capture the observed TPO stress-strain behavior including the rate-dependent initial linear elastic response; temperature, strain rate, and deformation state-dependent yield; temperature and deformation state-dependent strain hardening; and pronounced thermal softening effects at high (impact) strain rates. The model has also been examined for its ability to predict the response in plane strain compression based on material parameters chosen to capture the uniaxial compression response. The model is predictive of the initial strain rate dependent stiffness, yield, and strain hardening responses in plane strain. Such predictive capability demonstrates the versatility with which this model captures the three-dimensional anisotropic nature of TPO stress-strain behavior.

Internal Variable Modeling of the Creep of Monolithic Ceramics

1995 ◽  
Author(s):  
Charles S. White ◽  
Radwan M. Hazime
Keyword(s):  
Grain Boundaries ◽  
Elevated Temperatures ◽  
Internal Variable ◽  
Complete Characterization ◽  
Tensile Creep ◽  
Internal Variables ◽  
Variable Model ◽  
The Road ◽  
Monolithic Ceramics

Abstract Ceramics are assuming an important role for use in power generation. One of the road blocks is a complete characterization of the deformation and life of advanced ceramics at elevated temperatures. Substantial high temperature creep testing has been conducted in recent years. Most commonly, Norton’s law for deformation and the Monkman-Grant relationship for failure have been used to correlate test data. In this paper, internal variable modeling is discussed as an alternative to Norton’s Law/Monkman-Grant. Through the use of internal variables, micromodeling of the important mechanisms can be extended to the macroscopic behavior. Also, the effects of simultaneous or competing phenomena can be considered. An example is the growth of lenticular cavities on the two grain boundaries of certain silicon nitrides while the grain boundaries are crystallizing. The results of a preliminary internal variable model for HIPed silicon nitride is presented and compared with tensile creep experiments.

STRAIN RATE–DEPENDENT BEHAVIOR OF UNCURED RUBBER: EXPERIMENTAL INVESTIGATION AND CONSTITUTIVE MODELING

2022 ◽  
Author(s):  
Sanghyeub Kim ◽  
Thomas Berger ◽  
Michael Kaliske
Keyword(s):  
Strain Rate ◽  
Evolution Equations ◽  
Cyclic Creep ◽  
Tensile Tests ◽  
Internal Variables ◽  
Creep Tests ◽  
Rate Dependent ◽  
Nonlinear Viscosity ◽  
Dependent Behavior ◽  
Building Process

ABSTRACT The strain rate dependence of uncured rubber is investigated through a series of tensile tests (monotonic, multistep relaxation, cyclic creep tests) at different strain rates. In addition, loading/unloading tests in which the strain rate is varied every cycle are carried out to observe their dependence on the deformation history. A strain rate–dependent viscoelastic–viscoplastic constitutive model is proposed with the nonlinear viscosity and process-dependent recovery properties observed in the test results. Those properties are implemented by introducing evolution equations for additional internal variables. The identified material parameters capture the experiments qualitatively well. The proposed model is also evaluated by finite element simulations of the building process of a tire, followed by the in-molding.

Strain Rate Sensitivity of Commercial Aluminum Alloys

2019 ◽  
Author(s):  
R.C. Picu
Keyword(s):  
Plastic Deformation ◽  
Strain Rate ◽  
Strain Rate Sensitivity ◽  
Solid Solutions ◽  
Elevated Temperatures ◽  
Rate Sensitivity ◽  
Solute Diffusion ◽  
Rate Dependent ◽  
Ultrafine Grained ◽  
Using Data

This article presents a review of the strain rate-dependent mechanical behavior of aluminum and its commercial alloys. The importance of strain rate sensitivity (SRS) stems from its relation with ductility and formability. Plastic deformation is stable and localization less likely in alloys with higher SRS. After discussing the basic formulation used to interpret experimental data, the methods used to measure the SRS parameter are presented. This is followed by a brief review of the main mechanisms that render the flow stress sensitive to the deformation rate, including mechanisms leading to positive and negative SRS. The generic dependence of the SRS parameter on the strain, temperature, and strain rate are further presented using data for pure Al. The effect of alloying is analyzed in the context of solid solutions and precipitated commercial alloys. Results on solid solutions are discussed separately at low and elevated temperatures in order to evidence the role of solute diffusion on SRS. This article ends with a brief discussion of the grain size dependence of SRS, with emphasis on recent efforts to produce nanocrystalline and ultrafine-grained materials by severe plastic deformation.

Determination of Critical Stress for Dynamic Recrystallization of a High-Mn Austenitic TWIP Steel Micro-Alloyed with Vanadium

2016 ◽  
Vol 1812 ◽  
pp. 41-46
Author(s):  
Elvira García-Mora ◽  
Ignacio Mejía ◽  
Francisco Reyes-Calderón ◽  
José M. Cabrera
Keyword(s):  
Dynamic Recrystallization ◽  
Strain Rate ◽  
Strain Hardening ◽  
Critical Stress ◽  
Twip Steel ◽  
True Strain ◽  
Compression Tests ◽  
Hollomon Parameter ◽  
Strain And Strain Rate ◽  
Twip Steels

ABSTRACTWhen high strength and high ductility are required, the Twinning Induced Plasticity steels are an excellent choice. Their mechanical advantages are perfectly known in the automotive industry. Then, they are currently deeply studied. During the deformation at high temperature, TWIP steel experiences dynamic recrystallization. This mechanism results from dislocation interactions, and it depends of temperature, stress, strain, and strain rate. Experimental data give the maximum stress reached by the material, but the critical stress which determinates the DRX onset must be calculated from the strain hardening rate. Both stress and strain change simultaneously, and this variation gives the analytic data to determine σc, which is located at the inflection point of θ-σ plot. The main purpose of this paper was to study how the chemical composition and the experimental parameters (temperature and strain rate) affect the DRX, by the calculation and analysis of the σc values. Hot compression tests were applied to a pair of TWIP steels to compare the DRX onset and its relationship with the vanadium addition. The experimental variables were temperature and strain rate. The true stress–true strain plots were used to calculate σc by cutting data up to a previous point before the σp value, then, a polynomial fit and derivation were applied. The Zener-Hollomon parameter (Z) versus the stresses (peak and critical) plots show how the micro-alloying element vanadium improves the strain hardening in the analyzed TWIP steels.

Hot deformation behavior and dynamic recrystallization constitutive modeling of Al–Cu–Mg powder compacts processed by extrusion at elevated temperatures

2020 ◽  
pp. 146442072097132
Author(s):  
Katti Bharath ◽  
Asit Kumar Khanra ◽  
MJ Davidson
Keyword(s):  
Powder Metallurgy ◽  
Flow Stress ◽  
Dynamic Recrystallization ◽  
Strain Rate ◽  
Relative Density ◽  
Constitutive Equations ◽  
Hot Deformation ◽  
Deformation Behavior ◽  
Peak Flow ◽  
Hot Deformation Behavior

The deformation behavior of Al–Cu–Mg sintered preforms has been investigated by extrusion in the temperature range of 450–550°C and strain rate range of 0.1–0.3 s−1, respectively. The aim of this study is to analyze the effect of initial preform relative density on the hot deformation behavior and to model and predict the flow stress of extruded samples using constitutive equations. The true stress–strain curves exhibit three stages of deformation, which represent work hardening, dynamic recovery, and dynamic recrystallization during deformation at different temperatures, strain rates, and initial preform relative densities of 70%, 80%, and 90%, respectively. The results show that the flow stress values are influenced by initial preform relative density, deformation temperature, and strain rate. Microstructural examination of extruded specimens has been performed by optical microscopy and scanning electron microscopy. Arrhenius-type constitutive equations are developed to predict the flow stress of hot-extruded powder metallurgy processed aluminum alloy (Al–4%Cu–0.5%Mg). Zener–Hollomon parameter is used to explain the relationship between peak flow stress, temperature, and strain rate in an exponential equation containing the deformation activation energy and material constants. Subsequently, the statistical indicators correlation coefficient ( R) and the average absolute relative error are assessed to confirm the validity of constitutive equations. The results indicate the experimental and predicted peak flow stress values are in good agreement, which indicate the accuracy and reliability of the developed model for powder metallurgy processed Al–4%Cu–0.5%Mg preforms.

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