Ludwigson Equation Research Articles

Influence of Prior Fatigue Damage on Tensile Flow Behaviour of Type 316L(N) Austenitic Stainless Steel

The present study aims to understand the influence of prior fatigue loading on the tensile flow behaviour of austenitic 316L(N) stainless steel. The steel specimens were subjected to prior fatigue damage up to various fatigue life fractions. i.e., 5, 10, 30 and 50% of the fatigue life at a strain amplitude of ±0.6% and a nominal strain rate of 3×10-3 s-1 and 300, 823 and 873 K. Subsequently the specimens were subjected to tensile deformation at the above temperature and strain rate and the tensile flow behaviour was evaluated in terms of Hollomon, Swift, Ludwigson and Voce equations. At 300 K, the flow curves of the specimens of as-received, and prior fatigue damaged were well described by the Ludwigson equation, whereas at 823 and 873 K, both Voce and Ludwigson equation could be used to describe the flow behaviour of the specimens. Increase of work hardening with the increase in prior fatigue damage at 823 and 873 K is attributed to the occurrence of dynamic strain ageing (DSA). At 300 K, a three-stage work hardening behaviour with a prominent stage II athermal hardening was observed for all the specimens, whereas at higher temperatures, only the as-received specimen exhibited the three-stage work hardening behaviour. At 823 and 873 K, a two-stage work hardening behaviour, comprising of a transient stage followed by stage III was observed for the specimens subjected to prior fatigue damage up to various fatigue life fractions. The inter relationship between the work hardening parameters of Ludwigson was rationalised in terms of the linearly inverse relationship between the work hardening rate and plastic strain rates.

The effect of ageing on tensile behaviour, mode I and mixed mode I/III fracture toughness of 7010 aluminium alloy

Abstract The effect of ageing on the tensile behaviour, mode I fracture toughness and mixed mode I/III fracture toughness of 7010 aluminium alloy plate was investigated. It was found that the 0.2% proof stress in this alloy increased from the under-aged temper to the peak-aged (T6) temper and then subsequently decreased in the over-aged (T73) temper. On the other hand, the ductility exhibited a monotonic decrease from the under-aged temper to the over-aged temper. The Ludwigson equation was found to best describe the flow behaviour in all the three ageing tempers. Under mode I as well as mixed mode I/III loading, the highest fracture toughness was seen in the under-aged temper, whereas the fracture toughness values in the peak-aged and over-aged tempers were comparable.

Studying Deformation Behaviors in Austenitic Stainless Steels within a Temperature Range of 143 K < T < 420 K

This work deals with studying staging and macroscopic strain localization in austenitic stainless steel 12Kh18N9T within a temperature range of 143 K < T < 420 K. The visualization and evolution of macroscopic localized plastic deformation bands at different stages of work hardening were carried out by the method of the double-exposure speckle photography (DESP), which allows registering displacement fields with a high accuracy by tracing changes on the surface of the material under study and then comparing the specklograms recorded during uniaxial tension. The shape of the tensile curves σ(ε) undergoes a significant change with a decreasing temperature due to the γ-α'-phase transformation induced by plastic deformation. The processing of the deformation curves of the steel samples made it possible to distinguish the following stages of strain hardening, i.e. the stage of linear hardening and jerky flow stage. A comparative analysis of the design diagrams (with the introduction of additional parameters of the Ludwigson equation) and experimental diagrams of tension of steel 12Kh18N9T for different temperatures is carried out. The analysis of local strains distributions showed that at the stage of linear work hardening, a mobile system of plastic strain localization centers is observed. The temperature dependence of the parameters of plastic deformation localization at the stages of linear work hardening has been established. Unlike the linear hardening, the jerky flow possesses the propagation of single plastic strain fronts that occur one after another through the sample due to the γ-α' phase transition and the Portevin-Le Chatelier effect. It was found that at the jerky flow stage, which is the final stage before the destruction of the sample, the centers of deformation localization do not merge, leading to the neck formation.

Tensile Flow Analysis of Austenitic Type 316LN Stainless Steel: Effect of Nitrogen Content

The tensile flow analysis of austenitic type 316LN stainless steel with nitrogen level ranging from 0.07 to 0.22 wt.% is studied at the temperatures from 300 to 1123 K and at a strain rate of 3 × 10−3 s−1 using Ludwigson flow relationship. It has been observed that Ludwigson flow relationship provided an appropriate description for the σ-ep behavior and subsequent work/strain hardening response which displayed three independent temperature regions of ambient (300 K), intermediate (523-923 K) and elevated temperature (≥ 923 K). The anomalous response in terms of stress–strain data grouping together in a narrow band and appearance of peaks/plateau in Ludwigson parameters in the intermediate temperature region (523-923 K) is attributed to the manifestation of dynamic strain aging (DSA). The parameters such as strain hardening exponent (n1) and coefficient (K1), transition stress (σL) and transition strain (ɛL) increased with increasing the amount of nitrogen that in turn diminished the propensity to dynamic recovery. The applicability of Ludwigson equation permitted to predict the flow stress response and yield strength as well as ultimate tensile strength values with one-to-one correlation in comparison with the experimental data.

Effect of Bimodal Grain Size Distribution on the Strain Hardening Behavior of a Medium-Entropy Alloy

The evolution of strain hardening behavior of the Fe50(CoCrMnNi)50 medium-entropy alloy as a function of the fraction of recrystallized microstructure and the grain size was studied using the Hollomon and Ludwigson equations. The specimens under study were partially recrystallized, fully recrystallized with ultrafine-grained microstructure, and fully recrystallized with coarse grains. The yield strength decreases steadily as the fraction of recrystallized microstructure and grain size increases due to the recovery process and the Hall–Petch effect. Interestingly, the bimodal grain distribution was found to have a significant impact on strain hardening during plastic deformation. For instance, the highest ultimate tensile strength was exhibited by a 0.97 μm specimen, which was observed to contain a bimodal grain distribution. Furthermore, using the Ludwigson equation, the effect of the bimodal grain distribution was established from the behavior of K2 and n1 curves. These curves tend to show very high values in the specimens with a bimodal grain distribution compared to those that show a homogenous grain distribution. Additionally, the bimodal grain distribution contributes to the extensive Lüders strain observed in the 0.97 μm specimen, which induces a significant deviation of the Hollomon equation at lower strains.

A “new” empirical equation to describe the strain hardening behavior of steels and other metallic materials

The strain hardening behavior of five different steels of varying carbon content, ranging between 0.007 and 0.71%, such as an Interstitial free (IF), a microalloyed, and a low, medium and high carbon steels, have been critically examined using tensile loading. The test results for the IF and the microalloyed steels having only ferritic structure, exhibit a two-stage strain hardening behavior. On the other hand, the low, medium, and high carbon steels having a two-phase microstructure (ferrite-pearlite/cementite) exhibit three-stage strain hardening. A number of existing and frequently used empirical equations, such as the Hollomon, Ludwik, Ludwigson and Voce were used to explain the strain-hardening characteristics of these steels; however, none of the above could fully and satisfactorily explain the flow behavior especially at high strain region. In order to remedy this deficiency, a modification to Ludwigson equation has been suggested by the authors by introducing an extra exponential term to the existing Ludwigson equation. The modelled flow curves, on the basis of this proposed new equation, could explain the strain hardening behavior of these steels very satisfactorily over the entire strain range. In addition, the excellent matching of the flow curves, based on the proposed equation with the experimental flow curves of a number of metallic materials, as diverse as multiphase steel, TRIP-assisted steel, dual-phase steel, stainless steel (316L), Mg-alloy, Ti-alloy and a high entropy (Cantor) alloy, indicates its wider applicability.

Effect of Temperature on Tensile flow behaviour of Nimonic C-263 alloy

Tensile properties, deformation and flow behaviour of Ni base superalloy, Nimonic C-263, was studied at modified solutionizing and aging condition, in the temperature range of 25-650 oC and at a strain rate of 1.3×10-3S-1. The presence of Dynamic strain aging (DSA) behaviour was observed at a wide temperature regime of 200 oC to 600 oC. Also an anomalous variation in tensile properties was observed in the temperature ranges between 400 oC to 650 oC. The yield strength (Y.S) was decreased slightly with temperature up to 300 oC and from there, there is a peak in YS at 400 oC with minimum(%) of elongation and strain hardening rate is maximum at 400 oC. An attempt has been done in this paper, to Co-relate the mechanical properties and flow behaviour of the alloy with microstructral studies and flow equations at different temperatures. It is observed that Ludwigson equation provides the best fit at high temperature, at 650 oC, compared to Hollomon and Ludwik equations.

Characterization and modeling of the mechanical behavior of high silicon ductile iron

This paper investigates the effect of the solidification conditions and silicon content on the mechanical properties of ductile iron and presents empirical models for predicting the tensile behavior based on the microstructural characterizations. Two ductile iron grades of GJS-500-7 and GJS-500-14 were cast with silicon content of 2.36% and 3.71%, respectively. The cast geometry consisted of six plates with different thicknesses that provided different cooling rates during the solidification. Microstructure analysis, tensile and hardness tests were performed on the as-cast material. Tensile behavior was characterized by the Ludwigson equation. The tensile fracture surfaces were analyzed to quantify the fraction of porosity. The results showed that graphite content, graphite nodule count, ferrite fraction and yield strength were increased by increasing the silicon content. A higher silicon content resulted in lower work hardening exponent and strength coefficient on the Ludwigson equation. The results for 0.2% offset yield and the Ludwigson equation parameters were modeled based on microstructural characteristics, with influence of silicon content as the main contributing factor. The models were implemented into a casting process simulation to enable prediction of microstructure-based tensile behavior. A good agreement was obtained between measured and simulated tensile behavior, validating the predictions of simulation in cast components with similar microstructural characteristics.

Plastic deformation and fracture behaviour of high-nitrogen nickel-free austenitic stainless steel

ABSTRACTThe plastic deformation and fracture behaviour of a high-nitrogen nickel-free austenitic stainless steel were examined by performing tensile testing at room temperature and at a wide strain rate range. The tensile testing demonstrated that this steel shows a significant strain rate dependence of the strength and ductility. With increasing strain rate from 10−4 to 1 s−1, the yield strength increases from 673 to 801 MPa and the ultimate tensile strength increases from 958 to 1003 MPa; the uniform elongation decreases from 75 to 44% and the total elongation decreases from 86 to 58%. The analysis of the stress–strain curve by using the Ludwigson equation showed that this steel exhibits a two-stage strain hardening behaviour, the strain hardening exponents at low and high strain regions ( and ) and the transition strain () decrease with increasing strain rate. Based on the analysis results of the stress–strain curve, the transmission electron microscopy characterisation of the microstructure and the scanning electron microscopy observation of the deformation and fracture surfaces, the significant strain rate dependence of the strength and ductility of this steel was discussed and connected with the variation in the dislocation activity with strain rate.

Evolution of microstructure and tensile properties during solution treatment of nickel-based UNS N10276 alloy

Influence of solution treatment (ST) on microstructure evolution and mechanical behavior of nickel-based UNS N10276 alloy were investigated by a variety of techniques such as optical metallography, scanning electron microscopy, transmission electron microscopy and tensile testing. The average grain size remained almost unaffected by hold time at 1050°C, resulting from the pinning effect of secondary phase on the grain boundary. A sharp increase in grain size and grain growth rate was observed with increasing solutionizing temperature up to 1150°C irrespective of hold time, which was mainly related to the complete dissolution of precipitated particles as well as to the enhanced diffusion of solute atoms. The rate of grain growth reduced with extending solutionizing time and decreasing temperature. Flow curve parameters based on Ludwigson equation for the alloy after heat treatment were determined. The dissolution of secondary phase particles, combined with the grain growth, increased the work hardening exponent (n) and dramatically improved the tensile ductilities (TD), but reduced the tensile strengths (TS) to large extent. Both TD and n monotonically increased with increasing solutionizing temperature and prolonging hold time as the alloy was solutionized beyond 1100°C, while the TS and strain hardening rate (SHR) showed a reverse varying trend to that of TD and n. As the heat treatment was performed at 1050°C, tensile properties were not monotonically changing with hold time, which is associated with the increase in amount of precipitates and with the Ostwald ripening of precipitated particles. On the basis of Hall–Petch relation and experimental data, the mathematical relationships between tensile properties and grain size of the alloy solutionized at temperatures higher than 1150°C, were obtained by linear regression analysis. The Hall–Petch strengthening coefficient for yield strength of the UNS N10276 alloy was estimated to be 18.35MPamm1/2.

Strain rate and cold rolling dependence of tensile strength and ductility in high nitrogen nickel-free austenitic stainless steel**Project supported by the National Natural Science Foundations of China (Grant Nos. 51371089 and 51401083)

The tensile strength and ductility of a high nitrogen nickel-free austenitic stainless steel with solution and cold rolling treatment were investigated by performing tensile tests at different strain rates and at room temperature. The tensile tests demonstrated that this steel exhibits a significant strain rate and cold rolling dependence of the tensile strength and ductility. With the increase of the strain rate from 10−4 s−1 to 1 s−1, the yield strength and ultimate tensile strength increase and the uniform elongation and total elongation decrease. The analysis of the double logarithmic stress—strain curves showed that this steel exhibits a two-stage strain hardening behavior, which can be well examined and analyzed by using the Ludwigson equation. The strain hardening exponents at low and high strain regions (n2 and n1) and the transition strain (εL) decrease with increasing strain rate and the increase of cold rolling RA. Based on the analysis results of the stress–strain curves, the transmission electron microscopy characterization of the microstructure and the scanning electron microscopy observation of the deformation surfaces, the significant strain rate and cold rolling dependence of the strength and ductility of this steel were discussed and connected with the variation in the work hardening and dislocation activity with strain rate and cold rolling.

Analysis of flow behaviour of iron and steel using power law models

The tensile flow behaviour of mechanically milled iron and a low-alloy steel are analysed using power law models that describe the evolution of stress–strain during a tensile test. A new analytical power law equation is presented that is shown to be superior to the Hollomon and Ludwigson equation to describe the room temperature flow properties of the low-alloy steel over a wide strain rate range. It not only provides a better fit to the homogenous flow curve of the low-alloy steel but also predicts the maximum strength without the prior knowledge of the uniform strain. The new equation also fitted the flow curves of mechanically milled iron with grain sizes in nano to micrometer range.

Influence of Dynamic Strain Aging on Tensile Deformation Behavior of Alloy 617

Abstract To investigate the dynamic strain aging (DSA) behavior of Alloy 617, high-temperature tensile tests were carried out with strain rates variations of 10 −3 /s, 10 −4 /s, and 10 −5 /s from 24°C to 950°C. Five flow relationships, Hollomon, Ludwik, Swift, Ludwigson, and Voce, were applied to describe the tensile true stress–strain curves, and the DSA region was defined. In describing the tensile curves, Ludwigson's equation was superior to the other equations, and the DSA region was adequately defined by this equation as plateaus at intermediate temperatures from 200°C to 700°C. It was identified that Alloy 617 is dominated by three types of serrations, known as Types D, A+B, and C. The activation energy values for each serration type were obtained by the Arrhenius equation. By using the obtained activation energy values, the serrated yielding map and the DSA mechanism were drawn and manifested. In addition, the relationship between the tensile strength and strain rate at higher temperatures above 700°C was found to be closely related to the amounts of slip lines. In the scanning electron microscope (SEM) fractographs, there was a significant difference at the low, intermediate, and high temperatures, but almost the same to the three strain rates.

Plastic flow behavior and its relationship to tensile mechanical properties of high nitrogen nickel-free austenitic stainless steel

Plastic flow behavior of high nitrogen nickel-free austenitic stainless steel with solution, hot rolling and cold rolling treatments were examined and analyzed by using the Ludwigson equation. It was shown that this stainless steel exhibit a two-stage strain hardening behavior, which can be well described by the Ludwigson equation. Based on the Ludwigson equation and the Considère instability criterion, two equations of the uniform plastic strain and the stress ratio (true ultimate tensile strength/true yield strength) to the strain hardening exponents were derived. Based on these two equations, the dependence of the strength and ductility on the true yield strength and strain hardening behavior was discussed and connected with the observation results of microstructures.

Effects of temperature and strain rate on tensile stress–strain and workhardening behaviour of P92 ferritic steel

Detailed analysis on true stress σ–true plastic strain ϵ data indicated that the tensile flow and workhardening behaviour of P92 ferritic steel can be described most accurately by the combination of Ludwigson and Hollomon relations at strain rates ranging from 3·16×10−5 to 1·26×10−3 s−1 over the temperature range of 300–923 K. At room and intermediate temperatures, the Ludwigson equation follows the σ–ϵ data closely, whereas at high temperatures, the Ludwigson equation reduces to the Hollomon relation. The variations in σ–ϵ, workhardening parameters and θ–σ with temperature exhibited three distinct temperature regimes. At intermediate temperatures, anomalous variations in σ–ϵ, workhardening parameters and θ–σ with respect to temperature and strain rate have been observed. At high temperatures, the dominance of recovery is reflected in the rapid decrease in flow stress and workhardening parameters associated with Ludwigson/Hollomon relations with increasing temperature and decreasing strain rate.

Tensile stress–strain and work hardening behaviour of P9 steel for wrapper application in sodium cooled fast reactors

Tensile flow behaviour of P9 steel with different silicon content has been examined in the framework of Hollomon, Ludwik, Swift, Ludwigson and Voce relationships for a wide temperature range (300–873 K) at a strain rate of 1.3 × 10 −3 s −1. Ludwigson equation described true stress ( σ)–true plastic strain ( ε) data most accurately in the range 300–723 K. At high temperatures (773–873 K), Ludwigson equation reduces to Hollomon equation. The variations of instantaneous work hardening rate ( θ = dσ/ dε) and θσ with stress indicated two-stage work hardening behaviour. True stress–true plastic strain, flow parameters, θ vs. σ and θσ vs. σ with respect to temperature exhibited three distinct temperature regimes and displayed anomalous behaviour due to dynamic strain ageing at intermediate temperatures. Rapid decrease in flow stress and flow parameters, and rapid shift in θ– σ and θσ– σ towards lower stresses with increase in temperature indicated dominance of dynamic recovery at high temperatures.

Incorporating predicted local mechanical behaviour of cast components into finite element simulations

A software which enables the incorporation of local variations in both elastic and plastic mechanical behaviour predicted by a casting process simulation into a Finite Element Method (FEM) simulation is presented. The software uses a piecewise linearization of the Hollomon or the Ludwigson equation to characterize plastic mechanical behaviour of the material on an element level throughout a component. The accuracy obtained in the linearization is investigated, and the performance of the software is studied using different input parameters. The applicability of the software is verified and demonstrated on a ductile iron component, and a simulation strategy for cast components denoted a closed chain of simulations for cast components is discussed.

Comparing Three Equations Used for Modeling the Tensile Flow Behavior of Compacted Graphite Cast Irons at Elevated Temperatures

A comparison between three constituent relationships used to approximate the plastic part of a tensile test curve was performed on compacted graphite cast iron (CGI) samples at temperatures between room temperature and 873 K (600 °C). The investigated relationships were the Hollomon, Ludwigson, and Voce equations. The investigated CGI materials were alloyed with four different amounts of molybdenum, and each chemical composition was cast with three different solidification rates. The two coefficients in the Hollomon equation \( \sigma_{\text{H}} = K_{\text{H}} \times \varepsilon^{{n_{\text{H}} }} \) showed a temperature dependence, where the strength coefficient KH was temperature stable for temperatures up to 573 K (300 °C) and the strain-hardening exponent nH showed a maximum value at about 473 K (200 °C). Both coefficients were affected by an altered metal matrix and by increased nodularity, and they showed a slight increased value with reduced pearlite interlamellar spacing. Ludwigson added an exponential term \( e^{{K_{\text{L}} + n_{\text{L}} \times \varepsilon }} , \) including two new coefficients to the Hollomon equation, to adjust and improve the approximation. The main purpose of KL was to adjust the stress value at zero plastic strain and was affected little by the metal matrix constituents and microstructure features. The value of nL was greatly dependent on the total plastic strain in which small plastic strains resulted in larger nL values, whereas large plastic strains resulted in smaller values. The deformation behavior was similar for all samples; hence, the total plastic strain also had a large influence on whether the adjustment term was positive or negative as a consequence of how nH was chosen. Compared with the Hollomon and Ludwigson equations, the Voce equation \( \sigma_{\text{V}} = \sigma_{S} - \left( {\sigma_{S} - \sigma_{1} } \right)e^{{n_{\text{V}} \times \varepsilon }} \) included coefficients representing an initial stress value σ1 and a saturation stress value σS. The initial stress values and the saturation stress values showed great linear correlations with yield strength values at 0.2 pct elongation and ultimate tensile strength, respectively. The values of both these coefficients were reduced with increasing temperature but had a plateau or even a slight increase between about 473 K and 573 K (200 °C and 300 °C). nV was reduced constantly with increasing temperature and was affected by the total plastic strain values in the same way as nL. The overall best approximation of the stress values was made by the Ludwigson equation followed by the Hollomon equation and last by the Voce equation. The downside with the Ludwigson equation was that the correction term either could be positive or negative, which made it harder to use as a general equation to approximate stress values, compared with the Hollomon and Voce equations.

Tensile deformation behaviour of AISI 316L(N) SS

Tensile deformation behaviour of AISI 316L(N) stainless steel was investigated over the range 298 to 1023 K. The yield and tensile strength were found to meet the minimum values mandated by RCC-MR, the French design code for fast reactors. The tensile flow behaviour was modelled using Ludwigson and Voce equation. Ludwigson equation could model the flow behaviour adequately before the onset of saturation at temperatures >823 K. Voce equation could model the flow behaviour between 298 and 1023 K. The physical basis for the variation with respect to strain rate and temperature of the constants obtained by modelling the flow curve using Ludwigson and Voce equation is discussed.

The effect of ageing on tensile behaviour, mode I and mixed mode I/III fracture toughness of 7010 aluminium alloy

Abstract The effect of ageing on the tensile behaviour, mode I fracture toughness and mixed mode I/III fracture toughness of 7010 aluminium alloy plate was investigated. It was found that the 0.2% proof stress in this alloy increased from the under-aged temper to the peak-aged (T6) temper and then subsequently decreased in the over-aged (T73) temper. On the other hand, the ductility exhibited a monotonic decrease from the under-aged temper to the over-aged temper. The Ludwigson equation was found to best describe the flow behaviour in all the three ageing tempers. Under mode I as well as mixed mode I/III loading, the highest fracture toughness was seen in the under-aged temper, whereas the fracture toughness values in the peak-aged and over-aged tempers were comparable.