Voce Equation Research Articles

Constitutive equations optimized for determining strengths of metallic alloys

Abstract We investigate compatibilities of three constitutive equations, the Hollomon, the Swift, and the Voce equations for determination of yield and ultimate tensile strengths based on tensile true stress–strain curves of 27 metal alloys including those with power-law type and linear-type strain-hardening. We analyze each constitutive equation in terms of yield strength determined by the intercept of the linear elastic loading curve and plastic flow curve and ultimate tensile strength evaluated by the concept of instability in tension. We found that the describing plastic flow is very sensitive in determination of the yield strength and tensile strength from parameters of constitutive equation. Voce equation gives estimate yield strength and tensile strength better than Hollomon and Swift equations.

Combination between Voce formalism and improved Kocks–Mecking approach to model small strains of flow curves at high temperatures

Abstract Tensile test curves of an austenitic stainless steel (AISI 316L) are described through the Voce equation in combination with the kinetic approach to strain hardening analysis proposed by Kocks and Mecking (KM) at temperatures ranging from 700 to 1000 °C with strain rates between 10 −2 and 10 −5 s −1 . The KM approach to strain hardening analysis is used to find the Voce parameters (saturation stress σ V , relaxation strain e C , back-extrapolated flow stress to zero strain σ o ) from the strain hardening rate d σ /d e vs. the flow stress σ . The obtained Voce equations can well describe the flow curves only at large strains in Stage III of strain hardening, while a significant discrepancy occurs at small strains. In order to reduce this discrepancy an improved KM approach is here proposed. The original KM approach assumes that in Stage III of strain hardening the mean free path of mobile dislocations Λ is defined as β / ρ with β constant and ρ the total dislocation density. Λ , and in turn β , are linked to the dislocation cell-pattern typical of Stage III. The proposed improvement of the KM approach consists of assuming that β is not constant at small strains, where dislocations are not organised in cell-pattern, but homogeneously distributed in the material. Based on TEM observations reporting dislocation distributions at small and at high strains, it is proposed that β varies from an initial value at small strains to achieve at high strains the equilibrium value predicted by the original KM approach in Stage III of strain hardening. This assumption provides very good modelling flow curves also at small strains.

Kocks–Mecking approach to tensile work hardening behaviour of normalised and tempered and post-weld heat-treated modified 9Cr–1Mo steel

Detailed analysis performed on modified 9Cr–1Mo steel in normalised and tempered (N+T) and three different post-weld heat treatment (PWHT) conditions indicated that the work hardening behaviour can be adequately described by Kocks–Mecking approach in the temperature range 300–873 K. The steel in all test conditions exhibited two-stage work hardening behaviour in the variation of instantaneous work hardening rate (θ) with true stress (where θ = dσ/dε, σ is true stress and ε is true plastic strain). The variations of work hardening parameters and θ–σ with temperature exhibited three distinct temperature regimes. The influence of additional PWHTs is discussed in terms of the systematic variations in flow and work hardening parameters due to microstructural softening observed with increasing PWHT temperature in modified 9Cr–1Mo steel. Comparison of work hardening parameters obtained using Voce equation and Kocks–Mecking approach demonstrated an analogy between two approaches for the steel.

Modelling Tensile Curves of AISI 316L at High Temperatures Starting from Strain Hardening Analysis

The tensile curves of AISI 316L deformed at temperatures between 700 and 1000 °C in the strain rates range 10-5-10-2 s-1 are modelled with the Voce equation, starting from strain hardening analysis. The parameters, needed to draw the Voce equation, are the saturation stress σV, the critical strain eC and the stress σo, that respectively define the height of the flow curve, the velocity to achieve σV and namely the back-extrapolated flow stress to zero strain. A two-parameter model of strain hardening recently proposed [ is used to analyze the strain hardening rate, dσ/de, vs. the flow stress, σ. Through this analysis, σV, eC and the thermal activation of plastic flow s are obtained. In fact, the two-parameter model assumes that s and the total dislocation density ρ are the only two parameters needed to describe strain hardening. It has been reported [ that the parameter s can be parameterised in terms of strain rate and temperature and, furthermore, relationships between σV, σo, eC and s can be established. At this stage, the Voce equation can reproduce the experimental tensile curves at the explored temperatures and strain rates. However, the obtained Voce equations can well describe the tensile curves at large strains, while significant discrepancy occurs at small strains [. Preliminary results of an improved model based on two coupled differential equations with physical meaning are reported to correct this discrepancy at low strains.

Influence of temperature and post weld heat treatment on tensile stress–strain and work hardening behaviour of modified 9Cr–1Mo steel

Abstract True stress (σ)–true plastic strain (e) and work hardening behaviour of modified 9Cr–1Mo steel in normalised and tempered (N + T), and three different post weld heat treatment (PWHT) conditions have been examined in the framework of Hollomon, Ludwigson and Voce relationships in the temperature range 300–873 K. The analysis indicated that the σ–e behaviour is described accurately by the combination of Ludwigson and Hollomon equations. Alternatively, Voce equation provides an adequate description of σ–e behaviour in the range 300–873 K. The variations of work hardening parameters associated with Ludwigson/Hollomon and Voce relationships with temperature exhibited three distinct temperature regimes displaying signatures of dynamic strain ageing at intermediate temperatures and dominance of dynamic recovery at high temperatures. The influence of additional PWHTs is discussed in terms of the systematic variations in flow and work hardening parameters due to microstructural softening with increasing PWHT temperature observed for the steel.

Recent developments in modelling of microhardness saturation during SPD processing of metals and alloys

Ultrafine-grained and even nanostructured materials can be fabricated using severe plastic deformation to ultra-high strains in equal-channel angular pressing (ECAP), high-pressure torsion (HPT), machining and their combinations, such as machining of ECAP specimens, HPT of ECAP billets and HPT of machining chips. This report presents recent results of investigations of the microstructures and microtextures of pure copper, nickel and aluminium subjected to different deformation processes to ultimately high imposed strains. A comparison of the microstructure, dislocation density and microhardness developed during combinations of different strain paths is performed. All characteristics were analysed by X-ray, transmission and scanning electron microscopy, and electron backscatter diffraction (EBSD). The influence of different processing routes is discussed in terms of the accumulated strain and microstructure refinement. The saturation in grain refinement is examined with reference to the recovery taking place during ultra-high strain deformation. A phenomenological model based on the Voce equation is applied for fitting parameters based on the experimental data and this is suggested for a prediction of microhardness evolution for pure metals (Ag, Au) and Cu-based (Zn, Al) alloys.

Analysis of Tensile Stress-Strain and Work-Hardening Behavior in 9Cr-1Mo Ferritic Steel

Detailed analysis on tensile true stress (σ)-true plastic strain (e) and work-hardening behavior of 9Cr-1Mo steel have been performed in the framework of the Voce relationship and Kocks-Mecking approach for wide range of temperatures, 300 K to 873 K (27 °C to 600 °C) and strain rates (6.33 × 10−5 to 6.33 × 10−3 s−1). At all test conditions, σ-e data were adequately described by the Voce equation. 9Cr-1Mo steel exhibited two-stage work-hardening behavior characterized by a rapid decrease in instantaneous work-hardening rate (θ = dσ/de) with stress at low stresses (transient stage) followed by a gradual decrease in θ at high stresses (stage III). The variations of work-hardening parameters and θ-σ as a function of temperature and strain rate exhibited three distinct temperature regimes. Both work-hardening parameters and θ-σ displayed signatures of dynamic strain aging at intermediate temperatures and dominance of dynamic recovery at high temperatures. Excellent correlations have been obtained between work-hardening parameters evaluated using the Voce relationship and the respective tensile properties. A comparison of work-hardening parameters obtained using the Voce equation and Kocks-Mecking approach suggested an analogy between the two for the steel.

On the Work-Hardening of AA 3104-H19 Aluminum Alloy

The aim of this study is to investigate the work-hardening of AA 3104-H19 aluminum alloy. Uniaxial tensile tests were carried out in both the rolling and transverse directions. In the past, some authors have used such uniaxial tensile test data and the Hollomon equation fitted to them to predict the forming limit curves. In a previous study, we showed experimentally that the Hollomon equation exponent, typically 0.07, used in such limit strain predictions is too high and should be approximately 0.04. Other authors have obviously used such a high value because they have compensated for the reduction of limit strains near the plane strain by increasing the Hollomon equation exponent. The reduction of limit strains near the plane strain occurs, as the Marciniak-Kuczynski theory, which the aforementioned authors have used, assumes a local inhomogeneity or a thinner section in the sheet to explain the local necking in stretch forming. In this study, it is shown experimentally that the Hollomon equation exponent in both the rolling and transverse directions is very close to 0.04. It is shown further that the Hollomon or Voce equations do not describe very well the work-hardening of the AA 3104-H19 alloy in either the rolling or transverse direction.

Comparative tensile flow and work hardening behaviour of thin section and forged thick section 9Cr–1Mo ferritic steel in the framework of Voce equation and Kocks–Mecking approach

Abstract Detailed analysis indicated that true stress ( σ )-true plastic strain ( e ) and work hardening behaviour of 9Cr–1Mo steel in two different forms, i.e. 20 mm plate and thick section tubeplate forging can be adequately described by Voce equation and Kocks–Mecking approach in the temperature range 300–873 K. The steel exhibited two-stage work hardening in the variations of instantaneous work hardening rate ( θ = d σ /d e ) and θσ with stress. The variations in σ – e , work hardening parameters, θ – σ and θσ – σ with temperature exhibited three distinct temperature regimes. Good correlations between the predicted tensile properties using Voce equation and the respective experimental values along with analogy between Voce equation and Kocks–Mecking approach have been demonstrated for the steel. The differences in σ – e , work hardening parameters, θ – σ and θσ–σ observed between plate and tubeplate forging have been ascribed to the effects associated with initial microstructures for the two product forms of the steel.

Influence of Temperature on Tensile Stress–Strain and Work Hardening Behaviour of Forged Thick Section 9Cr–1Mo Ferritic Steel

True stress (σ)–true plastic strain (e) behaviour examined for temperatures ranging from 300 to 873 K suggested that a combination of Ludwigson and Hollomon equations describes σ–e behaviour most accurately in forged thick section 9Cr–1Mo steel. Alternatively, the flow behaviour for temperature range 300–873 K can be described adequately using single Voce equation. Good correlations have been obtained between work hardening parameters evaluated using Voce equation and the respective tensile properties. The variations of instantaneous work hardening rate (θ = dσ/de) and θσ with stress displayed two-stage behaviour characterised by a rapid decrease at low stresses (transient stage) followed by a gradual decrease at high stresses (stage-III). The variations of work hardening parameters, θ versus σ and θσ versus σ with temperature exhibited three distinct temperature regimes along with signatures of dynamic strain ageing at intermediate temperatures and dominance of dynamic recovery at high temperatures. Comparison of work hardening parameters obtained using Voce equation and Kocks–Mecking approach suggested an analogy between the two for the steel.

Applicability of Kocks–Mecking approach for tensile work hardening in P9 steel

Work hardening behaviour of P9 steel in the temperature range 300–873 K has been examined in the framework of Kocks–Mecking (K–M) approach. At all temperatures, P9 steel exhibited two-stage work hardening behaviour characterised by a rapid decrease in instantaneous work hardening rate (i.e. θ = dσ/dϵ, where σ is the true stress and ϵ is the true plastic strain) with stress at low stresses (transient stage) followed by a gradual decrease at high stresses (stage III). Stage III work hardening of P9 steel was adequately described by K–M approach. The variations of work hardening parameters associated with K–M approach for stage III with temperature indicated three distinct temperature regimes. At all temperatures, good correlations between the respective work hardening parameters evaluated using K–M approach and from Voce equation and its derivative have been obtained for P9 steel.

Effect of alloying elements on stage-III work-hardening behaviour of Al–Zn–Mg(–Cu) alloys

Abstract Four commercial Al–Zn–Mg(–Cu) alloys have been investigated with respect to the age-hardening response and how the stage-III work-hardening behaviour is influenced by the alloying elements. Tensile specimens of the as-cast and homogenized alloys have been exposed to a solution heat treatment and a two-step age-hardening treatment with varying time at the final temperature. The work-hardening parameters obtained by fitting an extended Voce equation to the experimental stress–strain curves have been analysed and discussed. Addition of Cu was found to increase the strength, whereas Mg seemed to be a limiting factor in age-hardening potential. Accelerated precipitation kinetics was observed when the Zn content was increased.

Influence of strain rate and temperature on tensile properties and flow behaviour of a reduced activation ferritic–martensitic steel

Tensile strength and flow behaviour of a Reduced Activation Ferritic–Martensitic (RAFM) steel (9Cr–1W–0.06Ta–0.22V–0.08C) have been investigated over a temperature range of 300–873 K at different strain rates. Tensile strength of the steel decreased with temperature and increased with strain rate except at intermediate temperatures. Negative strain rate sensitivity of flow stress of the steel at intermediate temperatures revealed the occurrence of dynamic strain ageing in the steel, even though no serrated flow was observed. The tensile flow behaviour of the material was well represented by the Voce strain hardening equation for all the test conditions. Temperature and strain rate dependence of the various parameters of Voce equation were interpreted with the possible deformation mechanisms. The equivalence between the saturation stress at a given strain rate in tensile test and steady state deformation rate at a given stress in creep test was found to be satisfied by the RAFM steel.

Flow Curve Modelling of an Austenitic Stainless Steel at High Temperatures Starting from the One-Parameter Model of Strain Hardening

The flow curves of an austenitic stainless steel deformed at temperatures 700-1000°C with strain rates 10-5-10-2 s-1 were modelled with the Voce equation. The parameters needed to draw the Voce equation, are the saturation stress σV that defines the height of the flow curve, the critical strain εC that defines the velocity to achieve σV, and the stress σo, namely the back-extrapolated flow stress to zero strain. A modified strain hardening analysis based on the one-parameter model was used to analyze the strain hardening rate dσ/dε vs. the flow stress σ in order to obtain σV and εC. The modified approach was based on the assumption that the dislocation multiplication component of strain hardening was temperature and strain rate dependent through the thermal activation term s of flow stress. A parameter s’ proportional to s was obtained from the strain hardening analysis and a relationship between s’ and temperature and strain rate was found. Relationships between σV, σo, εC and s’ were finally established and at this stage the Voce equation could reproduce the experimental flow curves at any imposed deformation conditions of temperature and strain rate.

Tensile flow and work hardening behaviour of 9Cr–1Mo ferritic steel in the frame work of Voce relationship

Abstract Tensile stress–strain and work hardening behaviour of 9Cr–1Mo steel, containing three levels of silicon content (i.e., 0.24, 0.42 and 0.60 wt.%) have been examined in the temperature range 300–873 K in the framework of strain hardening law proposed by Voce. True stress–true plastic strain data for all test conditions were adequately described by Voce equation. The variations of work hardening parameters with temperature exhibited three different temperature regimes and displayed good correlations with the respective tensile properties of the steel. The variations of instantaneous work hardening rate with stress exhibited two-stage behaviour characterised by a rapid decrease in work hardening rate (transient stage) at low stresses followed by a gradual decrease (stage-III) at high stresses. Both work hardening parameters and instantaneous work hardening rate exhibited signatures of dynamic strain ageing at intermediate temperatures and dominance of dynamic recovery at high temperatures.

Mechanical Behavior of a Rephosphorized Steel for Car Body Applications: Effects of Temperature, Strain Rate, and Pretreatment

Temperature and strain rate effects on the mechanical behavior of commercial rephosphorized, interstitial free steel have been investigated by uniaxial tensile testing, covering temperatures ranging from −60°C to +100°C and strain rates from 1×10−4 s−1 to 1×102 s−1 encompassing most conditions experienced in automotive crash situations. The effect of prestraining to 3.5% with or without successive annealing at 180°C for 30 min has also been evaluated. These treatments were used to simulate pressing of the plates and the paint-bake cycle in the production of car bodies. Yield and ultimate tensile strengths, ductility including uniform and total elongation and area reduction, thermal softening effect at high strain rate, and strain rate sensitivity of stress were determined and discussed in all cases. It was found that the Voce equation [σ=σs−(σs−σ0)exp(ε/ε0)] can be fitted to the experimental true stress-true plastic strain data with good precision. The parameter values in this equation were evaluated and discussed. Furthermore, temperature and strain rate effects were examined in terms of thermal and athermal components of the flow stresses. Finally, a thermal activation analysis was performed.

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.

Work-hardening behaviour of a heat-treatable AA7108 aluminium alloy deformed to intermediate strains by compression

This work is an experimental study of ageing and work-hardening considering various heat treatments of an AA7108 aluminium alloy in the as-cast and homogenized condition. Specimens have been exposed first to a solution heat treatment and a then to a two-step age-hardening to obtain different tempers. These tempers have been tested in an upsetting test applying state-of-the-art cold lubrication allowing intermediate strains to be reached before the onset of barrelling. The measured work-hardening response has been analyzed by fitting an extended Voce equation to the experimental stress–strain curves. With increasing overageing the stage III part of the stress–strain curve shrinks, and for the most overaged condition studied here, the stress–strain curve has reached a linear stage IV already at a strain of about 0.3. Interestingly, the slope of the stage IV curve then is lower than commonly reported for Al alloys. The microstructure and texture of the deformed material have been further investigated for a selection of tempers, and their influence on the work-hardening behaviour has been discussed.

On estimating the fracture stress and elongation of Al–7%Si–0.3%Mg alloy castings with single pores

Abstract An elastic–plastic fracture-mechanics approach was followed to estimate the fracture stress of Al–7%Si–0.3%Mg alloy castings as a function of size and shape of pores in castings, by using a combination of load limit and linear elastic fracture mechanics. Estimated elongation was then calculated from estimated fracture stress values by using the Voce equation for true stress-strain relationship and the Kocks–Mecking work hardening model. Estimated fracture stress and elongation values compared well with experimental findings of Surappa et al. [Scripta Metall. 20 (1986) 1281]. The methodology and results are discussed in the paper.

Effect of heat treatment on the strain hardening behaviour of an Al–Zn–Mg alloy

Abstract The strain hardening of the Al-5.2Zn-1.2Mg alloy AA7030 in different conditions has been investigated. The dislocation related strengthening was described by the Voce equation and the quantitative effects of different superposition laws have been studied. All tempers, except the over-aged, suffer from plastic instability before the geometric Considère condition is reached. This was attributed primarily to the occurrence of shear failure due to dynamic strain ageing effects. For plastic strain levels below approximately 8 %, preceding the plastic instability, the results show that the saturation stress in the under and peak-aged tempers is independent of heat treatment, whereas the kinetics of structure formation differs. Using a linear superposition law in the under and peak aged tempers has negligible effects, whereas the use of Pythagorean addition for the over-aged temper significantly decreases the goodness-of-fit, although not to unacceptable levels. Furthermore, the naturally aged temper shows strong influence of dynamic precipitation.